---
abstract: 'Let $\mathfrak{g}$ be a complex semisimple Lie algebra. We give a description of characters of irreducible Whittaker modules for $\mathfrak{g}$ with any infinitesimal character, along with a Kazhdan-Lusztig algorithm for computing them. This generalizes Milicic-Soergel's and Romanov's results for integral infinitesimal characters. As a special case, we recover the non-integral Kazhdan-Lusztig conjecture for category $\mathcal{O}$.'
address: 'Department of Mathematics, University of Utah, Salt Lake City, UT 84112'
author:
- Qixian Zhao
bibliography:
- 'References.bib'
title: Characters of Irreducible Whittaker modules for Complex Semisimple Lie Algebras
---

Introduction {#sec:intro}
============

Let ${\mathfrak{g}}$ be a complex semisimple Lie algebra, ${\mathfrak{h}}$ a Cartan subalgebra, ${\mathfrak{b}}$ a Borel subalgebra containing ${\mathfrak{h}}$, and ${\mathfrak{n}}$ the nilpotent radical of ${\mathfrak{b}}$. Let $G$ be a Zariski-connected algebraic group with Lie algebra ${\mathfrak{g}}$, and let $H$, $B$, $N$ be the respective subgroups corresponding to ${\mathfrak{h}}$, ${\mathfrak{b}}$ and ${\mathfrak{n}}$. Let $\Sigma$ be the root system of $({\mathfrak{g}},{\mathfrak{h}})$, and $W$ the Weyl group of $\Sigma$. Let ${\mathcal{U}}({\mathfrak{g}})$ be the universal enveloping algebra of ${\mathfrak{g}}$, and let ${\mathcal{Z}}({\mathfrak{g}})$ be the center of ${\mathcal{U}}({\mathfrak{g}})$. By an infinitesimal character, we mean a ${\mathbb{C}}$-algebra homomorphism from ${\mathcal{Z}}({\mathfrak{g}})$ to ${\mathbb{C}}$. Via the Harish-Chandra homomorphism, any infinitesimal character corresponds to a unique $W$-orbit in ${\mathfrak{h}}^*$.

The category of Whittaker modules, denoted by ${\mathcal{N}}$, is the full subcategory of all ${\mathfrak{g}}$-modules consisting of those that are finitely generated over ${\mathfrak{g}}$, locally finite over ${\mathfrak{n}}$, and locally finite over ${\mathcal{Z}}({\mathfrak{g}})$. Whittaker modules for real groups have been studied, for example in the work of Casselman-Hecht-Milicic @Casselman-Hecht-Milicic:Whittaker_real_groups. Over a complex semisimple Lie algebra, previous contributors include Kostant @Kostant:Whittaker who studied the "non-degenerate" case, McDowell @McDowell:Whittaker who constructed and studied standard Whittaker modules, Milicic-Soergel who reproved Kostant's results with geometry in @Milicic-Soergel:Whittaker_geometric and obtained a description of the composition factors of standard Whittaker modules and their multiplicities in the case of integral infinitesimal characters in @Milicic-Soergel:Whittaker_algebraic, and Romanov @Romanov:Whittaker who established a Kazhdan-Lusztig algorithm for Whittaker modules with regular integral infinitesimal characters.

It is shown in @McDowell:Whittaker that objects in ${\mathcal{N}}$ have finite length. By a standard linear algebra argument, irreducible objects in ${\mathcal{N}}$ land in full subcategories ${\mathcal{N}}_{\theta,\eta}$ consisting of modules such that (1) they admit an infinitesimal character given by the $W$-orbit $\theta$ in ${\mathfrak{h}}^*$, and (2) for each $\xi \in {\mathfrak{n}}$, $\xi - \eta(\xi)$ acts locally nilpotently for a character $\eta: {\mathfrak{n}}\to {\mathbb{C}}$. The main goal of this paper is

To describe characters of irreducible objects in ${\mathcal{N}}_{\theta,\eta}$ for any $\theta$ and $\eta$.

We will be using the character theory of Whittaker modules developed in @Romanov:Whittaker. As a byproduct, we obtain a Kazhdan-Lusztig algorithm for Whittaker modules. Our main theorems generalize Milicic-Soergel's and Romanov's results.

The main result
---------------

The precise answer to this problem is contained in Theorem \[thm:multiplicity\] and \[thm:multiplicity\_singular\]. To state the theorem, we need to introduce more notation. Let $\Sigma^+ \subset \Sigma$ be the set of positive roots defined by ${\mathfrak{b}}$, and let $\Pi \subset \Sigma^+$ be the set of simple roots. Let $\Theta = \{ \alpha \in \Pi \mid \eta|_{{\mathfrak{g}}_\alpha} \neq 0\}$, where ${\mathfrak{g}}_\alpha$ is the $\alpha$-root space in ${\mathfrak{g}}$. Let $W_\Theta \subseteq W$ be the parabolic subgroup corresponding to $\Theta$. Let $\lambda \in {\mathfrak{h}}^*$ be an antidominant element in $\theta$, that is, $\lambda \in \theta$ and $\alpha^\vee(\lambda) \not\in {\mathbb{Z}}_{> 0}$ for all $\alpha \in \Pi$, where $\alpha^\vee$ denotes the dual root to $\alpha$. Let $W_\lambda = \{w \in W \mid w \lambda - \lambda \in {\mathbb{Z}}\cdot \Sigma\}$, the integral Weyl group of $\lambda$, which is the Weyl group of the integral root subsystem $\Sigma_\lambda = \{\alpha \in \Sigma \mid \alpha^\vee(\lambda) \in {\mathbb{Z}}\} \subset \Sigma$. Finally, we take a set of positive roots in $\Sigma_\lambda$ by $\Sigma_\lambda^+ = \Sigma_\lambda \cap \Sigma^+$, and let $\Pi_\lambda \subseteq \Sigma_\lambda^+$ be the corresponding set of simple roots. Note that roots in $\Pi_\lambda$ may not be simple in $\Sigma^+$.

It is known (by the work of McDowell) that irreducible objects in ${\mathcal{N}}_{\theta,\eta}$ are parametrized by right $W_\Theta$-cosets in $W$. For a right $W_\Theta$-coset $C$, writing $w^C$ for the unique longest element in $C$, the irreducible object corresponding to $C$ is denoted by $L(w^C \lambda,\eta)$. It is the unique irreducible quotient of a "standard module", denoted by $M(w^C \lambda,\eta)$; see \[sec:Whittaker\_prelim\] for the precise definition. Characters of standard modules were explicitly calculated by Romanov. Hence, we can describe characters of irreducible objects $L(w^C \lambda,\eta)$ by expressing them as linear combinations of characters of standard modules $M(w^D \lambda,\eta)$ for various right $W_\Theta$-cosets $D$'s.

Not all $M(w^D \lambda,\eta)$'s appear in the character of $L(w^C \lambda,\eta)$. First, $M(w^D \lambda,\eta)$ can appear only if $D$ and $C$ are in the same double $(W_\Theta,W_\lambda)$-coset. Suppose $C$ is contained in a double coset $W_\Theta u W_\lambda$, where $u$ is the unique shortest element in this double coset (which exists by \[thm:cross-section\_db\_coset\]). We look at the left $W_\lambda$-coset $u W_\lambda$, and take the intersections of $u W_\lambda$ with various right $W_\Theta$-cosets in $W_\Theta u W_\lambda$. This is a partition of $u W_\lambda$. Left-translating back into $W_\lambda$, we obtain a partition of $W_\lambda$, which coincides with the partition of $W_\lambda$ given by right cosets of the parabolic subgroup $W_{\lambda,\Theta_\lambda^u}$ of $W_\lambda$ corresponding to some subset of simple roots $\Theta_\lambda^u = u{^{-1}}\Sigma_\Theta \cap \Pi_\lambda \subseteq \Pi_\lambda$ (\[lem:int\_Whittaker\_model\]). This defines a map from the set of right $W_\Theta$-cosets in $W_\Theta u W_\lambda$ to the set of right $W_{\lambda,\Theta_\lambda^u}$-cosets in $W_\lambda$, i.e. from $W_\Theta \backslash W_\Theta u W_\lambda$ to $W_{\lambda,\Theta_\lambda^u} \backslash W_\lambda$. We denote this map by $r_\lambda$ (\[lem:right\_coset\_partition\]). We say that $W_{\lambda,\Theta_\lambda^u} \backslash W_\lambda$ is the *integral model* for $W_\Theta \backslash W_\Theta u W_\lambda$ since, as will be apparent from the main theorem, $W_{\lambda,\Theta_\lambda^u} \backslash W_\lambda$ controls the characters of $L(w^C \lambda,\eta)$ for $C \in W_\Theta \backslash W_\Theta u W_\lambda$. Recall that there is a partial order on $W_\Theta \backslash W$ inherited from the restriction of Bruhat order to the set of longest elements in right $W_\Theta$-cosets. We denote the partial order on $W_{\lambda,\Theta_\lambda^u} \backslash W_\lambda$ by ${\leqslant}_{u,\lambda}$. Then $M(w^D \lambda,\eta)$ appears in the character of $L(w^C \lambda,\eta)$ only if $D$ is contained in $W_\Theta u W_\lambda$, and $r_\lambda D {\leqslant}_{u,\lambda} r_\lambda C$.

The precise coefficient of $\operatorname{ch}M(w^D \lambda,\eta)$ in $\operatorname{ch}L(w^C \lambda,\eta)$ is described using so called parabolic Kazhdan-Lusztig polynomials for $(W_\lambda,\Pi_\lambda,\Theta_\lambda^u)$ (see \[subsec:KL\_poly\] for the precise definition of these polynomials and a comparison with the version defined in @Soergel:KL_Tilting). These are polynomials $P_{E_1',E_2'}^{u,\lambda}$ in a variable $q$ indexed by pairs of right $W_{\lambda,\Theta_\lambda^u}$-cosets $(E_1',E_2')$ with $E_2' {\leqslant}_{u,\lambda} E_1'$. In particular, we have the polynomials $P_{r_\lambda C, r_\lambda D}^{u,\lambda}$. The character formula for regular infinitesimal characters is the following.

\[Regular case, \[thm:multiplicity\]\] Let $\theta$ be a regular Weyl group orbit in ${\mathfrak{h}}^*$ (i.e. $\alpha^\vee(\lambda) \neq 0$ for any root $\alpha$ and any $\lambda \in \theta$), and let $\lambda \in \theta$ be antidominant. For any $C \in W_\Theta \backslash W$, let $W_\Theta u W_\lambda$ be the double $(W_\Theta,W_\lambda)$-coset containing $C$, where $u$ is the unique shortest element in this double coset. Then $$\operatorname{ch}L(w^C\lambda,\eta) = 
        \sum_{\substack{D \in W_\Theta \backslash W_\Theta u W_\lambda\\                r_\lambda D {\leqslant}_{u,\lambda} r_\lambda C}} 
        P_{r_\lambda C,r_\lambda D}^{u,\lambda}(-1) \operatorname{ch}M(w^D \lambda,\eta),$$ where $\operatorname{ch}$ is the character map, and the $P_{r_\lambda C,r_\lambda D}^{u,\lambda}$'s are parabolic Kazhdan-Lusztig polynomials for $(W_\lambda,\Pi_\lambda,\Theta_\lambda^u)$ as defined in \[def:parabolic\_KL\_poly\].

The formula for singular infinitesimal characters (\[thm:multiplicity\_singular\]) can be obtained with minor modifications.

Viewing this as an equality in the Grothendieck group $K {\mathcal{N}}_{\theta,\eta}$ of ${\mathcal{N}}_{\theta,\eta}$, we obtain the change of basis matrix from the irreducible modules to standard modules, which is upper-triangular under suitable choices of orderings on the bases. Inverting this matrix, we obtain information on multiplicities of irreducible modules in standard modules. When $\lambda$ is integral, this recovers the description of multiplicities proven by Milicic-Soergel and Romanov. At another extreme, when $\eta = 0$ (i.e. $\Theta = \varnothing$), we recover the Kazhdan-Lusztig conjecture for regular non-integral blocks in Bernstein-Gelfand-Gelfand's category ${\mathcal{O}}$ as described in, say, @Kashiwara-Tanisaki:Non-int_KL (with a different normalization of the Kazhdan-Lusztig polynomials).

The proof of the theorem is based on an algorithm (Theorem \[thm:KL\_alg\]) for finding what Romanov called "Whittaker Kazhdan-Lusztig polynomials", which is a set of polynomials in a variable $q$ whose specialization to $q = -1$ gives us the coefficients in the character formula, analogous to the role of ordinary Kazhdan-Lusztig polynomials for Verma modules. This algorithm is suggested by Milicic. There are three main ingredients in the algorithm. First, by Beilinson-Bernstein's localization theorem, Whittaker modules correspond to a class of weakly $N$-equivariant ${\mathcal{D}}$-modules on the flag variety $X$ of ${\mathfrak{g}}$, known as twisted Harish-Chandra sheaves. Under this setting, the Whittaker Kazhdan-Lusztig polynomials will be given by Poincaré polynomials of higher $!$-inverse images of irreducible twisted Harish-Chandra sheaves along inclusion maps of Schubert cells. The algorithm is proven by induction on the dimension of the support of irreducible ${\mathcal{D}}$-modules. The second main ingredient is the $U$-functor. This is a geometric version of Vogan's $U$-functor in @Vogan:irred_char_I. It is used in conjunction with the decomposition theorem of holonomic ${\mathcal{D}}$-modules @Mochizuki:Decomp to increase dimension of the supports. This part is modified from the algorithm in @Milicic:Localization [V.2] for proving Kazhdan-Lusztig conjecture for principal block of category ${\mathcal{O}}$; it is also almost identical to the one in @Romanov:Whittaker. The last ingredient of the proof is the non-integral intertwining functors. These are used to fill in the role of $U$-functors when the latter fail due to non-integrality of some simple roots.

Outline of the paper
--------------------

The paper is organized as follows. In \[sec:Whittaker\_prelim\] we briefly recall basic facts about Whittaker modules and character theory. \[sec:int\_sys\] and \[sec:db\_coset\] are devoted to studying the structure of left $W_\lambda$-cosets and double $(W_\Theta,W_\lambda)$-cosets in the Weyl group. These results (some of which are probably known) are necessary to formulate the main theorem, and some are used directly in the proof. In \[sec:geom\], we first review basic facts about Beilinson-Bernstein's localization and twisted Harish-Chandra sheaves. Then, we study the effect of non-integral intertwining functors on irreducible ${\mathcal{D}}$-modules. Results in this section will be vital for the use of intertwining functors in the main algorithm. \[sec:KL\] contains the statement and the proof of the algorithm. The character formula is established in \[sec:character\_formula\]. Lastly, in \[sec:examples\], we provide an example on the $A_3$ root system.

Acknowledgements
----------------

I would like to thank D. Milicic for suggesting the algorithm, as well as for his mentorship, guidance, and valuable discussions. I also want to thank A. Romanov for helpful conversations.

Preliminaries on Whittaker modules {#sec:Whittaker_prelim}
==================================

In this section, we briefly review basic facts about Whittaker modules and character theory without proof. References include @Kostant:Whittaker, @McDowell:Whittaker, @Milicic-Soergel:Whittaker_geometric, @Milicic:Localization, and @Romanov:Whittaker.

As mentioned in \[sec:intro\], the category ${\mathcal{N}}$ of Whittaker modules is defined to be the full subcategory of all ${\mathfrak{g}}$-modules consisting of objects that are finitely generated over ${\mathfrak{g}}$, locally finite over ${\mathfrak{n}}$, and locally finite over ${\mathcal{Z}}({\mathfrak{g}})$. If we insist that ${\mathcal{Z}}({\mathfrak{g}})$ acts via a character given by a Weyl group orbit $\theta$, we obtain the full subcategory ${\mathcal{N}}_\theta$. If instead we require $\xi - \eta(\xi)$ acts locally nilpotently for all $\xi \in {\mathfrak{n}}$ for some character $\eta: {\mathfrak{n}}\to {\mathbb{C}}$, we obtain ${\mathcal{N}}_\eta$. Set ${\mathcal{N}}_{\theta,\eta} = {\mathcal{N}}_\theta \cap {\mathcal{N}}_\eta$. Any object in ${\mathcal{N}}$ has finite length, and any irreducible object is contained in one of the ${\mathcal{N}}_{\theta,\eta}$'s. Each $\eta$ defines a subset $\Theta$ of simple roots by $$\Theta = \{ \alpha \in \Pi \mid \eta |_{{\mathfrak{g}}_\alpha} \neq 0\} \subseteq \Pi.$$ Different $\eta$'s define different subcategories ${\mathcal{N}}_{\theta,\eta}$'s of ${\mathcal{N}}$, but the categorical structure of ${\mathcal{N}}_{\theta,\eta}$ are the same whenever two $\eta$'s give the same $\Theta$. If $\Theta = \Pi$, we say that $\eta$ is non-degenerate, in which case the category ${\mathcal{N}}_{\theta,\eta}$ is semisimple with one irreducible object, and ${\mathcal{N}}_\eta$ is equivalent to the category of finite dimensional ${\mathcal{Z}}({\mathfrak{g}})$-modules. If $\Theta = \varnothing$ (i.e. $\eta = 0$), ${\mathcal{N}}_{\theta,0}$ recovers a well-studied category containing Verma modules.

Next, we define the standard Whittaker modules. These are analogous to Verma modules in category ${\mathcal{O}}$. Recall that we have chosen a Borel subalgebra ${\mathfrak{b}}$, its nilpotent radical ${\mathfrak{n}}$, and a Cartan subalgebra ${\mathfrak{h}}$ with ${\mathfrak{b}}= {\mathfrak{n}}\rtimes {\mathfrak{h}}$, and $\Sigma \supset \Sigma^+ \supset \Pi$ are the set of roots of $({\mathfrak{g}},{\mathfrak{h}})$, a set of positive roots defined by ${\mathfrak{b}}$, and the corresponding set of simple roots. Let $\bar {\mathfrak{n}}$ be the nilpotent radical of the Borel subalgebra opposite to ${\mathfrak{b}}$. Let $\Sigma_\Theta \subset \Sigma$ be the subsystem generated by $\Theta$. Let $\Sigma_\Theta^+ = \Sigma^+ \cap \Sigma_\Theta$. Let $\rho_\Theta$ be the half-sum of roots in $\Sigma_\Theta^+$, and $\rho$ the half-sum of roots in $\Sigma^+$. Let $W_\Theta \subseteq W$ be the subgroup generated by simple reflections with respect to roots in $\Theta$, which is isomorphic to the Weyl group of $\Sigma_\Theta$. Let ${\mathfrak{p}}_\Theta \supset {\mathfrak{b}}$ be the parabolic subalgebra of ${\mathfrak{g}}$ corresponding to $\Theta$, ${\mathfrak{p}}_\Theta = {\mathfrak{l}}_\Theta \oplus {\mathfrak{u}}_\Theta$ its $\operatorname{ad}{\mathfrak{h}}$-stable Levi decomposition, and ${\mathfrak{l}}_\Theta = \bar {\mathfrak{n}}_\Theta \oplus {\mathfrak{h}}\oplus {\mathfrak{n}}_\Theta$ the restriction of the triangular decomposition ${\mathfrak{g}}= \bar {\mathfrak{n}}\oplus {\mathfrak{h}}\oplus {\mathfrak{n}}$ of ${\mathfrak{g}}$ to ${\mathfrak{l}}_\Theta$. Let ${\mathcal{Z}}({\mathfrak{l}}_\Theta)$ be the center of the enveloping algebra ${\mathcal{U}}({\mathfrak{l}}_\Theta)$ of ${\mathfrak{l}}_\Theta$. Consider the Harish-Chandra homomorphism ${\mathcal{Z}}({\mathfrak{l}}_\Theta) \to {\mathcal{U}}({\mathfrak{h}})$, twisted such that its image is $W_\Theta$-invariant. Any $\lambda \in {\mathfrak{h}}^*$ defines a map ${\mathcal{U}}({\mathfrak{h}}) \to {\mathbb{C}}$ and hence a character ${\mathcal{Z}}({\mathfrak{l}}_\Theta) \to {\mathbb{C}}$ by pre-composing with the Harish-Chandra homomorphism. We denote this character by $\chi_{\Theta,\lambda}$. Define the ${\mathfrak{l}}_\Theta$-module $$Y(\lambda,\eta) = {\mathcal{U}}({\mathfrak{l}}_\Theta) \dotimes_{{\mathcal{Z}}({\mathfrak{l}}_\Theta) \dotimes_{\mathbb{C}}{\mathcal{U}}({\mathfrak{n}}_\Theta)} {\mathbb{C}}_{\chi_{\Theta,\lambda},\eta},$$ that is, a module generated by a single vector $v$ on which ${\mathcal{Z}}({\mathfrak{l}}_\Theta)$ acts by the character $\chi_{\Theta,\lambda}$ and ${\mathfrak{n}}$ acts by the character $\eta$. This module is an irreducible ${\mathfrak{l}}_\Theta$-module. Letting ${\mathfrak{u}}_\Theta$ act trivially, we can view $Y(\lambda,\eta)$ as a ${\mathfrak{p}}_\Theta$-module. The standard Whittaker module of ${\mathfrak{g}}$ is defined to be $$M(\lambda,\eta) = {\mathcal{U}}({\mathfrak{g}}) \dotimes_{{\mathcal{U}}({\mathfrak{p}}_\Theta)} Y(\lambda-\rho+\rho_\Theta,\eta).$$ By construction, $M(\lambda,\eta)$ admits an infinitesimal character given by the Weyl group orbit $\theta$ of $\lambda$. Also, $M(\lambda_1,\eta) \cong M(\lambda_2,\eta)$ if and only if $\lambda_1$ and $\lambda_2$ are in the same $W_\Theta$-orbit. It is shown by McDowell that each $M(\lambda,\eta)$ has a unique irreducible quotient, which is denoted by $L(\lambda,\eta)$, and any irreducible object arises in this way. Hence, $L(\lambda,\eta)$ also has infinitesimal character given by $\theta$, both $M(\lambda,\eta)$ and $L(\lambda,\eta)$ live in ${\mathcal{N}}_{\theta,\eta}$.

Romanov developed in @Romanov:Whittaker [2.2] a character theory on ${\mathcal{N}}_{\theta,\eta}$. This is a map on objects of ${\mathcal{N}}_{\theta,\eta}$ that factors through and is injective on the Grothendieck group $K {\mathcal{N}}_{\theta,\eta}$. The characters of standard modules are computed explicitly in *loc. cit.*

The integral subsystem {#sec:int_sys}
======================

In this section, we collect some known results on the integral root subsystem. Most results here are known to experts. We provide the details here for the sake of the reader.

Let $\lambda \in {\mathfrak{h}}^*$. Let $\Sigma_\lambda = \{\alpha \in \Sigma \mid \alpha^\vee(\lambda) \in {\mathbb{Z}}\}$ be the integral subsystem, with Weyl group $W_\lambda$ identified with the subgroup $\{w \in W \mid w \lambda - \lambda \in {\mathbb{Z}}\cdot \Sigma\}$ of $W$. Roots in $\Sigma_\lambda$ are said to be *integral to $\lambda$*, and those not in $\Sigma_\lambda$ are said to be *non-integral to $\lambda$*. Let $\Sigma_\lambda^+ = \Sigma^+ \cap \Sigma_\lambda$ be a set of positive roots in $\Sigma_\lambda$, and $\Pi_\lambda \subseteq \Sigma_\lambda^+$ the corresponding set of simple roots. We will write ${\leqslant}_\lambda$ for the Bruhat order on $W_\lambda$ determined by reflections of roots in $\Pi_\lambda$.

A cross-section of $W/W_\lambda$
--------------------------------

For any $u \in W$, define the set $$\Sigma_u^+ = \{ \alpha \in \Sigma^+ \mid u \alpha \in - \Sigma^+\} = \Sigma^+ \cap (-u{^{-1}}\Sigma^+).$$ Set $$A_\lambda = \{ u \in W \mid \Sigma_u^+ \cap \Sigma_\lambda = \varnothing\}.$$

The following is well-known.

\[lem:cross-section\] $A_\lambda$ is a cross-section of $W/W_\lambda$.

This proof is copied verbatim from Milicic's unpublished notes. Let $u \in W$. Observe that $$\Sigma_u^+ \cap \Sigma_\lambda 
        = \Sigma^+ \cap (-u{^{-1}}\Sigma^+) \cap \Sigma_\lambda 
        = (\Sigma^+ \cap \Sigma_\lambda) \cap (-u{^{-1}}\Sigma^+)
        = \Sigma_\lambda^+ \cap (-u{^{-1}}\Sigma^+).$$ Hence $\Sigma_u^+ \cap \Sigma_\lambda = \varnothing \iff \Sigma_\lambda^+ \subseteq u{^{-1}}\Sigma^+$, and $$\label{eqn:A_lambda_alt}
        A_\lambda = \{u \in W \mid \Sigma_\lambda^+ \subseteq u{^{-1}}\Sigma^+\}.$$

We first show that any left $W_\lambda$-coset has a representative in $A_\lambda$. Take any $w \in W$. Then $w{^{-1}}\Sigma^+$ is a set of positive roots in $\Sigma$. Hence $\Sigma_\lambda \cap w{^{-1}}\Sigma^+$ is a set of positive roots in $\Sigma_\lambda$. Therefore there is an element $v \in W_\lambda$ such that $v(\Sigma_\lambda \cap w{^{-1}}\Sigma^+) = \Sigma_\lambda^+$, or equivalently $\Sigma_\lambda \cap vw{^{-1}}\Sigma^+ = \Sigma_\lambda^+$. In particular $\Sigma_\lambda^+ \subseteq vw{^{-1}}\Sigma^+$, and hence $wv{^{-1}}\in A_\lambda$. As a result $w \in A_\lambda v \subseteq A_\lambda W_\lambda$. Thus $W = A_\lambda W_\lambda$, and any left $W_\lambda$-coset has a representative in $A_\lambda$.

Now suppose $u_1,u_2 \in A_\lambda$ are in the same left $W_\lambda$-coset. Then there is $v \in W_\lambda$ with $u_1 = u_2 v$. This implies $$\Sigma_\lambda^+ 
        = \Sigma_\lambda \cap u_1{^{-1}}\Sigma^+ 
        = \Sigma_\lambda \cap v{^{-1}}u_2{^{-1}}\Sigma^+ 
        = v{^{-1}}( \Sigma_\lambda \cap u_2{^{-1}}\Sigma^+) 
        = v{^{-1}}\Sigma_\lambda^+.$$ Since $W_\lambda$ acts simply transitively on the set of sets of positive roots of $\Sigma_\lambda$, we have $v = 1$ and $u_1 = u_2$. Thus $A_\lambda$ is a cross-section of $W/W_\lambda$.

Here are some elementary properties that are easy to verify. We provide proofs for completeness.

\[lem:non-int\_refl\_subsys\] Let $\beta$ be a simple root and let $u \in W$.

1.  $u \Sigma_\lambda = \Sigma_{u \lambda}$;

2.  if $u \in A_\lambda$, $u \Sigma_\lambda^+ = \Sigma_{u \lambda}^+$;

3.  if $u \in A_\lambda$, $u \Pi_\lambda = \Pi_{u \lambda}$;

4.  $u W_\lambda u{^{-1}}= W_{u \lambda}$;

5.  if $s_\beta \in A_\lambda$ and $u \in A_\lambda$, $us_\beta \in A_{s_\beta\lambda}$.

6.  $s_\beta \in A_\lambda$ if and only if $\beta \in \Pi-\Pi_\lambda$.

(a): for any $\alpha \in \Sigma_\lambda$, $(u \alpha)^\vee (u\lambda) = \alpha^\vee( u{^{-1}}u \lambda) = \alpha^\vee(\lambda) \in {\mathbb{Z}}$. Hence $u \alpha \in \Sigma_{u\lambda}$ and $u \Sigma_\lambda \subseteq \Sigma_{u\lambda}$. Since both sets have the same size, equality holds.

(b): from (\[eqn:A\_lambda\_alt\]), we know $u \Sigma_\lambda^+ \subseteq \Sigma^+$. Hence $$u \Sigma_\lambda^+ 
        = u \Sigma_\lambda \cap \Sigma^+ 
        = \Sigma_{u \lambda} \cap \Sigma^+ 
        = \Sigma_{u \lambda}^+.$$

(c): elements in $\Pi_\lambda$ and $\Pi_{u\lambda}$ can be characterized by not being sums of other elements of $\Sigma_\lambda^+$ and $\Sigma_{u \lambda}^+$, respectively. Since $u: \Sigma_\lambda^+ \to \Sigma_{u \lambda}^+$ commutes with sums, it must send $\Pi_\lambda$ to $\Pi_{u \lambda}$.

(d): for any $w \in W_\lambda$, $$(u w u{^{-1}}) u \lambda - u \lambda 
        = u (w \lambda - \lambda)
        \in u ({\mathbb{Z}}\cdot \Sigma) = {\mathbb{Z}}\cdot \Sigma.$$ Hence $u W_\lambda u{^{-1}}\subseteq W_{u \lambda}$. Since both sides have the same size, equality holds.

(e): observe $$\begin{aligned}
        \Sigma_{us_\beta}^+ \cap \Sigma_{s_\beta\lambda}\
        &= (\Sigma^+ \cap (-(us_\beta){^{-1}}\Sigma^+)) \cap \Sigma_{s_\beta\lambda}\\
        &= (\Sigma^+ \cap \Sigma_{s_\beta \lambda}) \cap (-(us_\beta){^{-1}}\Sigma^+)\\
        &= \Sigma_{s_\beta\lambda}^+ \cap (-(us_\beta){^{-1}}\Sigma^+)\\
        &= s_\beta \Sigma_\lambda^+ \cap (-(us_\beta){^{-1}}\Sigma^+).
    \end{aligned}$$ Hence $$\begin{aligned}
        us_\beta \in A_{s_\beta \lambda}
        &\iff \Sigma_{us_\beta}^+ \cap \Sigma_{s_\beta\lambda}  = \varnothing\\
        &\iff s_\beta \Sigma_\lambda^+ \cap (-(us_\beta){^{-1}}\Sigma^+) = \varnothing\\
        &\iff s_\beta \Sigma_\lambda^+ \subseteq (us_\beta){^{-1}}\Sigma^+ = s_\beta u{^{-1}}\Sigma^+\\
        &\iff u\Sigma_\lambda^+ \subseteq \Sigma^+\\
        &\iff \Sigma_u^+ \cap \Sigma_\lambda = \varnothing \iff u \in A_\lambda
    \end{aligned}$$ which is true by assumption.

(f): if $s_\beta \in A_\lambda$, then since $A_\lambda$ is a cross-section of $W/W_\lambda$ and $s_\beta \neq 1$, we see $s_\beta \not\in W_\lambda$. Hence $$-\beta^\vee(\lambda)\beta = (\lambda - \beta^\vee(\lambda)\beta) - \lambda = s_\beta \lambda - \lambda \not\in {\mathbb{Z}}\cdot \Sigma,$$ and $\beta^\vee(\lambda) \not\in {\mathbb{Z}}$. Therefore $\beta \not\in \Pi_\lambda$. Suppose $\beta \not\in \Pi_\lambda$. Since the only positive root moved out of $\Sigma^+$ by $s_\beta$ is $\beta$, and $\beta$ is not in $\Sigma_\lambda$, we see that $s_\beta \Sigma_\lambda^+ \subseteq \Sigma^+$. This implies $s_\beta \in A_\lambda$ by (\[eqn:A\_lambda\_alt\]).

In particular, (c) and (d) implies that conjugation by $u \in A_\lambda$ sends simple reflections in $W_\lambda$ to simple reflections in $W_{u \lambda}$. This implies the following corollary. Recall that ${\leqslant}_\lambda$ denotes the Bruhat order on $(W_\lambda,\Pi_\lambda)$.

\[lem:Bruhat\_order\_conj\] Let $u \in A_\lambda$. Then conjugation by $u$ is an isomorphism of posets $$(W_\lambda,{\leqslant}_\lambda) {\xrightarrow{\,\raisebox{-1.2ex}[0ex][1ex]{${\mathrel{\overset{}{\scalebox{1}[1]{$\sim$}}}}$\,}}}(W_{u\lambda},{\leqslant}_{u\lambda}).$$

Restrictions of Bruhat order to left $W_\lambda$-cosets
-------------------------------------------------------

The next result (\[lem:u\_smallest\]) concerns the Bruhat order when restricted to cosets $u W_\lambda$. Namely, left multiplication by an element in $A_\lambda$ is a map from $W_\lambda$ to $W$ that preserves the Bruhat orders.

\[lem:test\_Bruhat\_order\] Let $w$, $s_\alpha \in W$ with $\alpha \in \Sigma^+$. Let $u \in W$ such that $u\alpha \in \Sigma^+$. Let $\mu$ be a regular antidominant integral weight, that is an element in ${\mathfrak{h}}^*$ such that $\beta^\vee(\mu) \in {\mathbb{Z}}_{<0}$ for all $\beta \in \Sigma^+$. Then $$u s_\alpha w < u w \iff u s_\alpha w \mu < u w \mu.$$ Here the left hand side is the Bruhat order, and the right hand side means that $u w \mu - u s_\alpha w \mu$ is nonzero and is a non-negative sum of simple roots.

We rewrite $$u s_\alpha w \mu = u w \mu - \alpha^\vee(w\mu) u \alpha = u w\mu - (w{^{-1}}\alpha)^\vee(\mu) u \alpha.$$ Hence $$\begin{aligned}
        u s_\alpha w \mu < u w \mu
        &\iff (w{^{-1}}\alpha)^\vee(\mu) u \alpha > 0 \nonumber\\
        &\iff (w{^{-1}}\alpha)^\vee(\mu) > 0 \label{eqn:test_Bruhat_order_a}\\
        &\iff w{^{-1}}\alpha \not\in \Sigma^+ \label{eqn:test_Bruhat_order_b}\\
        &\iff (uw){^{-1}}(u\alpha) \not\in \Sigma^+ \nonumber\\
        &\iff \ell(s_{u\alpha} uw) < \ell(uw) \label{eqn:test_Bruhat_order_d}\\
        &\iff s_{u\alpha} u w < u w, \nonumber
    \end{aligned}$$ where (\[eqn:test\_Bruhat\_order\_a\]) is because $u \alpha \in \Sigma^+$, (\[eqn:test\_Bruhat\_order\_b\]) is because $\mu$ is regular antidominant, (\[eqn:test\_Bruhat\_order\_d\]) is by @Bourbaki:Lie_4-6 [VI.1.6 Proposition 17(ii)], and the last equivalence is by the definition of Bruhat order. Finally, observe that $s_{u\alpha} uw = u s_\alpha u{^{-1}}uw = u s_\alpha w$ by \[lem:non-int\_refl\_subsys\](d). Thus $u s_\alpha w < uw$ as desired.

\[lem:Bruhat\_order\_pre\] Let $w$, $s_\alpha \in W_\lambda$ with $\alpha \in \Sigma_\lambda^+$, and let $u \in A_\lambda$. Suppose $s_\alpha w <_\lambda w$. Then $u s_\alpha w < u w$.

Let $\mu$ be a regular antidominant integral weight in ${\mathfrak{h}}^*$. Consider the projection ${\mathfrak{h}}^* {\twoheadrightarrow}\operatorname{span} \Sigma_\lambda$ along $\bigcap_{\alpha \in \Pi_\lambda} \ker \alpha^\vee$. Let $\bar\mu$ denote the image of $\mu$ under this projection, then $\mu = \bar\mu + \nu$ for some $\nu \in \bigcap_{\alpha \in \Pi_\lambda} \ker \alpha^\vee$, and $\bar\mu$ is regular antidominant integral weight for the integral subsystem $\Sigma_\lambda$ and the set of simple roots $\Pi_\lambda$.

By \[lem:test\_Bruhat\_order\], $$\begin{aligned}
        &s_\alpha w <_\lambda w\\
        &\iff s_\alpha w \bar\mu <_\lambda w \bar\mu\\
        &\iff s_\alpha w \bar\mu + \sum_{\alpha_i \in \Pi_\lambda} a_i \alpha_i = w \bar\mu \text{ for some } a_i \in {\mathbb{Z}}_{{\geqslant}0} \text{ not all zero}\\
        &\iff s_\alpha w \bar\mu + \nu + \sum_{\alpha_i \in \Pi_\lambda} a_i \alpha_i = w \bar\mu + \nu \text{ for some } a_i \in {\mathbb{Z}}_{{\geqslant}0} \text{ not all zero}.
    \end{aligned}$$ Since $s_\alpha w$ and $w$ are generated by the $s_{\alpha_i}$'s, $\alpha_i \in \Pi_\lambda$, and since $\nu$ is fixed by each $s_{\alpha_i}$, we see that $s_\alpha w \nu = w \nu = \nu$. Hence the above condition is equivalent to $$s_\alpha w \mu + \sum_{\alpha_i \in \Pi_\lambda} a_i \alpha_i = w \mu \text{ for some } a_i \in {\mathbb{Z}}_{{\geqslant}0} \text{ not all zero}.$$ Applying $u$ to both sides we get $$u s_\alpha w \mu + \sum_{\alpha_i \in \Pi_\lambda} a_i u \alpha_i = u w \mu \text{ for some } a_i {\geqslant}0 \text{ not all zero}$$ which implies, by the fact that $u \alpha_i \in u \Sigma_\lambda^+ \subseteq \Sigma^+$, $$u s_\alpha w \mu < u w \mu$$ and hence $u s_\alpha w < uw$ by \[lem:test\_Bruhat\_order\].

\[lem:Bruhat\_order\] Let $v,w \in W_\lambda$ and $v {\leqslant}_\lambda w$. Then for any $u \in A_\lambda$, $uv {\leqslant}uw$.

If equality holds, then the statement is trivial. Otherwise, by the definition of Bruhat order, there exist $\alpha_1,\ldots,\alpha_k \in \Sigma_\lambda^+$ such that $$v = s_{\alpha_k} \cdots s_{\alpha_1} w 
        <_\lambda \cdots <_\lambda
        s_{\alpha_1} w <_\lambda w.$$ Apply \[lem:Bruhat\_order\_pre\] to each inequality, we see $$uv = u s_{\alpha_k} \cdots s_{\alpha_1} w 
        < \cdots <
        u s_{\alpha_1} w < u w$$ as desired.

\[lem:u\_smallest\] For any $u \in A_\lambda$, $u$ is the unique smallest element in $u W_\lambda$ with respect to the restriction of Bruhat order to $u A_\lambda$.

The next lemma is analogous to a similar statement for parabolic subgroups (\[lem:Theta\_prelim\]), which we will need in a few occasions. The proof is a standard argument using the lifting property @Bjorner-Brenti:Coxeter [2.2.7].

\[lem:u\_in\_db\_coset\] Let $\alpha \in \Pi$, and $u \in A_\lambda$. Then either $s_\alpha u \in A_\lambda$, or $s_\alpha u \in u W_\lambda$.

Suppose $s_\alpha u \not\in u W_\lambda$. Then $s_\alpha u$ is in a different left $W_\lambda$-coset, i.e. $s_\alpha u \in u' W_\lambda$ for some $u' \in A_\lambda$ with $u' \neq u$. So there exists some $v \in W_\lambda$ such that $s_\alpha u = u' v$. We need to show that $v = 1$.

Write $w_1 \vartriangleleft w_2$ when $w_1 < w_2$ and $\ell(w_1) = \ell(w_2) -1$. From the relation $s_\alpha u = u' v$, either $u'v \vartriangleleft u$ or $u'v \vartriangleright u$. Also $s_\alpha uv{^{-1}}= u'$, so either $u' \vartriangleleft uv{^{-1}}$ or $u' \vartriangleright uv{^{-1}}$. From \[lem:u\_smallest\], we also know $u' {\leqslant}u'v$ and $u {\leqslant}uv{^{-1}}$. We have the following four possibilities.

1.  $\begin{tikzcd}[row sep = small, column sep = small]
                    u' \ar[r, phantom, "\vartriangleright" description] & uv{^{-1}}\ar[d, phantom,  "\rotatebox{-90}{${\geqslant}$}" description]\\
                    u'v \ar[u, phantom,  "\rotatebox{90}{${\geqslant}$}" description] & u \ar[l, phantom,  "\vartriangleleft" description]
                \end{tikzcd}$. This is impossible since it implies $u > u$.

2.  $\begin{tikzcd}[row sep = small, column sep = small]
                    u' \ar[r, phantom, "\vartriangleright" description] & uv{^{-1}}\ar[d, phantom,  "\rotatebox{-90}{${\geqslant}$}" description]\\
                    u'v \ar[u, phantom,  "\rotatebox{90}{${\geqslant}$}" description] & u \ar[l, phantom,  "\vartriangleright" description]
                \end{tikzcd}$. If $u'v > u'$, then from $u'v > u' \vartriangleright uv{^{-1}}{\geqslant}u$ we see that $\ell(u'v) {\geqslant}\ell(u) + 2$, which violates $u'v \vartriangleright u$. Therefore we must have $u'v = u'$ and hence $v = 1$.

3.  $\begin{tikzcd}[row sep = small, column sep = small]
                    u' \ar[r, phantom, "\vartriangleleft" description] & uv{^{-1}}\ar[d, phantom,  "\rotatebox{-90}{${\geqslant}$}" description]\\
                    u'v \ar[u, phantom,  "\rotatebox{90}{${\geqslant}$}" description] & u \ar[l, phantom,  "\vartriangleleft" description]
                \end{tikzcd}$. Same argument as in (b) shows that $v = 1$.

4.  $\begin{tikzcd}[row sep = small, column sep = small]
                    u' \ar[r, phantom, "\vartriangleleft" description] & uv{^{-1}}\ar[d, phantom,  "\rotatebox{-90}{${\geqslant}$}" description]\\
                    u'v \ar[u, phantom,  "\rotatebox{90}{${\geqslant}$}" description] & u \ar[l, phantom,  "\vartriangleright" description]
                \end{tikzcd}$. Let $k = \ell(uv{^{-1}}) - \ell(u)$. Then $$\begin{aligned}
                \ell(u'v) 
                &{\geqslant}\ell(u') \\
                &= \ell(uv{^{-1}}) -1\\
                &= \ell(u) + k-1\\
                &= \ell(u'v) -1 + k-1\\
                &= \ell(u'v) + k-2
            \end{aligned}$$ and $0 {\leqslant}k {\leqslant}2$. If $k = 2$, then $\ell(u') = \ell(u'v)$ and $v = 1$. If $k = 0$, then $\ell(u) = \ell(uv{^{-1}})$, whence $v = 1$. Suppose $k = 1$. Applying the lifting property (see *loc. cit.*) twice, we see $u' {\leqslant}u$ and $u {\leqslant}u'$. So $u' = u$, contradicting our assumption for $u'$. Therefore we must have $v = 1$.

Thus $v = 1$ in all cases, as desired.

Double cosets in the Weyl group {#sec:db_coset}
===============================

In this section, we examine the structure of double $(W_\Theta,W_\lambda)$-cosets in the Weyl group. In \[subsec:db\_coset\_xsec\], we construct a cross-section $A_{\Theta,\lambda}$ of $W_\Theta \backslash W / W_\lambda$ consisting of the unique shortest elements in each double coset (\[thm:cross-section\_db\_coset\]). Next, we show in \[subsec:int\_model\] that, if one looks at the partition of $W_\Theta \backslash W$ given by double cosets $W_\Theta \backslash W / W_\lambda$, then each block in this partition corresponds to a right coset in $W_\lambda$ of a parabolic subgroup of $W_\lambda$, called the "integral model" for this double coset. As mentioned in \[sec:intro\], the parabolic Kazhdan-Lusztig polynomials for $(W_\lambda,\Pi_\lambda)$ with respect to this parabolic subgroup describes the multiplicities of Whittaker modules indexed by right $W_\Theta$-cosets in this double coset. Lastly, in \[subsec:lem\_induction\], we prove a lemma which enables a key induction step in \[subsec:(4)\].

Throughout the section, $\lambda$ denotes an arbitrary element in ${\mathfrak{h}}^*$, and $\Theta$ is any subset of $\Pi$. Results in this section may be known. We include proofs for completeness.

Notations and preliminaries on $W_\Theta \backslash W$
------------------------------------------------------

We first recall some well-known facts of right $W_\Theta$-cosets and partial orders without proof. Write $w_\Theta \in W_\Theta$ for the longest element. The set $${}^\Theta W = \{ w \in W \mid w{^{-1}}\Theta \subseteq - \Sigma^+\}$$ is a cross-section of $W_\Theta \backslash W$ consisting of the longest elements in each coset, and $$\label{eqn:w_Theta_Theta_W}
    w_\Theta {}^\Theta W = \{w \in W \mid w{^{-1}}\Theta \subseteq \Sigma^+\}$$ is a cross-section consisting of the shortest elements in each coset. For a right $W_\Theta$-coset $C$, we write $w^C$ for the corresponding element in ${}^\Theta W$. The restriction of Bruhat order on the set ${}^\Theta W$ defines a partial order on $W_\Theta \backslash W$, which we shall denote by "${\leqslant}$". We will use the phrase "the length of $C$" to refer to the length of the element $w^C$. If ${\Theta_\lambda^u}$ is a subset of $\Pi_\lambda$, we write "${\leqslant}_{u,\lambda}$" for the partial order on $W_{\lambda,{\Theta_\lambda^u}} \backslash W_\lambda$.

The following facts will be used throughout this section.

\[lem:W\_Theta\_permutes\_roots\] Any element in $W_\Theta$ permutes positive roots outside $\Sigma_\Theta^+$, that is, it permutes the set $\Sigma^+ - \Sigma_\Theta^+$.

\[lem:Theta\_prelim\] Let $C$ be a right $W_\Theta$-coset and $\alpha \in \Pi$. Then exactly one of the following happens.

1.  $C s_\alpha > C$. In this case $w^{C s_\alpha} = w^C s_\alpha$, and for any $w \in C$, $w s_\alpha > w$.

2.  $C s_\alpha = C$.

3.  $C s_\alpha < C$. In this case $w^{C s_\alpha} = w^C s_\alpha$, and for any $w \in C$, $w s_\alpha < w$.

Moreover, the identity coset $W_\Theta$ is the only right $W_\Theta$-coset $C$ such that $C s_\alpha {\geqslant}C$ for all $\alpha \in \Pi$.

\[lem:Bruhat\_order\_rcoset\] Let $C$, $D$ be two right $W_\Theta$-cosets. Let $v \in D$, $w \in C$. If $v {\leqslant}w$, then $D {\leqslant}C$.

A cross-section of $W_\Theta \backslash W / W_\lambda$ {#subsec:db_coset_xsec}
------------------------------------------------------

Define the set $A_{\Theta,\lambda} = A_\lambda \cap (w_\Theta {}^\Theta W)$. We will show (in \[thm:cross-section\_db\_coset\]) that this is a cross-section of $W_\Theta \backslash W / W_\lambda$ consisting of the unique shortest elements in each double coset. Later results, as well as the main theorem of the paper, will often be formulated using this set.

\[lem:u\_in\_same\_right\_coset\] Let $u$, $u' \in A_\lambda$. Suppose $u$ and $u'$ are in the same double $(W_\Theta,W_\lambda)$-coset. Then $u$ and $u'$ are contained in the same right $W_\Theta$-coset.

The case $u = u'$ is trivial, so we may assume $u \neq u'$. By assumption, $u' = wuv{^{-1}}$ for some $w \in W_\Theta$ and $v \in W_\lambda$. We will do induction on $\ell(w)$.

Consider the case $\ell(w) = 1$. Then $w = s_\alpha$ for some $\alpha \in \Theta$, and $s_\alpha u = u' v$. By \[lem:u\_in\_db\_coset\] $v = 1$. Hence $u' = s_\alpha u \in W_\Theta u$, and $u$ and $u'$ are in the same right $W_\Theta$-coset.

Consider $\ell(w) = k > 1$. Write $w = s_\alpha w'$ for some $\alpha \in \Theta$ and some $w' \in W_\Theta$ so that $\ell(w) = \ell(w') + 1$. Then $w' u = (s_\alpha u') v$. We have two possibilities.

1.  $s_\alpha u' \in u' W_\lambda$. Then there exists $v' \in W_\lambda$ such that $s_\alpha u' = u' v'$, so $w' u (v' v){^{-1}}= u'$. Since $\ell(w') {\leqslant}k-1$, by the induction assumption, $u$ and $u'$ are in the same right $W_\Theta$-coset.

2.  $s_\alpha u' \not\in u' W_\lambda$. Then by \[lem:u\_in\_db\_coset\], $s_\alpha u' \in A_\lambda$. From the equation $w' u v{^{-1}}= s_\alpha u'$, $\ell(w') {\leqslant}k-1$ and the induction assumption, we see that $u$ and $s_\alpha u'$ are in the same right $W_\Theta$-coset. Since $s_\alpha u'$ and $u'$ are in the same right $W_\Theta$-coset, so are $u$ and $u'$.

\[lem:unique\_smallest\_right\_coset\] Consider any double coset $W_\Theta w W_\lambda$ in $W$.

1.  $W_\Theta w W_\lambda$ contains a unique smallest right $W_\Theta$-coset $C$, in the sense that $C {\leqslant}C'$ for any $C' \in W_\Theta \backslash W_\Theta w W_\lambda$.

2.  $A_\lambda \cap (W_\Theta w W_\lambda) \subseteq C$.

3.  The unique shortest element in $C$ is in $A_\lambda$.

By \[lem:u\_in\_same\_right\_coset\], there exists a right $W_\Theta$-coset $C$, contained in $W_\Theta w W_\lambda$, such that $A_\lambda \cap (W_\Theta w W_\lambda) \subseteq C$. Let $y$ be the unique shortest element in $C$. Say $y \in u A_\lambda$ for some $u \in A_\lambda$. Then $u {\leqslant}y$ by \[lem:u\_smallest\]. If $y \neq u$, we will have $u < y$, and hence by minimality of $y$, $u$ is in a different right $W_\Theta$-coset than $y$, contradicting the construction of $C$. Hence we must have $y = u$, i.e. the unique shortest element in $C$ is in $A_\lambda$. Lastly, for any other right $W_\Theta$-coset $C'$ in $W_\Theta w W_\lambda$, let $y' $ be its unique shortest element. $y'$ is contained in one of the left $W_\lambda$-cosets, say $y' \in u' W_\lambda$ for some $u' \in A_\lambda$. Then $u' {\leqslant}y'$ by \[lem:u\_smallest\]. Also $u' \neq y'$ (otherwise $C' \ni y' = u' \in C$ which would imply $C = C'$). Hence $u' < y'$. Therefore $C < C'$ by \[lem:Bruhat\_order\_rcoset\]. Thus $C$ is the unique smallest right $W_\Theta$-coset in $W_\Theta w W_\lambda$.

\[thm:cross-section\_db\_coset\] $A_{\Theta,\lambda} = A_\lambda \cap (w_\Theta {}^\Theta W)$ is a cross-section of $W_\Theta \backslash W / W_\lambda$ consisting of the unique shortest elements in each double coset. For each $u \in A_{\Theta,\lambda}$, $W_\Theta u$ is the unique smallest right $W_\Theta$-coset in $W_\Theta u W_\lambda$.

Take any double coset $W_\Theta w W_\lambda$. By \[lem:unique\_smallest\_right\_coset\](e), if we take the shortest element $u$ in the smallest right $W_\Theta$-coset in this double coset, then $u \in A_\lambda$. Hence $u \in A_\lambda \cap (w_\Theta {}^\Theta W)$. On the other hand, by \[lem:unique\_smallest\_right\_coset\](b), any other right $W_\Theta$-coset in $W_\Theta w W_\lambda$ has empty intersection with $A_\lambda$. Therefore $A_\lambda \cap (w_\Theta {}^\Theta W) \cap W_\Theta w W_\lambda = \{u\}$. This shows that $A_\lambda \cap (w_\Theta {}^\Theta W)$ is a cross-section.

Integral models {#subsec:int_model}
---------------

In this part, we study the "smallest" left $W_\lambda$-coset contained in $W_\Theta u W_\lambda$.

Let $u \in A_{\Theta,\lambda}$. Then $\Sigma_\Theta \cap \Pi_{u\lambda}$ is a set of simple roots for the root system $\Sigma_\Theta \cap \Sigma_{u\lambda}$.

Let $\beta \in \Sigma_\Theta \cap \Sigma_{u\lambda}$. Write $\beta$ as a ${\mathbb{Z}}_{{\geqslant}0}$-linear combination in terms of reflections of roots in $\Pi_{u\lambda}$. If one of the summands is from $\Pi_{u\lambda} - \Sigma_\Theta$, then writing $\beta$ as a sum of reflections of roots in $\Pi$, there is a summand that comes from $\Pi - \Theta$. This implies $\beta \not\in \Sigma_\Theta$, a contradiction. Hence $\beta$ is a sum of reflections of roots from $\Sigma_\Theta \cap \Pi_{u\lambda}$. Therefore $\Sigma_\Theta \cap \Pi_{u \lambda}$ spans $\Sigma_\Theta \cap \Sigma_{u\lambda}$. Since $\Sigma_\Theta \cap \Pi_{u\lambda}$ is a subset of simple roots in $\Sigma_{u\lambda}$, roots contained in $\Sigma_\Theta \cap \Pi_{u \lambda}$ remain simple in $\Sigma_\Theta \cap \Sigma_{u\lambda}$. Thus $\Sigma_\Theta \cap \Pi_{u \lambda}$ is a set of simple roots for $\Sigma_\Theta \cap \Sigma_{u\lambda}$.

Write $W_{u\lambda, \Sigma_\Theta \cap \Pi_{u \lambda}}$ for the parabolic subgroup of $W_{u\lambda}$ corresponding to $\Sigma_\Theta \cap \Pi_{u \lambda}$. Then $W_{u\lambda, \Sigma_\Theta \cap \Pi_{u \lambda}}$ is the Weyl group of $\Sigma_\Theta \cap \Sigma_{u\lambda}$ and is a subgroup of $W_\Theta \cap W_{u\lambda}$.

\[lem:int\_Whittaker\_model\] For any $u \in A_{\Theta,\lambda}$, $W_\Theta \cap W_{u\lambda} = W_{u \lambda, \Sigma_\Theta \cap \Pi_{u \lambda}}$. In particular, $W_\Theta \cap W_{u\lambda}$ is a parabolic subgroup of $W_{u \lambda}$.

$W_{u \lambda, \Sigma_\Theta \cap \Pi_{u \lambda}}$ is certainly contained in $W_\Theta \cap W_{u\lambda}$. Let $w \in W_\Theta \cap W_{u \lambda}$. Being in $W_\Theta$, $w$ permutes roots in $\Sigma_\Theta$; being in $W_{u\lambda}$, $w$ permutes roots in $\Sigma_{u\lambda}$. Hence $w$ permutes roots in $\Sigma_\Theta \cap \Sigma_{u\lambda}$, and it sends the set $\Sigma^+ \cap (\Sigma_\Theta \cap \Sigma_{u \lambda})$ of positive roots in $\Sigma_\Theta \cap \Sigma_{u \lambda}$ to another set of positive roots $w \Sigma^+ \cap (\Sigma_\Theta \cap \Sigma_{u \lambda})$. Since $W_{u\lambda, \Sigma_\Theta \cap \Pi_{u \lambda}}$ is the Weyl group of $\Sigma_\Theta \cap \Sigma_{u\lambda}$, there exists a unique element $v \in W_{u\lambda, \Sigma_\Theta \cap \Pi_{u \lambda}}$ that sends $w \Sigma^+ \cap (\Sigma_\Theta \cap \Sigma_{u \lambda})$ back to $\Sigma^+ \cap (\Sigma_\Theta \cap \Sigma_{u \lambda})$. Hence $vw$ permutes $\Sigma^+ \cap (\Sigma_\Theta \cap \Sigma_{u \lambda}) = \Sigma_{u\lambda}^+ \cap \Sigma_\Theta^+$. On the other hand, since $vw \in W_\Theta$, by \[lem:W\_Theta\_permutes\_roots\] it permutes $\Sigma^+ - \Sigma_\Theta^+$; $vw$ is also in $W_{u \lambda}$, so it permutes $\Sigma_{u\lambda}$. Hence, it permutes $(\Sigma^+ - \Sigma_\Theta^+) \cap \Sigma_{u\lambda} = \Sigma_{u\lambda}^+ - \Sigma_\Theta^+$. As a result, $vw$ permutes $$\big( \Sigma_{u\lambda}^+ \cap \Sigma_\Theta^+ \big) \cup \big( \Sigma_{u\lambda}^+ - \Sigma_\Theta^+ \big) 
        = \Sigma_{u\lambda}^+.$$ Since $W_{u\lambda}$ acts simply transitively on the set of sets of positive roots in $\Sigma_{u\lambda}$, we must have $vw = 1$. Therefore $w = v{^{-1}}\in W_{u\lambda,\Sigma_\Theta \cap \Pi_{u \lambda}}$. Thus $W_\Theta \cap W_{u\lambda} = W_{u \lambda, \Sigma_\Theta \cap \Pi_{u \lambda}}$, as desired.

For $u \in A_{\Theta,\lambda}$, write $$\Theta_\lambda^u = u{^{-1}}(\Sigma_\Theta \cap \Pi_{u\lambda}) = u{^{-1}}\Sigma_\Theta \cap \Pi_\lambda.$$ Since $\Sigma_\Theta \cap \Pi_{u\lambda}$ is a subset of simple roots in $\Sigma_{u\lambda}$, ${\Theta_\lambda^u}$ is a subset of simple roots in $u{^{-1}}\Sigma_{u \lambda} = \Sigma_\lambda$. Write $W_{\lambda,{\Theta_\lambda^u}}$ for the parabolic subgroup of $W_\lambda$ corresponding to ${\Theta_\lambda^u}$.

\[thm:int\_Whittaker\_model\] Let $u \in A_{\Theta,\lambda}$. The left-multiplication-by-$u$ map $$W_\lambda {\xrightarrow{\,\raisebox{-1.2ex}[0ex][1ex]{${\mathrel{\overset{}{\scalebox{1}[1]{$\sim$}}}}$\,}}}u W_\lambda$$ sends right $W_{\lambda,{\Theta_\lambda^u}}$-cosets in $W_\lambda$ to the subsets $C \cap u W_\lambda$ for right $W_\Theta$-cosets $C$ in $W_\Theta u W_\lambda$. All sets of the form $C \cap u W_\lambda$ are obtained in this way.

Moreover, this map preserves the partial orders on cosets: if $C',D' \in W_{\lambda,{\Theta_\lambda^u}} \backslash W_\lambda$ are sent to $C \cap u W_\lambda$ and $D \cap u W_\lambda$, respectively, then $D' {\leqslant}_{u,\lambda} C'$ implies $D {\leqslant}C$.

Consider the smallest right $W_\Theta$-coset $W_\Theta u$ of $W_\Theta u W_\lambda$. $$\begin{aligned}
        W_\Theta u \cap u W_\lambda 
        &= (W_\Theta \cap u W_\lambda u{^{-1}}) u\\
        &= (W_\Theta \cap W_{u\lambda} )u\\
        &= W_{u\lambda, \Sigma_\Theta \cap \Pi_{u \lambda}} u\\
        &= W_{u\lambda, u{\Theta_\lambda^u}} u\\
        &= (u W_{\lambda,{\Theta_\lambda^u}} u{^{-1}}) u\\
        &= u W_{\lambda ,{\Theta_\lambda^u}}.
    \end{aligned}$$ Hence left multiplication by $u$ sends $W_{\lambda ,{\Theta_\lambda^u}}$ to $W_\Theta u \cap u W_\lambda$. Since left multiplication by $u$ commutes with right multiplication by elements of $W_\lambda$, it sends right $W_{\lambda,{\Theta_\lambda^u}}$-cosets in $W_\lambda$ to right $W_\lambda$-translates of $W_\Theta u \cap u W_\lambda$, which gives us $C \cap u W_\lambda$ for various right $W_\Theta$-cosets $C$ in $W_\Theta u W_\lambda$. Moreover, any right $W_\Theta$-coset $C$ in $W_\Theta u W_\lambda$ is obtained as a right $W_\lambda$-translation of $W_\Theta u$, hence the intersection $C \cap u W_\lambda$ is necessarily the image of a right $W_{\lambda,{\Theta_\lambda^u}}$-coset.

To show that this map is order preserving, take two right $W_{\lambda,{\Theta_\lambda^u}}$-cosets $C'$ and $D'$ such that $D' {\leqslant}_{u,\lambda} C'$. This means that the ${\leqslant}_\lambda$-longest elements $v^{D'}$, $v^{C'}$ of $D'$ and $C'$ satisfy $v^{D'} {\leqslant}_\lambda v^{C'}$. Since left multiplication by $u$ preserves Bruhat orders (\[lem:Bruhat\_order\]), $u v^{D'} {\leqslant}u v^{C'}$. Therefore $D {\leqslant}C$ by \[lem:Bruhat\_order\_rcoset\].

\[lem:right\_coset\_partition\] As $u$ ranges over $A_{\Theta,\lambda}$, left multiplication by $W_\Theta u$ defines a bijection $$r_\lambda{^{-1}}: \bigcup_{u \in A_{\Theta,\lambda}} W_{\lambda, {\Theta_\lambda^u}} \backslash W_\lambda {\xrightarrow{\,\raisebox{-1.2ex}[0ex][1ex]{${\mathrel{\overset{}{\scalebox{1}[1]{$\sim$}}}}$\,}}}W_\Theta \backslash W$$ which is order-preserving when restricted to each $W_{\lambda, {\Theta_\lambda^u}} \backslash W_\lambda$ and commutes with right multiplication by $W_\lambda$. The image of $W_{\lambda, {\Theta_\lambda^u}} \backslash W_\lambda$ equals $W_\Theta \backslash W_\Theta u W_\lambda$.

We will write $r_\lambda: W_\Theta \backslash W \to \bigcup_{u \in A_{\Theta,\lambda}} W_{\lambda, {\Theta_\lambda^u}} \backslash W_\lambda$ for the inverse map.

The map $r_\lambda$ plays an important role towards our goal. Each nonzero standard (resp. irreducible) Whittaker module corresponds to a right $W_\Theta$-cosets $C$ contained in some double coset $W_\Theta u W_\lambda$. Each such $C$ corresponds, through the map $r_\lambda$, to a right $W_{\lambda,{\Theta_\lambda^u}}$-coset in $W_\lambda$, which in turn corresponds to a standard (resp. irreducible) Whittaker module for the data $(W_\lambda, \Pi_\lambda, {\Theta_\lambda^u})$. In \[sec:KL\], we will see that this correspondence preserves Whittaker Kazhnda-Lusztig polynomials. By @Romanov:Whittaker, Whittaker Kazhdan-Lusztig polynomials for Whittaker modules for the data $(W_\lambda, \Pi_\lambda, {\Theta_\lambda^u})$ are given by parabolic Kazhdan-Lusztig polynomials. In this way, we can describe characters of $L(w^C \lambda,\eta)$, $C \in W_\Theta \backslash W_\Theta u W_\lambda$ using parabolic Kazhdan-Lusztig polynomials for $(W_\lambda,\Pi_\lambda,\Theta_\lambda^u)$.

We also need to understand how the map $r_\lambda$ behaves under right multiplication by a non-integral simple reflection. This reflects the effect of non-integral intertwining functors which will be defined in \[subsec:intertwining\_functors\].

\[lem:Is\_pres\_lowest\_db\_coset\] Let $u \in A_{\Theta,\lambda}$, $\beta \in \Pi - \Pi_\lambda$. Then $W_\Theta u s_\beta$ is the smallest right $W_\Theta$-coset in $W_\Theta u s_\beta W_{s_\beta \lambda} = W_\Theta u W_\lambda s_\beta$.

By \[lem:non-int\_refl\_subsys\](e)(f), $u s_\beta \in A_{s_\beta \lambda}$. Hence the claim follows from \[lem:unique\_smallest\_right\_coset\].

Let $u \in A_{\Theta,\lambda}$, $\beta \in \Pi - \Pi_\lambda$. If $u' \in A_{\Theta,s_\beta \lambda}$ denotes the unique element such that $W_\Theta u' W_{s_\beta \lambda} = W_\Theta us_\beta W_{s_\beta \lambda}$, then $W_\Theta u' = W_\Theta u s_\beta$.

It is easy to see that $u' = u s_\beta$ if $W_\Theta u s_\beta \neq W_\Theta u$ and $u' = u$ if $W_\Theta u s_\beta = W_\Theta u$. But we will not need this fact.

\[lem:Is\_right\_coset\] Let $u \in A_{\Theta,\lambda}$, $\beta \in \Pi - \Pi_\lambda$. Let $u' \in A_{\Theta,s_\beta \lambda}$ be the unique element such that $W_\Theta u' W_{s_\beta \lambda} = W_\Theta u s_\beta W_{s_\beta \lambda}$. Then conjugation by $s_\beta$ is a bijection $$s_\beta (-) s_\beta : W_{s_\beta \lambda, \Theta_{s_\beta \lambda}^{u'}} \backslash W_{s_\beta \lambda} {\xrightarrow{\,\raisebox{-1.2ex}[0ex][1ex]{${\mathrel{\overset{}{\scalebox{1}[1]{$\sim$}}}}$\,}}}W_{\lambda,\Theta_\lambda^u} \backslash W_\lambda$$ that preserves the partial orders on right cosets. Moreover, the following diagram commutes $$\label{eqn:Is_right_coset}
        \begin{tikzcd}
            W_{s_\beta \lambda, \Theta_{s_\beta \lambda}^{u'}} \backslash W_{s_\beta \lambda} \ar[d,"r_{s_\beta \lambda}{^{-1}}"] \ar[r,"s_\beta (-) s_\beta"] &
            W_{\lambda,\Theta_\lambda^u} \backslash W_\lambda \ar[d,"r_\lambda{^{-1}}"]\\
            W_\Theta \backslash W \ar[r," (-) s_\beta"] & W_\Theta \backslash W
        \end{tikzcd}.$$ In particular, for any $C$, $D \in W_\Theta \backslash W_\Theta u' W_{s_\beta \lambda}$, $$r_{s_\beta\lambda} D {\leqslant}_{u',s_\beta \lambda} r_{s_\beta\lambda} C \iff r_\lambda (D s_\beta) {\leqslant}_{u,\lambda} r_\lambda (C s_\beta).$$

By \[lem:Bruhat\_order\_conj\], conjugation by $s_\beta$ is an isomorphism of groups and posets between $(W_{s_\beta \lambda},{\leqslant}_{s_\beta\lambda})$ and $(W_\lambda,{\leqslant}_\lambda)$. By the preceding corollary, there exists $w \in W_\Theta$ such that $w u' = u s_\beta$. Therefore $$\begin{aligned}
        s_\beta \Theta_\lambda^u
        &= s_\beta (u{^{-1}}\Sigma_\Theta \cap \Pi_\lambda)\\
        &= (us_\beta){^{-1}}\Sigma_\Theta \cap s_\beta \Pi_\lambda\\
        &= (wu'){^{-1}}\Sigma_\Theta \cap \Pi_{s_\beta \lambda}\\
        &= u'{}{^{-1}}(w{^{-1}}\Sigma_\Theta) \cap \Pi_{s_\beta \lambda}\\
        &= u'{}{^{-1}}\Sigma_\Theta \cap \Pi_{s_\beta \lambda}\\
        &= \Theta_{s_\beta \lambda}^{u'}.
    \end{aligned}$$ Hence conjugation by $s_\beta$ sends $W_{s_\beta \lambda, \Theta_{s_\beta \lambda}^{u'}}$ to $W_{\lambda,\Theta_\lambda^u}$ and therefore induces a bijection from $W_{s_\beta \lambda,\Theta_{s_\beta \lambda}^{u'}} \backslash W_{s_\beta \lambda}$ to $W_{\lambda,\Theta_\lambda^u} \backslash W_\lambda$. Furthermore, since conjugation by $s_\beta$ preserves Bruhat orders, it also preserves the partial orders on right cosets.

To check that the diagram commutes, take any $D' \in W_{s_\beta \lambda, \Theta_{s_\beta \lambda}^{u'}} \backslash W_{s_\beta \lambda}$. Along the top-right path, $D'$ is sent to $$W_\Theta u \cdot s_\beta D' s_\beta = W_\Theta w u' D' s_\beta = W_\Theta u' D' s_\beta,$$ which agrees with the image along the bottom-left path.

A technical lemma {#subsec:lem_induction}
-----------------

In the last part of this section, we prove a technical lemma that will be used in \[subsec:(4)\] in induction process.

\[lem:decrease\_of\_length\] Let $u \in A_{\Theta,\lambda}$ and $C \in W_\Theta \backslash W_\Theta u W_\lambda$. Suppose $C \neq W_\Theta u$. Then there exist $\alpha \in \Pi_\lambda$, $s {\geqslant}0$ and $\beta_1,\ldots,\beta_s \in \Pi$ such that, writing $z_0 = 1$, $z_i = s_{\beta_1} \cdots s_{\beta_i}$ and $z = z_s$, the following conditions hold:

1.  for any $0 {\leqslant}i {\leqslant}s-1$, $\beta_{i+1}$ is non-integral to $z_i{^{-1}}\lambda$;

2.  $z{^{-1}}\alpha \in \Pi \cap \Pi_{z{^{-1}}\lambda}$;

3.  $r_\lambda C s_\alpha <_{u,\lambda} r_\lambda C$;

4.  if $s > 0$, $Cz < C$;

5.  $C s_\alpha z = C z s_{z{^{-1}}\alpha} < C z$.

Since $C \neq W_\Theta u$, in particular $C \neq W_\Theta$, there exists a simple reflection $s_\gamma$ such that $C s_\gamma < C$.

If there exists $\alpha \in \Pi \cap \Pi_\lambda$ such that $C s_\alpha < C$, then this $\alpha$ together with $s = 0$ satisfies the requirement: (a) and (d) are void, while (b) and (e) are true by construction. We need to verify (c). Since $s_\alpha$ is simple in $(W_\lambda, \Pi_\lambda)$, we have three mutually exclusive possibilities: $r_\lambda  C s_\alpha <_{u,\lambda} r_\lambda C$, $r_\lambda  C s_\alpha = r_\lambda C$, or $r_\lambda  C s_\alpha >_{u,\lambda} r_\lambda C$. Since the map $r_\lambda{^{-1}}$ preserves the partial order, they imply $C s_\alpha < C$, $C s_\alpha = C$ and $C s_\alpha > C$, respectively. By our choice of $\alpha$, the last two possibilities cannot happen. Hence we must have $r_\lambda  C s_\alpha <_{u,\lambda} r_\lambda C$, which shows (c).

Suppose such $\alpha$ does not exist. Then any simple reflection that decreases the length of $C$ via right multiplication must be non-integral to $\lambda$. Let $s_{\beta_1}$, $\beta_1 \in \Pi - \Pi_\lambda$, be one of those. If there exists $\alpha' \in \Pi \cap \Pi_{s_{\beta_1} \lambda}$ with $C s_{\beta_1} s_{\alpha'} < C s_{\beta_1}$, we claim that $\alpha := s_{\beta_1} \alpha' \in s_{\beta_1} \Pi_{s_{\beta_1} \lambda} = \Pi_\lambda$, $s = 1$ and $\beta_1$ satisfy our requirements. (a) and (d) follows by our choice of $s_{\beta_1}$, (e) follows from the conditions on $\alpha'$. For (b), $$z{^{-1}}\alpha = s_{\beta_1} s_{\beta_1} \alpha' = \alpha' \in \Pi \cap \Pi_{s_{\beta_1} \lambda}$$ by definition of $z$ and $\alpha'$. It suffices to verify (c). We again have three possibilities: $r_\lambda C s_\alpha <_{u,\lambda} r_\lambda C$, $r_\lambda C s_\alpha = r_\lambda C$, or $r_\lambda C s_\alpha >_{u,\lambda} r_\lambda C$, which imply $C s_\alpha < C$, $C s_\alpha = C$ and $C s_\alpha > C$, respectively. To rule out some of the cases, we start with the inequality $$\begin{aligned}
        C > C s_{\beta_1} > C s_{\beta_1} s_{\alpha'} 
        = C s_{\beta_1} s_{s_{\beta_1} \alpha}
        = C s_{\beta_1} s_{\beta_1} s_\alpha s_{\beta_1}
        = C s_\alpha s_{\beta_1}.
    \end{aligned}$$ This implies $C s_\alpha < C$, and hence the cases $r_\lambda C s_\alpha = r_\lambda C$, and $r_\lambda C s_\alpha >_{u,\lambda} r_\lambda C$ cannot happen. Therefore $r_\lambda C s_\alpha <_{u,\lambda} r_\lambda C$ holds, and (c) is verified.

If such $\alpha'$ does not exist, then we can find $\beta_2,\ldots,\beta_s \in \Pi$ such that $C z_{i+1} < C z_i$ for all $1 {\leqslant}i {\leqslant}s-1$ until we get to a point where there exists $\alpha'' \in \Pi \cap \Pi_{z{^{-1}}\lambda}$ with $C z s_{\alpha''} < C z$ (termination of this process is proven in the next paragraph). We claim that $\alpha := z\alpha'' \in z \Pi_{z{^{-1}}\lambda} = \Pi_\lambda$, $s$ and $\beta_1,\ldots,\beta_s$ satisfy our requirements. The verification is essentially the same as in the previous case. (a), (b), (d) and (e) are satisfied by our choice of $\beta_i$'s and $\alpha''$. For (c), we have an inequality $$\label{eqn:decrease_of_length_step_a}
        C z > C z s_{\alpha''} = C z s_{z{^{-1}}\alpha} = C z z{^{-1}}s_\alpha z = C s_\alpha z$$ where $\ell(w^{Cz}) = \ell(w^C z) = \ell(w^C) - s$. Also $w^{C z s_{\alpha''}} = w^C z s_{\alpha''} = w^C s_\alpha z$. Hence $$\ell(w^C s_\alpha) 
        = \ell(w^C s_\alpha z z{^{-1}}) 
        = \ell(w^{C s_\alpha z} z{^{-1}}) 
        {\leqslant}\ell(w^{C s_\alpha z}) +s
        = \ell(w^{Cz}) -1 +s
        = \ell(w^C) -1 < \ell(w^C).$$ This rules out $C s_\alpha > C$ and hence the possibility $r_\lambda  C s_\alpha >_{u,\lambda} r_\lambda C$. We also cannot have $r_\lambda  C s_\alpha = r_\lambda C$ as it would imply $C s_\alpha = C$ and consequently $C s_\alpha z = Cz$, contradicting (\[eqn:decrease\_of\_length\_step\_a\]). Therefore $r_\lambda  C s_\alpha <_{u,\lambda} r_\lambda C$, and (c) is thus verified.

Lastly, let us show that this process of finding $\alpha''$ must terminate no later than when we get to $\ell(w^{Cz}) = \ell(w_\Theta)+1$. That is, we show that when $\ell(w^{Cz}) = \ell(w_\Theta)+1$, such an $\alpha''$ must exist. The condition $\ell(w^{Cz}) = \ell(w_\Theta)+1$ implies $C z = W_\Theta s_\gamma > W_\Theta$ for some simple reflection $s_\gamma$. If $\gamma \in \Pi - \Pi_{z{^{-1}}\lambda}$, then $s_\gamma \in A_{z{^{-1}}\lambda}$. Also, since $W_\Theta s_\gamma > W_\Theta$, any element of $W_\Theta s_\gamma$ must have length ${\geqslant}1$. Hence $s_\gamma$ is the shortest element of $W_\Theta s_\gamma$, i.e. $s_\gamma \in w_\Theta {}^\Theta W$. Therefore $s_\gamma \in A_{z{^{-1}}\lambda} \cap (w_\Theta {}^\Theta W) = A_{\Theta,z{^{-1}}\lambda}$. Since $C = W_\Theta s_\gamma z{^{-1}}$, by (repeatedly applying) \[lem:Is\_pres\_lowest\_db\_coset\], we see that $C$ is the smallest right $W_\Theta$-coset in the double $(W_\Theta,W_\lambda)$-coset containing it, that is, $C = W_\Theta u$. This contradicts our assumption on $C$. Therefore $\gamma \in \Pi \cap \Pi_\lambda$, and $\alpha'' = \gamma$ satisfies our requirement for $\alpha''$. Thus the process terminates.

Non-integral intertwining functors {#sec:geom}
==================================

In this section, we review basic facts on localizations and twisted sheaves, and prove some results on non-integral intertwining functors.

Preliminaries on localization and twisted sheaves {#subsec:geom_prelim}
-------------------------------------------------

References for facts below include @Beilinson-Bernstein:Localization, @Jantzen_Conj, @Milicic-Soergel:Whittaker_geometric, @Milicic:Localization, @Romanov:Whittaker.

Let $\theta$ be a Weyl group orbit in ${\mathfrak{h}}^*$, and let ${\mathfrak{m}}_\theta$ be the maximal ideal in ${\mathcal{Z}}({\mathfrak{g}})$ determined by the infinitesimal character corresponding to $\theta$. Let $X$ be the flag variety of ${\mathfrak{g}}$, viewed as the variety of Borel subalgebras of ${\mathfrak{g}}$. For a point $x \in X$, the corresponding Borel subalgebra is denoted by ${\mathfrak{b}}_x$. For each $\lambda \in \theta$, Beilinson and Bernstein constructed in @Beilinson-Bernstein:Localization a twisted sheaf of differential operators ${\mathcal{D}}_\lambda$ on $X$ whose global sections are equal to ${\mathcal{U}}_\theta := {\mathcal{U}}({\mathfrak{g}})/ {\mathcal{U}}({\mathfrak{g}}) {\mathfrak{m}}_\theta$. Here, we use the parametrization as in @Milicic:Localization, under which ${\mathcal{D}}_{-\rho} = {\mathcal{D}}_X$. Recall that $\lambda \in {\mathfrak{h}}^*$ is called antidominant regular if $\alpha^\vee(\lambda) \not\in {\mathbb{Z}}_{{\geqslant}0}$ for all $\alpha \in \Sigma^+$. If $\lambda$ is antidominant and regular, Beilinson and Bernstein showed that taking global sections is an equivalence of categories $$\Gamma(X,-): \operatorname{Mod}_{qc}({\mathcal{D}}_\lambda) \cong \operatorname{Mod}({\mathcal{U}}_\theta)$$ between the category of quasi-coherent ${\mathcal{D}}_\lambda$-modules and the category of ${\mathcal{U}}_\theta$-modules, and a quasi-inverse is given by the "localization functor" ${\mathcal{D}}_\lambda \dotimes_{{\mathcal{U}}_\theta} -$. The subcategory ${\mathcal{N}}_{\theta,\eta}$ of $\operatorname{Mod}({\mathcal{U}}_\theta)$ corresponds, under the above equivalence of categories, to the category $\operatorname{Mod}_{coh}({\mathcal{D}}_\lambda,N,\eta)$ of **$\eta$-twisted Harish-Chandra sheaves**. This is the full subcategory of all coherent ${\mathcal{D}}_\lambda$-modules consisting of weakly $N$-equivariant ${\mathcal{D}}_\lambda$-modules such that the differential of $N$-action and the action of ${\mathfrak{n}}\subset {\mathcal{D}}_\lambda$ differ by $\eta$.

If $\lambda$ is antidominant and singular (i.e. not regular), then we instead have an equivalence of categories $$\label{eqn:BB_equivalence_singular}
    \Gamma(X,-): {\mathscr{Q}}\operatorname{Mod}_{qc}({\mathcal{D}}_\lambda) \cong \operatorname{Mod}({\mathcal{U}}_\theta)$$ where ${\mathscr{Q}}\operatorname{Mod}_{qc}({\mathcal{D}}_\lambda)$ is the quotient of $\operatorname{Mod}_{qc}({\mathcal{D}}_\lambda)$ by objects with zero global section.

In our notation for holonomic ${\mathcal{D}}$-modules, for a morphism $f$ between smooth varieties, we have direct images $f_+$, $f_!$ and inverse images $f^+$, $f^!$. Here $f_+$ agrees with the one in @Borel:D-mods [VI.5], and also agrees with the $*$-direct image in the usual six-functor formalism. $f_!$ is the functor obtained by conjugating $f_+$ by holonomic duality. $f^!$ agrees with the one defined in @Borel:D-mods [VI.4]. When $f$ is a closed immersion of a subvariety, $H^0 f^! {\mathcal{V}}$ consists of sections of ${\mathcal{V}}$ supported in the subvariety. $f^+$ is a shift of $f^!$ by the relative dimension; forgetting the ${\mathcal{D}}$-module structures, $f^+$ agrees with the usual ${\mathcal{O}}$-module inverse image $f^*$. $\eta$-twisted Harish-Chandra sheaves are functorial with respect to all these operations.

Let $C(w)$, $w \in W$ be the Schubert cells (i.e. $N$-orbits) on $X$, with inclusion maps $i_w: C(w) \to X$. There exist nonzero $\eta$-twisted Harish-Chandra sheaves on $C(w)$ if and only if $w$ is the longest element in the right $W_\Theta$-coset that contains it. If this is the case, the category $\operatorname{Mod}_{coh}({\mathcal{D}}_{C(w)},N,\eta)$ is semisimple, in which the unique irreducible object, denoted by ${\mathcal{O}}_{C(w)}^\eta$, has ${\mathcal{O}}_{C(w)}$ as the underlying structure of an $N$-equivariant ${\mathcal{O}}_{C(w)}$-module, but with an $\eta$-twisted ${\mathcal{D}}_{C(w)}$-action. The ${\mathcal{D}}$-module direct images $${\mathcal{I}}(w,\lambda,\eta) = i_{w,+} {\mathcal{O}}_{C(w)}^\eta,\qquad
    {\mathcal{M}}(w,\lambda,\eta) = i_{w,!} {\mathcal{O}}_{C(w)}^\eta$$ are called the standard module and costandard module, respectively. The standard module ${\mathcal{I}}(w,\lambda,\eta)$ contains a unique irreducible submodule, denoted by ${\mathcal{L}}(w,\lambda,\eta)$, and ${\mathcal{L}}(w,\lambda,\eta)$ is the unique irreducible quotient of ${\mathcal{M}}(w,\lambda,\eta)$. These exhaust all irreducible objects in $\operatorname{Mod}_{coh}({\mathcal{D}}_\lambda,N,\eta)$. If $w$ is the longest element in a right $W_\Theta$-coset and $\lambda$ is antidominant, Beilinson-Bernstein's equivalence of categories sends ${\mathcal{M}}(w,\lambda,\eta)$ to $M(w\lambda,\eta)$ and ${\mathcal{L}}(w,\lambda,\eta)$ to either $L(w\lambda,\eta)$ or $0$. If $\lambda$ is furthermore regular, ${\mathcal{L}}(w,\lambda,\eta)$ is always sent to $L(w\lambda,\eta)$. This allows us to study Whittaker modules using geometry on $X$.

For certain reasons, ${\mathcal{I}}(w,\lambda,\eta)$ is more convenient to work with than ${\mathcal{M}}(w,\lambda,\eta)$. The holonomic duality ${\mathbb{D}}$ sends ${\mathcal{I}}(w,\lambda,\eta)$ and ${\mathcal{L}}(w,\lambda,\eta)$ to ${\mathcal{M}}(w,-\lambda,\eta)$ and ${\mathcal{L}}(w,-\lambda,\eta)$, respectively. Hence, we have the following flowchart $${\mathcal{N}}_{\theta,\eta} \xrightarrow{{\mathcal{D}}_\lambda\dotimes_{{\mathcal{U}}_\theta}-} \operatorname{Mod}_{coh}({\mathcal{D}}_\lambda,N,\eta) \xrightarrow{{\mathbb{D}}} \operatorname{Mod}_{coh}({\mathcal{D}}_{-\lambda},N,\eta),$$ $$L(w^C\lambda,\eta) \mapsto {\mathcal{L}}(w^C,\lambda,\eta) \mapsto {\mathcal{L}}(w^C,-\lambda,\eta),$$ $$M(w^D\lambda,\eta) \mapsto {\mathcal{M}}(w^C,\lambda,\eta) \mapsto {\mathcal{I}}(w^C,-\lambda,\eta).$$ Modules in $\operatorname{Mod}_{coh}({\mathcal{D}}_{-\lambda},N,\eta)$ are all holonomic and have finite length. Hence the set of irreducible objects form a basis for the Grothendieck group $K \operatorname{Mod}_{coh}({\mathcal{D}}_{-\lambda},N,\eta)$. A standard argument using pullback-pushforward adjunctions shows that the set of standard modules also form a basis for $K \operatorname{Mod}_{coh}({\mathcal{D}}_\lambda,N,\eta)$. Therefore, our goal of finding coefficients of $\operatorname{ch}M(w^D\lambda,\eta)$ in $\operatorname{ch}L(w^C\lambda,\eta)$ is translated to finding the change of bases matrix.

In the rest of the paper, we will use facts about ${\mathcal{D}}$-modules without citing references, including the triangle for closed-open immersions, base change theorem, and Kashiwara's equivalence of categories. These facts are contained in @Borel:D-mods, IV.8.3, 8.4 and 7.11, respectively.

Non-integral intertwining functors {#subsec:intertwining_functors}
----------------------------------

Let $\lambda \in {\mathfrak{h}}^*$ be arbitrary and let $\eta: {\mathfrak{n}}\to {\mathbb{C}}$ be any character. Recall that ${}^\Theta W$ denotes the set of longest elements in right $W_\Theta$-cosets. For a right $W_\Theta$-coset $C$, the corresponding element in ${}^\Theta W$ is denoted by $w^C$.

For any $w \in W$, let $Z_w$ denote the subset of $X \times X$ consists of pairs $(x,y)$ such that ${\mathfrak{b}}_x$ and ${\mathfrak{b}}_y$ are in relative position $w$. This means that for any common Cartan subalgebra ${\mathfrak{c}}$ and any representative of $w$ in $N_G({\mathfrak{c}})$ (also denoted by $w$), ${\mathfrak{b}}_x = \operatorname{Ad}w({\mathfrak{b}}_y)$. If $w$ is fixed, we write $$X \xleftarrow{p_1} Z_w \xrightarrow{p_2} X$$ for the two projections. For an integral weight $\mu \in {\mathfrak{h}}^*$, write ${\mathcal{O}}_X(\mu)$ for the $G$-homogeneous line bundle on $X$ where the ${\mathfrak{b}}$-action on the geometric fiber at $x_{\mathfrak{b}}\in X$ (the point on $X$ that corresponds to ${\mathfrak{b}}$) is given by $\mu$.

For $w \in W$ and $\lambda \in {\mathfrak{h}}^*$, the **intertwining functor** $LI_w$ is defined to be $$\begin{aligned}
        LI_w : D^b({\mathcal{D}}_\lambda) &\to D^b({\mathcal{D}}_{w\lambda}),\notag\\
        {\mathcal{F}}^\bullet &\mapsto p_{1+} \big( p_1^* {\mathcal{O}}_X(\rho-w\rho) \dotimes_{{\mathcal{O}}_{Z_w}} p_2^+ {\mathcal{F}}^\bullet \big)\notag\\
        &\qquad = {\mathcal{O}}_X(\rho-w\rho) \dotimes_{{\mathcal{O}}_X} p_{1+} p_2^+ {\mathcal{F}}^\bullet.
    \end{aligned}$$ Write $I_w$ for $H^0 LI_w$. It is shown in @Milicic:Localization [L.3] that $LI_w$ is the left derived functor of $I_w$.

For properties of intertwining functors including \[lem:non-int\_I\] below, readers can refer to *loc. cit.* In the definition, the tensor product with the line bundle is a twist which ensures that the functor lands in the correct category. The main property we will use is the fact that, when $w = s_\beta$ is a simple reflection non-integral to $\lambda$, $I_{s_\beta}$ is an equivalence of categories. When appropriately applied to ${\mathcal{D}}$-modules on the flag variety, $I_{s_\beta}$ enlarges the support of an irreducible module without losing information, which helps with induction on the dimension of the support.

\[lem:non-int\_I\] If $\beta \in \Pi - \Pi_\lambda$, then $I_{s_\beta}$ is an equivalence of categories $$I_{s_\beta} : \operatorname{Mod}_{qc}({\mathcal{D}}_\lambda) \cong \operatorname{Mod}_{qc}({\mathcal{D}}_{s_\beta\lambda})$$ whose quasi-inverse is $I_{s_\beta}$.

To use these functors for our purpose, we need to compute the action of intertwining functors on standard and irreducible modules. Romanov computed the following result for $Cs_\beta > C$. The main ingredients of the proof there are base change formula and projection formula for ${\mathcal{D}}$-modules.

\[<span>@Romanov:Whittaker [3.15]</span>\]\[lem:I\_moves\_std\_general\] Let $\beta \in \Pi$ and $C \in W_\Theta \backslash W$ such that $C s_\beta > C$. Then for any $\lambda \in {\mathfrak{h}}^*$, $$LI_{s_\beta} {\mathcal{I}}(w^C,\lambda,\eta) = {\mathcal{I}}(w^C s_\beta, s_\beta \lambda,\eta).$$

Combined with \[lem:non-int\_I\] we get

\[lem:I\_moves\_std\] Let $\beta \in \Pi - \Pi_\lambda$ and $C \in W_\Theta \backslash W$ such that $C s_\beta \neq C$. Then $$I_{s_\beta} {\mathcal{I}}(w^C,\lambda,\eta) = {\mathcal{I}}(w^C s_\beta, s_\beta \lambda, \eta).$$

Suppose $C s_\beta > C$, then the statement follows from \[lem:I\_moves\_std\_general\]. But since $I_{s_\beta}$ is an equivalence of categories with inverse $I_{s_\beta}$, $${\mathcal{I}}(w^C,\lambda,\eta) = I_{s_\beta} {\mathcal{I}}(w^C s_\beta, s_\beta \lambda, \eta).$$

It remains to consider the case $C s_\beta = C$.

For a simple root $\beta$, write $X_\beta$ for the partial flag variety of type $\beta$, and write $p_\beta: X \to X_\beta$ for the natural projection. This is a Zariski-local ${\mathbb{A}}^1$-fibration. $x$ and $y$ are contained in the same $p_\beta$-fiber (i.e. $p_\beta(x) = p_\beta(y)$) if and only if ${\mathfrak{b}}_x$ and ${\mathfrak{b}}_y$ are in relative position $1$ or $s_\beta$.

\[lem:S\] Let $C \in W_\Theta \backslash W$ and $\beta \in \Pi$ such that $C s_\beta = C$. Set $$S = \{ (x,y) \in C(w^C) \times C(w^C) \mid {\mathfrak{b}}_x \text{ and } {\mathfrak{b}}_y \text{ are in relative position } s_\beta\} \subset Z_{s_\beta}.$$ Write $$C(w^C) \xleftarrow{p_1|_S} S \xrightarrow{p_2|_S} C(w^C)$$ for the projections. Then $$(p_1|_S)_+ (p_2|_S)^+ {\mathcal{O}}_{C(w^C)}^\eta = {\mathcal{O}}_{C(w^C)}^\eta.$$

For convenience, write $w = w^C$, $p_1 = p_1|_S$ and $p_2 = p_2|_S$. Set $$S' = C(w) \times_{p_\beta(C(w))} C(w) = \{ (x,y) \in C(w) \times C(w) \mid p_\beta(x) = p_\beta(y) \}.$$ Then $S \subset S \cup \Delta_{C(w)} = S' \subset Z_{s_\beta}$, where $\Delta_{C(w)}$ denotes the diagonal. Write $$C(w) \xleftarrow{q_1} S' \xrightarrow{q_2} C(w)$$ for the projections, and $$\Delta_{C(w)} \xrightarrow{i_\Delta} S' \xleftarrow{i_S} S$$ for the inclusions. Then $i_\Delta$ is a closed immersion with relative dimension $1$, and $i_S$ is open. We have the following diagram $$\label{diag:S}
        \begin{tikzcd}
            S \ar[dr, "i_S", crossing over] \ar[ddr, bend right, "p_1"'] \ar[rrd, bend left, "p_2"] &[-5ex] \\[-2ex]
            & S' \ar[d, "q_1"] \ar[r, "q_2"] & C(w) \ar[d, "p_\beta"]\\
            & C(w) \ar[r, "p_\beta"] & p_\beta(C(w))
        \end{tikzcd}$$ where the bottom-right square is Cartesian.

Applying the usual closed-open triangle to $q_2^+ {\mathcal{O}}_{C(w)}^\eta$, we get $${    \begin{tikzcd}[column sep = tiny, row sep = small, ampersand replacement = \&]
    { i_{\Delta+} i_\Delta^! q_2^+ {\mathcal{O}}_{C(w)}^\eta } \ar[rr] \&\& { q_2^+ {\mathcal{O}}_{C(w)}^\eta } \ar[dl]\\
    \& { i_{S+} i_S^+ q_2^+ {\mathcal{O}}_{C(w)}^\eta} \ar[ul,"{[1]}"]
    \end{tikzcd}}.$$ Applying $q_{1+}$, we get $${    \begin{tikzcd}[column sep = tiny, row sep = small, ampersand replacement = \&]
    { q_{1+} i_{\Delta+} i_\Delta^! q_2^+ {\mathcal{O}}_{C(w)}^\eta } \ar[rr] \&\& { q_{1+} q_2^+ {\mathcal{O}}_{C(w)}^\eta } \ar[dl]\\
    \& { q_{1+} i_{S+} i_S^+ q_2^+ {\mathcal{O}}_{C(w)}^\eta} \ar[ul,"{[1]}"]
    \end{tikzcd}}.$$ Applying base change to the bottom-right square in (\[diag:S\]), $q_{1+} q_2^+ {\mathcal{O}}_{C(w)} \cong p_\beta^+ p_{\beta+} {\mathcal{O}}_{C(w)}^\eta$. Here $p_{\beta+} {\mathcal{O}}_{C(w)}^\eta$ is an $\eta$-twisted Harish-Chandra sheaf on $p_\beta(C(w))$. But $p_\beta(C(w))$ is isomorphic to $C(w s_\beta)$ as an $N$-variety via $p_\beta$, and since $w s_\beta$ is not the longest element in $W_\Theta w s_\beta = W_\Theta w$, we know there is no $\eta$-twisted Harish-Chandra sheaf on $C(w s_\beta)$ except $0$. Hence $p_{\beta+} {\mathcal{O}}_{C(w)}^\eta = 0$ and thus $q_{1+} q_2^+ {\mathcal{O}}_{C(w)} = 0$. As a result, $$q_{1+} i_{S+} i_S^+ q_2^+ {\mathcal{O}}_{C(w)}^\eta = q_{1+} i_{\Delta+} i_\Delta^! q_2^+ {\mathcal{O}}_{C(w)}^\eta [1].$$ We simplify both sides. For the left hand side, $i_s^+ q_2 ^+ = (q_2 \circ i_s)^+ = p_2^+$, and $q_{1+} i_{S+} = (q_1 \circ i_S)_+ = p_{1+}$, hence the left hand side equals $p_{1+} p_2^+ {\mathcal{O}}_{C(w)}^\eta$. For the right hand side, $q_{1+} i_{\Delta+} = (q_1 \circ i_\Delta)_+$ and $q_1 \circ i_\Delta$ is the projection $\Delta_{C(w)} \to C(w)$ along the first coordinate which is an $N$-equivariant isomorphism. Moreover, $$\begin{aligned}
        i_\Delta^! q_2^+ {\mathcal{O}}_{C(w)}^\eta [1]
        &= i_\Delta^+ q_2^+ {\mathcal{O}}_{C(w)}^\eta [1] [-1]\\
        &= (q_2 \circ i_\Delta)^+ {\mathcal{O}}_{C(w)}^\eta,
    \end{aligned}$$ and $q_2 \circ i_\Delta$ is the projection $\Delta_{C(w)} \to C(w)$ along the second coordinate, also an $N$-equivariant isomorphism. Thus $$p_{1+} p_2^+ {\mathcal{O}}_{C(w)}^\eta
        = (q_1 \circ i_\Delta)_+ (q_2 \circ i_\Delta)^+ {\mathcal{O}}_{C(w)}^\eta 
        = {\mathcal{O}}_{C(w)}^\eta.$$

\[lem:two\_cells\] Let $s_\beta \in \Pi$ and $C \in W_\Theta \backslash W$ such that $C s_\beta = C$. Write $\iota_{w^C}: C(w^C) {\hookrightarrow}C(w^C) \cup C(w^C s_\beta)$ for the inclusion. Then for any ${\mathcal{F}}\in \operatorname{Mod}_{coh}({\mathcal{D}}_{C(w^C) \cup C(w^C s_\beta)},N,\eta)$, $${\mathcal{F}}= \iota_{w^C+} \iota_{w^C}^! {\mathcal{F}}= ( \iota_{w^C +} {\mathcal{O}}_{C(w^C)}^\eta )^{\oplus \operatorname{rank}_{\mathcal{O}}\iota_{w^C}^! {\mathcal{F}}}.$$

Write $w = w^C$. The assumption implies that $w s_\beta \in C$, $w s_\beta < w$, and that $C(w)$ and $C(w s_\beta)$ are open and closed in $C(w) \cup C(w s_\beta)$, respectively.

Since the category of $\eta$-twisted Harish-Chandra sheaves on $C(w)$ is semisimple, $\iota_w^! {\mathcal{F}}$ is a direct sum of copies of ${\mathcal{O}}_{C(w)}^\eta$. This implies the second equality. For the first equality, adjunction gives a map $$\label{eqn:iota_w_adjunction}
        {\mathcal{F}}\to \iota_{w+} \iota_w^! {\mathcal{F}}$$ whose kernel and cokernel are supported on $C(w s_\beta)$, which are equal to direct images of $\eta$-twisted Harish-Chandra sheaves on $C(w s_\beta)$ by Kashiwara's equivalence. But $w s_\beta$ is not the longest element in $C$, so there is no such module on $C(w s_\beta)$ except zero. Hence (\[eqn:iota\_w\_adjunction\]) is an isomorphism, which establishes the first equality.

\[lem:I\_fixes\_std\] Let $C \in W_\Theta \backslash W$, $\beta \in \Pi$ such that $C s_\beta = C$. Then $$LI_{s_\beta} {\mathcal{I}}(w^C,\lambda,\eta) = {\mathcal{I}}(w^C, s_\beta \lambda,\eta).$$

Write $w = w^C$. Let $$F = Z_{s_\beta} \times_{p_2,X,i_w} C(w) = \{ (x,y) \in X \times C(w) \mid {\mathfrak{b}}_x \text{ and } {\mathfrak{b}}_y \text{ are in relative position } s_\beta \}.$$ and let $S$ be as in \[lem:S\]. Then $S$ is a subvariety of $F$. It's easy to see that $$p_1(F) = \{x \in X \mid {\,\exists\,}y \in C(w) \text{ such that } {\mathfrak{b}}_x \text{ and } {\mathfrak{b}}_y \text{ are in relative position } s_\beta \} = C(w) \cup C(w s_\beta).$$ So we have the following diagram $$\label{diag:SF}
        \begin{tikzcd}
            &[-5ex] S \ar[dl,"p_1|_S"'] \ar[dr, "a_S"] & [-4ex]\\
            C(w) \ar[dr, "\iota_w"] & & F \ar[dl, "p_1|_F"'] \ar[dr, "i_F"'] \ar[rr, equal] & & F \ar[dl, "i_F"] \ar[dr, "p_2|_F"]\\
            & C(w) \cup C(w s_\beta) \ar[dr, "j_w"] && Z_{s_\beta} \ar[dl, "p_1"'] \ar[dr, "p_2"] && C(w) \ar[dl, "i_w"']\\
            && X && X
        \end{tikzcd}$$ The right-most square is Cartesian by definition of $F$. The top-left square is also Cartesian, i.e. $S$ is the preimage of $C(w)$ along $p_1: F \to C(w) \cup C(w s_\beta)$. Using base change, $$\begin{aligned}
        LI_{s_\beta} {\mathcal{I}}(w,\lambda,\eta)
        &= {\mathcal{O}}_X(\rho - s_\beta \rho) \dotimes_{{\mathcal{O}}_X} p_{1+} p_2^+ i_{w+} {\mathcal{O}}_{C(w)}^\eta \nonumber\\
        &= {\mathcal{O}}_X(\rho - s_\beta \rho) \dotimes_{{\mathcal{O}}_X} p_{1+} i_{F+} (p_2|_F)^+ {\mathcal{O}}_{C(w)}^\eta \nonumber\\
        &= {\mathcal{O}}_X(\rho - s_\beta \rho) \dotimes_{{\mathcal{O}}_X} j_{w+} (p_1|_F)_+ (p_2|_F)^+ {\mathcal{O}}_{C(w)}^\eta. \label{eqn:LI(std)_step_a}
    \end{aligned}$$ We claim that $(p_1|_F)_+ (p_2|_F)^+ {\mathcal{O}}_{C(w)}^\eta = \iota_{w+} {\mathcal{O}}_{C(w)}^\eta$. By \[lem:two\_cells\], $$(p_1|_F)_+ (p_2|_F)^+ {\mathcal{O}}_{C(w)}^\eta = \iota_{w+} \iota_w^! (p_1|_F)_+ (p_2|_F)^+ {\mathcal{O}}_{C(w)}^\eta.$$ Apply base change using the top-left square in (\[diag:SF\]), $$\iota_w^! (p_1|_F)_+ (p_2|_F)^+ {\mathcal{O}}_{C(w)}^\eta 
        = (p_1|_S)_+ a_S^! (p_2|_F)^+ {\mathcal{O}}_{C(w)}^\eta
        = (p_1|_S)_+ a_S^+ (p_2|_F)^+ {\mathcal{O}}_{C(w)}^\eta$$ Note that $p_2|_F \circ a_S = p_2|_S$. Hence, by \[lem:S\], the sheaf in the above equation equals ${\mathcal{O}}_{C(w)}^\eta$. Therefore $$(p_1|_F)_+ (p_2|_F)^+ {\mathcal{O}}_{C(w)}^\eta = \iota_{w+} \iota_w^! (p_1|_F)_+ (p_2|_F)^+ {\mathcal{O}}_{C(w)}^\eta = \iota_{w+} {\mathcal{O}}_{C(w)}^\eta$$ as claimed. As a result, $$\begin{aligned}
        (\ref{eqn:LI(std)_step_a}) 
        &= {\mathcal{O}}_X(\rho - s_\beta \rho) \dotimes_{{\mathcal{O}}_X} j_{w+} \iota_{w+} {\mathcal{O}}_{C(w)}^\eta\\
        &= {\mathcal{O}}_X(\rho - s_\beta \rho) \dotimes_{{\mathcal{O}}_X} i_{w+} {\mathcal{O}}_{C(w)}^\eta\\
        &= {\mathcal{I}}(w, s_\beta \lambda, \eta)    
    \end{aligned}$$ which proves the proposition.

\[lem:I\_on\_std\] Let $\beta \in \Pi - \Pi_\lambda$. Let $C \in W_\Theta \backslash W$. Then $$\begin{aligned}
        I_{s_\beta} {\mathcal{I}}(w^C,\lambda,\eta) &= {\mathcal{I}}(w^{C s_\beta}, s_\beta\lambda, \eta),\\
        I_{s_\beta} {\mathcal{L}}(w^C,\lambda,\eta) &= {\mathcal{L}}(w^{C s_\beta}, s_\beta\lambda, \eta)
    \end{aligned}$$ (note that we have $w^{C s_\beta}$ instead of $w^C s_\beta$ on the right hand sides).

The statement about standard modules is the combination of \[lem:I\_moves\_std\] and \[lem:I\_fixes\_std\]. Since $I_{s_\beta}$ is an equivalence of categories, it must send the unique irreducible submodule of ${\mathcal{I}}(w^C,\lambda,\eta)$ to the unique irreducible submodule of ${\mathcal{I}}(w^{C s_\beta}, s_\beta\lambda, \eta)$, i.e. it must send ${\mathcal{L}}(w^C,\lambda,\eta)$ to ${\mathcal{L}}(w^{C s_\beta}, s_\beta\lambda, \eta)$.

Next, we show that non-integral intertwining functors also preserves pullback of irreducible modules to strata.

\[thm:Is\_pullback\] Let $\beta \in \Pi - \Pi_\lambda$, $C,D \in W_\Theta \backslash W$ and $p \in {\mathbb{Z}}$. Then $$\operatorname{rank}_{\mathcal{O}}H^p i_{w^D}^! {\mathcal{L}}(w^C, \lambda,\eta) = \operatorname{rank}_{\mathcal{O}}H^p i_{w^{D s_\beta}}^! {\mathcal{L}}(w^{C s_\beta}, s_\beta \lambda, \eta).$$

The proof we give below uses the same tools as in the previous proposition. An alternative proof goes as follows. Using semisimplicity of $\operatorname{Mod}_{coh}({\mathcal{D}}_{C(w^C)},N,\eta)$, inverse-image and direct-image adjunction, and some facts on derived categories of highest weight categories, one can express $\operatorname{rank}_{\mathcal{O}}H^p i_{w^D}^! {\mathcal{L}}(w^C, \lambda,\eta)$ as the dimension of the $p$-th ${\mathcal{D}}_\lambda$-module $\operatorname{Ext}$ group of ${\mathcal{M}}(w^C,\lambda,\eta)$ and ${\mathcal{L}}(w^C,\lambda,\eta)$. Since $I_{s_\beta}$ is an equivalence of categories, the proposition follows from the fact that $I_{s_\beta}$ induces an isomorphism on $\operatorname{Ext}$-groups.

Write $w = w^D$.

There are two cases, $D s_\beta \neq D$ or $D s_\beta = D$. Consider the first case. Assume $D s_\beta < D$. Then $w^{D s_\beta} = w^D s_\beta = w s_\beta$. Let $$F = C(w s_\beta) \times_{i_{w s_\beta}, X, p_1} Z_{s_\beta} = \{ (x,y) \in C(w s_\beta) \times X \mid {\mathfrak{b}}_x \text{ and } {\mathfrak{b}}_y \text{ are in relative position } s_\beta \}.$$ Then the second projection $p_2|_F: F \to X$ induces an isomorphism of $F$ onto $C(w)$, and we have the following commuting diagram $$\begin{tikzcd}
            &[-1ex] F \ar[dl, "p_1|_F"'] \ar[dr, "i_F"'] \ar[rr, equal] & & F \ar[dl,"i_F"] \ar[dr, "p_2|_F", "\cong"']\\
            C(w s_\beta) \ar[dr, "i_{w s_\beta}"'] && Z_{s_\beta} \ar[dl, "p_1"'] \ar[dr, "p_2"] && C(w) \ar[dl, "i_w"]\\
            & X && X
        \end{tikzcd}$$ where the left square is Cartesian. Using base change, one computes $$\begin{aligned}
        i_{w^{D s_\beta}}^! {\mathcal{L}}(w^{C s_\beta} ,s_\beta \lambda, \eta) 
        &= i_{w s_\beta}^! I_{s_\beta} {\mathcal{L}}(w^C, \lambda,\eta) \nonumber\\
        &= i_{w s_\beta}^! p_{1+} p_2^+ {\mathcal{L}}(w^C, \lambda,\eta) \label{eqn:rk_pullback_step_a}\\
        &= (p_1|_F)_+ i_F^! p_2^+ {\mathcal{L}}(w^C, \lambda,\eta) \nonumber\\
        &= (p_1|_F)_+ i_F^! p_2^! {\mathcal{L}}(w^C, \lambda,\eta) [-1] \nonumber\\
        &= (p_1|_F)_+ (p_2|_F)^! i_w^! {\mathcal{L}}(w^C, \lambda,\eta) [-1].\label{eqn:rk_pullback_step_b}
    \end{aligned}$$ Here in (\[eqn:rk\_pullback\_step\_a\]) we did not write the twist by the line bundle ${\mathcal{O}}_X(\rho-w\rho)$ because there is no twist on $C(w s_\beta)$. Since $\operatorname{Mod}_{coh}({\mathcal{D}}_{C(w)},N,\eta)$ is semisimple, we can write $$\begin{aligned}
        i_w^! {\mathcal{L}}(w^C, \lambda,\eta) 
        &= \bigoplus_{p \in {\mathbb{Z}}} \Big( H^p i_w^! {\mathcal{L}}(w^C, \lambda,\eta) [-p] \Big)\\
        &= \bigoplus_{p \in {\mathbb{Z}}} \Big( {\mathcal{O}}_{C(w)}^\eta {}^{\oplus \operatorname{rank}_{\mathcal{O}}H^p i_w^! {\mathcal{L}}(w^C, \lambda,\eta)} [-p] \Big).
    \end{aligned}$$ Therefore, using the fact that $p_2|_F$ is an isomorphism, $$\label{eqn:rk_pullback_step_c}
        (\ref{eqn:rk_pullback_step_b})
        = \bigoplus_{p \in {\mathbb{Z}}} \Big( (p_1|_F)_+ {\mathcal{O}}_F^\eta {}^{\oplus \operatorname{rank}_{\mathcal{O}}H^p i_w^! {\mathcal{L}}(w^C, \lambda,\eta)} [-p-1] \Big).$$ To compute $(p_1|_F)_+ {\mathcal{O}}_F^\eta$, we use the fact that a map of homogeneous spaces of a unipotent group is isomorphic to a coordinate projection of affine spaces, that is, we have the following commutative diagram where all maps are $N$-equivariant, for some $N$-actions on ${\mathbb{A}}^1 \times {\mathbb{A}}^{\ell(w s_\beta)}$ and ${\mathbb{A}}^{\ell(w s_\beta)}$: $$\begin{tikzcd}
            F \ar[d,"\cong"'] \ar[r, "p_1|_F"] & C(w s_\beta) \ar[d,"\cong"]\\
            {\mathbb{A}}^1 \times {\mathbb{A}}^{\ell(w s_\beta)} \ar[r,"pr_1"] & {\mathbb{A}}^{\ell(w s_\beta)}
        \end{tikzcd}.$$ So it suffices to compute $pr_{1+} {\mathcal{O}}_{{\mathbb{A}}^1 \times {\mathbb{A}}^{\ell(w s_\beta)}}^\eta$. Since $pr_1$ is a coordinate projection, $pr_1^+ {\mathcal{O}}_{{\mathbb{A}}^{\ell(w s_\beta)}}^\eta = {\mathcal{O}}_{{\mathbb{A}}^1} \boxtimes {\mathcal{O}}_{{\mathbb{A}}^{\ell(w s_\beta)}}^\eta$ (we remark that, without the assumption of $D s_\beta \neq D$, $w s_\beta$ and $w$ can be in the same right $W_\Theta$-coset, in which case ${\mathcal{O}}_{{\mathbb{A}}^{\ell(w s_\beta)}}^\eta$ does not exist). On the other hand, by functoriality of $\eta$-twisted Harish-Chandra sheaves, we must have $pr_1^+ {\mathcal{O}}_{{\mathbb{A}}^{\ell(w s_\beta)}}^\eta = {\mathcal{O}}_{{\mathbb{A}}^1 \times {\mathbb{A}}^{\ell(w s_\beta)}}^\eta$. We conclude that $${\mathcal{O}}_{{\mathbb{A}}^1 \times {\mathbb{A}}^{\ell(w s_\beta)}}^\eta = {\mathcal{O}}_{{\mathbb{A}}^1} \boxtimes {\mathcal{O}}_{{\mathbb{A}}^{\ell(w s_\beta)}}^\eta.$$ As a result, writing $p: {\mathbb{A}}^1 \to \{*\}$ for the unique morphism to a point, $$\begin{aligned}
        pr_{1+} {\mathcal{O}}_{{\mathbb{A}}^1 \times {\mathbb{A}}^{\ell(w s_\beta)}}^\eta
        &= pr_{1+} ( {\mathcal{O}}_{{\mathbb{A}}^1} \boxtimes {\mathcal{O}}_{{\mathbb{A}}^{\ell(w s_\beta)}}^\eta )\\
        &= (p_+ {\mathcal{O}}_{{\mathbb{A}}^1}) \boxtimes \big( (\operatorname{Id}_{{\mathbb{A}}^{\ell(w s_\beta)}})_+ {\mathcal{O}}_{{\mathbb{A}}^{\ell(w s_\beta)}}^\eta \big) \\
        &= {\mathbb{C}}[1] \boxtimes {\mathcal{O}}_{{\mathbb{A}}^{\ell(w s_\beta)}}^\eta\\
        &= {\mathcal{O}}_{{\mathbb{A}}^{\ell(w s_\beta)}}[1].
    \end{aligned}$$ Therefore, $$(p_1|_F)_+ {\mathcal{O}}_F^\eta = {\mathcal{O}}_{C(w s_\beta)}^\eta [1]$$ and hence $$\begin{aligned}
        i_{w^{D s_\beta}}^! {\mathcal{L}}(w^{C s_\beta} ,s_\beta \lambda, \eta) 
        &= (\ref{eqn:rk_pullback_step_c})\\
        &= \bigoplus_{p \in {\mathbb{Z}}} \Big( ({\mathcal{O}}_{C(w s_\beta)}^\eta[1])^{\oplus \operatorname{rank}_{\mathcal{O}}H^p i_w^! {\mathcal{L}}(w^C, \lambda,\eta)} [-p-1] \Big)\\
        &= \bigoplus_{p \in {\mathbb{Z}}} \Big( {\mathcal{O}}_{C(w s_\beta)}^\eta {}^{\oplus \operatorname{rank}_{\mathcal{O}}H^p i_w^! {\mathcal{L}}(w^C, \lambda,\eta)} [-p] \Big).
    \end{aligned}$$ Thus $$\begin{aligned}
        \operatorname{rank}_{\mathcal{O}}H^p i_{w^{D s_\beta}}^! {\mathcal{L}}(w^{C s_\beta} ,s_\beta \lambda, \eta)
        &= \operatorname{rank}_{\mathcal{O}}{\mathcal{O}}_{C(w s_\beta)}^\eta {}^{\oplus \operatorname{rank}_{\mathcal{O}}H^p i_w^! {\mathcal{L}}(w^C, \lambda,\eta)}\\
        &= \operatorname{rank}_{\mathcal{O}}H^p i_w^! {\mathcal{L}}(w^C, \lambda,\eta).
    \end{aligned}$$

Now consider the case $D s_\beta = D$. In this case $w^{D s_\beta} = w^D = w$. Set $$F = C(w) \times_{i_w, X, p_1} Z_{s_\beta} 
        = \{ (x,y) \in C(w) \times X \mid {\mathfrak{b}}_x \text{ and } {\mathfrak{b}}_y \text{ are in relative position } s_\beta \}$$ and set $S$ as in \[lem:S\], viewed as a subvariety of $F$. Then the following diagram commutes $$\begin{tikzcd}
            &&&&[-4ex] S \ar[dl, "b_S"'] \ar[dr, "p_2|_S"] &[-5ex]\\
            & F \ar[dl, "p_1|_F"'] \ar[dr, "i_F"'] \ar[rr,equal] && F \ar[dl, "i_F"] \ar[dr, "p_2|_F"] && C(w) \ar[dl, "\iota_w"']\\
            C(w) \ar[dr, "i_w"'] && Z_{s_\beta} \ar[dl, "p_1"'] \ar[dr, "p_2"] && C(w) \cup C(w s_\beta) \ar[dl, "j_w"]\\
            & X && X
        \end{tikzcd}$$ where the left-most square and the top-right square are Cartesian. Using base change, $$\begin{aligned}
        i_{w^{D s_\beta}}^! {\mathcal{L}}(w^{C s_\beta} ,s_\beta \lambda, \eta)
        &= i_w^! I_{s_\beta} {\mathcal{L}}(w^C, \lambda,\eta) \nonumber\\
        &= i_w^! p_{1+} p_2^+ {\mathcal{L}}(w^C, \lambda,\eta) \nonumber\\
        &= (p_1|_F)_+ i_F^! p_2^! {\mathcal{L}}(w^C, \lambda,\eta) [-1] \nonumber\\
        &= (p_1|_F)_+ (p_2|_F)^! j_w^! {\mathcal{L}}(w^C, \lambda,\eta) [-1].\label{eqn:rk_pullback_step_d}
    \end{aligned}$$ By \[lem:two\_cells\], $$j_w^! {\mathcal{L}}(w^C, \lambda,\eta) = \iota_{w+} \iota_w^! j_w^! {\mathcal{L}}(w^C, \lambda,\eta).$$ Hence $$\begin{aligned}
        (\ref{eqn:rk_pullback_step_d})
        &= (p_1|_F)_+ (p_2|_F)^! \iota_{w+} \iota_w^! j_w^! {\mathcal{L}}(w^C, \lambda,\eta) [-1] \nonumber\\
        &= (p_1|_F)_+ b_{S+} (p_2|_S)^! i_w^! {\mathcal{L}}(w^C, \lambda,\eta) [-1] \nonumber\\
        &= (p_1|_F)_+ b_{S+} (p_2|_S)^+ i_w^! {\mathcal{L}}(w^C, \lambda,\eta). \label{eqn:rk_pullback_step_e}
    \end{aligned}$$ Here $p_1|_F \circ b_S = p_1|_S$. Also $i_w^! {\mathcal{L}}(w^C, \lambda,\eta)$ is a direct sum of ${\mathcal{O}}_{C(w)}^\eta$ in various degrees. Hence by \[lem:S\], $$\begin{aligned}
        \operatorname{rank}_{\mathcal{O}}H^p i_{w^{D s_\beta}}^! {\mathcal{L}}(w^{C s_\beta} ,s_\beta \lambda, \eta)
        &= \operatorname{rank}_{\mathcal{O}}H^p (\ref{eqn:rk_pullback_step_e})\\
        &= \operatorname{rank}_{\mathcal{O}}H^p i_w^! {\mathcal{L}}(w^C, \lambda,\eta).
    \end{aligned}$$

Main algorithm and main results {#sec:KL}
===============================

In this section, we formulate an algorithm for computing a set of polynomials in $q$ indexed by pairs of right $W_\Theta$-cosets whose evaluation at $q = -1$ leads to the character formula for irreducible modules. This is in the same spirit as ordinary Kazhdan-Lusztig polynomials for category ${\mathcal{O}}$. The algorithm we will prove is suggested by D. Milicic.

In \[subsec:KL\_poly\], we define the parabolic Kazhdan-Lusztig polynomials, the module ${\mathcal{H}}_\Theta$, and related notations. The main algorithm is contained in \[subsec:algorithm\]. Proof of the algorithm is divided into subsections, from \[subsec:algorithm\] to \[subsec:(4)\].

The parabolic Hecke algebra and parabolic Kazhdan-Lusztig polynomials {#subsec:KL_poly}
---------------------------------------------------------------------

Let us define or recall the objects that will be used to formulate the algorithm.

Recall the sets $A_{\Theta,\lambda} = A_\lambda \cap (w_\Theta {}^\Theta W)$ and ${\Theta_\lambda^u} \subseteq \Pi_\lambda$ defined in \[subsec:db\_coset\_xsec\] and \[subsec:int\_model\]. Recall also that we have a partial order on $W_\Theta \backslash W$ inherited from the Bruhat order on ${}^\Theta W$, denoted by ${\leqslant}$. Similarly, we have a partial order on $W_{\lambda,{\Theta_\lambda^u}} \backslash W_\lambda$ which we denote by ${\leqslant}_{u,\lambda}$.

Let $u$ denote an element in $A_{\Theta,\lambda}$. Let ${\mathcal{H}}_{{\Theta_\lambda^u},\lambda}$ be the free ${\mathbb{Z}}[q,q{^{-1}}]$-module with basis $\delta_{C'}$, $C' \in W_{\lambda,{\Theta_\lambda^u}} \backslash W_\lambda$. For any $\alpha \in \Pi_\lambda$, define a ${\mathbb{Z}}[q,q{^{-1}}]$-linear operator on ${\mathcal{H}}_{{\Theta_\lambda^u},\lambda}$ by $$T_\alpha^{u,\lambda} (\delta_{C'}) =
    \begin{cases}
        q \delta_{C'} + \delta_{C' s_\alpha} & \text{if } C' s_\alpha >_{u,\lambda} C';\\
        0 & \text{if } C' s_\alpha = C';\\
        q{^{-1}}\delta_{C'} + \delta_{C' s_\alpha} & \text{if } C' s_\alpha <_{u,\lambda} C'.
    \end{cases}$$ Let ${\mathcal{H}}_\Theta$ be the free ${\mathbb{Z}}[q,q{^{-1}}]$-module with basis $\delta_C$, $C \in W_\Theta \backslash W$, and $T_\alpha$, $\alpha \in \Pi$ is defined in the same way as $T_\alpha^{u,\lambda}$, replacing $C'$ by $C$ and $>_{u,\lambda}$, $<_{u,\lambda}$ by $>$, $<$, respectively. The module ${\mathcal{H}}_\Theta$ is supposed to be the underlying module of the parabolic Hecke algebra, and $T_\alpha$ is supposed to encode information of the parabolic Kazhdan-Lusztig basis. However, we will not explicitly use the multiplicative structure on ${\mathcal{H}}_\Theta$.

We will use a left action of $W$ on ${\mathcal{H}}_\Theta$ defined by $w \cdot \delta_C = \delta_{wC}$. Similarly, a right action of $W$ on ${\mathcal{H}}_\Theta$ is defined by $\delta_C \cdot w = \delta_{C w}$. We will simply write $w \delta_C$, $\delta_C w$ for the actions, omitting the dots. $w(-)w{^{-1}}$ then denotes the simultaneous action of $w$ on the left and $w{^{-1}}$ on the right. By \[lem:Is\_right\_coset\], $s_\beta (-) s_\beta$ defines a bijection $$s_\beta(-) s_\beta : W_{\lambda,\Theta_\lambda^u} \backslash W_\lambda {\xrightarrow{\,\raisebox{-1.2ex}[0ex][1ex]{${\mathrel{\overset{}{\scalebox{1}[1]{$\sim$}}}}$\,}}}W_{s_\beta \lambda, \Theta_{s_\beta \lambda}^{u'}} \backslash W_{s_\beta \lambda}$$ where $u' \in A_{\Theta,s_\beta \lambda}$ is the unique element such that $W_\Theta u' W_{s_\beta \lambda} = W_\Theta u s_\beta W_{s_\beta \lambda}$. We extend this to an isomorphism $$s_\beta (-) s_\beta : {\mathcal{H}}_{{\Theta_\lambda^u},\lambda} {\xrightarrow{\,\raisebox{-1.2ex}[0ex][1ex]{${\mathrel{\overset{}{\scalebox{1}[1]{$\sim$}}}}$\,}}}{\mathcal{H}}_{\Theta_{s_\beta\lambda}^{u'},s_\beta \lambda},\quad
    \delta_{C'} \mapsto \delta_{s_\beta C' s_\beta}.$$

Recall that we have a bijection $$r_\lambda : W_\Theta \backslash W \to \bigcup_{u \in A_{\Theta,\lambda}} W_{\lambda, {\Theta_\lambda^u}} \backslash W_\lambda$$ defined in \[lem:right\_coset\_partition\]. We extend $r_\lambda$ to a map $$r_\lambda: {\mathcal{H}}_\Theta {\xrightarrow{\,\raisebox{-1.2ex}[0ex][1ex]{${\mathrel{\overset{}{\scalebox{1}[1]{$\sim$}}}}$\,}}}\bigoplus_{u \in A_{\Theta,\lambda}} {\mathcal{H}}_{{\Theta_\lambda^u},\lambda},\quad
    \delta_C \mapsto \delta_{r_\lambda C}.$$

The following theorem, proved in @Romanov:Whittaker [5.1], defines a set of polynomials indexed by pairs of right cosets. It is verified in *op. cit.* Remark 6.4 that these polynomials agree with the **parabolic Kazhdan-Lusztig polynomials**. More precisely, in the case $W_\lambda = W$ and $\Theta_\lambda^u = \Theta$, the polynomials $P_{C'D'}$ are equal to the polynomials $n_{y,x}$ in @Soergel:KL_Tilting [3.2.1] for $x = w_\Theta w^{C'}$, $y = w_\Theta w^{D'}$.

For a right coset $E' \in W_{\lambda,{\Theta_\lambda^u}} \backslash W_\lambda$, we write $(W_{\lambda,{\Theta_\lambda^u}} \backslash W_\lambda)_{{\leqslant}_{u,\lambda} E'}$ for the set of those cosets $C'$ such that $C' {\leqslant}_{u,\lambda} E'$.

\[Parabolic Kazhdan-Lusztig polynomials for $(W_\lambda,\Pi_\lambda,\Theta_\lambda^u)$\]\[def:parabolic\_KL\_poly\] For any $E' \in W_{\lambda,{\Theta_\lambda^u}} \backslash W_\lambda$, there exists a unique set of polynomials $\{P_{C'D'}^{u,\lambda}\} \subset q {\mathbb{Z}}[q]$ indexed by $\{(C',D') \in (W_{\lambda,\Theta_\lambda^u} \backslash W_\lambda)_{{\leqslant}_{u,\lambda} E'} \times (W_{\lambda,\Theta_\lambda^u} \backslash W_\lambda)_{{\leqslant}_{u,\lambda} E'} \mid D' <_{u,\lambda} C'\}$ such that the function $$\psi_{u,\lambda}: (W_{\lambda,{\Theta_\lambda^u}} \backslash W_\lambda)_{{\leqslant}_{u,\lambda} E'} {\xrightarrow{\;\;\;\;\;}}{\mathcal{H}}_{{\Theta_\lambda^u},\lambda},\quad
        C' \mapsto \delta_{C'} + \sum_{D' <_{u,\lambda} C'} P_{C'D'}^{u,\lambda} \delta_{D'}$$ satisfies the following property: for any $C' \in W_{\lambda,{\Theta_\lambda^u}} \backslash W_\lambda$ with $C' \neq W_{\lambda,\Theta_\lambda^u}$, there exist $\alpha \in \Pi_\lambda$ and $c_{D'} \in {\mathbb{Z}}$ such that $C' s_\alpha <_{u,\lambda} C'$ and $$\label{eqn:parabolic_KL_poly}
        T_\alpha^{u,\lambda} (\psi_{u,\lambda}(C' s_\alpha)) = \sum_{D' {\leqslant}_{u,\lambda} C'} c_{D'} \psi_{u,\lambda}(D').$$ Moreover, the polynomials $P_{C'D'}^{u,\lambda}$ do not depend on the choice of $E'$.[^1]

Set $P_{C'C'}^{u,\lambda} = 1$ for all $C' \in W_{\lambda,\Theta_\lambda^u} \backslash W_\lambda$. When $\Theta_\lambda^u = \varnothing$, we sometimes write $\{P_{wv}^\lambda\}_{w,v \in W_\lambda}$ for these polynomials. When $W_\lambda = W$ (which implies $\Theta_\lambda^u = \Theta$), these polynomials will be denoted by $\{P_{CD}\}_{C,D \in W_\Theta \backslash W}$.

As shown in @Romanov:Whittaker [6], the above theorem is equivalent to the existence and uniqueness of the usual parabolic Kazhdan-Lusztig basis. Therefore, this theorem can be proven in an entirely combinatorial fashion. However the proof as appeared in *op. cit.* is geometric using ${\mathcal{D}}$-modules on the flag variety. In fact it is the geometric construction that connects these polynomials with the multiplicity problem of Whittaker modules. The same idea is used earlier by Milicic in his unpublished notes @Milicic:Localization and will also be used in our algorithm.

Main algorithm {#subsec:algorithm}
--------------

Define the map $$\nu: \operatorname{Obj} \operatorname{Mod}_{coh}({\mathcal{D}}_\lambda,N,\eta) {\xrightarrow{\;\;\;\;\;}}{\mathcal{H}}_\Theta,\quad
    {\mathcal{F}}\mapsto \sum_{D \in W_\Theta \backslash W} \sum_{p \in {\mathbb{Z}}} \big(\operatorname{rank}_{\mathcal{O}}H^p i_{w^D}^! {\mathcal{F}}\big) q^p \delta_D.$$ Clearly, this map can be defined for any complexes of ${\mathcal{D}}_\lambda$-modules with $\eta$-twisted $N$-equivariant cohomologies. The following easy property of $\nu$ is immediate:

\[lem:nu\_std\] For any $C \in W_\Theta \backslash W$, $\lambda \in {\mathfrak{h}}^*$ and $\eta: {\mathfrak{n}}\to {\mathbb{C}}$, $$\nu({\mathcal{I}}(w^C,\lambda,\eta)) = \delta_C.$$

Let $D \in W_\Theta \backslash W$. Then $i_{w^D}^! {\mathcal{I}}(w^C,\lambda,\eta) = i_{w^D}^! i_{w^C+} {\mathcal{O}}_{C(w^C)}^\eta$. If $C = D$, this is ${\mathcal{O}}_{C(w^C)}^\eta$ by Kashiwara's theorem. Otherwise, this is $0$ by base change. Hence the claim follows by the definition of $\nu$.

\[Kazhdan-Lusztig Algorithm for Whittaker modules\]\[thm:KL\_alg\] Fix a character $\eta: {\mathfrak{n}}\to {\mathbb{C}}$. For any $\lambda \in {\mathfrak{h}}^*$, there exists a unique function $$\varphi_\lambda: W_\Theta \backslash W {\xrightarrow{\;\;\;\;\;}}{\mathcal{H}}_\Theta$$ such that for any $C \in W_\Theta \backslash W$, if we write $u$ for the unique element in $A_{\Theta,\lambda}$ such that $C$ is contained in $W_\Theta u W_\lambda$, the following conditions hold:

1.  for some $P_{r_\lambda C,r_\lambda D}^{u,\lambda} \in q {\mathbb{Z}}[q]$, $$\varphi_\lambda(C) = \delta_C + 
                \sum_{\substack{D \in W_\Theta \backslash W_\Theta u W_\lambda\\                            r_\lambda D <_{u,\lambda} r_\lambda C}} P_{r_\lambda C,r_\lambda D}^{u,\lambda} \delta_D.$$

2.  for any $\alpha \in \Pi \cap \Pi_\lambda$ with $C s_\alpha < C$, there exist $c_D \in {\mathbb{Z}}$ such that $$T_\alpha( \varphi_\lambda (C s_\alpha) ) =      
                \sum_{\substack{D \in W_\Theta \backslash W_\Theta u W_\lambda \\ r_\lambda D {\leqslant}_{u,\lambda} r_\lambda C}} c_D \varphi_\lambda( D)$$

3.  for any $\beta \in \Pi - \Pi_\lambda$ such that $C s_\beta < C$, $$\varphi_{s_\beta \lambda} (C s_\beta) = \varphi_\lambda (C) s_\beta$$ (recall that the action ${\mathcal{H}}_\Theta {\righttoleftarrow}W$ is given by $\delta_C \cdot w = \delta_{C w}$).

4.  The polynomials $P_{r_\lambda C,r_\lambda D}^{u,\lambda}$ are parabolic Kazhdan-Lusztig polynomials for $(W_\lambda,\Pi_\lambda, {\Theta_\lambda^u})$.

Moreover, the map $\varphi_\lambda$ is given by $$\varphi_\lambda(C) = \nu({\mathcal{L}}(w^C,\lambda,\eta)).$$

This algorithm is suggested by Milicic. It is necessary that the theorem works for any $\lambda$. In the proof, we will often translate between different $\lambda$'s. The character formula of irreducible Whittaker modules, which is stated in \[sec:character\_formula\], will be a direct consequence of this algorithm. See \[sec:intro\] for a discussion of the geometric idea behind this algorithm.

Let us start the proof of the theorem. Uniqueness is determined by (1), (4) and the uniqueness of parabolic Kazhdan-Lusztig polynomials. For existence, we will show that $\varphi_\lambda(C) = \nu({\mathcal{L}}(w^C,\lambda,\eta))$ satisfies the requirements (1)-(4) by induction on $\ell(w^C)$.

Consider the base case $\ell(w^C) = \ell(w_\Theta)$, that is, $C = W_\Theta$, $w^C = w_\Theta$. The argument for this case in the same as in @Romanov:Whittaker. Recall that $B \supseteq H$ are the Borel subgroup and the maximal torus corresponding to ${\mathfrak{b}}$ and ${\mathfrak{h}}$, respectively. Write $P_\Theta \supseteq B$ for the parabolic subgroup of $G$ of type $\Theta$. Let $P_\Theta = L_\Theta \ltimes N_\Theta$ for the Levi decomposition of $P_\Theta$ such that $L_\Theta \supseteq H$, and let $S_\Theta = L_\Theta/Z(L_\Theta)$. Let $O$ be the unique closed $P_\Theta$-orbit in $X$ with inclusion maps $i_{w_\Theta,O}: C(w_\Theta) {\hookrightarrow}O$ and $i_O : O {\hookrightarrow}X$. Then $O$ is isomorphic to the flag variety of $S_\Theta$ in which $C(w_\Theta)$ is the open Schubert cell, and $O$ is a closed subvariety of $X$. Write $${\mathcal{I}}(w_\Theta,\lambda,\eta) = i_{w_\Theta +} {\mathcal{O}}_{C(w_\Theta)}^\eta 
    = i_{O+} i_{w_\Theta,O +} {\mathcal{O}}_{C(w_\Theta)}^\eta.$$ By @Milicic-Soergel:Whittaker_geometric [5], $i_{w_\Theta,O +} {\mathcal{O}}_{C(w_\Theta)}^\eta$ is an irreducible ${\mathcal{D}}$-module on $O$. Since $i_O$ is a closed immersion, by Kashiwara's equivalence of categories, $i_{O+}$ preserves irreducibility. Hence ${\mathcal{I}}(w_\Theta,\lambda,\eta)$ is irreducible, and ${\mathcal{I}}(w_\Theta,\lambda,\eta) = {\mathcal{L}}(w_\Theta,\lambda,\eta)$. As a result $$\nu({\mathcal{L}}(w_\Theta,\lambda,\eta))
    = \nu({\mathcal{I}}(w_\Theta,\lambda,\eta))
    = \delta_{W_\Theta}$$ by \[lem:nu\_std\]. Therefore, the function $\varphi_\lambda(C)$ satisfies (1) for $C = W_\Theta$. The conditions (2)-(4) are void. This completes the base case.

Now consider the case $\ell(w^C) = k > \ell(w_\Theta)$. The verification of (1)-(4) for $C$ is divided into subsections.

Inductive step: verification of \[thm:KL\_alg\](3) for $\ell(w^C) = k$ {#subsec:(3)}
----------------------------------------------------------------------

Assume $\beta \in \Pi - \Pi_\lambda$ is such that $C s_\beta < C$. By definition, $$\begin{aligned}
    \varphi_\lambda(C) s_\beta 
    &= \Big( \sum_{D \in W_\Theta \backslash W} \sum_{p \in {\mathbb{Z}}} \big(\operatorname{rank}_{\mathcal{O}}H^p i_{w^D}^! {\mathcal{L}}(w^C,\lambda,\eta)\big) q^p \delta_D \Big) s_\beta\\
    &= \sum_{D \in W_\Theta \backslash W} \sum_{p \in {\mathbb{Z}}} \big(\operatorname{rank}_{\mathcal{O}}H^p i_{w^D}^! {\mathcal{L}}(w^C,\lambda,\eta)\big) q^p \delta_{D s_\beta}\end{aligned}$$ and $$\begin{aligned}
    \varphi_{s_\beta \lambda} (C s_\beta) 
    &= \sum_{D \in W_\Theta \backslash W} \sum_{p \in {\mathbb{Z}}} \big(\operatorname{rank}_{\mathcal{O}}H^p i_{w^D}^! {\mathcal{L}}(w^{C s_\beta},s_\beta\lambda, \eta)\big) q^p \delta_D\\
    &= \sum_{D \in W_\Theta \backslash W} \sum_{p \in {\mathbb{Z}}} \big(\operatorname{rank}_{\mathcal{O}}H^p i_{w^{D s_\beta}}^! {\mathcal{L}}(w^{C s_\beta},s_\beta\lambda, \eta)\big) q^p \delta_{D s_\beta}\end{aligned}$$ where in the last equality we rearranged the sum by the bijection $W_\Theta \backslash W {\xrightarrow{\raisebox{-1.2ex}[0ex][1ex]{$\sim$}}}W_\Theta \backslash W$, $D \mapsto D s_\beta$. Hence it suffices to show that, for any $D \in W_\Theta \backslash W$ and any $p \in {\mathbb{Z}}$, $$\operatorname{rank}_{\mathcal{O}}H^p i_{w^D}^! {\mathcal{L}}(w^C,\lambda,\eta)
    = \operatorname{rank}_{\mathcal{O}}H^p i_{w^{D s_\beta}}^! {\mathcal{L}}(w^{C s_\beta},s_\beta\lambda, \eta).$$ This follows by \[thm:Is\_pullback\].

Inductive step: verification of \[thm:KL\_alg\](2) for $\ell(w^C) = k$
----------------------------------------------------------------------

This part of the argument is modified from @Milicic:Localization [V.2] and is almost identical to @Romanov:Whittaker [5]. Instead of reproving all the details, we briefly review Romanov's argument and point out the necessary changes for our situation.

Let $\alpha \in \Pi \cap \Pi_\lambda$ be such that $C s_\alpha < C$. Let $X_\alpha$ be the partial flag variety with natural projection $p_\alpha: X \to X_\alpha$. For any $D \in W_\Theta \backslash W$ such that $w^D s_\alpha < w^D$, write $Z_{w^D} = C(w^D) \cup C(w^D s_\alpha)$ (not to be confused with the variety $Z_w \subseteq X \times X$ in the definition of intertwining functors) with inclusion map $j_{w^D}: Z_{w^D} {\hookrightarrow}X$[^2]. The main tools in *loc. cit.* are the functor $$U_\alpha := p_\alpha^+ p_{\alpha_+}$$ and the long exact sequence $$\begin{gathered}
\label{eqn:LES_applied}
    \cdots \to 
    ({\mathcal{O}}_{Z_{w^D}}^\eta)^{\oplus \operatorname{rank}_{\mathcal{O}}H^p i_{w^D}^! {\mathcal{L}}(w^C s_\alpha,\lambda,\eta)} \to
    ({\mathcal{O}}_{Z_{w^D}}^\eta)^{\oplus \operatorname{rank}_{\mathcal{O}}H^p i_{{w^D} s_\alpha}^! {\mathcal{L}}(w^C s_\alpha,\lambda,\eta)} \to
    H^p j_{w^D}^! U_\alpha {\mathcal{L}}(w^C s_\alpha,\lambda,\eta) \\ \to
    ({\mathcal{O}}_{Z_{w^D}}^\eta)^{\oplus \operatorname{rank}_{\mathcal{O}}H^{p+1} i_{w^D}^! {\mathcal{L}}(w^C s_\alpha,\lambda,\eta)} \to
    ({\mathcal{O}}_{Z_{w^D}}^\eta)^{\oplus \operatorname{rank}_{\mathcal{O}}H^{p+1} i_{{w^D} s_\alpha}^! {\mathcal{L}}(w^C s_\alpha,\lambda,\eta)} \to \cdots\end{gathered}$$ (the long exact sequence comes from the distinguished triangle of the inclusion of the two cells in $Z_{w^D}$). Using the parity degree vanishing (*op. cit.* 5.3) of parabolic Kazhdan-Lusztig polynomials, Romanov was able to split the above long exact sequence into short exact sequences, implying a collection of equalities on rank of pullbacks. Applying these equalities to the definition of $\nu(U_\alpha {\mathcal{L}}(w^C s_\alpha,\lambda,\eta))$ to show that $$\label{eqn:U_T_comparison}
    \nu(U_\alpha {\mathcal{L}}(w^C s_\alpha,\lambda,\eta)) = T_\alpha( \nu ({\mathcal{L}}(w^C s_\alpha,\lambda,\eta)))$$ which implies, by the decomposition theorem of holonomic ${\mathcal{D}}$-modules @Mochizuki:Decomp and semisimplicity of $U_\alpha$ @Milicic:Localization [V.2.7], that for some $c_D \in {\mathbb{Z}}$, $$\label{eqn:Romanov_T_alpha}
    T_\alpha (\varphi_\lambda(C s_\alpha)) = \sum_{D {\leqslant}C} c_D \varphi_\lambda (D)$$ for any integral $\lambda$.

Roughly the same argument applies to our situation and produces the same comparison (\[eqn:U\_T\_comparison\]). The change happens at the last step: compared with (\[eqn:Romanov\_T\_alpha\]), the right hand side of our equation \[thm:KL\_alg\](2) is only summing over right $W_\Theta$-cosets $D$'s such that $D$ is in the same double $(W_\Theta,W_\lambda)$-coset as $C$ and such that $r_\lambda D {\leqslant}_{u,\lambda} r_\lambda C$. To get this restricted sum, we apply the induction assumption to $C s_\alpha$. Specifically, \[thm:KL\_alg\](1) for $C s_\alpha$ compared with the definition of $\varphi_\lambda( C s_\alpha) = \nu( {\mathcal{L}}(w^C s_\alpha,\lambda,\eta))$ implies that $$i_{w^D}^! {\mathcal{L}}( w^C s_\alpha,\lambda,\eta) = 0$$ whenever $D$ is either not in the same double $(W_\Theta,W_\lambda)$-coset as $C$, or in the same double coset as $C$ but $r_\lambda D \not{\leqslant}_{u,\lambda} r_\lambda C$. Combined with (\[eqn:LES\_applied\]), we see $j_{w^D}^! U_\alpha {\mathcal{L}}(w^C s_\alpha,\lambda,\eta) = 0$, and so are any of its direct summands. Since $i_{w^D}^! {\mathcal{L}}(w^D,\lambda,\eta) = {\mathcal{O}}_{C(w^D)}^\eta \neq 0$, in particular $j_{w^D}^! {\mathcal{L}}(w^D,\lambda,\eta) \neq 0$, we conclude that ${\mathcal{L}}(w^D,\lambda,\eta)$ is not a direct summand of $U_\alpha {\mathcal{L}}(w^C s_\alpha,\lambda,\eta)$. We record this separately as a lemma for later use.

\[lem:U\_alpha\_summands\] Let $\alpha \in \Pi\cap \Pi_\lambda$ and $C \in W_\Theta \backslash W$ be such that $C s_\alpha < C$. If $D \in W_\Theta \backslash W$ is either not in the same double $(W_\Theta,W_\lambda)$-coset as $C$, or in the same double coset as $C$ but $r_\lambda D \not{\leqslant}_{u,\lambda} r_\lambda C$, then $j_{w^D}^! U_\alpha {\mathcal{L}}(w^C s_\alpha,\lambda,\eta) = 0$. In particular, $$U_\alpha {\mathcal{L}}(w^C s_\alpha,\lambda,\eta) = \bigoplus_{\substack{D \in W_\Theta \backslash W_\Theta u W_\lambda\\ r_\lambda D {\leqslant}_{u,\lambda} r_\lambda C}} {\mathcal{L}}(w^D, \lambda,\eta)^{\oplus c_D}$$ for some $c_D \in {\mathbb{Z}}$.

Therefore $$\begin{aligned}
    T_\alpha( \varphi_\lambda(C s_\alpha))
    &= \nu( U_\alpha {\mathcal{L}}(w^C s_\alpha,\lambda,\eta) ) \label{eqn:compare_U_T_1}\\
    &= \nu\Big( \bigoplus_{\substack{D \in W_\Theta \backslash W_\Theta u W_\lambda\\ r_\lambda D {\leqslant}_{u,\lambda} r_\lambda C}} {\mathcal{L}}(w^D, \lambda,\eta)^{\oplus c_D} \Big) \\
    &= \sum_{\substack{D \in W_\Theta \backslash W_\Theta u W_\lambda\\ r_\lambda D {\leqslant}_{u,\lambda} r_\lambda C}} c_D \nu ({\mathcal{L}}(w^D, \lambda,\eta)) \\
    &= \sum_{\substack{D \in W_\Theta \backslash W_\Theta u W_\lambda\\ r_\lambda D {\leqslant}_{u,\lambda} r_\lambda C}} c_D \varphi_\lambda(D) \label{eqn:compare_U_T_5}.\end{aligned}$$ and \[thm:KL\_alg\](3) is verified for $C$.

Inductive step: verification of \[thm:KL\_alg\](1) for $\ell(w^C) = k$ {#subsec:(1)}
----------------------------------------------------------------------

Suppose there exists $\beta \in \Pi - \Pi_\lambda$ such that $C s_\beta < C$. The induction assumption applies to $C s_\beta$. Hence, from \[thm:KL\_alg\](1) for $C s_\beta$ and $s_\beta \lambda$, we obtain $$\varphi_{s_\beta \lambda} (C s_\beta) 
    = \delta_{C s_\beta} 
    + \sum_{\substack{  D \in W_\Theta \backslash W_\Theta u' W_{s_\beta \lambda}\\ r_{s_\beta \lambda} D \; <_{u',s_\beta \lambda} \;  r_{s_\beta \lambda} (C s_\beta)}} 
    P_{r_{s_\beta \lambda} (C s_\beta), r_{s_\beta \lambda} D}^{u',s_\beta\lambda} \delta_D,$$ for some $P_{r_{s_\beta \lambda} (C s_\beta), r_{s_\beta \lambda} D}^{u',s_\beta\lambda} \in q {\mathbb{Z}}[q]$, where $u'$ is the unique element in $A_{\Theta,s_\beta \lambda}$ such that $C s_\beta \in W_\Theta \backslash W_\Theta u' W_{s_\beta \lambda}$. Applying \[thm:KL\_alg\](3) for $C$, $$\varphi_\lambda(C) 
    = \varphi_{s_\beta \lambda} (C s_\beta) s_\beta
    = \delta_C 
    + \sum_{\substack{  D \in W_\Theta \backslash W_\Theta u' W_{s_\beta \lambda}\\ r_{s_\beta \lambda} D \; <_{u',s_\beta \lambda}\;  r_{s_\beta \lambda} (C s_\beta)}} 
    P_{r_{s_\beta \lambda} (C s_\beta), r_{s_\beta \lambda} D}^{u',s_\beta\lambda} \delta_{D s_\beta}.$$ By \[lem:Is\_pres\_lowest\_db\_coset\] and its corollary, there exists $w \in W_\Theta$ with $w u' = u s_\beta$. Hence $$\begin{aligned}
    W_\Theta u' W_{s_\beta \lambda} 
    &= W_\Theta w u' s_\beta W_\lambda s_\beta\\
    &= W_\Theta u s_\beta s_\beta W_\lambda s_\beta\\
    &= W_\Theta u W_\lambda s_\beta,\end{aligned}$$ and we see that $D \in W_\Theta \backslash W_\Theta u' W_{s_\beta \lambda}$ if and only if $D s_\beta \in W_\Theta \backslash W_\Theta u W_\lambda$. By \[lem:Is\_right\_coset\], $$r_{s_\beta \lambda} D <_{u',s_\beta \lambda} r_{s_\beta \lambda} (C s_\beta) \iff r_\lambda (D s_\beta) <_{u,\lambda} r_\lambda C.$$ Hence $$\begin{aligned}
    \varphi_\lambda(C) &= \delta_C 
    + \sum_{\substack{  D s_\beta \in W_\Theta \backslash W_\Theta u W_\lambda \\ r_\lambda (D s_\beta) <_{u,\lambda} r_\lambda C}}
     P_{r_{s_\beta \lambda} (C s_\beta), r_{s_\beta \lambda} D}^{u',s_\beta\lambda} \delta_{D s_\beta}\end{aligned}$$ for some $P_{r_{s_\beta \lambda} C s_\beta, r_{s_\beta \lambda} D}^{u',s_\beta\lambda} \in q {\mathbb{Z}}[q]$, and \[thm:KL\_alg\](1) holds for $C$ in this case.

If such $\beta$ does not exist, then there exists a simple integral root $\alpha$ with $C s_\alpha < C$. Recall the definition of $\varphi_\lambda(C)$: $$\varphi_\lambda(C) = \sum_{D \in W_\Theta \backslash W} \sum_{p \in {\mathbb{Z}}} 
    \big( \operatorname{rank}_{\mathcal{O}}H^p i_{w^D}^! {\mathcal{L}}(w^C,\lambda,\eta) \big) q^p \delta_D.$$ Since ${\mathcal{L}}(w^C,\lambda,\eta)$ is supported on the closure of $C(w^C)$, $\operatorname{rank}_{\mathcal{O}}H^p i_{w^D}^! {\mathcal{L}}(w^C,\lambda,\eta) = 0$ for those $D$'s with $D \not{\leqslant}C$.

We will use information about the $U_\alpha$-functor to get a more precise information on which terms vanish in the sum. By semisimplicity of the functor $U_\alpha$, $U_\alpha {\mathcal{L}}(w^C s_\alpha,\lambda,\eta)$ is a direct sum of some irreducibles ${\mathcal{L}}(w^D,\lambda,\eta)$. For those ${\mathcal{L}}(w^D,\lambda,\eta)$'s that appear, $j_{w^D}^! {\mathcal{L}}(w^C,\lambda,\eta)$ is a direct summand of $j_{w^D}^! U_\alpha {\mathcal{L}}(w^C s_\alpha,\lambda,\eta)$. However, we have seen that if $D$ is either

1.  not in the same double $(W_\Theta,W_\lambda)$-coset as $C$, or

2.  in the same double coset as $C$ but $r_\lambda D \not{\leqslant}_{u,\lambda} r_\lambda C$,

then $j_{w^D}^! U_\alpha {\mathcal{L}}(w^C s_\alpha,\lambda,\eta) = 0$ (see \[lem:U\_alpha\_summands\]). Therefore $j_{w^D}^! {\mathcal{L}}(w^C,\lambda,\eta) = 0$ for those $D$'s of type (i) or (ii). Further pulling back to $C(w^D)$, we see that $i_{w^D}^! {\mathcal{L}}(w^C,\lambda,\eta) = 0$ for those $D$'s of type (i) or (ii). In other words, $$\varphi_\lambda(C) = \sum_{\substack{D\in W_\Theta \backslash W_\Theta u W_\lambda\\ r_\lambda D {\leqslant}_{u,\lambda} r_\lambda C}} \sum_{p \in {\mathbb{Z}}} \big( \operatorname{rank}_{\mathcal{O}}H^p i_{w^D}^! {\mathcal{L}}(w^C,\lambda,\eta) \big) q^p \delta_D.$$ It suffices to compute the remaining coefficients. Consider $D = C$. Since ${\mathcal{I}}(w^C,\lambda,\eta)$ is a direct image, it contains no section supported in $\partial C(w^C)$ except $0$. The same holds for ${\mathcal{L}}(w^C,\lambda,\eta)$ being a submodule of ${\mathcal{I}}(w^C,\lambda,\eta)$. Hence ${\mathcal{L}}(w^C,\lambda,\eta)|_{X - \partial C(w^C)}$ is a nonzero submodule of ${\mathcal{I}}(w^C,\lambda,\eta)|_{X - \partial C(w^C)}$. But ${\mathcal{I}}(w^C,\lambda,\eta)|_{X - \partial C(w^C)}$ is irreducible by Kashiwara's equivalence of categories for the closed immersion $C(w^C) {\hookrightarrow}X - \partial C(w^C)$, so ${\mathcal{L}}(w^C,\lambda,\eta)|_{X - \partial C(w^C)} = {\mathcal{I}}(w^C,\lambda,\eta)|_{X - \partial C(w^C)}$, and their further pullback to $C(w^C)$ is ${\mathcal{O}}_{C(w^C)}^\eta$. Hence $$\sum_{p \in {\mathbb{Z}}} \big( \operatorname{rank}_{\mathcal{O}}H^p i_{w^C}^! {\mathcal{L}}(w^C,\lambda,\eta) \big) q^p = 1.$$ When $D < C$, we have $H^0 i_{w^D}^! {\mathcal{L}}(w^C,\lambda,\eta) = 0$ since $H^0 i_{w^D}^!$ takes sections supported in $C(w^D)$ and $C(w^D) \subseteq \partial C(w^C)$. Hence $$\sum_{p \in {\mathbb{Z}}} \big( \operatorname{rank}_{\mathcal{O}}H^p i_{w^D}^! {\mathcal{L}}(w^C,\lambda,\eta) \big) q^p \in q {\mathbb{Z}}[q].$$ Thus \[thm:KL\_alg\](1) holds for $C$.

Inductive step: verification of \[thm:KL\_alg\](4) for $\ell(w^C) = k$ {#subsec:(4)}
----------------------------------------------------------------------

Based on our definition of parabolic Kazhdan-Lusztig polynomials \[def:parabolic\_KL\_poly\], we need to find $\alpha \in \Pi_\lambda$ such that $r_\lambda C s_\alpha <_{u,\lambda} r_\lambda C$ and equation (\[eqn:parabolic\_KL\_poly\]) holds for the function $$\psi_{u,\lambda}(r_\lambda C) := r_\lambda \varphi_\lambda(C).$$ If $\alpha$ can be chosen to be in $\Pi \cap \Pi_\lambda$, then (\[eqn:parabolic\_KL\_poly\]) follows from \[thm:KL\_alg\](2) for $C$. Otherwise, we will need to use non-integral intertwining functors to move $\alpha$ to some simple root $z{^{-1}}\alpha$, using \[lem:decrease\_of\_length\]. To translate the result back to $\alpha$, we need the following lemmas.

Let $\alpha \in \Pi \cap \Pi_\lambda$ and $u \in A_{\Theta,\lambda}$. Then $$r_\lambda \circ T_\alpha = T_\alpha^{u,\lambda} \circ r_\lambda$$ as maps from $r_\lambda{^{-1}}{\mathcal{H}}_{{\Theta_\lambda^u},\lambda} \subseteq {\mathcal{H}}_\Theta$ to ${\mathcal{H}}_{{\Theta_\lambda^u},\lambda}$.

It suffices to check equality on $\delta_C$ for any $C \in W_\Theta \backslash W_\Theta u W_\lambda$. The left hand side is $$\begin{aligned}
        r_\lambda T_\alpha( \delta_C )
        &= 
        \begin{cases}
            r_\lambda (q{^{-1}}\delta_C + \delta_{C s_\alpha}), & \text{if } C s_\alpha < C,\\
            0, & \text{if } C s_\alpha = C,\\
            r_\lambda(q \delta_C + \delta_{Cs_\alpha}), & \text{if } C s_\alpha > C,
        \end{cases}\\
        &= 
        \begin{cases}
            q{^{-1}}\delta_{r_\lambda C} + \delta_{r_\lambda C s_\alpha}, & \text{if } C s_\alpha < C,\\
            0, & \text{if } C s_\alpha = C,\\
            q \delta_{r_\lambda C} + \delta_{r_\lambda Cs_\alpha}, & \text{if } C s_\alpha > C.
        \end{cases}
    \end{aligned}$$ The right hand side is $$T_\alpha^{u,\lambda} (r_\lambda \delta_C) = T_\alpha^{u,\lambda} (\delta_{r_\lambda C}) =
        \begin{cases}
            q{^{-1}}\delta_{r_\lambda C} + \delta_{r_\lambda C s_\alpha}, & \text{if } r_\lambda C s_\alpha <_{u,\lambda} r_\lambda  C,\\
            0, & \text{if } r_\lambda C s_\alpha = r_\lambda C,\\
            q \delta_{r_\lambda C} + \delta_{r_\lambda Cs_\alpha}, & \text{if } r_\lambda C s_\alpha >_{u,\lambda} r_\lambda C.
        \end{cases}$$ Hence it suffices to show that the conditions in the above cases correspond. This is true as $r_\lambda{^{-1}}$ preserves the partial orders on right cosets.

Let $\alpha \in \Pi \cap \Pi_\lambda$, $\beta \in \Pi - \Pi_\lambda$ and $u \in A_{\Theta,\lambda}$. Let $u' \in A_{\Theta,s_\beta\lambda}$ be the unique element such that $W_\Theta u s_\beta W_{s_\beta\lambda} = W_\Theta u' W_{s_\beta \lambda}$. Then $$(s_\beta (-) s_\beta ) \circ T_\alpha^{u,\lambda} = T_{s_\beta \alpha}^{u',s_\beta\lambda} \circ (s_\beta (-) s_\beta)$$ as maps from ${\mathcal{H}}_{{\Theta_\lambda^u},\lambda}$ to ${\mathcal{H}}_{\Theta_{s_\beta\lambda}^{u'},s_\beta \lambda}$, where $s_\beta (-) s_\beta$ denotes conjugation by $s_\beta$.

It suffices to check equality on $\delta_{C'}$ where $C' \in W_{\lambda,\Theta_\lambda^u} \backslash W_\lambda$. The left hand side applied to $\delta_{C'}$ reads $$s_\beta T_\alpha^{u,\lambda} (\delta_{C'}) s_\beta =
        \begin{cases}
            q{^{-1}}\delta_{s_\beta C' s_\beta} + \delta_{s_\beta C' s_\alpha s_\beta}, & \text{if } C' s_\alpha <_{u,\lambda} C',\\
            0, & \text{if } C' s_\alpha = C',\\
            q \delta_{s_\beta C' s_\beta} + \delta_{s_\beta C' s_\alpha s_\beta}, & \text{if } C' s_\alpha >_{u,\lambda} C'.
        \end{cases}$$ The right hand side reads $$\begin{aligned}
        T_{s_\beta \alpha}^{u',s_\beta \lambda} (s_\beta \delta_{C'} s_\beta) 
        &= T_{s_\beta \alpha}^{u',s_\beta \lambda} (\delta_{s_\beta C' s_\beta})\\
        &= 
        \begin{cases}
            q{^{-1}}\delta_{s_\beta C' s_\beta} + \delta_{s_\beta C' s_\beta s_{s_\beta \alpha}}, & s_\beta C' s_\beta <_{u',s_\beta\lambda} s_\beta C' s_\beta s_{s_\beta \alpha},\\
            0, & s_\beta C' s_\beta = s_\beta C' s_\beta s_{s_\beta \alpha},\\
            q \delta_{s_\beta C' s_\beta} + \delta_{s_\beta C' s_\beta s_{s_\beta \alpha}}, & s_\beta C' s_\beta >_{u',s_\beta\lambda} s_\beta C' s_\beta s_{s_\beta \alpha}.
        \end{cases}     
    \end{aligned}$$ Since $s_{s_\beta \alpha} = s_\beta s_\alpha s_\beta$, the above expression can be simplified as $$T_{s_\beta \alpha}^{u',s_\beta \lambda} (s_\beta \delta_{C'} s_\beta) =
        \begin{cases}
            q{^{-1}}\delta_{s_\beta C' s_\beta} + \delta_{s_\beta C' s_\alpha s_\beta}, & \text{if } s_\beta C' s_\beta <_{u',s_\beta\lambda} s_\beta C' s_\alpha s_\beta,\\
            0, & \text{if } s_\beta C' s_\beta = s_\beta C' s_\alpha s_\beta,\\
            q \delta_{s_\beta C' s_\beta} + \delta_{s_\beta C' s_\alpha s_\beta}, & \text{if }s_\beta C' s_\beta >_{u',s_\beta\lambda} s_\beta C' s_\alpha s_\beta.
        \end{cases}$$ By \[lem:Is\_right\_coset\], conjugation by $s_\beta$ preserves the partial orders on right cosets, i.e. $C' s_\alpha <_{u,\lambda} C'$ if and only if $s_\beta C' s_\beta <_{u',s_\beta\lambda} s_\beta C' s_\alpha s_\beta$ and similarly for "$=$" and "$>$". Hence $s_\beta T_\alpha^{u,\lambda} (\delta_{C'}) s_\beta = T_{s_\beta \alpha}^{u',s_\beta \lambda} (s_\beta \delta_{C'} s_\beta)$ as required.

Choose $\alpha \in \Pi_\lambda$, $s {\geqslant}0$ and $\beta_1,\ldots,\beta_s \in \Pi$ such that if we write $z_0 = 1$, $z_i = s_{\beta_1} \cdots s_{\beta_i}$ and $z = z_s$, the following conditions hold:

1.  for any $0 {\leqslant}i {\leqslant}s-1$, $\beta_{i+1}$ is non-integral to $z_i{^{-1}}\lambda$;

2.  $z{^{-1}}\alpha \in \Pi \cap \Pi_{z{^{-1}}\lambda}$;

3.  $r_\lambda C s_\alpha <_{u,\lambda} r_\lambda C$;

4.  if $s > 0$, $C z < C$;

5.  $C s_\alpha z = C z s_{z{^{-1}}\alpha} < C z$.

Such a choice exists by \[lem:decrease\_of\_length\]. By (d), the induction assumption applies to $C z$ for $z{^{-1}}\lambda$. In particular, if we apply \[thm:KL\_alg\](2) to $C z s_{z{^{-1}}\alpha} < C z$ and $z{^{-1}}\lambda$, we obtain the equation $$T_{z{^{-1}}\alpha} \big( \varphi_{z{^{-1}}\lambda}( C z s_{z{^{-1}}\alpha}) \big) 
    = \sum_{\substack{D\in W_\Theta \backslash W_\Theta u' W_{z{^{-1}}\lambda}\\ r_{z{^{-1}}\lambda} D \; {\leqslant}_{u',z{^{-1}}\lambda}\;  r_{z{^{-1}}\lambda} (Cz)}} c_D \varphi_{z{^{-1}}\lambda}(D)$$ where $u'$ is the unique element in $A_{\Theta,z{^{-1}}\lambda}$ such that $Cz \in W_\Theta \backslash W_\Theta u' W_{z{^{-1}}\lambda}$. Applying $r_{z{^{-1}}\lambda}$ to both sides, we get, using the first lemma above, $$\begin{aligned}
    T_{z{^{-1}}\alpha}^{u',z{^{-1}}\lambda} \big( r_{z{^{-1}}\lambda} \varphi_{z{^{-1}}\lambda}( C z s_{z{^{-1}}\alpha}) \big) 
    &= r_{z{^{-1}}\lambda} T_{z{^{-1}}\alpha} \big( \varphi_{z{^{-1}}\lambda}( C z s_{z{^{-1}}\alpha}) \big) \\
    &= \sum_{\substack{D\in W_\Theta \backslash W_\Theta u' W_{z{^{-1}}\lambda}\\ r_{z{^{-1}}\lambda} D \;{\leqslant}_{u',z{^{-1}}\lambda}\; r_{z{^{-1}}\lambda} (Cz)}} c_D r_{z{^{-1}}\lambda} \varphi_{z{^{-1}}\lambda}(D).\end{aligned}$$ Now applying $z(-)z{^{-1}}$ to both sides, we obtain $$\label{eqn:(4)_a}
    z T_{z{^{-1}}\alpha}^{u',z{^{-1}}\lambda} \big( r_{z{^{-1}}\lambda} \varphi_{z{^{-1}}\lambda}( C z s_{z{^{-1}}\alpha}) \big) z{^{-1}}= \sum_{\substack{D\in W_\Theta \backslash W_\Theta u' W_{z{^{-1}}\lambda}\\ r_{z{^{-1}}\lambda} D \;{\leqslant}_{u',z{^{-1}}\lambda}\; r_{z{^{-1}}\lambda} (Cz)}} c_D z \big( r_{z{^{-1}}\lambda} \varphi_{z{^{-1}}\lambda}(D) \big) z{^{-1}}.$$ By repeatedly applying the second lemma, the left hand side equals $$T_\alpha^{u,\lambda} \big( z r_{z{^{-1}}\lambda} \varphi_{z{^{-1}}\lambda}( C z s_{z{^{-1}}\alpha}) z{^{-1}}\big).$$ Using the commutative diagram (\[eqn:Is\_right\_coset\]) and rewriting $C z s_{z{^{-1}}\alpha} = C s_\alpha z$, this equals $$T_\alpha^{u,\lambda} \big( r_\lambda (\varphi_{z{^{-1}}\lambda}(C s_\alpha z) z{^{-1}}) \big).$$ Then, using \[thm:KL\_alg\](3) for $C s_\alpha$, $\varphi_{z{^{-1}}\lambda} (C s_\alpha z) z {^{-1}}= \varphi_\lambda(C s_\alpha z z{^{-1}}) =  \varphi_\lambda( C s_\alpha)$. Therefore, the left hand side of (\[eqn:(4)\_a\]) equals $$T_\alpha^{u,\lambda} \big( r_\lambda \varphi_\lambda( C s_\alpha) \big)
    = T_\alpha^{u,\lambda} (\psi_{u,\lambda}(r_\lambda C) ).$$ Similarly, for the right hand side of (\[eqn:(4)\_a\]), applying (\[eqn:Is\_right\_coset\]) and \[thm:KL\_alg\](3) for $D$, we get $$(\ref{eqn:(4)_a}) 
    = \sum_{\substack{D\in W_\Theta \backslash W_\Theta u' W_{z{^{-1}}\lambda}\\ r_{z{^{-1}}\lambda} D \;{\leqslant}_{u',z{^{-1}}\lambda}\; r_{z{^{-1}}\lambda} (Cz)}} c_D r_\lambda \varphi_\lambda(Dz{^{-1}})
    = \sum_{\substack{D\in W_\Theta \backslash W_\Theta u' W_{z{^{-1}}\lambda}\\ r_{z{^{-1}}\lambda} D \;{\leqslant}_{u',z{^{-1}}\lambda}\; r_{z{^{-1}}\lambda} (Cz)}} c_D \psi_{u,\lambda}(r_\lambda (D z{^{-1}})).$$ As in the first part of \[subsec:(1)\], we can rewrite the subscript of the sum. There is an element $w \in W_\Theta$ such that $wu' = u z$ by \[lem:Is\_pres\_lowest\_db\_coset\]. Hence $$\begin{aligned}
    W_\Theta u' W_{z{^{-1}}\lambda} 
    &= W_\Theta wu' z{^{-1}}W_\lambda z\\
    &= W_\Theta uz z{^{-1}}W_\lambda z\\
    &= W_\Theta u W_\lambda z,\end{aligned}$$ and $D \in W_\Theta \backslash W_\Theta u' W_{z{^{-1}}\lambda}$ if and only if $D z{^{-1}}\in W_\Theta \backslash W_\Theta u W_\lambda$. Moreover, by \[lem:Is\_right\_coset\], $$r_{z{^{-1}}\lambda} D {\leqslant}_{u',z{^{-1}}\lambda} r_{z{^{-1}}\lambda} Cz \iff r_\lambda (D z{^{-1}}) {\leqslant}_{u,\lambda} r_\lambda C.$$ Hence (\[eqn:(4)\_a\]) becomes $$T_\alpha^{u,\lambda} (\psi_{u,\lambda}(r_\lambda C) )
    = \sum_{\substack{ Dz{^{-1}}\in W_\Theta \backslash W_\Theta u W_\lambda\\ r_\lambda (D z{^{-1}}) {\leqslant}_{u,\lambda} r_\lambda C}} c_D \psi_{u,\lambda}(r_\lambda (D z{^{-1}})).$$ Therefore, $\alpha \in \Pi_\lambda$ is such that $r_\lambda C s_\alpha <_{u,\lambda} r_\lambda C$ and equation (\[eqn:parabolic\_KL\_poly\]) holds for $C s_\alpha$. By \[def:parabolic\_KL\_poly\], the polynomials $P_{r_\lambda C, r_\lambda D}^{u,\lambda}$ are parabolic Kazhdan-Lusztig polynomials for $(W_\lambda,\Pi_\lambda,\Theta_\lambda^u)$. Thus \[thm:KL\_alg\](4) holds for $C$.

This completes the proof of \[thm:KL\_alg\].

Character formula for irreducible modules {#sec:character_formula}
=========================================

Regular case {#subsec:character_formula}
------------

By standard arguments, the algorithm \[thm:KL\_alg\] leads to a character formula for irreducible Whittaker modules for regular infinitesimal characters. The arguments in this part are standard.

Recall that $K\operatorname{Mod}_{coh}({\mathcal{D}}_\lambda,N,\eta)$ denotes the Grothendieck group of $\operatorname{Mod}_{coh}({\mathcal{D}}_\lambda,N,\eta)$, $[{\mathcal{F}}]$ denotes the class of ${\mathcal{F}}$ in the Grothendeick group. Let $\lambda$ be antidominant regular. As explained in \[subsec:geom\_prelim\], under holonomic duality and Beilinson-Bernstein's localization, $K \operatorname{Mod}_{coh}({\mathcal{D}}_{-\lambda},N,\eta)$ is isomorphic to $K {\mathcal{N}}_{\theta,\eta}$ where $[{\mathcal{I}}(w^C,-\lambda,\eta)]$ is sent to the class $[M(w^C \lambda,\eta)]$ of the standard Whittaker module $M(w^C \lambda,\eta)$ in $K{\mathcal{N}}_{\theta,\eta}$, and $[{\mathcal{L}}(w^C,-\lambda,\eta)]$ is sent to $[L(w^C \lambda,\eta)]$. The character map $\operatorname{ch}$ on ${\mathcal{N}}_{\theta,\eta}$ factors through the Grothendieck group, and the character formula for $M(w^C \lambda,\eta)$ is known. Hence, to find a character formula for $L(w^C\lambda,\eta)$, it suffices to express $[L(w^C\lambda,\eta)]$ as a linear combination of the $[M(w^D\lambda,\eta)]$'s, and the latter can be achieved by writing $[{\mathcal{L}}(w^C,-\lambda,\eta)]$ as a linear combination of the $[{\mathcal{I}}(w^D,-\lambda,\eta)]$'s.

To relate our algorithm with the Grothendieck group, recall the map $\nu$ used to define the function $\varphi_\lambda$: $$\nu: \operatorname{Obj} \operatorname{Mod}_{coh}({\mathcal{D}}_\lambda,N,\eta) {\xrightarrow{\;\;\;\;\;}}{\mathcal{H}}_\Theta,\quad
    {\mathcal{F}}\mapsto \sum_{D \in W_\Theta \backslash W} \sum_{p \in {\mathbb{Z}}} \big(\operatorname{rank}_{\mathcal{O}}H^p i_{w^D}^! {\mathcal{F}}\big) q^p \delta_D.$$ If we evaluate at $q = -1$, then the sum $\sum_{p \in {\mathbb{Z}}} (-1)^p \operatorname{rank}_{\mathcal{O}}H^p i_{w^D}^! {\mathcal{F}}$ is additive with respect to short exact sequences. Hence $\nu|_{q=-1}$ factors through the Grothendieck group, that is, we obtain a map $$\nu|_{q=-1}: K\operatorname{Mod}_{coh}({\mathcal{D}}_\lambda,N,\eta) {\xrightarrow{\;\;\;\;\;}}{\mathcal{H}}_\Theta|_{q=-1}.$$ For $C$, $D \in W_\Theta \backslash W$, write $[{\mathcal{L}}(w^C,\lambda,\eta):{\mathcal{I}}(w^D,\lambda,\eta)]$ for the coefficient of $[{\mathcal{I}}(w^D,\lambda,\eta)]$ in the expression of $[{\mathcal{L}}(w^C,\lambda,\eta)]$, that is, $$[{\mathcal{L}}(w^C,\lambda,\eta)] = \sum_{D \in W_\Theta \backslash W} [{\mathcal{L}}(w^C,\lambda,\eta):{\mathcal{I}}(w^D,\lambda,\eta)] [{\mathcal{I}}(w^D,\lambda,\eta)].$$ Then, if $u \in A_{\Theta,\lambda}$ and $C \in W_\Theta \backslash W_\Theta u W_\lambda$, from \[lem:nu\_std\] and \[thm:KL\_alg\](1), $$\begin{aligned}
    & \delta_C + 
    \sum_{\substack{D \in W_\Theta \backslash W_\Theta u W_\lambda\\            r_\lambda D <_{u,\lambda} r_\lambda C}} 
        P_{r_\lambda C,r_\lambda D}^{u,\lambda}(-1) \delta_D\\
    =& \varphi_\lambda(C)|_{q=-1} \\
    =& \nu({\mathcal{L}}(w^C,\lambda,\eta))|_{q=-1}\\
    =& \nu \left( \sum_{D \in W_\Theta \backslash W} [{\mathcal{L}}(w^C,\lambda,\eta):{\mathcal{I}}(w^D,\lambda,\eta)] [{\mathcal{I}}(w^D,\lambda,\eta)] \right)|_{q=-1}\\
    =& \sum_{D \in W_\Theta \backslash W} [{\mathcal{L}}(w^C,\lambda,\eta):{\mathcal{I}}(w^D,\lambda,\eta)] \nu({\mathcal{I}}(w^D,\lambda,\eta))|_{q=-1}\\
    =& \sum_{D \in W_\Theta \backslash W} [{\mathcal{L}}(w^C,\lambda,\eta):{\mathcal{I}}(w^D,\lambda,\eta)] \delta_D.\end{aligned}$$ Comparing the coefficients on both sides, we obtain the following description. Recall our convention that $P_{r_\lambda C,r_\lambda C}^{u,\lambda} = 1$.

\[thm:multiplicity\_D-mods\] Let $\lambda \in {\mathfrak{h}}^*$ and let $\eta: {\mathfrak{n}}\to {\mathbb{C}}$ be any character. Let $C$, $D \in W_\Theta \backslash W$. Let $u \in A_{\Theta,\lambda}$ be such that $C \subseteq W_\Theta u W_\lambda$. Then $$[{\mathcal{L}}(w^C,\lambda,\eta):{\mathcal{I}}(w^D,\lambda,\eta)] =
        \begin{cases}
            P_{r_\lambda C,r_\lambda D}^{u,\lambda}(-1), & \text{if } D \in W_\Theta \backslash W_\Theta u W_\lambda \text{ and } r_\lambda D {\leqslant}_{u,\lambda} r_\lambda C,\\
            0, & \text{otherwise},
        \end{cases}$$ where the $P_{r_\lambda C,r_\lambda D}^{u,\lambda}$'s are parabolic Kazhdan-Lusztig polynomials for $(W_\lambda,\Pi_\lambda,\Theta_\lambda^u)$ as defined in \[def:parabolic\_KL\_poly\].

We can apply this to $-\lambda$. Since $W_\lambda$ and $W_{-\lambda}$ are equal as subgroups of $W$, the parabolic Kazhdan-Lusztig polynomials for $W_\lambda$ and $W_{-\lambda}$ are the same. Passing through holonomic duality, Beilinson-Bernstein's equivalence of categories and the character map, we obtain

\[Character formula for irreducible Whittaker modules: Regular case\] \[thm:multiplicity\] Let $\lambda \in {\mathfrak{h}}^*$ be antidominant and regular. Let $\eta: {\mathfrak{n}}\to {\mathbb{C}}$ be any character. For any $C \in W_\Theta \backslash W$, let $u \in A_{\Theta,\lambda}$ be the unique element such that $C \subseteq W_\Theta u W_\lambda$. Then $$        \operatorname{ch}L(w^C\lambda,\eta) = 
        \sum_{\substack{D \in W_\Theta \backslash W_\Theta u W_\lambda\\                r_\lambda D {\leqslant}_{u,\lambda} r_\lambda C}} 
            P_{r_\lambda C,r_\lambda D}^{u,\lambda}(-1) \operatorname{ch}M(w^D \lambda,\eta),$$ where the $P_{r_\lambda C,r_\lambda D}^{u,\lambda}$'s are parabolic Kazhdan-Lusztig polynomials for $(W_\lambda,\Pi_\lambda,\Theta_\lambda^u)$ as defined in \[def:parabolic\_KL\_poly\].

When $\lambda$ is integral, we have a simpler description, which we state separately.

\[Character formula for irreducible Whittaker modules: Regular integral case\] Let $\lambda \in {\mathfrak{h}}^*$ be antidominant, regular and integral. Let $\eta: {\mathfrak{n}}\to {\mathbb{C}}$ be any character. For any $C \in W_\Theta \backslash W$, $$\operatorname{ch}L(w^C\lambda,\eta) = 
        \sum_{\substack{D \in W_\Theta \backslash W\\ D {\leqslant}C}}
            P_{CD}(-1) \operatorname{ch}M(w^D \lambda,\eta),$$ where the $P_{CD}$'s are parabolic Kazhdan-Lusztig polynomials for $(W,\Pi,\Theta)$ as defined in \[def:parabolic\_KL\_poly\].

Inverting the matrix $(P_{CD}(-1))_{C,D}$, we recover the description in @Milicic-Soergel:Whittaker_algebraic and @Romanov:Whittaker of multiplicities of irreducible Whittaker modules in standard Whittaker modules with antidominant regular integral infinitesimal characters.

At another extreme, when $\eta = 0$ (i.e. $\Theta = \varnothing$), we recover the well-known non-integral Kazhdan-Lusztig conjecture for Bernstein-Gelfand-Gelfand's category ${\mathcal{O}}$.

\[Kazhdan-Lusztig conjecture for category ${\mathcal{O}}$\] Let $\lambda \in {\mathfrak{h}}^*$ be antidominant and regular. For any $w \in W$, $$\operatorname{ch}L(w\lambda) = 
        \sum_{\substack{v \in W_\lambda\\ v {\leqslant}_\lambda w}} P_{wv}^\lambda(-1) \operatorname{ch}M(v\lambda),$$ where the $P_{wv}^\lambda$'s are Kazhdan-Lusztig polynomials for $(W_\lambda,\Pi_\lambda, \varnothing)$ as defined in \[def:parabolic\_KL\_poly\], $M(v\lambda)$ is the Verma module of highest weight $v\lambda - \rho$, and $L(w \lambda)$ is the unique irreducible quotient of $M(w \lambda)$.

Singular case {#subsec:characer_formula_singular}
-------------

The singular case is conceptually not much harder than the regular case.

Let $\lambda \in {\mathfrak{h}}^*$ be antidominant and singular. The identification $\Gamma(X,{\mathcal{M}}(w^D,\lambda,\eta)) = M(w^D \lambda,\eta)$ still holds for singular $\lambda$. As in the regular case, we can apply \[thm:multiplicity\_D-mods\] to $-\lambda$, then apply the holonomic duality ${\mathbb{D}}$ and the global section functor $\Gamma(X,-)$. This produces the equality $$\label{eqn:character_formula_pre}
    \operatorname{ch}\Gamma(X,{\mathcal{L}}(w^C,\lambda,\eta)) = 
    \sum_{\substack{D \in W_\Theta \backslash W_\Theta u W_\lambda\\            r_\lambda D {\leqslant}_{u,\lambda} r_\lambda C}} 
    P_{r_\lambda C,r_\lambda D}^{u,\lambda}(-1) \operatorname{ch}M(w^D \lambda,\eta).$$ However, $\Gamma(X,{\mathcal{L}}(w^C,\lambda,\eta))$ could be zero, and the $M(w^D \lambda,\eta)$'s could coincide for different $D$'s. Therefore, it suffices to describe which $M(w^D \lambda,\eta)$'s coincide and which $\Gamma(X,{\mathcal{L}}(w^C,\lambda,\eta))$'s are zero.

The first question has an easy answer. Recall that for $C,D \in W_\Theta \backslash W$, $M(w^D \lambda,\eta) = M(w^C \lambda,\eta)$ if and only if $W_\Theta w^D \lambda = W_\Theta w^C \lambda$. Let $W^\lambda$ be the stabilizer of $\lambda$ in $W$. Then the above condition is equivalent to $W_\Theta w^D W^\lambda = W_\Theta w^C W^\lambda$, i.e. that $C$ and $D$ are in the same double $(W_\Theta,W^\lambda)$-coset.

\[lem:std\_mods\_coincide\] Let $\lambda \in {\mathfrak{h}}^*$ be antidominant and let $\eta :{\mathfrak{n}}\to {\mathbb{C}}$ be a character. The following are equivalent:

1.  $M(w^C \lambda,\eta) = M(w^D \lambda,\eta)$;

2.  $\Gamma(X,{\mathcal{M}}(w^C,\lambda,\eta)) = \Gamma(X,{\mathcal{M}}(w^D,\lambda,\eta))$;

3.  $C$ and $D$ are in the same double $(W_\Theta,W^\lambda)$-coset.

Therefore, for a fixed standard Whittaker module $M$, there is a unique double coset $W_\Theta v W^\lambda$ such that $\Gamma(X,{\mathcal{M}}(w^D,\lambda,\eta)) = M$ for all $D \in W_\Theta \backslash W_\Theta v W^\lambda$.

The following proposition answers the second question.

\[lem:irred\_vanishing\_singular\] Let $\lambda \in {\mathfrak{h}}^*$ be antidominant and let $\eta :{\mathfrak{n}}\to {\mathbb{C}}$ be a character. Let $v \in W$. Then the set $W_\Theta \backslash W_\Theta v W^\lambda$ of right $W_\Theta$-cosets contains a unique smallest element $C$. Furthermore,

1.  $\Gamma(X, {\mathcal{L}}(w^C,\lambda,\eta)) = L(w^C \lambda,\eta) \neq 0$; and

2.  $\Gamma(X,{\mathcal{L}}(w^D,\lambda,\eta)) = 0$ for any $D \in W_\Theta \backslash W_\Theta v W^\lambda$ not equal to $C$.

In other words, for a fixed standard Whittaker module $M$, among all the costandard ${\mathcal{D}}_\lambda$-modules that realizes $M$, the irreducible quotient of the one with the smallest support realizes the unique irreducible submodule of $M$.

Write $M = M(v\lambda,\eta)$ and $L = L(v\lambda,\eta)$.

First, there is one and at most one $D$ in $W_\Theta \backslash W$ with $\Gamma(X,{\mathcal{L}}(w^D,\lambda,\eta)) = L$. This is because, by the theory of localization, there is a unique irreducible ${\mathcal{D}}_\lambda$-module ${\mathcal{V}}$ with $\Gamma(X,{\mathcal{V}}) = L$ (${\mathcal{V}}$ is the unique irreducible quotient of ${\mathcal{D}}_\lambda \dotimes_{{\mathcal{U}}_\theta} L$). By the classification of irreducible twisted Harish-Chandra sheaves, ${\mathcal{V}}$ equals to ${\mathcal{L}}(w^D,\lambda,\eta)$ for a single $D \in W_\Theta \backslash W$.

Since ${\mathcal{L}}(w^D,\lambda,\eta)$ is the unique irreducible quotient of ${\mathcal{M}}(w^D,\lambda,\eta)$ and $\Gamma(X,-)$ is exact on ${\mathcal{D}}_\lambda$-modules, $\Gamma(X,{\mathcal{L}}(w^D,\lambda,\eta))$ equals the unique irreducible quotient $L(w^D\lambda,\eta)$ of $M(w^D\lambda,\eta)$. Therefore, the equality $L(v \lambda,\eta) = \Gamma(X,{\mathcal{V}}) = \Gamma(X,{\mathcal{L}}(w^D,\lambda,\eta)) = L(w^D \lambda,\eta)$ implies $M(w^C \lambda,\eta) = M(w^D \lambda,\eta)$ which forces our $D$ to be in the double coset $W_\Theta v W^\lambda$.

It remains to show that such a $D$ is minimum in $W_\Theta \backslash W_\Theta v W^\lambda$. Let $C$ be a minimal element in $W_\Theta \backslash W_\Theta v W^\lambda$. The composition factors of ${\mathcal{M}}(w^C,\lambda,\eta)$ consist of some ${\mathcal{L}}(w^E,\lambda,\eta)$'s with $E {\leqslant}C$. Taking global sections, we see that the composition factors of $M = \Gamma(X,{\mathcal{M}}(w^C,\lambda,\eta))$ consist of some $\Gamma(X,{\mathcal{L}}(w^E,\lambda,\eta))$'s that are nonzero and with $E {\leqslant}C$. On the other hand, $L = \Gamma(X,{\mathcal{L}}(w^D,\lambda,\eta))$ is a composition factor of $M$. Hence $\Gamma(X, {\mathcal{L}}(w^D,\lambda,\eta)) = \Gamma(X,{\mathcal{L}}(w^E,\lambda,\eta))$ for some $E {\leqslant}C$. By the same uniqueness statement appeared in the preceding paragraph, ${\mathcal{L}}(w^D,\lambda,\eta) = {\mathcal{L}}(w^E,\lambda,\eta)$ and hence $D = E {\leqslant}C$. By the minimality of $C$, $D = C$. Thus $C=D$ is the minimum element in $W_\Theta \backslash W_\Theta v W^\lambda$ and $\Gamma(X,{\mathcal{L}}(w^C,\lambda,\eta)) = L$.

Let $c \in {\mathbb{C}}$ be a nonzero number such that $c \alpha^\vee(\lambda) \not\in {\mathbb{Z}}$ for all $\alpha \in \Sigma$ with $\alpha^\vee(\lambda) \neq 0$. Then $W^\lambda = W_{c\lambda}$. Hence by \[thm:cross-section\_db\_coset\], the set $$A_\Theta^\lambda := A_{c\lambda} \cap (w_\Theta {}^\Theta W)$$ is a cross-section of $W_\Theta \backslash W / W^\lambda$ consisting of the unique shortest elements in each double coset. \[lem:irred\_vanishing\_singular\] can be rephrased as follows.

\[lem:irred\_vanishing\_singular'\] Let $\lambda \in {\mathfrak{h}}^*$ be antidominant and let $\eta :{\mathfrak{n}}\to {\mathbb{C}}$ be a character. Let $C \in W_\Theta \backslash W$. The following are equivalent:

1.  $C = W_\Theta v$ for some $v \in A_\Theta^\lambda$;

2.  $\Gamma(X,{\mathcal{L}}(w^C,\lambda,\eta)) \neq 0$;

3.  $\Gamma(X,{\mathcal{L}}(w^C,\lambda,\eta)) = L(w^C \lambda,\eta)$.

Using these observations, we can write down a character formula for general infinitesimal characters.

\[Character formula for irreducible Whittaker modules: General case\] \[thm:multiplicity\_singular\] Let $\lambda \in {\mathfrak{h}}^*$ be antidominant. Let $\eta: {\mathfrak{n}}\to {\mathbb{C}}$ be any character. For any $v \in A_\Theta^\lambda$, let $C = W_\Theta v$, and let $u \in A_{\Theta,\lambda}$ be the unique element such that $C \subseteq W_\Theta u W_\lambda$. Then $$        \operatorname{ch}L(v\lambda,\eta) = \operatorname{ch}L(w^C\lambda,\eta) = 
        \sum_{z \in A_\Theta^\lambda \cap (W_\Theta u W_\lambda)}
        \left(
        \sum_{\substack{                D \in W_\Theta \backslash W_\Theta z W^\lambda\\                r_\lambda D {\leqslant}_{u,\lambda} r_\lambda C}} 
        P_{r_\lambda C,r_\lambda D}^{u,\lambda}(-1) 
        \right)
         \operatorname{ch}M(z \lambda,\eta),$$ where the $P_{r_\lambda C,r_\lambda D}^{u,\lambda}$'s are parabolic Kazhdan-Lusztig polynomials for $(W_\lambda,\Pi_\lambda,\Theta_\lambda^u)$ as defined in \[def:parabolic\_KL\_poly\]. As $v$ ranges over $A_\Theta^\lambda$, $L(v \lambda,\eta)$ exhausts all irreducible objects in ${\mathcal{N}}_{\theta,\eta}$.

The right hand side is obtained by rearranging the right side of (\[eqn:character\_formula\_pre\]) based on \[lem:std\_mods\_coincide\]. The left hand side and the last statement (that those $L(v\lambda,\eta)$'s exhaust all irreducibles) follows from \[lem:irred\_vanishing\_singular'\] and Beilinson-Bernstein's equivalence of categories in the singular case (\[eqn:BB\_equivalence\_singular\]).

An example in $A_3$ {#sec:examples}
===================

The $A_3$ root system (pictured below) is the smallest example in which all nontrivial phenomena appear. To make the picture more readable, only the positive roots are connected to the origin. Here $\lambda$ can be chosen to be $\lambda = -m \rho + c(-\alpha + 2\beta + \gamma)$ for any nonzero number $c$ transcendental over ${\mathbb{Q}}$ and any large enough integer $m$ so that $\lambda$ is antidominant regular ($-\alpha + 2\beta +\gamma$ is a vector perpendicular to the plane spanned by $\alpha+\beta$ and $\gamma$).

$$\begin{tikzcd}[start anchor = real center, end anchor = real center, column sep = 2ex, row sep = 1ex]
        &[-1.5ex] &[-1.5ex] \phantom{\bullet} \ar[rrrrrr, dotted, no head] \ar[ddll, dotted, no head] \ar[dddddd, dotted, no head] & & & \bullet \ar[loop, phantom, "\beta + \gamma", distance = 1cm] & &[-1.5ex] &[-1.5ex] \phantom{\bullet} \ar[ddll, dotted, no head] \ar[dddddd, dotted, no head]\\
                & \bullet \ar[loop, phantom, "\beta", distance = 1cm] & & & & & & \bigodot \ar[loop, phantom, "\alpha + \beta + \gamma", distance = 1cm]\\
                \phantom{\bullet} \ar[rrrrrr, dotted, no head] \ar[dddddd, dotted, no head] & & & \bigodot \ar[loop, phantom, "\alpha + \beta", distance = 1cm] & & & \phantom{\bullet} \ar[dddddd, dotted, no head]\\
                & & \bullet & & & & & & \bigodot \ar[loop, phantom, "\gamma", distance = 1cm]\\
                & & & & \phantom{\bullet} \ar[llluuu, dash, equal] \ar[luu, dash] \ar[ruuuu, dash] \ar[rrruuu, dash] \ar[rrrru, dash, thick] \ar[drr, dash, equal]\\
                \bigodot & & & & & & \bullet \ar[loop, phantom, "\alpha", distance = 1cm]\\
                & & \phantom{\bullet} \ar[rrrrrr, dotted, no head] \ar[ddll, dotted, no head] & & & \bigodot & & & \phantom{\bullet} \ar[ddll, dotted, no head]\\
                & \bigodot & & & & & & \bullet\\
                \phantom{\bullet} \ar[rrrrrr, dotted, no head] & & & \bullet & & & \phantom{\bullet}
    \end{tikzcd}$$

In the above diagram, $\{\alpha,\beta,\gamma\}$ are simple roots. $\Theta = \{\beta,\gamma\}$ which are denoted by double lines in the picture. Roots in $\Sigma_\lambda$ are marked by $\bigodot$, and those not in $\Sigma_\lambda$ are marked by $\bullet$.

Below is a diagram of the Weyl group, arranged in a way so that elements in the same right $W_\Theta$-coset are grouped together and are connected by double lines. Elements surrounded by shapes are the longest elements in right $W_\Theta$-cosets. Elements that are crossed out are those in $W_\lambda$. There are two double $(W_\Theta,W_\lambda)$-cosets: elements in $W_\Theta s_\gamma s_\beta W_\lambda$ are underlined; elements in $W_\Theta W_\lambda$ are those that are not underlined.

$$\begin{tikzpicture}
        \matrix[column sep = 1ex, row sep = 2ex]{
            &[-3ex] \node[ellipse, draw] (abarba) {$w_0$}; &[-3ex] &[-5ex] &[-3ex] &[-2ex] &[-5ex] &[-5ex] &[-2ex] &[-5ex] &[-5ex]\\
                        \node (abrba) {$s_\alpha s_\beta s_\gamma s_\beta s_\alpha$}; \node[cross out, draw] {\phantom{sss}}; & & 
            \node (barba) {$s_\beta s_\alpha s_\gamma s_\beta s_\alpha$}; &&&&&&&
            \node[rectangle, draw] (abarb) {$\uline{s_\alpha s_\beta s_\alpha s_\gamma s_\beta}$};\\
                        \node (arba) {$s_\alpha s_\gamma s_\beta s_\alpha$}; \node[cross out, draw] {\phantom{sss}}; & &
            \node (brba) {$s_\beta s_\gamma s_\beta s_\alpha$}; & & & &
            \node[trapezium, trapezium left angle=70, trapezium right angle=-70, draw] (abar) {$s_\alpha s_\beta s_\alpha s_\gamma$}; \node[cross out, draw] {\phantom{sss}}; & &
            \node (abrb) {$\uline{s_\alpha s_\beta s_\gamma s_\beta}$}; & &
            \node (barb) {$\uline{s_\beta s_\alpha s_\gamma s_\beta}$};\\
                        & \node (rba) {$s_\gamma s_\beta s_\alpha$}; & & 
            \node[rectangle split, rectangle split horizontal, rectangle split parts=3, draw, minimum height=1.2\baselineskip, inner sep=1pt] (aba) {\nodepart{two}$s_\alpha s_\beta s_\alpha$}; \node[cross out, draw] {\phantom{sss}}; & &
            \node (abr) {$s_\alpha s_\beta s_\gamma$}; & &
            \node (bar) {$s_\beta s_\alpha s_\gamma$}; & 
            \node (arb) {$\uline{s_\alpha s_\gamma s_\beta}$}; & &
            \node (brb) {$\uline{s_\beta s_\gamma s_\beta}$};\\
                        & & \node (ab) {$s_\alpha s_\beta$}; & &
            \node (ba) {$s_\beta s_\alpha$}; &
            \node (ar) {$s_\alpha s_\gamma$}; & &
            \node (br) {$s_\beta s_\gamma$}; & &
            \node (rb) {$\uline{s_\gamma s_\beta}$};\\
                        & & \node (a) {$s_\alpha$}; & &
            \node (b) {$s_\beta$}; & &
            \node (r) {$s_\gamma$}; \node[cross out, draw] {\phantom{sss}};\\
                        & & & \node (1) {$1$}; \node[cross out, draw] {\phantom{sss}};\\
        };
        \draw [double equal sign distance] (1) to (a);
        \draw [double equal sign distance] (1) to (b);
        \draw [double equal sign distance] (a) to (ab);
        \draw [double equal sign distance] (b) to (ba);
        \draw [double equal sign distance] (a) to (ba);
        \draw [double equal sign distance] (b) to (ab);
        \draw [double equal sign distance] (ab) to (aba);
        \draw [double equal sign distance] (ba) to (aba);
                \draw [double equal sign distance] (r) to (ar);
        \draw [double equal sign distance] (r) to (br);
        \draw [double equal sign distance] (ar) to (abr);
        \draw [double equal sign distance] (br) to (bar);
        \draw [double equal sign distance] (ar) to (bar);
        \draw [double equal sign distance] (br) to (abr);
        \draw [double equal sign distance] (abr) to (abar);
        \draw [double equal sign distance] (bar) to (abar);
                \draw [double equal sign distance] (rb) to (arb);
        \draw [double equal sign distance] (rb) to (brb);
        \draw [double equal sign distance] (arb) to (abrb);
        \draw [double equal sign distance] (brb) to (barb);
        \draw [double equal sign distance] (arb) to (barb);
        \draw [double equal sign distance] (brb) to (abrb);
        \draw [double equal sign distance] (abrb) to (abarb);
        \draw [double equal sign distance] (barb) to (abarb);
                \draw [double equal sign distance] (rba) to (arba);
        \draw [double equal sign distance] (rba) to (brba);
        \draw [double equal sign distance] (arba) to (abrba);
        \draw [double equal sign distance] (brba) to (barba);
        \draw [double equal sign distance] (arba) to (barba);
        \draw [double equal sign distance] (brba) to (abrba);
        \draw [double equal sign distance] (abrba) to (abarba);
        \draw [double equal sign distance] (barba) to (abarba);
                            \end{tikzpicture}$$

Let's first look at the double coset $W_\Theta s_\gamma s_\beta W_\lambda$, with $u = s_\gamma s_\beta \in A_{\Theta,\lambda}$. It equals a single left $W_\lambda$-coset $s_\gamma s_\beta W_\lambda$ and a single right $W_\Theta$-coset $W_\Theta s_\gamma s_\beta$. Hence $$\Theta_\lambda^{s_\gamma s_\beta} = \Pi_\lambda,\quad
    W_{\lambda,\Theta_\lambda^{s_\gamma s_\beta}} = W_\lambda,$$ and $r_\lambda (W_\Theta s_\gamma s_\beta) = W_\lambda 1$, the unique right $W_\lambda$-coset in $W_\lambda$. Therefore $$\begin{aligned}
    \varphi_\lambda(W_\Theta s_\gamma s_\beta) &= \delta_{W_\Theta s_\gamma s_\beta},\\
    \operatorname{ch}L(s_\gamma s_\beta \lambda,\eta) &= \operatorname{ch}M(s_\gamma s_\beta \lambda,\eta).\end{aligned}$$

Now let's look at the other double coset $W_\Theta  W_\lambda$, with $u = 1$ and $$\Theta_\lambda^1 = \{\alpha+\beta\}, \quad
    W_{\lambda,\Theta_\lambda^1} = \{1, s_{\alpha+\beta}\}.$$ The root system $\Sigma_\lambda$ and a diagram for $(W_\lambda,\Pi_\lambda,\Theta_\lambda^1)$ are $$\begin{tikzcd}[start anchor = real center, end anchor = real center, column sep = 1ex, row sep = 5ex]
        & \bigodot \ar[dr, equal] \ar[loop, phantom, "\alpha+\beta", distance = 1cm] &  & \bigodot \ar[loop, phantom, "\alpha+\beta+\gamma", distance = 1cm] \ar[dl, dash]\\
        \bigodot \ar[rr, dash] && \phantom{\bigodot} \ar[dl, dash] \ar[dr, dash] \ar[rr, dash] && \bigodot \ar[loop, phantom, "\gamma", distance = 1cm]\\
        & \bigodot &  & \bigodot 
    \end{tikzcd}\qquad
    \begin{adjustbox}{trim = 0 2.3cm 0 0} 
        \begin{tikzpicture}
            \matrix[row sep = 1cm,column sep = 0.2cm]{
                & \node[ellipse, draw] (abr) {$s_{\alpha+\beta+\gamma}$};\\[-4ex]
                \node[trapezium, trapezium left angle=70, trapezium right angle=-70, draw] (ab-r) {$s_{\alpha+\beta} s_\gamma$}; &&                \node (r-ab) {$s_\gamma s_{\alpha+\beta}$};\\
                \node[rectangle split, rectangle split horizontal, rectangle split parts=3, draw, minimum height=1.2\baselineskip, inner sep=1pt] (ab) {\nodepart{two}$s_{\alpha+\beta}$}; &&                \node (r) {$s_\gamma$};\\[-3ex]
                & \node (1) {$1$};\\
            };
            \draw [double equal sign distance] (abr) to  (r-ab) ;
            \draw [double equal sign distance] (ab-r) to  (r) ;
            \draw [double equal sign distance] (ab) to  (1) ;
            \draw (abr) to (ab-r) ;
            \draw (ab-r) to (ab);
        \end{tikzpicture}.
    \end{adjustbox}$$ The map $r_\lambda$ restricted to $W_\Theta \backslash W_\Theta W_\lambda$ can be visualized as $$\begin{tikzpicture}
        \matrix[row sep = 1cm,column sep = 0.2cm]{
            & \node[ellipse, draw] (abr) {$s_{\alpha+\beta+\gamma}$};\\[-4ex]
            \node[trapezium, trapezium left angle=70, trapezium right angle=-70, draw] (ab-r) {$s_{\alpha+\beta} s_\gamma$}; &&            \node (r-ab) {$s_\gamma s_{\alpha+\beta}$};\\
            \node[rectangle split, rectangle split horizontal, rectangle split parts=3, draw, minimum height=1.2\baselineskip, inner sep=1pt] (ab) {\nodepart{two}$s_{\alpha+\beta}$}; &&            \node (r) {$s_\gamma$};\\[-3ex]
            & \node (1) {$1$};\\
        };
        \draw [double equal sign distance] (abr) to  (r-ab) ;
        \draw [double equal sign distance] (ab-r) to  (r) ;
        \draw [double equal sign distance] (ab) to  (1) ;
        \draw (abr) to (ab-r) ;
        \draw (ab-r) to (ab);
    \end{tikzpicture}
    \raisebox{10ex}{$\xleftarrow[\cong]{\quad r_\lambda \quad}$}
    \begin{adjustbox}{trim = 0 1cm 0 0}
        \begin{tikzpicture}
            \matrix[row sep = 1cm,column sep = 0.2cm]{
                &[-5ex] \node[ellipse, draw] (abarba) {$W_\Theta w_0$};\\[-2ex]
                \node[trapezium, trapezium left angle=70, trapezium right angle=-70, draw] (abar) {$W_\Theta s_\alpha s_\beta s_\alpha s_\gamma$};\\
                \node[rectangle split, rectangle split horizontal, rectangle split parts=3, draw, minimum height=1.2\baselineskip, inner sep=1pt] (aba) {\nodepart{two}$W_\Theta s_\alpha s_\beta s_\alpha$};\\
                \node {};\\
            };
                                \end{tikzpicture}
    \end{adjustbox}$$ where a coset on the right hand side is sent to the coset on the left with the same shape. The parabolic Kazhdan-Lusztig polynomials for $(W_\lambda,\Pi_\lambda,\Theta_\lambda^1)$ are $$\begin{tabu}{c|c|ccc}
        P_{C'D'}^{1,\lambda} & \text{\backslashbox{$C'$}{$D'$}} & W_{\lambda,\Theta_\lambda^1} s_{\alpha+\beta} & W_{\lambda,\Theta_\lambda^1} s_{\alpha+\beta} s_\gamma & W_{\lambda,\Theta_\lambda^1} s_{\alpha+\beta+\gamma}\\ \hline
        & W_{\lambda,\Theta_\lambda^1} s_{\alpha+\beta} & 1 & 0 & 0\\
        & W_{\lambda,\Theta_\lambda^1} s_{\alpha+\beta} s_\gamma & q & 1 & 0\\
        & W_{\lambda,\Theta_\lambda^1} s_{\alpha+\beta+\gamma} & 0 & q & 1
    \end{tabu}$$ Hence $$\begin{aligned}
    \varphi_\lambda(W_\Theta s_\alpha s_\beta s_\alpha)
    &= P_{(W_{\lambda,\Theta_\lambda^1} s_{\alpha+\beta}), (W_{\lambda,\Theta_\lambda^1} s_{\alpha+\beta})}^{1,\lambda} \delta_{W_\Theta s_\alpha s_\beta s_\alpha}\\ 
    &\qquad + P_{(W_{\lambda,\Theta_\lambda^1} s_{\alpha+\beta}), (W_{\lambda,\Theta_\lambda^1} s_{\alpha+\beta} s_\gamma)}^{1,\lambda} \delta_{W_\Theta s_\alpha s_\beta s_\alpha s_\gamma}\\
    &\qquad + P_{(W_{\lambda,\Theta_\lambda^1} s_{\alpha+\beta}), (W_{\lambda,\Theta_\lambda^1} s_{\alpha+\beta+\gamma})}^{1,\lambda} \delta_{W_\Theta w_0}\\
    &= \delta_{W_\Theta s_\alpha s_\beta s_\alpha},\\
        \varphi_\lambda(W_\Theta s_\alpha s_\beta s_\alpha s_\gamma) 
    &= P_{(W_{\lambda,\Theta_\lambda^1} s_{\alpha+\beta} s_\gamma), (W_{\lambda,\Theta_\lambda^1} s_{\alpha+\beta})}^{1,\lambda} \delta_{W_\Theta s_\alpha s_\beta s_\alpha}\\ 
    &\qquad + P_{(W_{\lambda,\Theta_\lambda^1} s_{\alpha+\beta} s_\gamma), (W_{\lambda,\Theta_\lambda^1} s_{\alpha+\beta} s_\gamma)}^{1,\lambda} \delta_{W_\Theta s_\alpha s_\beta s_\alpha s_\gamma}\\
    &\qquad + P_{(W_{\lambda,\Theta_\lambda^1} s_{\alpha+\beta} s_\gamma), (W_{\lambda,\Theta_\lambda^1} s_{\alpha+\beta+\gamma})}^{1,\lambda} \delta_{W_\Theta w_0}\\
    &= q \delta_{W_\Theta s_\alpha s_\beta s_\alpha} + \delta_{W_\Theta s_\alpha s_\beta s_\alpha s_\gamma},\\
        \varphi_\lambda(W_\Theta w_0) 
    &= P_{(W_{\lambda,\Theta_\lambda^1} s_{\alpha+\beta+\gamma}), (W_{\lambda,\Theta_\lambda^1} s_{\alpha+\beta})}^{1,\lambda} \delta_{W_\Theta s_\alpha s_\beta s_\alpha}\\ 
    &\qquad + P_{(W_{\lambda,\Theta_\lambda^1} s_{\alpha+\beta+\gamma}), (W_{\lambda,\Theta_\lambda^1} s_{\alpha+\beta} s_\gamma)}^{1,\lambda} \delta_{W_\Theta s_\alpha s_\beta s_\alpha s_\gamma}\\
    &\qquad + P_{(W_{\lambda,\Theta_\lambda^1} s_{\alpha+\beta+\gamma}), (W_{\lambda,\Theta_\lambda^1} s_{\alpha+\beta+\gamma})}^{1,\lambda} \delta_{W_\Theta w_0}\\
    &= q \delta_{W_\Theta s_\alpha s_\beta s_\alpha s_\gamma} + \delta_{W_\Theta  w_0}.\end{aligned}$$ Specializing to $q = -1$, we get $$\begin{aligned}
    \operatorname{ch}L(s_\alpha s_\beta s_\alpha \lambda,\eta) 
    &= \operatorname{ch}M(s_\alpha s_\beta s_\alpha \lambda,\eta),\\
    \operatorname{ch}L( s_\alpha s_\beta s_\alpha s_\gamma \lambda,\eta)
    &= -\operatorname{ch}M(s_\alpha s_\beta s_\alpha \lambda,\eta) + \operatorname{ch}M( s_\alpha s_\beta s_\alpha s_\gamma \lambda,\eta),\\
    \operatorname{ch}L(w_0 \lambda,\eta)
    &= -\operatorname{ch}M( s_\alpha s_\beta s_\alpha s_\gamma \lambda,\eta) + \operatorname{ch}M(w_0 \lambda,\eta).\end{aligned}$$

[^1]: Romanov actually denotes the map by $\varphi$. We reserve the notation $\varphi$ to be used in the main algorithm \[thm:KL\_alg\].

[^2]: In @Romanov:Whittaker [5] $Z_{w^D}$ is denoted by $Z_\alpha$, and $j_{w^D}$ is denoted simply by $j$.

{'source': 'arXiv_out/2112'}