Motivic Chern classes of Schubert cells, Hecke algebras, and applications to Casselman's problem
Paolo Aluffi Leonardo C. Mihalcea Jörg Schürmann Changjian Su

Motivic Chern classes]Motivic Chern classes of Schubert cells,
Hecke algebras, and applications to Casselman's problem




Mathematics Department, 
Florida State University,
Tallahassee FL 32306

aluffi@math.fsu.edu



Department of Mathematics, 
Virginia Tech University, 
Blacksburg, VA 24061

lmihalce@vt.edu


Mathematisches Institut, Universität Münster, Germany
jschuerm@uni-muenster.de


Department of Mathematics, University of Toronto, 
Toronto, ON, M5S, 2E4, Canada
changjiansu@gmail.com
[2010]Primary 14C17, 20C08, 14M15; Secondary 17B10, 14N15, 33D80

P. Aluffi was supported in part by NSA Award H98230-16-1-0016
and a Simons Collaboration Grant; L. C. Mihalcea was supported in part
by NSA Young Investigator Award H98320-16-1-0013 and a Simons
Collaboration Grant; J. Schürmann was funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under
Germany's Excellence Strategy – EXC 2044-390685587, Mathematics
Münster: Dynamics – Geometry – Structure

 
Motivic Chern classes are elements in the -theory of an algebraic
variety X, depending on an extra parameter y. They are determined
by functoriality and a normalization property for smooth X. In this
paper we calculate the motivic Chern classes of Schubert cells in the
(equivariant) -theory of flag manifolds G/B. We show that the
motivic class of a Schubert cell is determined recursively by the
Demazure-Lusztig operators in the Hecke algebra of the Weyl group of
G, starting from the class of a point. The resulting classes are
conjectured to satisfy a positivity property. We use the recursions to
give a new proof that they are equivalent to certain -theoretic
stable envelopes recently defined by Okounkov and collaborators, thus
recovering results of Fehér, Rimányi and Weber. The Hecke
algebra action on the -theory of the Langlands dual flag manifold
matches the Hecke action on the Iwahori invariants of the principal
series representation associated to an unramified character for a
group over a nonarchimedean local field. This gives a correspondence
identifying the
duals of the motivic Chern classes to the
standard basis in the Iwahori invariants, and the fixed point basis to
Casselman's basis. We apply this correspondence to prove two
conjectures of Bump, Nakasuji and Naruse concerning factorizations and
holomorphy properties of the coefficients in the transition matrix
between the standard and the Casselman's basis.



Résumé.
Les classes de Chern motiviques sont des éléments de la
-théorie d'une variété algébri­que X, qui dépendent
d'un paramètre supplémentaire y. Elles sont déterminées par la
fonctorialité et une propriété de normalisation pour X lisse.
Dans cet article, nous calculons les classes de Chern motiviques des
cellules de Schubert dans la -théorie (équivariante) des
variétés de drapeaux G/B. Nous montrons que la classe motivique
d'une cellule de Schubert est détérminée récursivement grâce
aux opérateurs de Demazure-Lusztig de l'algèbre de
Hecke du groupe de Weyl de G, à partir de la classe d'un point. 
Nous conjecturons que les classes obtenues satisfont une propriété
de positivité.  Nous utilisons nos récurrences pour obtenir une
nouvelle preuve du fait que les classes sont équivalentes à
certaines enveloppes stables définies récemment en -théorie
par Okounkov et ses collaborateurs, retrouvant ainsi un résultat de 
Fehér, Rimányi, et Weber. L'action de l'algèbre de Hecke sur la
-théorie de la variété de drapeaux du dual de Langlands
coïncide avec l'action de Hecke sur les invariants d'Iwahori de la
représentation par série principale associée à un caractère non
ramifié pour un groupe sur un corps local non archimédien.  Cela
induit une correspondance identifiant les duaux des
classes de Chern motiviques à la base standard du module des
invariants d'Iwahori, et la base des points fixes à la base de
Casselman. Nous appliquons ce résultat pour démontrer deux
conjectures dues à Bump, Nakasuji et Naruse concernant les factorisations
et les propriétés d'holomorphie des coefficients de la matrice de
transition entre la base standard et la base de Casselman.


[
    Changjian Su
    December 30, 2023
=====================







§ INTRODUCTION
 
Let X be a complex algebraic variety, and let (/X) be the
(relative) Grothendieck group of varieties over X. It consists of
classes of morphisms [f: Z → X] modulo the scissors relations;
cf. <cit.> and <ref>
below. Brasselet, Schürmann and
Yokura <cit.> defined the motivic Chern transformation _y: (/X) → K(X)[y] with
values in the -theory group of coherent sheaves in X to which
one adjoins a formal variable y. The transformation _y is a
group homomorphism, it is functorial with respect to proper
push-forwards, and if X is smooth, it satisfies the normalization
condition

    _y[𝕀_X: X → X] = ∑ [∧^j T^*(X)] y^j .
 
Here [∧^j T^*(X)] is the -theory class of the bundle of
degree j differential forms on X. If Z ⊆ X is a
constructible subset, we denote by _y(Z):= _y[Z ↪
X] ∈ K(X)[y] the motivic Chern class of Z. Because _y is a
group homomorphism, it follows that if X =  Z_i is a
disjoint union of constructible subsets, then _y(X) = ∑_y(Z_i).  As explained
in <cit.>, the motivic Chern
class _y(Z) is related by a Hirzebruch-Riemann-Roch type statement
to the Chern-Schwartz-MacPherson (CSM) class (Z) in the homology 
of X. We recall that the existence and functoriality properties of this 
CSM class were conjectured by Deligne and Grothendieck and proved by 
Robert MacPherson (<cit.>).
Earlier, Marie-Hélène Schwartz had independently established a theory
of Chern classes for singular varieties, using obstruction theory
(<cit.>). Jean-Paul Brasselet and Schwartz proved
that the two classes coincide via the Alexander isomorphism (<cit.>).
Both the
motivic and the CSM classes give a functorial way to attach
-theory, respectively (co)homology classes, to constructible
subsets, and both satisfy the usual motivic relations. There is also an
equivariant version of the motivic Chern class transformation, which
uses equivariant varieties and morphisms, and has values in the
appropriate equivariant -theory group. Its definition was given
in <cit.>, following closely the approach
of <cit.>.  

We take this opportunity to provide further details on the
construction and properties of equivariant motivic Chern classes, such
as functoriality and a Verdier-Riemann-Roch formula; see <ref>
below. However, the main goals of this paper are to build the
computational foundations for the study of the (equivariant) motivic
Chern classes of Schubert cells in the generalized flag manifolds, and
to relate this to the representation theory of p-adic groups. Our
main application consists of formulas for the transition coefficients
between the standard and the Casselman bases of the module of Iwahori
invariants of the principal series representation, in terms of
localizations of motivic Chern classes.

Let G be a complex, semisimple, linear algebraic group, and B a
Borel subgroup.  By functoriality, the (equivariant) motivic Chern
classes of Schubert cells in G/B determine those in any flag
manifold G/P, where P is a parabolic subgroup. Based on previously
discovered features of the CSM classes of Schubert
cells <cit.>,
it was expected that the motivic classes would be closely related to
objects which appear in geometric representation theory. We prove in
this paper that the motivic Chern classes of Schubert cells are
recursively determined by the Demazure-Lusztig operators which appear
in early works of Lusztig on Hecke algebras <cit.>.
Further, the motivic classes of Schubert cells are equivalent (in a
precise sense) to the -theoretic stable envelopes defined by
Okounkov and collaborators in <cit.>. This equivalence was proved recently by
Fehér, Rimányi and Weber <cit.>; our
approach, based on comparing the Demazure-Lusztig recursions to the
recursions for the stable envelopes found by Su, Zhao and Zhong
in <cit.> gives another proof of this result. Via this
equivalence, the motivic Chern classes can be considered as natural
analogues of the Schubert classes in the -theory of the cotangent
bundle of G/B.
 
As in the authors' previous work on CSM classes <cit.>, 
the connections to Hecke algebras and -theoretic stable envelopes
yield remarkable identities among (duals of)
motivic Chern classes. We use these identities to prove two
conjectures of Bump, Nakasuji and
Naruse <cit.> about the
coefficients in the transition matrix between the Casselman's basis
and the standard basis in the Iwahori-invariant space of the principal
series representation for an unramified character for a group over a
non archimedean local field.

We present next a more extensive description of our results.



 §.§ Statement of results
 
Let G be a complex, semisimple, linear algebraic group, and fix B,
B^- a pair of opposite Borel subgroups of G. Denote by T:= B ∩
B^- the maximal torus, by W:= N_G(T)/T the Weyl group, and by X:=
G/B the (generalized) flag variety. For each Weyl group element w
∈ W consider the Schubert cell X(w)^∘:= B w B/B, a subvariety
of (complex) dimension ℓ(w). The opposite Schubert cell
Y(w)^∘:= B^- w B/B has complex codimension ℓ(w). The
closures X(w) and Y(w) of these cells are the Schubert varieties.
Let _w, respectively ^w be the -theoretic Schubert
classes associated to the structure sheaves of X(w), respectively
Y(w). The equivariant -theory ring of X, denoted by K_T(X),
is an algebra over K_T()=R(T)—the representation ring of
T—and it has an R(T)-basis given by the Schubert classes
_w (or ^w), where w varies in the Weyl group W.

If E is an equivariant vector bundle over X, we denote by [E]
its class in K_T(X), and by λ_y(E) the class

    λ_y(E) = ∑ [∧^i E] y^i ∈ K_T(X) [y] .
 
For a T-stable subvariety Ω⊆ X recall the notation

    _y(Ω):= _y[ Ω↪ X] ∈ K_T(X)[y] .
 
Our first main result is a recursive formula to calculate
_y(X(w)^∘), the (equivariant) motivic Chern class of the
Schubert cell. For each simple positive root α_i, consider the
Demazure operator ∂_i: K_T(X) → K_T(X)
(<cit.>,
<cit.>);
this is a
K_T()-linear endomorphism.  Extend ∂_i linearly with
respect to y, and define the Demazure-Lusztig (DL) operators
_i, ^∨_i: K_T(X)[y] → K_T(X)[y] by

    _i: = λ_y(_α_i) ∂_i 
    - 𝕀;   ^∨_i: = ∂_i 
    λ_y(_α_i) - 𝕀,
 
where _α_i = G ×^B _α_i is the 
equivariant line bundle whose fiber over 1.B has weight 
α_i.  The operator
^∨_i appeared classically in Lusztig's study of Hecke
algebras <cit.>, and _i appeared recently in
related
works <cit.>. The two
operators are adjoint to each other via the -theoretic
intersection pairing; see <ref> below. Our first main
result is the following (cf. Theorem <ref>).

 
Let w ∈ W and let s_i be a simple reflection such that
ℓ(ws_i) > ℓ(w). Then

    _y(X(ws_i)^∘) = _i( _y(X(w)^∘)) .
 
 

Using the (equivariant) -theoretic Chevalley
formula <cit.> to
multiply by classes of line bundles, the DL operators give a recursive
formula to calculate the motivic Chern classes, starting from the
class of a point.

Theorem <ref> generalizes the analogous result
from <cit.> where it was proved that the CSM
classes of Schubert cells are recursively determined by operators in
the degenerate Hecke algebra.  The proof of Theorem <ref>
relies on the calculations of push-forwards of classes from the
Bott-Samelson desingularizations of Schubert varieties.

To illustrate the result, we list below the non-equivariant motivic
Chern classes of Schubert cells in (3):= _3()/B,
the manifold parametrizing flags in ^3. In this case, the Weyl
group is the symmetric group S_3, generated by simple reflections
s_1 and s_2, and w_0 = s_1 s_2 s_1 is the longest element.

    _y(X(𝕀))    = _𝕀; 
    _y(X(s_1)^∘)    = (1+y) _s_1 - (1+2y) _𝕀;
    _y(X(s_2)^∘)    = (1+y) _s_2 - (1+2y) _𝕀;
    _y(X(s_1 s_2)^∘)    = (1+y)^2 _s_1 s_2 - (1+y) (1+2y)
    _s_1 - (1+y)(1+3y) _s_2 + (5y^2+ 5y+1) _𝕀;
    _y(X(s_2 s_1)^∘)    = (1+y)^2 _s_2 s_1 - (1+y) (1+2y)
    _s_2 - (1+y)(1+3y) _s_1 + (5y^2 + 5y+1) _𝕀;
    _y(X(w_0)^∘)    = (1+y)^3 _w_0 - (1+y)^2 (1+2y)
    (_s_1s_2 + _s_2 s_1) + 
        (1+y)(5 y^2 + 4 y +1)(_s_1  + _s_2) 
    - (8y^3+11y^2+ 5y+1) _𝕀.
One observes in these examples, and one can also prove it in general,
that the specialization y ↦ 0 in _y(X(w)^∘) yields the
(push-forward to G/B of the)
ideal sheaf of the boundary of the Schubert variety X(w). The
Schubert class _w is obtained if one takes y ↦ 0 in a
recursion given by a renormalization of the inverse of the dual operator
^∨_i; see Example <ref>. In fact, Theorem <ref> generalizes the
well-known fact from Schubert Calculus that the Schubert classes
_w are obtained recursively by the Demazure operators
∂_i. These and other combinatorial properties of the motivic
Chern classes will be studied in a continuation to this paper.

A remarkable feature in the examples listed above is a positivity
property. Based on substantial computer evidence we make the following
positivity conjecture:

 
Consider the Schubert expansion 

    _y(X(w)^∘) =  ∑ c(w;u) _u ∈ K_T(X)[y] .

Then the coefficients c(w;u) ∈ K_T()[y] satisfy
(-1)^ℓ(w)-ℓ(u)c(w,u)∈_≥ 0[y][e^-α_1, …
  , e^-α_r], where α_i are the positive simple
roots. In particular, in the non-equivariant case,

    (-1)^ℓ(w)-ℓ(u)c(w;u) ∈_≥ 0[y]  .



In type A, a similar positivity property was also conjectured
in <cit.>, along with a log concavity property. In the
non-equivariant case, 
it is conjectured in <cit.> 
that the polynomials

    (-1)^ℓ(w) - ℓ(u) c(w;u)/(1+y)^ℓ(u)
 
are log-concave. In cohomology, Aluffi and Mihalcea conjectured that CSM
classes of Schubert cells are 
positive <cit.>. For 
Grassmannians, this was established
by J. Huh <cit.>; a few special cases were settled
earlier in <cit.>. In full G/P generality, and in the
non-equivariant case, the conjecture was recently proved
in <cit.>, using the theory of characteristic cycles in
the cotangent bundle of G/B. There is also a stronger version of
this conjecture, which claims that in addition c(w;u)≠ 0 whenever
u ≤ w. Huh's proof shows this, and also establishes (implicitly)
the equivariant version for Grassmannians. The statement of the
conjecture is reminiscent of the positivity in (equivariant)
-theory proved by Buch <cit.>, Brion <cit.>,
and Anderson, Griffeth and
Miller <cit.>.

Let _y^∨(Y(v)^∘) be the classes obtained by applying the
(inverse) dual operators (^∨_i)^-1 to ^w_0
(instead of _i to _𝕀); see
Definition <ref>. We prove (Theorem <ref>)
that for every u, v∈ W,

    ⟨_y(X(u)^∘), _y^∨(Y(v)^∘)⟩ 
    = δ_u,v(-y)^ℓ(u)- G/B∏_α>0
    (1+ye^-α) ,
 
where ⟨· , ·⟩ is the -theoretic
intersection pairing. This orthogonality property, which we call
`Hecke duality', mirrors the similar orthogonality of
Chern-Schwartz-MacPherson (CSM) classes proved by the authors
in <cit.>.



 §.§ Applications to Casselman's problem
 
The main application in this paper is to use the Hecke algebra action
on motivic Chern classes of Schubert cells to prove two conjectures of
Bump, Nakasuji and
Naruse <cit.> about properties
of certain coefficients of the transition matrix between two natural
bases of the Iwahori invariant part of the principal series
representation. We briefly recall below the relevant history,
definitions and the results; the details and complete proofs are given
in <ref> below.

Let τ be an unramified character for a split reductive Chevalley
group (F) over a nonarchimedean local field F with finite
residue field _q'.  The principal series representation is the
induced representation I(τ):=_(F)^(F)
(τ). We consider its submodule I(τ)^I of Iwahori invariants
of I(τ); this is a Hecke module, with an additive basis indexed
by the Weyl group W. There are two important bases: the standard basis, given by the characteristic
functions φ_w, and the Casselman basis { f_w },
defined using certain intertwiners; see (<ref>). Casselman's problem <cit.> is to find the
transition matrix between the two bases:

    φ_u = ∑ a_u,w f_w .

As observed by Bump and Nakasuji <cit.>, it is better to consider
the basis ψ_u := ∑_w ≥ uφ_w and the expansion

    ψ_u = ∑m̃_u,w f_w .
 
By Möbius inversion, the problems of finding either of the
transition matrices are equivalent. Recent solutions to the
Casselman's problem were obtained by Naruse and
Nakasuji <cit.>, using the Yang-Baxter basis in the
Hecke algebra introduced by Lascoux, Leclerc and
Thibon <cit.>, and by Su, Zhao and Zhong <cit.>,
by means of the theory of stable envelopes developed
in <cit.>. The
-theoretic stable envelopes are certain classes in the
equivariant -theory of the cotangent bundle T^*(G/B), indexed
by the Weyl group elements; see <ref> below. Su, Zhao and
Zhong proved that the Hecke algebra action on the basis of stable
envelopes coincides with the Hecke algebra action on the standard
basis φ_w. Under their correspondence, the Hecke action on the
Casselman's basis fits with the Hecke action on the fixed point basis
in equivariant -theory.

Fehér, Rimányi and Weber <cit.> 
observed that motivic Chern classes and
-theoretic stable envelopes are closely related; see also
<ref> below. Therefore it is not a surprise that one can
recover the Hecke correspondence from <cit.> using motivic
Chern classes. The advantage of this point of view is that motivic
Chern classes satisfy strong functoriality properties, and this will
allow us to obtain additional properties of the coefficients
m̃_u,v.

Let  be the Langlands dual of .
It turns out that the Hecke module I(τ)^I is more naturally
related to the equivariant -theory K_(/)
for the Langlands dual flag manifold.
Let ι_w be the fixed point basis in K_(/),
and let b_w be the multiple of ι_w determined by the
localization 

    b_w|_w =_y^∨(Y(w)^∘)|_w

at the fixed point e_w. The formula for _y^∨(Y(w)^∘)|_w
∈ K_()[y^-1] is explicit; see
Proposition <ref>. From this formula it follows that
the elements b_w are in the localized ring

    K_(/)_[y^-1]:= 
    K_(/)[y^-1] ⊗_K_()(K_()) ,
 
where  denotes the fraction field.  We show 
(Theorem <ref>)
that for a
sufficiently general τ there is an isomorphism of Hecke modules
Ψ: K_(/)_[y,y^-1]
⊗_K_()_τ→ I(τ)^I such that

    Ψ (_y^∨(Y(w)^∘) ⊗ 1) =  φ_w;   Ψ(b_w ⊗ 1)=  f_w;   Ψ(y) = -q' ,

with q'=|_q'| the number of elements in the finite residue
field _q', and _τ the one-dimensional
K_()-module obtained by evaluation at τ. Using this
result, we prove that m̃_u,w = m_u,w(τ), where
m_u,w are the coefficients in the expansion

    _y^∨(Y(u)) := ∑_w≥ u_y^∨(Y(w)^∘) 
    = ∑ m_u,w b_w   ∈  
    K_(/)_[y^-1] .
 
Implicit in this is that the coefficients m_u,w may be regarded as
complex valued functions defined on a certain Zariski open subset of
the dual torus .

The Hecke isomorphism Ψ provides a `dictionary', translating all
statements about m̃_u,w into statements about m_u,w,
which have geometric meaning. The key result for the
representation theoretic applications is that the coefficients
m_u,w are given by localization
(cf. Proposition <ref> below):

 
(a) For every w≥ u∈ W, the coefficient  m_u,w equals 

    m_u,w = (_y(Y(u))|_w/_y(Y(w)^∘)|_w)^∨,
 
where ∨ is the operator mapping e^λ↦ e^-λ
for e^λ∈ K_T() and y ↦ y^-1.

(b) Assume that Y(u) is smooth at the fixed point e_w and denote
by (N_Y(w) Y(u))_w the normal space at e_w in Y(u), regarded
as a trivial (but not equivariantly trivial) vector bundle. Then

    m_u,w = λ_y^-1 ((N_Y(w) Y(u))_w)/λ_-1 ((N_Y(w) Y(u))_w).



In particular, the entries 
m_1,w
are obtained from the motivic
Chern class of the full flag variety _y(Y(𝕀))=_y(G/B) =
λ_y(T^*(G/B)), and one recovers the (geometric version of the)
classical Gindikin-Karpelevich formula, proved by
Langlands <cit.>:

    m_1,w = ∏1 +y^-1 e^α/1 - e^α,
 
where the product of over positive roots α such that
w^-1(α) < 0. Let

    S(u,w):={α∈ R^+ | u≤ s_α w < w}.

Our main application is the following factorization formula for
m_u,v, see Theorem <ref>
below:

 
 
For every u≤ w∈ W,

    m_u,w=∏_α∈ S(u,w)1+y^-1e^α/1-e^α,

if and only if the Schubert variety Y(u) is smooth at the torus
fixed point e_w.
 

This is the geometric analogue of a conjecture of Bump and
Nakasuji <cit.> for simply laced types,
generalized to all types by Naruse <cit.>, and
further analyzed by Nakasuji and Naruse <cit.>.
While this paper was in preparation, Naruse informed us that he also
obtained an (unpublished) proof of the implication assuming
factorization. Both Naruse's and our proofs are based on Kumar's
cohomological criterion for smoothness of Schubert
varieties <cit.>; Naruse used Hecke algebra calculations;
ours relies on properties of motivic Chern classes. The original
conjecture of Bump and Nakasuji from <cit.> was stated in terms
of conditions under which certain Kazhdan-Lusztig polynomials P_w_0
  w^-1, w_0 u^-1 equal 1; we explain the equivalence to the
statement above (in simply laced types) and discuss further this
conjecture in sections <ref> and
<ref>.

A second conjecture refers to a holomorphy property. In relation to
Kazhdan-Lusztig theory, Bump and Nakasuji <cit.>
defined the coefficients

    r̃_u,w := ∑_u ≤ x ≤ w (-1)^ℓ(x) 
    - ℓ(u)m̃_x,w,
 
where the bar operator replaces q' by q'^-1. Using
Möbius inversion it follows that the coefficients a_u,w 
from (<ref>) satisfy a̅_u,w=
r̃_u,w. Geometrically, 
these correspond to the coefficients
r_u,w obtained from the expansion

    _y^∨(Y(u)^∘) = ∑r_u,w b_w   ∈  K_T(G/B)_[y^-1] ,
 
where f̅(y) := f(y^-1). We prove the following result 
(cf. Theorem <ref>), which answers affirmatively
Conjecture 1 from <cit.>.
 
Let u ≤ w be two Weyl group elements. Then the functions

    ∏_α∈ S(u,w)(1-e^α)r_u,w   , 
    ∏_α∈ S(u,w)(1-e^α)m_u,w

are holomorphic on the torus T.


Both Theorems <ref> and <ref>
are consequences of Theorem <ref>. The proof of the
latter requires a second orthogonality property between motivic Chern
classes and their duals, proven by means of the connection with the
the theory of -theoretic
stable envelopes. From this orthogonality we deduce the following key
formula, proved in Theorem <ref>:

    _y^∨(Y(u)^∘) = ∏_α>0(1+ye^-α) 
    (_y(Y(u)^∘))/λ_y(T^*(G/B))
 
as elements in the appropriate localized -theory ring,
where [E]= (-1)^ X [E^∨] ⊗ [∧^ X
  T^*(X)] is the (equivariant) Grothendieck-Serre duality
operator, with X=G/B.The class
    (_y(Ω))/λ_y(T^*(G/B)) may be regarded
    as the motivic analogue of the Segre-MacPherson class
    c_*(_Ω)/c(T(G/B)),
cf. Definition <ref>.
The proof requires a precise relationship between the motivic Chern classes and
stable envelopes. If ι: X → T^*_X is the zero section, then
our statement is that (roughly)

    (ι^*(_+(w))) = N(q) _-q^-1(X(w)^∘) ,
 
where _+(w) is a stable envelope, N(q) is a normalization
parameter, and q=-y^-1 is determined from the dilation action of
^* on the fibers of the cotangent bundle. The precise statement is
given in Theorem <ref>.

Formula (<ref>) is part of a more general paradigm,
stemming from the classical works of Sabbah <cit.> and
Ginzburg <cit.>, relating intersection theory
on the cotangent bundle to that of characteristic classes of singular
varieties. For instance, the (co)homological analogues of the motivic
Chern classes of Schubert cells—the CSM classes—are equivalent to
Maulik and Okounkov's cohomological stable
  envelopes <cit.>. This statement, observed
by Rimányi and Varchenko <cit.>, and by the
authors in <cit.>, is a consequence of the fact that both
the stable envelopes and the CSM classes are determined by certain
interpolation conditions obtained from equivariant localization;
cf. Weber's article <cit.>. The relation to stable envelopes was
recently extended to -theory by Fehér, Rimányi and
Weber <cit.> (see also <cit.>). They showed
that the motivic Chern classes of the Schubert cells satisfy the same
localization conditions as the -theoretic stable envelopes
appearing in papers by Okounkov and
Smirnov <cit.> for a
particular choice of parameters. (The result
from <cit.> is more general, involving the motivic
Chern classes for orbits in a space with finitely many orbits under a
group action.) We reprove this result by comparing the
Demazure-Lusztig type recursions for motivic Chern classes to the
recursions for the stable envelopes found by Su, Zhao and Zhong
in <cit.>. We also discuss the relation between the motivic
Chern class and various choices of parameters for the -theoretic
stable envelopes, which might be of independent interest;
see <ref> below.

Acknowledgments. 
Part of this work was performed at MSRI during
the special semester in `Enumerative geometry beyond numbers'; LM and
CS would like to thank the organizers of the workshops of the special
MSRI semester for providing support to participate to this program,
and to MSRI for providing a stimulating environment. LM would like to
thank Anders Buch, Daniel Orr, Richard Rimányi, Mark Shimozono and
Andrzej Weber for stimulating discussions. CS thanks IHES for
providing excellent research environment and Joel Kamnitzer, Shrawan
Kumar, Andrei Okounkov and Eric Vasserot for useful
discussions. 
Special thanks are due to Hiroshi Naruse for informing us
of the refined Bump-Nakasuji-Naruse conjecture from
Theorem <ref>. This happened while CS and LM were
attending the `International Festival in Schubert Calculus',
Guangzhou, China, November 2017; we wish to thank the organizers of
this conference for providing support and an environment conducive to
research. PA also thanks the University of Toronto for the hospitality.
 Finally, all authors wish to thank the anonymous referees, for the careful 
reading and for many suggestions which improved the exposition and helped streamline 
some of the arguments.




§ SCHUBERT VARIETIES AND THEIR BOTT-SAMELSON RESOLUTIONS

In this section we recall the basic definitions and facts about the
Bott-Samelson resolution of Schubert varieties. These will be used in
the next section to perform the calculation of the motivic Chern class
of a Schubert cell. Our main references
are <cit.> and <cit.>.

Let G be a complex semisimple linear algebraic group. Fix a Borel 
subgroup B
and let T := B ∩ B^- be the maximal torus, where B^- denotes
the opposite Borel subgroup. Let W:=N_G(T)/T be the Weyl group, and
ℓ:W → the associated length function. Denote by
w_0 the longest element in W; then B^- = w_0 B w_0. Let also
Δ := {α_1, … , α_r }⊆ R^+ denote the
set of simple roots included in the set of positive roots for
(G,B). The simple reflection for the root α_i ∈Δ is
denoted by s_i, and P_i denotes the corresponding minimal
parabolic subgroup.

Let X:=G/B be the flag variety. It has stratifications by Schubert
cells X(w)^∘:= BwB/B and by opposite Schubert cells
Y(w)^∘:=B^- w B/B.  The closures X(w):= X(w)^∘
and Y(w):=Y(w)^∘ are the Schubert varieties. Note
that Y(w)=w_0 X(w_0w). With these definitions, _ X(w) =
_ Y(w) = ℓ(w). The Weyl group W admits a partial
ordering, called the Bruhat ordering, defined by u ≤ v if and only
if X(u) ⊆ X(v).

We recall next the definition of the Bott-Samelson resolution of a
Schubert variety, following <cit.>
and <cit.>.  Fix w ∈ W and a 
decomposition of w, i.e., a sequence (i_1, … , i_k) such that
w=s_i_1·…· s_i_k.
If ℓ(w) = k, then this decomposition is called reduced.
This data determines a tower Z of ^1-bundles and a
birational map θ: Z → X(w) as follows.

If the word is empty, then define Z:= =B/B↪ G/B. 
In general assume we have
constructed Z':= Z_i_1, …, i_k-1 and the map θ': Z'
→ X(w')→ G/B, for w'= s_i_1⋯ s_i_k-1.

Define Z=Z_i_1,…, i_k so that the diagram

    @R=10pt@C=10pt
    Z [rr]^-θ[dd]_-π       G/B [dd]^-p_i_k
       □
    
    Z' [rr]^-p_i_k∘ θ'       G/P_i_k

is a fiber square; the morphism p_i_k is the natural projection.
In fact, p_i_k: G/B → G/P_i_k is the projectivization of a 
homogeneous rank-2 vector bundle, hence so is π: Z→ Z'.
From this construction it follows that Z is a smooth
projective variety of dimension k.

The Bott-Samelson variety Z is equipped with a simple normal
crossing (SNC) divisor ∂ Z, constructed inductively as follows.  If
Z=, then ∂ Z=∅. In general,
the map π admits a section σ, defined as
the unique map Z'→ Z making the following diagram commute:

    @C=30pt
    Z' @.>[dr]|-∃!σ@/^1pc/[drr]^θ'@/_1pc/[ddr]_𝕀_Z'
        Z [r]^-θ[d]_-π    G/B [d]^-p_i_k
        Z' [r]^-p_i_k∘ θ'    G/P_i_k

In particular, θ'=θ∘σ.
We let D_k:=σ(Z'), and then
the SNC divisor on Z is defined by

    ∂ Z = π^-1(∂ Z') ∪ D_k
 
where ∂ Z' is the inductively constructed SNC divisor on Z'.
The following result is well known, see e.g.,
<cit.>. 

 
If s_i_1… s_i_k is a reduced word for w, then
the image of the composition θ= _1 ∘ θ_1:
Z_i_1,…,i_k→ G/B is the Schubert variety X(w). Moreover,
θ^-1(X(w) ∖ X(w)^∘) = ∂ Z_i_1,…,i_k
and the restriction map

    θ: Z_i_1, …,i_k∖∂ Z_i_1,…,i_k→ X(w)^∘
 
is an isomorphism. 

The proposition implies that the Bott-Samelson variety
Z_i_1,…,i_k is a log-resolution of the Schubert variety
X(w).






§ EQUIVARIANT -THEORY OF FLAG MANIFOLDS AND DEMAZURE-LUSZTIG OPERATORS
 
In this section we recall the definition and basic properties of
equivariant -theory of flag manifolds, and of certain
Demazure-Lusztig operators acting on equivariant -theory. This
setup is well-known from the theory of Hecke algebras, see
e.g., <cit.> and <cit.>.



 §.§ Equivariant -theory
 
Let X be a smooth, quasi-projective algebraic variety endowed with a
T-action. The (algebraic) equivariant -theory ring K_T(X) is
the ring generated by symbols [E], where E → X is an equivariant
vector bundle, modulo the relations [E]=[E_1]+[E_2] for all short
exact sequences 0 → E_1 → E → E_2 → 0 of equivariant vector
bundles. The ring addition is given by direct sums, and multiplication
by tensor products. Since X is smooth, every (equivariant) coherent
sheaf has a finite resolution by (equivariant) vector bundles
<cit.>, and K_T(X)
coincides with the Grothendieck group of (equivariant) coherent
sheaves on X. The ring K_T(X) is an algebra over the Laurent
polynomial ring K_T() =[e^± t_1, … , e^± t_r] where
e^t_i are characters corresponding to a basis of the Lie algebra
of T; alternatively K_T() may be viewed as the representation
ring R(T) of T.

In our situation X=G/B and T acts on X by left
multiplication. Since X is smooth, the ring K_T(X) coincides with
the Grothendieck group of T-linearized coherent sheaves on X.
There is a pairing, called the -theoretic intersection pairing,

    ⟨ - , - ⟩ :K_T(X) ⊗
    K_T(X) → K_T() = R(T)
 
defined on classes [E], [F] of vector bundles by

    ⟨ [E], [F] ⟩ := ∫_X E ⊗ F = χ(X; E ⊗ F)
    .
 
Here χ(X; -) is the (equivariant) Euler characteristic, i.e.,
the virtual character

    χ(X;-) = ∑ (-1)^i  _T (H^i(X; -)) .


Let _w:=[_X(w)] be the Grothendieck class determined by
the structure sheaf of X(w) (a coherent sheaf), and similarly
^w:= [_Y(w)]. The equivariant -theory ring has
K_T()-bases {_w }_w ∈ W and {^w }_w ∈
  W. Let ∂ X(w) :=X(w) ∖ X(w)^∘ be the boundary
of the Schubert variety X(w), and similarly ∂ Y(w) the
boundary of Y(w). It is known that the dual bases of {_w } and {^w } are given by the classes of the ideal sheaves
^w:= [_Y(w)(-∂ Y(w))], respectively _w:=
[_X(w)(-∂ X(w))], i.e.,

    ⟨_u , ^v ⟩ =
    ⟨^u, _v ⟩ = δ_u,v.
 
See e.g., <cit.> for the
non-equivariant case; the same proof works equivariantly. See
also <cit.>.  In fact,

    _w= ∑_v ≤ w_v 
       and   _w
     = ∑_v ≤ w (-1)^ℓ(w) - ℓ(v)_v

(<cit.>).
We will also need that

    ⟨_u, ^v ⟩ =
    
    0    if  u< v 
    
    1    if  u≥ v 
    ;

this is proved in e.g., <cit.>.



 §.§ Demazure-Lusztig (DL) operators
 
Fix a simple root α_i ∈Δ and the corresponding minimal
parabolic subgroup P_i ⊆ G. 
Consider the diagram

    @C=50pt
    G/B ×_G/P_i G/B
    [r]^-_1[d]^_2    G/B [d]^p_i
     
    G/B [r]^p_i    G/P_i

The Demazure operator ∂_i: K_T(X) → K_T(X)
<cit.> is defined by

    ∂_i:= (p_i)^* (p_i)_* = (_1)_* _2^* .

Since G/B→ G/P_i is a projective bundle, (p_i)_*(p_i)^* is the identity,
and it follows that ∂_i^2 = ∂_i.
The operator ∂_i satisfies (e.g., from <cit.>)

    ∂_i(_w) = 
    _ws_i    if  ws_i>w ; 
    _w     otherwise .

One can verify that for v∈ W, represented by a reduced word s_i_1⋯
s_i_k, the operator

    ∂_v:=∂_i_1∘⋯∘∂_i_k

is independent of the chosen reduced word; cf. <cit.>.  With this definition we have that if
ℓ(uv^-1)=ℓ(u)+ℓ(v^-1), then
∂_v(_u)=_uv^-1.  Since p_i is G-equivariant
and Y(w)=w_0 X(w_0w), it follows easily that
∂_i(^w)=^ws_i if ws_i<w and
∂_i(^w) = ^w otherwise.

Fix an indeterminate y. The λ_y-class of a vector bundle E
is the class

    λ_y(E):= ∑_k [∧^k E] y^k ∈ K_T(X)[y] .
 
The λ_y-class is multiplicative, i.e., if 0 → E_1 → E →
E_2 → 0 is a short exact sequence of equivariant vector bundles,
then λ_y(E) = λ_y(E_1) λ_y(E_2) as elements in
K_T(X)[y]. We refer to the
books <cit.> for
details in the non-equivariant case. The equivariant case involves no
additional subtleties.

We define next the main operators used in this paper. 
 
Let α_i ∈Δ be a simple root. Define the operators 

    _i: = λ_y(T^*_p_i) ∂_i - 𝕀;   ^∨_i: = ∂_i λ_y(T^*_p_i) - 𝕀.
 
 
The operators _i and ^∨_i are K_T()[y]-module
endomorphisms of K_T(X)[y]. We will occasionally work in
K_T(X)[y^± 1] and regard these as K_T()[y^± 1]-module
endomorphisms.

 
The operator ^∨_i was defined by Lusztig
<cit.> in relation to affine Hecke algebras and
equivariant -theory of flag varieties. (Lusztig worked in
topological equivariant -theory, but since X=G/B has a
T-invariant algebraic cell-decomposition by Schubert cells, the
algebraic and topological equivariant -theories of X are
naturally
isomorphic <cit.>[Proposition 5.5.6,
  p. 272].) As we shall see below, the `dual' operators _i
arises naturally in the study of motivic Chern classes of Schubert
cells. In an algebraic form, the operators _i appeared recently in
<cit.> and 
 <cit.>, in relation to Whittaker functions.


 
The operators _i and ^∨_i are 
adjoint to each other. That is, for every a, b ∈ K_T(X), 

    ⟨_i(a), b ⟩ = ⟨ a , 
    ^∨_i(b) ⟩.
 
The same equality holds for a, b ∈ K_T(X)[y^± 1], if one
extends the pairing bilinearly in y.


  
The identity is self adjoint and ∂_i is also self adjoint.
Indeed, by the projection formula

    ⟨∂_i (a), b ⟩ = ∫_G/B p_i^*(p_i)_*(a) · b
    = ∫_G/P_i (p_i)_*(a)· (p_i)_*(b) ,
 
and the last expression is symmetric in a,b. It remains to show that
coefficient of y in both sides is the same, i.e., ⟨ T^*_p_i∂_i (a), b ⟩ = ⟨ a, ∂_i T^*_p_i (b)
⟩.  We calculate

    ⟨ T^*_p_i∂_i (a) , b ⟩   = 
    ∫_G/B T^*_p_i  p_i^*((p_i)_* a) · b = 
    ∫_G/P_i (p_i)_*(a) · (p_i)_*( T^*_p_i· b)  
       = ∫_G/B a · p_i^*(p_i)_*( T^*_p_i· b) = 
    ⟨ a, ∂_i(T^*_p_i· b) ⟩.
 


According to Lusztig's result (<cit.>),
the operators ^∨_i satisfy the braid relations and the
quadratic relations defining the Hecke algebra H_W(-y) of the Weyl
group W with parameter -y. (In the language of this paper, the
variable q from <cit.> satisfies q=-y.)  Since the
 -theoretic pairing is non-degenerate,
Lemma <ref> implies that both sets of operators
_i and ^∨_i satisfy the same identities. We
record this next.

 
The operators _i and ^∨_i satisfy the braid relations
for the Weyl group W. For each simple root α_i ∈Δ the
following quadratic relations hold:

    (_i + 𝕀) (_i + y) = (^∨_i + 𝕀) 
    (^∨_i + y) = 0 .



From these relations it follows that the operators _i and
^∨_i are invertible in K_T(X)[y^± 1]:

    _i^-1 = - 1/y_i - 1+y/y𝕀  ;    (^∨_i)^-1 
    = -1/y^∨_i - 1+y/y𝕀.

Since the operators _i, ^∨_i satisfy the braid
relations, we may define

    _v := _i_1⋯_i_k  ,  ^∨_v := ^∨_i_1⋯^∨_i_k

for v∈ W represented by a reduced word s_i_1⋯ s_i_k.

The cohomological versions of the Demazure-Lusztig operators, which
appear in the study of degenerate Hecke algebras
<cit.> are self inverse (i.e.,
(_i^coh)^2 = 𝕀) and therefore satisfy the relations of the
group algebra [W] of the Weyl group. Obviously, this is not true
in -theory, due to (<ref>).  However, the multiplication
of Demazure-Lusztig operators behaves rather nicely, as shown in the
following proposition.

 
Let u, v ∈ W be two Weyl group elements. Then 

    _u ·_v^-1 = c_uv^-1(y)
    _uv^-1 + ∑_w < uv^-1 c_w(y) _w ,

where c_uv^-1 and c_w(y) are rational functions in y
determined by u,v. Further, if ℓ(uv^-1) = ℓ(u)
+ℓ(v^-1), then c_uv^-1(y) = (-y)^-ℓ(v).  The same
statements hold for the multiplication ^∨_u ·
(^∨_v)^-1 of the dual operators.
 
To prove Proposition <ref> we need the following lemma.

 
Let u,v ∈ W be two Weyl group elements, and let s be a root
reflection such that us>u and vs < v. Then usv^-1 < uv^-1. 


 
(Cf. <cit.>.)
Let s = s_α for some positive root α.  By hypothesis
us_α>u and vs_α<v, hence u(α)>0 and
v(α)<0. Since uv^-1(-v(α))<0, it follows that
uv^-1s_v(α)<uv^-1.  On the other hand we have
s_v(α) = v s_α v^-1; therefore
uv^-1s_v(α)=us_α v^-1, and we are done.


 
We use ascending induction on ℓ(v) ≥ 0. The statement is clear
if ℓ(v) =0. For ℓ(v) >0 write v = v' s_k where ℓ(v')
< ℓ(v). By definition, _u ·_v^-1 = _u ·_k^-1·_v'^-1. We have two cases: u s_k < u
and us_k>u. Consider first the situation us_k <u. Then u has a
reduced decomposition ending in s_k, i.e., u = u' s_k and
ℓ(u') < ℓ(u). Then

    _u ·_k^-1·_v'^-1 =
    _u'·_k ·_k^-1·_v'^-1 = _u'·_v'^-1,
 
and since v' < v the result is known by induction. Assume next that
us_k >u. Using equation (<ref>) we obtain

    _u ·_v^-1 = _u ·_k^-1·_v'^-1 = - 1/y_u · (_k + y+1) ·_v'^-1
    = - 1/y_us_k·_v'^-1 -
    1+y/y_u ·_v'^-1.
 
By induction, the leading term of _us_k·_v'^-1
is _us_k v'^-1=_uv^-1 and the leading term of _u ·_v'^-1 is _u v'^-1. Observe that
vs_k = v'<v.  From Lemma <ref> we obtain that
uv'^-1=us_kv^-1<uv^-1, and this concludes the proof of the
existence of an expression (<ref>) as stated. If
ℓ(u)+ℓ(v^-1)=ℓ(uv^-1), then arguing again by ascending
induction on ℓ(v) gives, with notation as above,

    _u ·_v^-1 
    = - 1/y_us_k·_v'^-1 + ⋯
    = - 1/y1/(-y)^ℓ(v')_uv^-1 + ⋯ ,

proving that c(uv^-1)=(-y)^-ℓ(v) as claimed.

The statements for the dual operators are proved in the same way.




 §.§ Actions on Schubert and fixed point bases
 
We will need several formulas concerning the action of the
Demazure-Lusztig operators on Schubert classes and on the classes
determined by the torus fixed points. For instance, the definition and
a standard localization argument
(cf. Lemma <ref>(a) below) imply that, for
every w ∈ W,

    _i( _w) = 
    
    (1+y e^-wα_i) _ws_i + l.o.t.     if  ws_i>w;
     
    y e^wα_i_w + l.o.t.    if  ws_i<w.

where l.o.t. (lower order terms) stands for a sum of terms P(y, e^t)
_u with u<ws_i on the first branch and u <w on the second
branch. A similar formula holds for the dual operator:

    ^∨_i (^w) = (1+ y e^w_0w(α_i)) ^ws_i +
    l.o.t.   if  ws_i < w ,
 
where now the l.o.t. consist of multiples of ^v for v > ws_i.
Consider next the localized equivariant -theory ring

    K_T(G/B) ↪  K_T(G/B)_ 
    := K_T(G/B) ⊗_K_T()( K_T())  ,

where  denotes the fraction field. The Weyl group elements
w∈ W are in bijection with the torus fixed points e_w:=wB ∈
G/B. Let ι_w:=[_e_w] ∈ K_T(G/B)_ be the class of
the structure sheaf of e_w. By the localization theorem, the classes
ι_w form a basis for the localized equivariant -theory
ring; we call this the fixed point basis. For a weight λ
consider the G-equivariant line bundle _λ := G ×^B
_λ with character λ in the fiber over 1.B.[
Observe that our notation is opposite to the one 
in <cit.>.]
For example, the relative cotangent bundle T^*_p_i for the projection
p_i is isomorphic to _α_i.  We need the following
lemma.

 
The following formulas hold in K_T(G/B)_[y^± 1]:

(a) For every weight λ, _λ·ι_w =
e^w λι_w;

(b) For every simple root α_i, 

    ∂_i (ι_w) =1/1-e^wα_iι_w
    +1/1-e^-wα_iι_ws_i;


(c) The action of the operator _i on the fixed point basis is
given by the following formula

    _i(ι_w)=-1+y/1-e^-wα_iι_w
    +1+ye^-wα_i/1-e^-wα_iι_ws_i.


(d) The action of the adjoint operator ^∨_i is given by


    ^∨_i(ι_w)=-1+y/1-e^-wα_iι_w
    +1+ye^wα_i/1-e^-wα_iι_ws_i.


(e) The action of the inverse operator (^∨_i)^-1is given
by

    (^∨_i)^-1(ι_w)=- 1+y^-1/1- e^wα_iι_w-y^-1+e^wα_i/1-e^-wα_iι_ws_i.



 
Part (a) is a standard localization calculation, based on the fact
that the fiber of ℒ_λ over the fixed point e_w is
a one-dimensional T-module of weight w(λ).  For part (b),
notice that ∂_i (ι_w) = [_p_i^-1(e_w)], where
p_i:G/B → G/P_i is the projection, and (abusing notation) e_w
also denotes the corresponding fixed point in G/P_i. The fiber
p_i^-1(e_w)≅^1 equals w.[X(s_i)], the w-translate of
the Schubert curve.  It contains only two T-fixed points:
e_w and e_ws_i.  It follows that [_p_i^-1(e_w)] = a
ι_w + b ι_ws_i for suitable a, b.  By the projection
formula we can regard this as an expansion in the localized
equivariant -theory of the fiber itself. By the localization
theorem (see e.g., <cit.>, and also <cit.>), 

    1 = a · (ι_w)|_e_w =a (1 - e^w α_i) ;    
    1 =b·  (ι_ws_i)|_e_ws_i = b (1 - e^- w α_i) .
   
Then the statement in (b)
follows.  Part (c) follows from (a) and (b), applied to
_i(ι_w)=(1+y_α_i)∂_i(ι_w)-ι_w.
Formula (d) for the adjoint operator follows similarly.  Finally, part
(e) follows because (^∨_i)^-1=-y^-1(^∨_i+y+1)
(cf. equation (<ref>)).


We also record the action of several specializations of the
Demazure-Lusztig operators.

 
(a) The specializations at y=0 satisfy

    (_i)_y=0 = (^∨_i)_y=0 = ∂_i-𝕀.
 
Further, the following holds for every w ∈ W:

    (∂_i - 𝕀) (_w )= 
    _ws_i    if  ws_i > w 
     
    - _w    if  ws_i < w  .

(b) Let w ∈ W. Then the specializations at y=-1 satisfy

    (_i)_y=-1 (ι_w) = ι_ws_i ;    
    (^∨_i)_y=-1 (ι_w)= 1- e^w α_i/1- e^-wα_iι_ws_i = -e^w α_iι_ws_i.

In other words, this specialization is compatible with the right Weyl
group multiplication.


 
Part (a) is an easy exercise using
the definition of _i and identities (<ref>) 
and (<ref>).
Part (b) is immediate 
from Lemma <ref>. 




§ MOTIVIC CHERN CLASSES
 We recall the definition
of the motivic Chern classes,
following <cit.>.  For now let
X be a quasi-projective, complex algebraic variety, with an action
of the complex torus T. First we recall the definition of the
(relative) motivic Grothendieck group ^T(/X) of varieties
over X, mostly following Looijenga's notes <cit.>
and Bittner's papers <cit.>. For
simplicity, we only consider the T-equivariant quasi-projective
context, in <cit.>), which is enough for all applications in
this paper.  The group ^T(/X) is the free abelian group
generated by symbols [f: Z → X] for isomorphism classes of
T-equivariant morphisms f : Z → X, where Z is a
quasi-projective T-variety, modulo the usual additivity relations

    [f: Z → X] = [f: U → X] + [f:Z  ∖ U → X]
 for U
⊆ Z an open invariant subvariety.  If X= then
^T(/) is a ring with the product given by the external
product of morphisms, and the groups ^T(/X) have a module
structure over ^T(/) also given by the external product.
For every equivariant morphism g:X → Y of quasi-projective
T-varieties there is a functorial push-forward g_!: ^T(/X)
→^T(/Y) (given by composition) and pull-back
g^*:^T(/Y) →^T(/X) (given by fiber product).
Finally there are external products 
    ×: ^T(/X)×^T(/X') →^T(/X× X');    [f]× [f']↦
    [f× f'],
 which are ^T(/)-bilinear and commute with
push-forward and pull-back.  In particular push-forward g_! and
pull-back g^* are ^T(/)-linear.

 
(Cf. <cit.>.)
For every variety X, similar functors can be defined on the ring of
constructible functions (X), and the Grothendieck ring
(/X) may be regarded as a motivic version of (X). In
fact, there is a map e: (/X) →(X) sending [f:Y →
  X] to f_!(_Y), where f_!(_Y) is defined using compactly
supported Euler characteristic of the fibers. The map e is a group
homomorphism, and if X= then e is a ring homomorphism. The
constructions extend equivariantly 
to ^T(/X) →^T(X), with
^T(X)⊆(X) the subgroup of T-invariant constructible functions.

 
 
Let X be a quasi-projective, non-singular, complex algebraic variety
with an action of the torus T. There exists a unique natural
transformation _y: ^T(/X) → K_T(X)[y] satisfying the
following properties:
 

  (1) It is functorial with respect to T-equivariant proper
  morphisms of non-singular, quasi-projective varieties.

  (2) 
It satisfies the normalization condition 

    _y[𝕀_X: X → X] = λ_y(T^*_X) = ∑ y^i [∧^i T^*_X]_T
    ∈ K_T(X)[y].


The transformation _y satisfies the following properties:


  (3) It commutes with external products:

    _y[f× f': Z× Z'→ X× X']=_y[f: Z→ X] ⊠_y[f': Z'→ X']  .



  (4) For every smooth, T-equivariant morphism π: X → Y of
  quasi-projective and non-singular algebraic varieties, and any [f:
    Z → Y] ∈^T(/Y), the following Verdier-Riemann-Roch
  (VRR) formula holds:

    λ_y(T^*_π) ·π^* _y[f:Z → Y] = _y[π^* f:Z
      ×_Y X → X] .

 


 
In the non-equivariant case all these statements are proved
in <cit.>. Fehér,
Rimányi and Weber used similar ideas to extend the motivic Chern
classes to the equivariant situation in <cit.>. They
proved that there is a well defined class

    _y[f:Z → X] ∈ K_T(X)[y]

for Z and X smooth, but omitted the details showing functoriality
properties, referring instead back
to <cit.>. For completeness, we
prove this theorem, using the definition (and well-definedness)
from <cit.>.  First one gets as
in <cit.> a tautological isomorphism

    ^T(/X)→^T(/X): [f:Z → X]↦ [f:Z → X] ,

where ^T(/X) is the corresponding (relative) motivic
Grothendieck group of smooth quasi-projective T-varieties
mapping to X, with the corresponding `additivity' relation only
asked for T-invariant open subsets U⊆ Z with Z∖
U a smooth closed subvariety of Z.  Here the inverse  image of [f:Z
  → X] ∈^T(/X) in ^T(/X) for Z possibly singular
is defined as

    [f:Z → X] ↦∑_i   [f:Z_i → X]

where Z=  Z_i is a decomposition into a finite disjoint
union of T-invariant (locally closed) smooth subvarieties
Z_i⊆ Z.  Using this presentation, _y is determined by
its value on ^T(/X): i.e., we have _y[f:Z → X] := ∑_i
 _y[f:Z_i → X] with notation as above. If U is a smooth
quasi-projective T-variety we consider the definition of _y[f: U
  → X] from <cit.> and show that it satisfies the
corresponding `additivity' relation. Let f: U → X be an
equivariant morphism with U quasi-projective and non-singular. Then
there exists a non-singular quasi-projective algebraic variety 
containing U such that the following are satisfied:
 

  *  admits an action of T and the inclusion i:U
  → is an open, T-equivariant embedding,


  * the boundary D:=∖ U =
  ⋃_i=1,…,s D_i is a T-invariant simple normal crossing
  divisor with smooth irreducible components D_i,


  * there exists a proper T-equivariant morphism :
  → X such that f = ∘ i
(such a morphism  is then also projective, by 
e.g., <cit.>).

This follows as in <cit.> from the
existence of a T-equivariant projective completion <cit.>,
<cit.> and  from 
equivariant resolution of singularities 
<cit.>.
 Then `additivity and normalization' forces

    _y[f: U → X]:=∑_I⊆{1,…,s} 
    (-1)^|I|f_I*λ_y(T^*D_I) ,

with D_I:=∩_i∈ I D_i (and D_∅:=) and f_I:=
|_D_I. By <cit.> the right-hand side is
independent of all choices. In particular _y satisfies the
normalization property from part (2) of the statement.  We show next
that the transformation _y satisfies the corresponding
`additivity' property. Let Y⊆ U be a closed T-invariant
subvariety.  By induction on the number of connected components of
Y, we can assume Y is connected. Then one can find as
in <cit.> a partial compactification of U and f
as before in such a way that the closure Y of Y in
 is smooth and Y has normal crossing with D. Then
Y and |_Y is such a partial
compactification of Y and f|_Y with the corresponding simple normal
crossing divisor Y∩ D. Let us now first assume that
Y is hypersurface in U, so that Y∪ D =
Y∪⋃_i=1,…,s D_i is a simple normal
crossing divisor in .  Then one can use  and  as a
partial compactification of f: U∖ Y→ X, with the
corresponding simple normal crossing divisor Y∪ D.
 Using in this context the definition given in (<ref>), one gets
 precisely the sought-for `additivity' property in the case of
a hypersurface Y in U.
 In general, let U be the
blow-up of  along Y, with exceptional divisor E
and D_i the strict transform of D_i (i=1,…,s),
as well as f̃: U→ X the induced proper
morphism. Then D_I is the blow-up of D_I along
Y_I:=Y∩ D_I with exceptional divisor
E_I:=E ∩D_I for I⊆{1,…,s}. As
observed in <cit.> and <cit.>, the key equality needed is the following `blow-up
relation',

    _y[f̃: D_I → X] -_y[f̃: E_I → X] =
    _y[: D_I → X] -_y[: Y_I → X] ;

the heart of its proof relies on vanishing of certain sheaf cohomology
groups proved in <cit.>. 
Using the blow-up
relations and the additivity from the hypersurface case for the partial
compactification X of U∖ Y, with corresponding
simple normal crossing divisor E ∪D,  proves the `additivity' property in general. This construction
shows that the transformation _y is determined by its image on
classes [f:Z → X] where Z is a non-singular, irreducible,
quasi-projective algebraic variety and f is a T-equivariant proper morphism.

To prove part (1) of the statement, i.e., functoriality with respect
to T-equivariant proper morphisms, observe that if g: X→ X' is a
proper equivariant morphism of smooth quasi-projective T-varieties,
then  and g∘ f:=g∘ is a partial
compactification of g∘ f: U→ X', with (g∘ f)_I=g∘
f_I such that

    _y[g∘ f: U → X'] =∑_I⊆{1,…,s} 
    (-1)^|I|g_*f_I*λ_y(T^*D_I)=g_*( _y [f: U →
      X])  .

With the construction of _y given above, the proofs of parts (3)
and (4) of the statement follow as in the non-equivariant case
of <cit.>, by
making all -theory classes and morphisms equivariant.


If one forgets the T-action, then the equivariant motivic Chern
class recovers the non-equivariant motivic Chern class from
<cit.>, either by its
construction, or by the properties (1)-(2) from
Theorem <ref> and the corresponding results
from <cit.>. Further,
Theorem <ref> and its proof work more generally for a
possibly singular, quasi-projective T-equivariant base variety X,
provided one works with the Grothendieck group K_0^T(X) of
T-equivariant coherent _X-modules; then one obtains _y:
^T(/X) → K_0^T(X)[y].

Since most of the time the variety X will be understood from the
context, for Z ⊆ X a (not necessarily closed) subvariety
we use the notation

    _y(Z) := _y[Z ↪ X] .
 
By functoriality, if Z ⊆ X is a smooth closed subvariety, then
_y[i:Z ↪ X]= i_*(λ_y(T^*Z) )
as elements in K_T(X). We will often suppress the push-forward
notation.

 There are some differences between the definition 
of the relative equivariant Grothendieck group of varieties in 
<cit.>, and hypotheses 
used therein, and those used in this paper. For instance, 
<cit.> use finite groups G with a
`good' action; we use a torus T in the complex quasi-projective
context, but can work similarly with a complex linear algebraic group
G. Bittner also divides by an additional `projective bundle
relation', stating that for a G-equivariant projective bundle
(V)→ Z over a relative G-variety Z→ X:

    [(V) → Z → X] = [^(V) -1× Z→ Z → X]  ,

where on the right-hand side G only acts on Z and X. This is 
not needed in this paper; we will show in future work
that the motivic Chern class also 
factorizes over this additional relation. Despite these differences, the 
proof of Theorem <ref> applies to all these contexts, 
following the ideas from loc. cit. and <cit.>. 
At the heart of the arguments is the fact that ^G(/X) ≃^G(/X), together with results on equivariant completion,  
equivariant resolution of singularities and an equivariant weak 
factorization theorem 
<cit.> 
as used in <cit.>. If one also divides by the 
'projective bundle relation', then one can also define a motivic 
duality involution on the Grothendieck group (localized at the class 
of the affine line), which commutes under the motivic Chern class
transformation with the Grothendieck-Serre duality involution. For a 
discussion of this involution see <cit.> and <cit.>; 
also cf. (<ref>) below.


The following general lemma is useful.
 
Let X_1, X_2 ⊆ X be three T-equivariant varieties. 
The following equalities hold in K_T(X)[y].

(a) The inclusion exclusion formula: 

    _y(X_1 ∪ X_2) = _y(X_1) +
    _y(X_2) - _y(X_1 ∩ X_2) .


(b) If X_1, X_2, X are smooth, and X_1,X_2 intersect transversally
(so that X_1∩ X_2 is also smooth), then

    _y (X_1 ∩ X_2) = _y(X_1) _y (X_2)/_y (X).


(c) More generally, (<ref>) holds if X_1, X_2 are unions of
smooth hypersurfaces such that X_1∪ X_2 is a divisor with simple
normal crossings.



 
The statement in (a) is immediate from the additivity property in the
Grothendieck group. Part (b) follows from standard exact sequences,
using that the normal bundle N_X_1 ∩ X_2 X is the restriction
to X_1 ∩ X_2 of N_X_1 X ⊕ N_X_2 X. Finally, part (c)
follows from repeated application of (a) and (b), using
inclusion-exclusion and induction on X
and on the number of components of X_1 ∪ X_2.
 



In part (a), the scheme structure on the union X_1∪ X_2 is
irrelevant; in fact, _y[Z↪ X]=
_y[Z_red↪ X] since both classes equal
_y(X)-_y(X∖ Z).   
In parts (b) and (c) we work in the ring of formal series in y, which 
allows us to invert the class _y(X) = 1 + ∑_k>0 y^k [∧^k T^*X];
the right-hand side must actually land in K_T(X)[y], since it equals the left-hand side.
When X=G/B, the inverse of _y(X) can also be calculated from
Remark <ref>.  


  An alternative formulation of 
Lemma <ref>(b) may be given in terms of motivic Segre classes 
_y(X_i) := _y(X_i)/_y(X): 

    _y(X_1 ∩ X_2) = _y(X_1) 
    ·_y(X_2) .

A statement generalizing this formula for X_i possibly singular and under 
a Whitney transversality assumption will be proved in upcoming work.
In the (co)homological case, i.e. after replacing the motivic Chern
classes by the CSM classes, the analogue of this statement
was proved in <cit.>.




§ MOTIVIC CHERN CLASSES OF SCHUBERT CELLS VIA DEMAZURE-LUSZTIG OPERATORS
 
In this section we calculate the motivic Chern classes of
Schubert cells in X=G/B, using Demazure-Lusztig operators.

We use the definitions and notation from <ref>. Fix a
word (i_1, … , i_k) and let Z:=Z_i_1, … , i_k and
Z':=Z_i_1, … , i_k-1 be the corresponding Bott-Samelson
varieties. Recall that we have determined a section σ:Z' → Z 
of the projection π:Z → Z', and let D:=D_k=σ(Z'). 
The `boundary' ∂ Z := π^-1(∂ Z') ∪ D is a
simple normal crossings divisor.
We have the diagram

    @R=12pt@C=12pt
    Z [rr]^-θ[dd]^π       G/B [dd]^p_i_k
       □
     
    Z' @/^1pc/[uu]^σ[rr]^-p_i_k∘ θ'       G/P_i_k

If w=s_i_1·…· s_i_k is a reduced decomposition,
then Z is a resolution of the Schubert variety X(w)
(Proposition <ref>). But note that the construction of
the Bott-Samelson variety for (i_1, … , i_k) can be carried out
even if the word is non-reduced.  Our main theorem is:

 
Let (i_1, … , i_k) be a (possibly non-reduced) word. Then 

    θ_* _y[Z∖∂ Z↪ Z] 
    = _i_kθ'_* _y[Z' ∖∂ Z'↪ Z'] ,
 
as elements in K_T(G/B)[y]. In particular, if w ∈ W and s_i is
a simple reflection such that ws_i > w, then

    _y[X(ws_i)^∘↪ G/B]= _i _y[X(w)^∘↪ G/B] .



 
The second claim follows from the first. Indeed, take any reduced word
(i_1,…, i_k-1) for w∈ W, so that (i_1,…,
i_k-1,i) is a reduced word for ws_i.  The restrictions θ:Z
 ∖∂ Z → X(ws_i)^∘ and θ':Z' ∖∂ Z' → X(w)^∘ are (equivariant) isomorphisms and, by
functoriality, θ_*_y[Z ∖∂ Z↪ Z] =
_y[X(ws_i)^∘↪ G/B] and θ'_*_y[Z'
∖∂ Z'↪ Z'] 
= _y[X(w)^∘↪ G/B].

We now prove the first assertion[We are especially grateful to one referee for
suggesting a simplification of our original argument.]. 
By the inductive construction of ∂ Z,

    _y[Z∖∂ Z↪ Z]
    =_y[π^-1(Z'∖∂ Z')↪ Z]
    -σ_* _y[Z'∖∂ Z'↪ Z'].

By the VRR formula in Theorem <ref> (4),

    _y[π^-1(Z'∖∂ Z')↪ Z]
    =λ_y(T^*_π) π^* _y[Z'∖∂ Z'↪ Z'].

Since T^*_π=θ^* T^*_p_i_k, using the projection formula and the 
base change formula θ_* π^*=(p_i_k)^*(p_i_k∘θ')_* gives

    θ_* _y[π^-1(Z'∖∂ Z')↪ Z]
       =λ_y(T^*_p_i_k) (p_i_k)^*(p_i_k)_* θ'_* 
    _y[Z'∖∂ Z'↪ Z'] 
       =(λ_y(T^*_p_i_k) ∂_i_k) θ'_* 
    _y[Z'∖∂ Z'↪ Z']

As θ∘σ = θ', we obtain

    θ_* _y[Z∖∂ Z↪ Z]
       =(λ_y(T^*_p_i_k) ∂_i_k - 𝕀) θ'_* 
    _y[Z'∖∂ Z'↪ Z']  
       =_i_kθ'_* 
    _y[Z'∖∂ Z'↪ Z']

as stated.


We record the following corollary.

 
Let w ∈ W and let s_i be a simple reflection. Then 

    _i (_y  (X(w)^∘)) =
    _y( X(ws_i)^∘)     if  ws_i>w ;
     
    -(y+1) _y(X(w)^∘ ) - y _y (X(ws_i )^∘ )     if  ws_i < w .

 

 
The identity from the ws_i > w branch was proved in
Theorem <ref>. Assume that ws_i <w. Then the same
result shows that _y(X(w)^∘) = _i (_y(X(ws_i)^∘)),
thus

    _i (_y (X(w)^∘)) = _i^2 (_y (X(ws_i)^∘)) .

By the quadratic relations from
Proposition <ref>, _i^2 = - (y+1)_i -
y ·𝕀. Now we apply the right-hand side to _y
(X(ws_i)^∘), using again Theorem <ref> and that
(ws_i) s_i > ws_i.



In particular, setting y=-1:

    _i|_y=-1 (_-1  (X(w)^∘)) =_-1  (X(ws_i)^∘)

regardless of whether ws_i precedes or follows w in the Bruhat order. 
Combined with Lemma <ref>, this implies that 

    _-1(X(w)^∘) = ι_w ,
 
the class of the fixed point e_w.
The corresponding statement holds for the CSM class in (co)homology,
cf. <cit.>.



Recall that if w=s_i_1⋯ s_i_k is a reduced decomposition, the 
operator _w := _i_1⋯_i_k is well-defined
(cf. (<ref>)). With this notation,

    _y (X(w)^∘)=_w^-1(_𝕀)

as a consequence of Theorem <ref>.


 
The equivariant motivic Chern classes for ℙ^1 are: 

    _y(X(𝕀)) = _𝕀;   _y(X(s)^∘) = (1+ e^-α_1 y)
    _ℙ^1 - (1 + (1+ e^-α_1)y) _𝕀.

To recover the non-equivariant classes from the equivariant ones one 
substitutes e^λ↦ 1 for each weight λ. 
For instance, the non-equivariant motivic Chern class of 
X(s)^∘⊂ℙ^1 is 

    _y(X(s)^∘) = (1+ y)_ℙ^1 - (1 + 2y) _𝕀.

(Note that this recovers the examples of the classes of X(s_1)^∘ 
and X(s_2)^∘ given in the introduction.)


 The equivariant motivic Chern classes for larger 
flag manifolds are much more complicated. For instance, the equivariant motivic 
Chern class of the Schubert cell X(s_1 s_2)^∘⊂(3) is 

    _y(X(s_1s_2)^∘) =     (1+ e^-α_1y)(1+ e^-(α_1 
    + α_2)y) _s_1 s_2 - 
        (1+ e^-α_1y)(1+(1+ e^-(α_1
    + α_2))y) _s_1  - 
        (1 + (1 + e^-α_1)(1+e^-α_2)y 
    + e^-α_2(1+ e^-α_1+ e^-2α_1)y^2)_s_2 + 
        (1 + (2+ e^-α_1 + e^-α_2 + e^-(α_1+α_2))y) _𝕀
    + 
        (1+ e^-α_1 + e^-α_2 + e^-(α_1+α_2) + 
    e^-(2 α_1 + α_2))y^2_𝕀.





 §.§ Motivic Chern classes in G/P
 
Let P ⊃ B be a parabolic subgroup containing B and let W^P
⊆ W be the subset of minimal length representatives for
W/W_P, the quotient of W by the subgroup W_P generated by the
reflections in P. For w W_P ∈ W/W_P, ℓ(wW_P) denotes the
length of the (unique) representative of wW_P in W^P. The Schubert
cells in G/P are X(wW_P)^∘:= BwP/P ⊆ G/P; then
X(wW_P)^∘≃^ℓ(wW_P). The natural projection
π: G/B → G/P sends X(w)^∘ to X(wW_P)^∘ and it is an
isomorphism if w ∈ W^P. From this and the functoriality of motivic
Chern classes it follows that ∀ w∈ W^P,

    π_* _y[X(w)^∘↪ G/B] = _y[X(wW_P)^∘↪ G/P] .
 

 
In fact, one can prove more: from <cit.> one obtains
that the restriction π|_X(w)^∘: X(w)^∘→ X(wW_P)^∘
is an equivariantly trivial fibration with fiber a Schubert cell of
dimension ℓ(w) - ℓ(wW_P) in π^-1(e_wW_P) ≃ P/B,
regarded as a homogeneous space for the Levi subgroup of P. It is
not difficult to show that this implies that for all w ∈ W,

    π_* _y[X(w)^∘↪ G/B] = (-y)^ℓ(w) - ℓ(w
      W_P)_y[X(wW_P)^∘↪ G/P] .
 
Details of the proof and applications to point counting in
characteristic p will be included in a continuation to this paper.






§ THE HECKE DUALITY FOR MOTIVIC CHERN CLASSES
 
It was proved in <cit.> that the Poincaré duals
of the CSM classes of Schubert cells are given by the operators which
are adjoint to the Hecke-type operators which determine the CSM
classes. The same phenomenon holds in the context of this paper, with
the same idea of proof. However, in this context the DL operators
satisfy the quadratic relations (<ref>), while in the
cohomological case studied in <cit.> the corresponding
operators are self-inverse.  This leads to somewhat more involved
calculations for motivic Chern classes. In analogy with the dual CSM
class from <cit.> we make the following
definition.

 
Let w ∈ W. The dual motivic Chern class is defined by 

    _y^∨(Y(w)^∘):=(^∨_w_0w)^-1(_y(Y(w_0)))
    =(^∨_w_0w)^-1(^w_0)∈ K_T(G/B)[y,y^-1] .


The name of this class is explained by the following theorem, which is
the -theoretic analogue of <cit.>.


For every u, v∈ W, 

    ⟨_y(X(u)^∘), _y^∨(Y(v)^∘)⟩ =
    δ_u,v(-y)^ℓ(u)- G/B∏_α>0(1+ye^-α)
    .



  Another interpretation of the dual class, in terms of 
Serre duality, is given in Theorem <ref> below.
In fact, we could have alternatively defined the dual class by this theorem, then 
proved that it is given by the operator from Definition <ref>. See also 
<cit.> for 
more about the relation between Serre duality and the Hecke involution on 
Demazure-Lusztig operators.

Also note that, geometrically,
the quantity ∏_α>0(1+ye^-α) equals
λ_y(T^*_w_0(G/B)), i.e., it is the λ_y class of the
fiber of the cotangent bundle at w_0.  


 
Using the definition of both flavors of motivic classes, and the fact
that _i and ^∨_i are adjoint to each other, we obtain
that

    ⟨_y(X(u)^∘), _y^∨(Y(v)^∘)⟩ = ⟨_u^-1 (_𝕀), (^∨_w_0v)^-1(^w_0)
    ⟩ = ⟨_𝕀, ^∨_u ·
    (^∨_w_0v)^-1(^w_0) ⟩.
 
By Proposition <ref>, 

    ^∨_u · (^∨_w_0v)^-1 = c_uv^-1 w_0(y)
    ^∨_uv^-1 w_0 + ∑_w < uv^-1 w_0 c_w(y)
    ^∨_w .
 
Since ^∨_w (^w_0) is a combination of Schubert 
classes ^w' such that w≤ w' and ⟨_𝕀,
^w⟩=δ_w, 𝕀, ⟨_𝕀, ^∨_u
· (^∨_w_0v)^-1(^w_0) ⟩ is 0 unless
uv^-1 w_0 = w_0, i.e., u=v. In this case, by (<ref>), 
the coefficient of ^𝕀 in
^∨_w_0(^w_0) is 
∏_α >0 (1+ y e^-α). 
By Proposition <ref>, the coefficient of
^∨_w_0 in ^∨_u · (^∨_w_0u)^-1 is
(-y)^-ℓ(w_0u). Therefore,

    ⟨_y(X(u)^∘), _y^∨(Y(u)^∘)⟩   =⟨_𝕀, ^∨_u · (^∨_w_0u)^-1(^w_0)
    ⟩
       =⟨_𝕀,(-y)^-ℓ(w_0u)T^∨_w_0(^w_0)
    ⟩
       =⟨_𝕀,(-y)^-ℓ(w_0u)∏_α >0 (1+ y
    e^-α)^𝕀⟩
       =(-y)^ℓ(u)- G/B∏_α >0 (1+ y e^-α),
 
concluding the proof.


 
It is natural to consider the normalized class 

    _y(Y(w)^∘):= (-y)^ G/B -
      ℓ(w)_y^∨(Y(w)^∘) .
 
The classes _y(Y(w)^∘) are given by the normalized
operator _i := ^∨_i + (1+y) 𝕀; cf. equation
(<ref>). The coefficients in the Schubert expansion of this class
are polynomial in y.  


 
The motivic Chern class for Schubert cells in (3) were listed in
the introduction.  The normalized dual motivic classes
_y(Y(w)^∘) for the Schubert cells in (3),
computed using (<ref>) and Definition <ref>, are:

    _y(Y(w_0))    = ^w_0; 
    _y(Y(s_1 s_2)^∘)    = (1+y) ^s_1s_2 + y
    ^w_0; 
    _y(Y(s_2 s_1)^∘)    = (1+y) ^s_2 s_1 + y
    ^w_0; 
    _y(Y(s_1)^∘)    = (1+y)^2 ^s_1 + y(1+y)
    ^s_1 s_2 + 2y (1+y)^s_2 s_1 + y^2 ^w_0;
    _y(Y(s_2)^∘)    = (1+y)^2 ^s_2 + 2y(1+y)
    ^s_1 s_2 + y (1+y)^s_2 s_1 + y^2 ^w_0;
    _y(Y(𝕀)^∘)    = (1+y)^3 ^𝕀 + y (1+y)^2
    (^s_1 + ^s_2) +2 y^2 (1+y)( ^s_1 s_2 
    + ^s_2 s_1) + y^3 ^w_0.

An algebra verification, using the fact that ⟨_u,
^v ⟩ = 1 if u ≥ v and ⟨_u, ^v
⟩ = 0 otherwise (cf. (<ref>)), shows that

    ⟨_y(X(u)^∘), _y(Y(v)^∘) ⟩ =
    (1+y)^(3)δ_u,v,
 
as prescribed by Theorem <ref>. 
(Here we are setting
the equivariant variables e^α to 1.)
 At this time we note
that the analogue of the positivity Conjecture <ref> is
false for the dual classes. For instance the coefficient of
^s_3 s_1 s_2 in the expansion of
_y(Y(𝕀)^∘) ∈ K((4)) equals y^2
(4y-1)(1+y)^3.  



In the next result we determine the action of the operators
^∨_i on the dual motivic classes.

 
Let w ∈ W be a Weyl group element and s_i a simple
reflection. Then the following equalities hold:

    a^∨_i (_y^∨(Y(w)^∘)) = 
    _y^∨(Y(ws_i)^∘)     if  ws_i > w 
     
    -(y+1) _y^∨(Y(w)^∘) - y _y^∨(Y(ws_i)^∘) 
        if  ws_i<w


    b
    (^∨_i)^-1 (_y^∨(Y(ws_i)^∘)) = 
    _y^∨(Y(w)^∘)     if  ws_i > w 
     
    -1/y_y^∨(Y(w)^∘) - y+1/y_y^∨(Y(ws_i)^∘)     if  ws_i<w



 
To prove part (a), consider first the case when ws_i> w. Then
w_0 ws_i < w_0 w, thus ^∨_w_0 w = ^∨_w_0w s_i^∨_i. By Definition <ref>,

    ^∨_i (_y^∨(Y(w)^∘)) =
    ^∨_i(^∨_w_0w)^-1(^w_0) =
    (^∨_w_0ws_i)^-1(^w_0) = _y^∨(Y(ws_i)^∘).
 
The situation when ws_i <w is treated as in the proof of
Corollary <ref>, using that ^∨_i satisfies the
quadratic relations from Proposition <ref>. Part
(b) follows from (a) by applying (^∨_i)^-1 to both sides.
 






§ THREE RECURSIONS FOR LOCALIZATIONS OF MOTIVIC CHERN CLASSES

In this section, we use the Demazure Lusztig operators _i to
obtain recursive relations for the ordinary and dual motivic Chern
classes of Schubert cells. These recursions will be used to compare
the motivic Chern classes both with stable envelopes and with
Casselman's basis. We also record a divisibility property for
localizations of motivic classes, to be used later in the proof of
Theorem <ref>.



 §.§ Recursions
 
Consider the localized equivariant -theory ring defined by 

    K_T(G/B) ↪  K_T(G/B)_ := K_T(G/B)
    ⊗_K_T()(K_T()) .

The Lefschetz fixed point formula in equivariant -theory
(see e.g., <cit.>) gives the expansion of
the motivic Chern classes in terms of the fixed point classes
ι_w, for every w ∈ W:

    _y(X(w)^∘)   =∑_u≤ w_y(X(w)^∘)|_uι_u/λ_-1(T_u^*(G/B))
       =∑_u≤
      w_y(X(w)^∘)|_uι_u/∏_α>0
    (1-e^uα)  ∈ K_T(G/B)_[y] .


The following three propositions give recursions formulas for various
flavors of motivic Chern classes. These will be used later to make the
connection with the Hecke algebra action on the principal series
representation. The similarity of the recursions can be explained by
the fact that they are related either by an automorphism of G/B or
by the involution exchanging the Demazure-Lusztig operators.


The localizations _y(X(w)^∘)|_u are uniquely determined by the
following conditions:


  (a) 
_y(X(w)^∘)|_u=0, unless u≤ w.

  (b)
If u=w:

    _y(X(w)^∘)|_w=∏_α>0,wα<0(1+ye^wα)
    ∏_α>0,wα>0(1-e^wα).


  (c) If ws_i>w, then

    _y(X(ws_i)^∘)|_u=-1+y/1-e^-uα_i_y(X(w)^∘)|_u+1+ye^uα_i/1-e^-uα_i_y(X(w)^∘)|_us_i.





Part (a) follows because the motivic class is supported on the
Schubert variety X(w). To prove part (b), observe that
_y(X(w)^∘)|_w=_y(X(w))|_w, by additivity and because
_y(X(v)^∘)|_w = 0 for v < w by part (a). Then

    _y(X(w)^∘)|_w=_y(X(w))|_w =
    λ_y(T^*_wX(w))λ_-1(N_w^∨),
 
where T^*_wX(w) and N_w^∨ are the fibers at the fixed point e_w
of the dual of the cotangent, respectively the conormal bundle for
X(w). (A more general result is proved in Theorem <ref>
below.) Part (c) follows by applying the operator _i to
Equation (<ref>) and taking the coefficients of
ι_u; this requires the action of _i on the fixed point
basis described in Lemma <ref>. Finally, the
uniqueness follows by induction on the length of w.


For later use, we also record the similar result for the motivic
Chern class of the opposite Schubert cells.

The localizations _y(Y(w)^∘)|_u are uniquely determined by the
following conditions:


  (a)
_y(Y(w)^∘)|_u=0, unless u≥ w.

  (b)
If u=w:

    _y(Y(w)^∘)|_w=∏_α>0,w(α)
      >0(1+ye^wα)∏_α>0,w(α) < 0(1-e^wα).


  (c)
If ws_i>w, then

    _y(Y(w)^∘)|_u=-1+y/1-e^-uα_i_y(Y(ws_i)^∘)|_u+1+ye^uα_i/1-e^-uα_i_y(Y(ws_i)^∘)|_us_i.




 
The left multiplication by w_0 induces an automorphism of G/B
sending X(w) to Y(w_0 w). This is not T-equivariant, but it is
equivariant with respect to the map T → T defined by t ↦ w_0 t
w_0. This induces an automorphism of K_T(G/B) and its localized
version, twisting the coefficients by w_0. Then the proposition
follows from Proposition <ref> above, by applying w_0.

Similar formulas hold for the dual classes
  _y^∨(Y(w)^∘):

The localizations _y^∨(Y(w)^∘)|_u are uniquely determined
by the following conditions:


  (a)
_y^∨(Y(w)^∘)|_u=0, unless u≥ w.

  (b)
If u=w: 

    _y^∨(Y(w)^∘)|_w=(-1)^ G/B-ℓ(w)∏_α>0,wα>0(y^-1+e^-wα)
    ∏_α>0,wα<0(1-e^wα) .


  (c)
If ws_i>w, then

    ^∨_y(Y(w)^∘)|_u=1+y^-1/e^uα_i-1^∨_y(Y(ws_i)^∘)|_u+y^-1+
    e^-uα_i/e^-uα_i-1^∨_y(Y(ws_α_i)^∘)|_us_α_i.




 
These formulae are regarded in K_T()[y^-1] ↪ (K_T())[y^-1].  The uniqueness follows directly from
induction. So we only need to show that _y^∨(Y(w)^∘)|_u
satisfies these properties. The support property follows because
(^∨_i)^-1 sends a Schubert class ^u into classes
supported on Y(u) ∪ Y(u s_i); then one applies
Proposition <ref>. To calculate the localization at
w, we use the duality from Theorem <ref> and 
the Lefschetz fixed point formula to obtain

    (-y)^ℓ(w) -  G/B∏_α > 0 (1+ ye^-α)     =
      ⟨_y(X(w)^∘), _y^∨(Y(w)^∘) ⟩
        = ∑_u ∈ W(_y(X(w)^∘) ·_y^∨(Y(w)^∘))|_u/∏_α>0 (1- e^u(α))·∫_G/Bι_u  .
 
The only non-zero contribution is for u=w, and the integral equals
1, thus

    _y^∨(Y(w)^∘)|_w = (-y)^ℓ(w) -  G/B∏_α > 0 
    (1+ye^-α)(1-e^w(α))/_y(X(w)^∘)|_w.
 
Part (b) follows from this and the localization from
Proposition <ref>. Part (c) follows as in
Proposition <ref>, using now
Proposition <ref> and part (e) of
Lemma <ref>.




 §.§ A divisibility property for localization coefficients
 
We record the following property which will be used in our
applications to p-adic groups.


For every w≤ u∈ W, the polynomial _y(Y(w)^∘)|_u ∈
K_T()[y] is divisible by

    ∏_α>0, uα>0(1+ye^uα)∏_α>0,w≰
      us_α<u(1-e^uα) .




By a general property of motivic classes proved
in <cit.>, the localization
coefficient _y(Y(w)^∘)|_u is divisible by
λ_y(T_u^*Y(u)^∘)=∏_α>0,uα>0
(1+ye^uα). As α varies in the set of positive roots,
the factors 1-e^uα and 1+ye^u α are relative prime
to each other. Then it remains to show that for every α>0 such
that w≰ us_α<u, the localization coefficient
_y(Y(w)^∘)|_u is divisible by 1-e^uα. Let C ≃^1 denote the T-stable curve connecting the fixed points
e_u and e_us_α. The T-weight of the tangent space T_u
C is -uα. By the K-theoretic analogue of the GKM conditions,
applied to _y(Y(w)^∘) (see e.g., 
<cit.>),
the difference

    _y(Y(w)^∘)|_u - _y(Y(w)^∘)|_us_α

is divisible by 1-e^u α. Then the claim follows because
_y(Y(w)^∘)|_us_α
=0, as the hypothesis w ≰ us_α implies that 
e_u s_α∉ Y(w).We thank A. Okounkov for 
comments leading to this proof. 






§ MOTIVIC CHERN CLASSES AND -THEORETIC STABLE ENVELOPES
 
In this section, we recall some basic properties of the
-theoretic stable basis of T^*(G/B), including a recursive
relation. Our main references are <cit.>. We compare the recursive
relation obtained in <cit.> to the one for motivic Chern
classes, and we deduce that the two objects are closely related. This
was also found by Féher and Rimányi and Weber
in <cit.> (see also <cit.>), using
interpolation techniques for motivic Chern classes. In cohomology, the
relation between stable envelopes and Chern-Schwartz-MacPherson
classes was noticed in <cit.> and
it was used in <cit.> to obtain a second `stable basis
duality'. In theorem <ref> we generalize this duality to
-theory.



 §.§ -theoretic stable envelopes
 
The cotangent bundle of G/B is the homogeneous bundle T^*(G/B) := G
×^B T^*_1.B(G/B), given by equivalence classes

    { [g,v]: (g,v) ∈ G × T^*_1.B(G/B)  and  (gb, v)
    ∼ (g, b.v), ∀ g ∈ G, b ∈ B };
 
here T^*_1.B(G/B) is the cotangent space at the identity with its
natural B-module structure. As before, let T be the maximal torus
in B, and consider the ^*-module  with character q^1/2. 
We let ^* act trivially on G/B and we consider the T
×^* action on the cotangent bundle defined by
(t,z).[g,v]=[tg,z^-2v]. In other words, T acts via its natural
left action; ^* acts such that the cotangent fibers get a weight
q^-1 and K_T ×^*() = K_T()[q^± 1/2].

The stable basis is a certain basis for the localized equivariant
-theory 

    K_T×^*(T^*(G/B))_
    :=K_T×^*(T^*(G/B))⊗_K_T×^*()(K_T×^*()),
 
where  means taking the fraction field. The basis elements are
called the stable envelopes {_, T^1/2,
  (w) | w∈ W} and were defined by Maulik and Okounkov in the
cohomological case. We recall their definition in -theory below,
following mainly Okounkov's lectures <cit.>
and <cit.>.

For a fixed Weyl group element, the definition of the stable envelope
_, T^1/2, (w) depends on three
parameters:


  * a chamber  in the Lie algebra of the maximal torus T, or
equivalently, a Borel subgroup of G.

  * a polarization T^1/2∈ K_T×^*(T^*(G/B)) of the
tangent bundle T(T^*(G/B)), i.e., 
a solution of the equation 

    T^1/2+q^-1(T^1/2)^∨=T(T^*(G/B))
 
in the ring K_T×^*(T^*(G/B))_.
The only polarizations utilized in this paper 
are T(G/B) and T^*(G/B). For every polarization
T^1/2, there is an opposite polarization defined as
T^1/2_=q^-1(T^1/2)^∨.


  * A sufficiently general fractional equivariant line bundle on G/B,
i.e. a general element ∈_T(T^*(G/B))⊗_,
called the slope of the stable envelope. The dependence on the
slope parameter is locally constant, in the following sense.

The choice of a maximal torus T ⊆ G determines a
decomposition of ( T)^* ⊗ into alcoves;
these are the complements of the affine hyperplanes H_α^∨, n
= {λ∈ ( T)^*⊗: ⟨λ ,
α^∨⟩ = n } as α^∨ varies in the set of
positive coroots, and n over the integers. The alcove structure is
independent the choice of a chamber (and hence of the Borel subgroup
B), and the stable envelopes are constant for fractional multiples
of (pull-backs of) line bundles _λ= G ×^B _λ
for weights λ in a given alcove.


The torus fixed point set (T^*(G/B))^T= (G/B)^T is in one-to-one
correspondence with the Weyl group W. For every w∈ W, we still use
e_w to denote the corresponding fixed point. For a chosen Weyl chamber
 in T, pick any cocharacter σ∈. 
The attracting set of w∈ W,  
also called the 
Białynicki-Birula cell in the literature, is defined as

    _(w)={x∈ T^*(G/B) |lim_z→ 0σ(z)· x=w }.
 
It is not difficult to show that _(w) is the
conormal space over the attracting variety in G/B for w; the
latter attracting variety is a Schubert cell in G/B.  Define a
partial order on the fixed point set W to be the (transitive 
closure of the) following relation:

    e_w≼_ e_v    if   _(v)∩ e_w≠∅.

Then the order determined by the positive (resp., negative) chamber is
the same as the Bruhat order (resp., the opposite Bruhat order).

Any chamber  determines a decomposition of the tangent
space N_w:= T_w(T^*(G/B)) as N_w=N_w,+⊕ N_w,- into
T-weight spaces which are positive and negative with respect to
 respectively. For every polarization T^1/2,
denote 
N_w∩ T^1/2|_w by N_w^1/2. 
Similarly, we have
N_w,+^1/2 and N_w,-^1/2. In particular,
N_w,-=N^1/2_w,-⊕
q^-1(N_w,+^1/2)^∨. Consequently, we have

    N_w,-- N_w^1/2=q^-1(N_w,+^1/2)^∨-
    N_w,+^1/2
 
as virtual vector bundles. The determinant bundle of the virtual
bundle N_w,-- N_w^1/2 is a complete square and its
square root will be denoted by ( N_w,-/
  N_w^1/2)^1/2;
cf. <cit.>. For instance, if we choose
the polarization T^1/2 = T(G/B), the positive chamber, and w=𝕀
then both N_𝕀^1/2 and N_𝕀, - have weights -
α, where α varies in the set of positive roots; in this
case the virtual bundle N_𝕀,-- N_𝕀^1/2=0. 

Let f:=∑_μ f_μ e^μ∈ K_T×^*
() be a Laurent polynomial, where e^μ∈ K_T() 
and f_μ∈[q^1/2,q^-1/2]. The Newton polytope
of f, denoted by _Tf, is

    _T f= ({μ| f_μ≠ 0})⊆
    X^*(T)⊗_,
 
 where X^*(T) denotes the character lattice of T.  The 
following theorem defines the -theoretic stable envelopes.

 <cit.>
For every chamber , a sufficiently general , and a
polarization T^1/2, there exists a unique map of
K_T×^*()-modules

    _,T^1/2,:K_T×^*((T^*(G/B))^T)→ K_T×^*(T^*(G/B))

such that for every w∈ W, the class
Γ:=_,T^1/2,(w) satisfies:


  * (Support) 
Γ⊆∪_z≼_
  w_(z);

  * (Normalization) 
Γ|_w=(-1)^ N_w,+^1/2(
  N_w,-/ N_w^1/2)^1/2__(w)|_w;

  * (Degree) For a fixed point e_u, identify
|_u with the character of the fiber of 
 over e_u. Then for every v≺_ w,

    _TΓ|_v⊆_T_,T^1/2,
    (v)|_v+ |_v-|_w .




The difference |_v-|_w in the degree condition implies
that the stable basis does not depend on the choice of the
linearization of .

Let + denote the chamber such that all the roots in B are positive
on it, and let - denote the opposite chamber. From now on we fix the
`fundamental slope' given by := _ρ⊗
1/N, where ρ is the sum of fundamental weights and N is a
large enough positive integer. Recall that ω_G/B :=_2
  ρ is the canonical bundle of G/B, therefore the slope  can also be thought as a (fractional version of a)
square root of the canonical line bundle. We will use the following
notation:

    _+(w):=_+, T(G/B), ()^-1(w), 
     and _-(w):=_-, T^*(G/B), (w).
 
The positive chamber and negative chamber stable basis are dual bases
in the localized equivariant ring, i.e.,

    ⟨_+(w), _-(u)⟩_T^*(G/B)=δ_w,u,

where ⟨·, ·⟩_T^*(G/B) is the equivariant
-theory pairing on T^*(G/B) defined via localization;
see <cit.>, <cit.>,
or <cit.>. We will study the pairing on K_T
  ×^*(T^*(G/B)) in more detail below,
in <ref>.



 §.§ Automorphisms
 
The stable envelopes for various triples of parameters can be related
to each other by automorphisms of the equivariant -theory ring
K_T ×^*(T^*(G/B)). We will use the following types of
automorphisms:

 

  a.  
the automorphism induced by the left Weyl group multiplication. Recall
that this induces an automorphism of K_T ×^*(T^*(G/B))
which twists the coefficients in K_T ×^*() by w. In
terms of localization, for every ∈
K_T×^*(T^*(G/B)), we have

    w()|_u=w(|_w^-1u).



  b. 
The duality automorphism, mapping [E] ↦ [E^∨], i.e., the
class of a vector bundle to its dual.  For []∈ K_T(G/B),
[]^∨ denotes the class obtained by taking the alternating
sum of duals in an equivariant resolution of  by vector
bundles. This automorphism also acts on K_T ×^*() by
taking e^λ↦ e^-λ and q^1/2↦
q^-1/2.

  c. 
The multiplication by the class of a line bundle.  We can fix an
integral weight λ∈ X^*(T) and a Borel subgroup B, and
consider the equivariant line bundle _λ= G ×^B
_λ. We will abuse notation and will denote with the same
symbol a line bundle on G/B and on its cotangent bundle.

  d. 
For the ring K_T×^*(G/B)=K_T(G/B)[q^1/2,q^-1/2], a
composition of the previous two automorphisms gives the
(equivariant) Grothendieck-Serre duality. This is an automorphism
 of K_T(G/B)[q^1/2,q^-1/2] defined as
follows: for every []∈ K_T(G/B),

    []:=[RHom(,ω^∙_G/B)]
    := ω_G/B^∙⊗ []^∨∈ K_T(G/B),
 
where ω^∙_G/B≃ω_G/B[ G/B] is the
(equivariant) dualizing complex of the flag variety; 
thus, [ω^∙_G/B]=(-1)^ G/B[_2 ρ].
Observe that

    ([]^∨)^∨ = [];    ([] ⊗ω_G/B^∙) = []^∨.

Extend the operation  to
K_T(G/B)[q^1/2,q^-1/2] by sending
q^1/2↦ q^-1/2.

The following lemma, proved in the appendix, records the effect of
these automorphisms on -theoretic stable envelopes.

 
(a) Let u,w ∈ W. Under the left Weyl group multiplication, 

    w. _, T^1/2, (u) = _w , w T^1/2,
      w.(wu) .
 
In particular, if both the polarization T^1/2 and the line bundle
 are G-equivariant, then

    w. _, T^1/2, (u) = _w , T^1/2,
      (wu) .


(b) The duality automorphism acts by sending q^1/2↦ q^-1/2 and

    (_,T^1/2, (w))^∨=q^-
          G/B/2_,T^1/2_,
        ^-1(w),

where T^1/2_:=q^-1(T^1/2)^∨
is the opposite polarization;
see <cit.>, i.e., this duality
changes the polarization and slope parameters to the opposite ones,
while keeping the chamber parameter invariant.

(c) Let ,
'∈_T(T^*(G/B)) be any equivariant line
bundles, let w ∈ W and a ∈ a rational number. Then

    _, T^1/2,a ⊗'(w) =  ('|_w)^-1' ⊗_, T^1/2,a (w) ,
 
as elements in K_T×^*(T^*(G/B))_.




 §.§ Recursions for stable envelopes
 
Because the ^*-fixed locus of the cotangent bundle is the zero
section (i.e. G/B), it follows that the torus fixed point locus
(T^*(G/B))^T ×^* coincides with the fixed locus (G/B)^T,
a discrete set indexed by the Weyl group W. Therefore the
equivariant -theory classes associated to fixed points form a
basis in the localized ring K_T ×^*(T^*(G/B))_. In
order to compare motivic Chern classes to stable envelopes, we need
the following result proved in <cit.>.


The restriction coefficients _-(w)|_u are uniquely
characterized by


  * _-(w)|_u=0, unless u≥ w.

  * _-(w)|_w=q^ℓ(w)/2∏_α>0,wα<0
(1-e^-wα)∏_α>0,wα>0(1-qe^-wα).

  * If ws_i>w, then

    q^1/2_-(w)|_u=(1-q)/1-e^-uα_i_-(ws_i)|_u+1-qe^-uα_i/1-e^uα_i_-(ws_i)|_us_i.




Applying parts (a) and (b) from Lemma <ref>, and from the
definitions of the stable envelopes, we obtain that for every u ∈ W,

    w_0 . (_- (u))^∨ = q^-  G/B/2_+(w_0 u) .
 

Then we immediately obtain the following analogue of
Proposition <ref>:


The localizations _+(w)|_u are uniquely characterized by the
following properties:


  * _+(w)|_u=0, unless u≤ w.

  * _+(w)|_w=q^ℓ(w)/2∏_α>0,wα<0(1-q^-1e^wα)
∏_α>0,wα>0(1-e^wα).

  * If ws_i>w, then

    q^1/2_+(ws_i)|_u=q-1/1-e^uα_i_+(w)|_u-e^uα_i-q/1-e^-uα_i_+(w)|_us_i.






 §.§ Motivic classes are pull-backs of stable envelopes
 
One of the key formulas in <cit.> shows that the dual CSM
class equals the Segre-Schwartz-MacPherson (SSM) class, up to a
normalization coefficient. The proof of that identity is based on a
transversality argument, which can be expressed either in terms of
(cohomological) stable basis elements or in terms of transversality of
characteristic cycles. The same phenomenon occurs in -theory.
Let i:G/B↪ T^*(G/B) be the inclusion of the zero
section into the cotangent bundle. Define

    _+'(w):=(i^*_+(w))=(-1)^
      G/B(i^*_+(w))^∨⊗ [_2ρ]∈
    K_T(G/B)[q^1/2 ,q^-1/2 ] .

The following result relates motivic Chern classes and stable
envelopes and it is the -theoretic analogue of the cohomological
results from <cit.>
and <cit.>. It is also equivalent to results
from <cit.>, where it is shown that motivic Chern
classes satisfy the same localization properties as the stable
envelopes for a certain triple of parameters; cf. Remark <ref>
below.

 
For every w∈ W, we have

    q^-ℓ(w)/2_+'(w)=_-q^-1(X(w)^∘)∈
    K_T(G/B)[q,q^-1].



 
We compare localization properties of the motivic Chern classes with
those for the Grothendieck-Serre dual of _+(w). We have that

    '_+(w)|_u=(-1)^
      G/Be^2uρ(_+(w)|_u)|_e^λ→ e^-λ,
      q^1/2→ q^-1/2.
 
Then the corresponding result from Proposition <ref> for
_+'(w) is that the localizations '_+(w)|_u are uniquely
characterized by the following properties


  * _+'(w)|_u=0, unless u≤ w.

  * _+'(w)|_w=q^ℓ(w)/2∏_α>0,wα<0
(1-q^-1e^wα)∏_α>0,wα>0(1-e^wα).

  * If ws_α_i>w, then

    q^-1/2_+'(ws_α_i)|_u=
    q^-1-1/1-e^-uα_i_+'(w)|_u
    +1-q^-1e^uα_i/1-e^-uα_i_+'(w)|_us_α_i.

 
Comparing this with localizations of motivic Chern classes from
Proposition <ref> finishes the proof.


A similar statement relates _-(w) to the motivic Chern class
of the opposite Schubert cells. We record the statement next; the
proof is essentially the same, and details are left to the reader.
Define

    _-'(w):=q^- G/Bi^*(_-(w))⊗ 
    [ω^∙_G/B] ∈ K_T(G/B)[q^1/2,
    q^-1/2].



For every w∈ W,

    q^ℓ(w)/2_-'(w) =_-q^-1(Y(w)^∘)∈
    K_T(G/B)[q,q^-1].



 
Using Theorem <ref> and Lemma <ref>, one can
show that for every w∈ W,

    q^-ℓ(w)/2i^*_+,T(G/B),(w)
    =_-q^-1(X(w)^∘),
 
where  is the fundamental slope. This is consistent
with the choices of parameters for stable envelopes
from <cit.>. In fact, a direct check of the
normalization and degree conditions shows that for any slope
ℒ,

    𝒟 (i^*( _+,T(G/B),ℒ)) =
    i^*(_+,T(G/B),(ℒ)^-1) .

Again we leave the proof details to the reader.  


 
In upcoming work we will prove that

    (-q)^-  G/B i^*( (i_w!_Y(w)^∘^H)) ⊗
      [ω^∙_G/B] = _-q^-1(Y(w)^∘) ,
 
where i_w: Y(w)^∘→ G/B is the inclusion, and (i_w!_Y(w)^∘^H) is the associated graded (or
  ^*-equivariant) sheaf on T^*(G/B) determined by the shifted
mixed Hodge module _Y(w)^∘^H; see <cit.>
(with our q ^-1
  corresponding to the parameter q in Tanisaki's paper). Since
i^* is an isomorphism, we deduce from Theorem <ref>
that:

    _-(w) = (-1)^ G/B(i_w!_Y(w)^∘^H) .
 
Since the cycle associated to the coherent sheaf ((i_w)_!
_Y(w)^∘) is the characteristic cycle of the constructible
function _Y(w)^∘, this equation can be seen as the
-theoretic generalization of the equivalence between
(cohomological) stable envelopes and characteristic cycles, indicated
by Maulik and Okounkov <cit.>;
see also <cit.> for a proof.




 §.§ Stable basis duality
 
As for the CSM classes, there are two sources for Poincaré type
dualities of the motivic Chern classes. The first is a consequence of
the existence of two adjoint Demazure-Lusztig operators. The second,
which has a geometric origin, uses the duality from
(<ref>) for the stable envelopes, on the cotangent
bundle. Given that the localization pairing on the cotangent bundle
can also be expressed in terms of a twisted Poincaré pairing on
the zero section, this leads to some remarkable identities among
motivic Chern classes. Recall from the equation (<ref>)
that the `opposite' stable envelopes are dual to each other with
respect to the -theoretic pairing on T^*(G/B), defined as
follows: for every ,∈ K_T×^*(T^*(G/B)),

    ⟨,⟩_T^*(G/B):=∑_w∈
      W[]|_w·[]|_w/∏_α>0
    (1-e^wα)(1-qe^-wα).

Recall that i:G/B↪ T^*(G/B) is the inclusion of the
zero section. By localization, the pairing in T^*(G/B) is related to
the ordinary 
pairing in the equivariant -theory 
of G/B:

    ⟨,⟩_T^*(G/B)
    =⟨ i^*,i^*/λ_-q(T(G/B))⟩.

Here the ordinary pairing in the equivariant -theory of
G/B is extended (by the equivariant projection formula) bilinearly
over ( K_T×^*()) to a pairing:

    ⟨ -,- ⟩ : K_T×^*(G/B)_× K_T×^*(G/B)_→( K_T×^*()) .

Moreover, λ_-q(T(G/B))∈ K_T×^*(G/B)
↪ K_T×^*(G/B)_ is invertible in the 
localized ring by the following observation.

 
A -theoretic analogue of <cit.> gives 

    λ_-q(T^*(G/B))λ_-q(T(G/B))=
    ∏_α>0(1-q e^α)(1-q e^-α ).

As in loc. cit., this follows by localization, because for all w
∈ W,

    λ_-q(T^*(G/B))|_w ·λ_-q(T(G/B))|_w=
    ∏_α>0(1-q e^α)(1-q e^-α )

as w permutes the set of roots.


We need the following lemma.


Let [], []∈ K_T×^*(G/B)_ such
that

    ⟨ [], [] ⟩=f(e^t,q^1/2)∈ K_T×^*()_.
 
Then

    ⟨([]), []^∨⟩ = ⟨ []^∨,
    ([]) ⟩ = f(e^-t,q^-1/2) = ( ⟨
        [], [] ⟩)^∨,
 
i.e., all weights are inverted by this operation.


  
By the definition of the equivariant Grothendieck-Serre duality
  operator, it suffices to prove the equality

    ⟨ []^∨, ([]) ⟩ = f(e^-t,q^-1/2) .

Applying the Lefschetz fixed point formula in equivariant
-theory <cit.> we obtain

    f(e^t,q^1/2)=
    ⟨ [], [] ⟩=∑_w[]|_w·
      []|_w/∏_α>0(1-e^wα).

Recall that [ω_G/B^∙] = 
(-1)^ G/B [ _2ρ]. Then

    ⟨([]), []^∨⟩   = (-1)^ G/B⟨ []^∨, []^∨⊗ 
    [_2ρ]⟩
       = (-1)^ G/B∑_w([]|_w)^∨([]|_w)^∨/∏_α>0
    (1-e^wα)e^2wρ
       =∑_w([]|_w)^∨([]|_w)^∨/∏_α>0(1-e^-wα)
       =f(e^-t , q^-1/2) .

The second-to-last equality holds because 2 ρ = ∑_α >0α, thus e^2 w (ρ) = ∏_α > 0 e^w(α), and
the last equality follows since the effect of taking (-)^∨ is to
invert the T and ^* weights.


 
Let u,w ∈ W and y = -q^-1. Then the following orthogonality
relation holds:

    ⟨_y(X(w)^∘), (_y(Y(u)^∘))/λ_y(T^*(G/B))(-y)^ G/B-ℓ(u)⟩=δ_w,u.
 
Equivalently,

    _y^∨(Y(u)^∘) = ∏_α>0(1+ye^-α)
    (_y(Y(u)^∘))/λ_y(T^*(G/B)) ∈ K_T×^*(G/B)_.



 
The idea is to use Theorem <ref> to express
_y(X(u)^∘) in terms of the Grothendieck-Serre dual, then use
Lemma <ref> to relate the pairing in the statement of the
theorem to the pairing between orthogonal stable envelopes. We start
by observing that

    λ_y(T^*(G/B)) = (λ_y^-1 T(G/B))^∨∈ K_T(G/B)[y,y^-1] ,

 and that by Theorem <ref> 

    (_y(Y(u)^∘))    = ( (-y)^ - ℓ(u)/2(-y)^ G/Bι^*(_-(u)) ⊗
         [ω_G/B^∙] ) 
       = (-y)^ℓ(u)/2 -  G/B (ι^*(_-(u)))^∨.
   
From this, the second term of the pairing equals 

    (_y(Y(u)^∘))/λ_y(T^*(G/B)) (-y)^
      G/B-ℓ(u)= (ι^*(_-(u))/λ_y^-1
      (T(G/B))  (-y)^ℓ(u)/2)^∨.
 
Then Theorem <ref>, Lemma <ref>, and
orthogonality of stable envelopes (Equation (<ref>))
imply that

    ⟨_y(X(w)^∘), (_y(Y(u)^∘))/λ_y(T^*(G/B))(-y)^ G/B-ℓ(u)⟩
       = ⟨ ( (-y)^-ℓ(w)/2ι^* _+(w)), 
    (ι^*(_-(u))/λ_y^-1 (T(G/B))  
    (-y)^ℓ(u)/2)^∨⟩
       =⟨ (-y)^ℓ(u)-ℓ(w)/2ι^* _+(w), 
    ι^* _-(u)/λ_y^-1 (T(G/B)) ⟩_y ↦ y^-1, e^t ↦ e^-t
       =(-y)^ℓ(w)-ℓ(u)/2⟨_+(w), _-(u)
      ⟩_T^*(G/B)|_y ↦ y^-1, e^t ↦ e^-t
       = δ_w,u,

where the second equality follows from Lemma <ref>, the
third equality follows from Equation (<ref>) and the
last one follows from Equation (<ref>). This proves the
first assertion.  The second assertion follows from the `Hecke
orthogonality' of motivic Chern classes, proved in
Theorem <ref>.

  The proof of the previous theorem depends 
in an essential way on the orthogonality of stable envelopes. This 
dependence can be removed by proving the transversality 
formula mentioned in Remark <ref>. This approach, based on the 
transversality formula from <cit.>,
was utilized in <cit.> to prove the cohomological case of 
Theorem <ref>.


Theorem <ref> justifies the definition of a dual motivic
Chern class of a Schubert variety:

 
Let w ∈ W. Define the dual motivic Chern class of a dual Schubert
variety by

    _y^∨(Y(w)):= ∑_u ≥ w_y^∨(Y(u)^∘) .



By Theorem <ref>,

    _y^∨(Y(w)) = ∏_α>0 (1 + y e^-α)/λ_y(T^*(G/B))(_y (Y(w)))  ∈ K_T×^*(G/B)_.
 
The class

    (_y (Y(w)))/λ_y(T^*(G/B)) ∈ K_T×^*(G/B)_
 
can be thought as a motivic Segre class, i.e., a -theoretic
analogue of the Segre-Schwartz-MacPherson class discussed
in <cit.>. More precisely, it will be the motivic Segre
class of the motivic dual _([Y(w)]) of the dual
Schubert variety Y(w), for an equivariant motivic duality _
(extending <cit.>),
with

    (_y (Y(w)))  = _y(_([Y(w)]))

as an equivariant extension 
of <cit.>.





§ SMOOTHNESS OF SCHUBERT VARIETIES AND LOCALIZATIONS OF MOTIVIC CHERN CLASSES
 
Among the main applications of this paper are properties about the
transition matrix between the standard and the Casselman's basis for
Chevalley groups over nonarchimedean local fields. The matrix
coefficients are rational functions, and of particular interest to us
are certain factorizations and polynomial properties of these
coefficients conjectured by Bump, Nakasuji, and Naruse; see section
<ref> below and
<cit.>. We will prove 
in <ref> that the transition matrix from the `Casselman
setting' corresponds to transition matrix between (dual) motivic Chern
classes of Schubert varieties and an appropriate normalization of the
fixed point basis. This motivates the study in this section of the
underlying `geometric' transition matrix between the motivic classes
and fixed point basis. The main result of this section is
Theorem <ref>
(Theorem <ref> from the introduction); a 
representation-theoretic version of this theorem 
will be proved in Theorem <ref> below.


 §.§ A smoothness criterion
 
In this section we prove a criterion for the smoothness of Schubert
varieties in terms of the motivic Chern classes. We need the following 
lemma, which is implicit in <cit.>.
 
(a) Let i:X ⊆ M be a 
closed embedding
of G-equivariant,
non-singular, quasi-projective, algebraic varieties, with 
N^∨_X M the conormal bundle of X inside M.
Then: 

    i^* _y[X → M] = λ_y(T^*_X) ⊗λ_-1(N^∨_X M) .

(b) Let X ⊆ M be a closed embedding
of T-equivariant,
algebraic varieties, and assume that M is smooth. Let p ∈ X be a
smooth T-fixed point, and j:V ⊆ M any T-invariant open set such
that p ∈ V and X':=V ∩ X is smooth (e.g., V:=M∖
  X_sing). Let ι_p: { p }→ M be the inclusion. Then: 

    ι_p^* _y[X → M] = λ_y (T^*_p X) ·λ_-1((N^∨_X M)_p) .


 
Part (a) follows from the functoriality of motivic Chern classes and the self-intersection formula in -theory
<cit.>: 

    i^* _y[X → M] = i^* i_* _y [𝕀_X] = _y[𝕀_X] ⊗λ_-1(N^∨_X M) = λ_y(T^*_X) ⊗λ_-1(N^∨_X M) .

Now let us prove (b).
Let ι_p': {p}→ V denote the embedding. Note that 

    ι_p^* _y[X → M] = (ι_p')^* j^* _y[X → M] =
    (ι_p')^* _y ( j^* [X → M] ) = (ι_p')^* _y[ X' → V] ,
 
where the second equality follows from the Verdier-Riemann-Roch
formula from Theorem <ref>, as j is an open embedding
(thus a smooth morphism), with relative tangent bundle equal to 1. 
Applying Part (a) to the closed
embedding X' ↪ V,
we obtain:

    _y[ X' → V] = λ_y(T^*_X') ⊗λ_-1(N^∨_X'
    V) .
 
The claim follows by pulling back via (ι'_p)^*, using that
(ι'_p)^* is a ring homomorphism in (equivariant) -theory.


We also need a variant of Kumar's cohomological criterion for smoothness
of Schubert varieties.

 
Let u,w be two Weyl group elements such that u ≤ w. Then the
Schubert variety Y(u) is smooth at e_w if and only if the
localization of the equivariant fundamental class [Y(u)]∈
A^T_*(G/B) in the equivariant Chow group is given by:

    [Y(u)|_w= ( ∏_β>0, u≰ s_β wβ) ∈ A_*^T()=[α_i |  i=1,…, r]  .

If Y(u) is smooth at e_w, then the torus weights of T_wY(u) are
{-wα|α>0,ws_α≥ u}.

The statement above is an equivalent, but different, 
formulation from that in <cit.>. We briefly indicate next the 
steps needed to bring it into the original formulation.
Consider the automorphism of the set R^+ of positive roots given by
α↦ -w_0(α). One checks that this is actually an
automorphism of the Dynkin diagram. It induces the automorphism w
↦ w_0 w w_0 of the Weyl group W, preserving the length and
the Bruhat order. It also induces an automorphism of G/B sending the
Schubert cell Y(w)^∘ to Y(w_0 w w_0)^∘. In particular,
Y(u) is smooth at e_w if and only if Y(w_0 u w_0) is smooth at
w_0 w w_0. Then

    Y(w_0uw_0)  is smooth at  e_w_0ww_0
    ⟺   X(uw_0)  is smooth at  e_ww_0
    ⟺   [Y(u)]|_w =∏_β>0, u≰ s_β wβ.

Here in the last equivalence, we used the original version of
Kumar's criterion, as stated in
<cit.>. (Notice that the term d_w,u in loc. cit. is equal to the localization [Y(u)]|_w, see <cit.> and <cit.>.)

 Let S'(u,w):={α > 0: u≤ w s_α< w}. An immediate
consequence of the theorem is that if Y(u) is smooth at e_w, then
the weights of the normal space 
(N_Y(w) Y(u))_w
= T_w (Y(u))/T_w
(Y(w)) of Y(w) at e_w in Y(u) are

    S(u,w):= {β >0: u ≤ s_β w < w } ={-w(α): α∈ S'(u,w) }.
 


The main theorem of this section is
the following We are thankful to a referee for suggesting the current formulation.:
 
Let u,w∈ W such that u≤ w. The following are equivalent:

(a) The opposite Schubert variety
Y(u) is smooth at the torus fixed point e_w.

(b) The localization of the motivic Chern class _y(Y(u)) at w is given by 

    _y(Y(u))|_w=∏_α>0,ws_α≥ u(1+ye^wα)
    ∏_α>0,u≰ ws_α(1-e^wα).


(c) The localization of the structure sheaf ^u∈
K_T(G/B) is given by:

    ^u|_w= ∏_α>0,u≰ ws_α(1-e^wα).

(d) The localization of the equivariant fundamental class [Y(u)]∈
A^T_*(G/B) in the equivariant Chow group is given by:

    [Y(u)]|_w= ∏_α>0,u≰ ws_α(-wα)=
    ∏_β>0, u≰ s_β wβ.



By Lemma <ref>(b) and the weight space description from 
Remark <ref>,
(a) implies (b). From the normalization property, the specialization
 _y=0(Y)=[_Y]∈_T(Y) if Y is smooth, and it follows from functoriality that
_y=0(Y)=[_Y]
if Y has rational singularities. This is the case
for Schubert varieties (<cit.>), so
_y=0(Y(u))=^u.
Hence, (c) follows from (b) by setting y=0.  Consider the `geometric'
equivariant Chern character

    _T:K_T(G/B) →A_*^T(G/B)

to an appropriate completion of the equivariant Chow group; see
<cit.>. From the definition of _T and by
<cit.> it follows that the top homological
degree term of the equivariant Chern character _T(^u) is the
equivariant fundamental class [Y(u)]_T. Together with the fact that
_T(e^λ) = 1+ λ + higher degree cohomological
terms, this implies that if (c) holds, then the top degree
term of _T(^u) localizes to

    [Y(u)]|_w= ∏_α>0, u ≰ ws_α (-w(α))=
    ∏_β>0, u≰ s_β wβ.
 
Here the second equality uses the change of variable β=-w(α) and the fact that w(α)<0 since ws_α <w. Thus, (d) holds.
Finally, (d) implies (a) by Kumar's Theorem <ref>.




 §.§ The geometric Bump-Nakasuji-Naruse 
conjecture
 

Motivated by the applications to
representation theory from <ref>, we study the
following problem. Define the element b_w ∈
K_T(G/B)_[y^-1] by the formula

    b_w:=(-1)^ G/B - ℓ(w)∏_α>0, wα>0y^-1
      +e^-wα/1-e^wαι_w.

Equivalently, b_w is the multiple of the fixed point basis element
ι_w which satisfies (b_w)|_w = _y^∨(Y(w)^∘)|_w.
Consider the expansion of the class _y^∨(Y(u)) from Definition
<ref>:

    _y^∨(Y(u)) = ∑ m_u,w b_w   ∈
    K_T(G/B)_[y^-1]  .

It is easy to see that
m_u,w=0 unless u ≤ w. For pairs
u≤ w∈ W, recall that S(u,w):={β∈ R^+ | u≤ s_β w
< w}.  The main result of this section is the following geometric
analogue of
a conjecture of Bump, Nakasuji, and Naruse
<cit.>.



For every u≤ w∈ W, 

    m_u,w=∏_α∈ S(u,w)1+y^-1e^α/1-e^α

if and only if the Schubert variety Y(u) is smooth at the torus
fixed point e_w.
 

The p-adic representation theory counterpart of this theorem will be
given in Theorem <ref> below.
The coefficients m_u,w calculate
the transition matrix between the `standard basis' and `Casselman's
basis' for the Iwahori invariants of the principal series
representation. The statement is a generalization of the original
Bump-Nakasuji conjecture, communicated to us by H. Naruse;
see <cit.>. In fact, Naruse informed us that he
obtained the implication of this theorem which assumes the
factorization. Naruse's proof of this implication, and ours, are both
based on Kumar's cohomological criterion for smoothness
(<cit.>; see Theorem <ref>).  Naruse's proof is
based on Hecke algebra calculations, while ours uses motivic Chern
classes.

After harmonizing conventions between this paper
and <cit.>, and passing to the `geometric'
version, the original conjecture states the following (see
<cit.> and <cit.>):

 
Let G be a complex, simply laced, reductive, linear algebraic group. 
Then the coefficient m_u,v satisfies the factorization in
(<ref>) if and only if 

    P_w_0 w^-1, w_0 u^-1 = 1 ,

where P_w_0 w^-1, w_0 u^-1 denotes the Kazhdan-Lusztig
polynomial.


We first prove this statement, assuming
Theorem <ref>.

 
Since the group G is simply laced, an unpublished result of D.
Peterson, re-proved by Carrell and Kuttler (see
e.g., <cit.> or
<cit.>) shows that the condition
that Y(u) is smooth at e_w is equivalent to Y(u) being
rationally smooth at e_w. For arbitrary G, rational smoothness is
equivalent to the fact that the Kazhdan-Lusztig polynomial P_w_0 u,
  w_0 w equals 1, by a theorem Kazhdan and Lusztig
<cit.>. By Theorem
<ref>, it remains to shows that P_w_0 w^-1,
  w_0 u^-1 = 1 if and only if P_w_0 w, w_0 u = 1. In turn,
this is equivalent to

    P_u, w = 1 ⟺ P_w_0 u^-1w_0, w_0 w^-1 w_0 = 1
    .
 
This is proved in the next lemma below.
 

 
Let G be a complex reductive linear algebraic group of arbitrary
Lie type. Then P_u, w = 1 if and only if P_w_0 u^-1w_0, w_0
w^-1 w_0 = 1.


 
We use a characterization of the condition that the Kazhdan-Lusztig
polynomials is equal to 1, proved in various generality by Deodhar,
Carrell and Peterson; see
<cit.>. Let  be the set of
(not necessarily simple) reflections in W. Then P_u,w = 1 if and
only if

    #{ r ∈: y < ry ≤ w } = ℓ(w) - ℓ(y) ,   ∀
    u ≤ y ≤ w .

It is well known that taking inverses, and conjugating by w_0 are
bijections of W which preserve both the length and the Bruhat order
of elements. Thus y < w if and only if w_0 y^-1 w_0 < w_0 w^-1
w_0 and ℓ(w) - ℓ(y) = ℓ(w_0 w^-1 w_0) - ℓ(w_0 y^-1
w_0). This finishes the proof.


We note that in general rational smoothness is different from
smoothness, therefore the statement from Corollary <ref>
does not generalize to non-simply laced case. The statement of
<cit.> is slightly different from the final
version stated in Corollary <ref> and in
<cit.>. The initial statement was analyzed by Lee,
Lenart and Liu in <cit.>, and they found that
under certain conditions on the reduced words of w and z the
factorization holds, but in general there are counterexamples. We
refer to <cit.>
for work closely related to <cit.>.

We now return to the proof of Theorem <ref>. The
key part is the following result, which may be of interest in its own
right.

 
(a) For every w≥ u∈ W, the coefficient m_u,w equals 

    m_u,w = (_y(Y(u))|_w/_y(Y(w)^∘)|_w)^∨,
 
and it is an element in K_T()_[y^- 1].

(b) Assume that Y(u) is smooth at e_w. Then 

    m_u,w = λ_y^-1 ((N_Y(w) Y(u))_w)/λ_-1 ((N_Y(w) Y(u))_w).

In particular, we obtain a geometric analogue of the
Langlands-Gindikin-Karpelevich formula <cit.>:

    m_1,w = ∏_α <0, w^-1 (α)>01 +y^-1 e^α/1 - e^α.



 
Localizing both sides of (<ref>) at the 
fixed point e_w gives _y^∨(Y(u))|_w = m_u,w·_y^∨(Y(w)^∘)|_w. Therefore,
using Theorem <ref> and
Definition <ref>, we obtain:

    m_u,w =_y^∨(Y(u))|_w/_y^∨(Y(w)^∘)|_w =
    (_y(Y(u))|_w)/(_y(Y(w)^∘)|_w) =
    (_y(Y(u))|_w/_y(Y(w)^∘)|_w)^∨.
 
This proves the first part of (a). The claim that the element m_u,w 
is in K_T()_[y^- 1] follows from Theorem <ref>
and Proposition <ref>(b), which show that 
∏_α > 0, w(α)>0 (1+ ye^wα) divides _y(Y(u))|_w.

For part (b), we use part (a)
and Theorem <ref>
to obtain

    m_u,w    = (_y(Y(u))|_w/_y(Y(w)^∘)|_w)^∨
        = (λ_y(T^*_w Y(u)) ·λ_-1((N_Y(u)^∨ (G/B))_w)/λ_y(T^*_w Y(w)) ·λ_-1((N_Y(w)^∨ (G/B))_w))^∨
        = λ_y^-1(T_w Y(u)) ·λ_-1((N_Y(u) (G/B))_w)/λ_y^-1(T_w Y(w)) ·λ_-1((N_Y(w) (G/B))_w)
        = λ_y^-1 ((N_Y(w) Y(u))_w)/λ_-1
      ((N_Y(w) Y(u))_w).
 
The last equality follows from the multiplicativity of the λ_y
class, and the short exact sequences

    0 → T_w Y(w) → T_w Y(u) → (N_Y(w)Y(u))_w → 0

and 

    0 → N_w,Y(u) (G/B) → N_w,Y(w) (G/B) → (N_Y(w)Y(u))_w 
    → 0  .

The case when u=1 follows from the description of the weights from
Theorem <ref>.
 

 
If Y(u) is smooth at e_w the claim follows from
Proposition <ref>(b), using the description of
appropriate weights from Remark <ref>.
Conversely, assume that

    m_u,w=∏_u≤ s_α w<w1+y^-1e^α/1-e^α.
 
Part (a) of Proposition <ref>, together with the
localization result from Proposition <ref> imply that

    _y(Y(u))|_w=∏_α>0,ws_α≥
      u(1+ye^wα)∏_α>0, u≰ ws_α(1-e^wα).
 
Therefore, Y(u) is smooth at e_w by
Theorem <ref>.






§ MOTIVIC CHERN CLASSES AND THE PRINCIPAL SERIES REPRESENTATION
 
The goal of this section is to establish an isomorphism of Hecke
modules between the Iwahori invariants of the unramified principal
series representations of a group over a non archimedean local field
and the (localized) equivariant -theory of the flag variety for
the complex Langlands dual group; see Theorem <ref>. A
similar relation was established recently in <cit.>, using the
equivariant -theory of the cotangent bundle and the stable
basis. The advantage of using motivic Chern classes is that their
functoriality properties will help get additional properties of this
correspondence. For instance, we use functoriality to relate
localization coefficients of the motivic Chern classes to coefficients
in the transition matrix between the standard basis and Casselman's basis (defined below). This will be applied to solve
some conjectures of Bump, Nakasuji and Naruse about the coefficients
in the transition matrix between the standard and the Casselman's
basis.



 §.§ Iwahori invariants of the principal series representation
 
We recall below the definition and properties of the two bases in the
Iwahori invariants of the principal series representation. The
literature in this subject uses several normalization conventions. We
will be consistent with the conventions used in the paper of
Reeder <cit.> and in <cit.>, because they
fit with our previous geometric calculations in this paper; these
conventions differ from those in <cit.>
or <cit.>, and when necessary we will explain the
differences. Let  be a split, reductive, Chevalley group
defined over ; see e.g., <cit.>.  Let =
≤ be a standard Borel subgroup containing a maximal
torus  and its unipotent radical . Let W:=
N_()/ be the Weyl group.  We will also consider the
Langlands dual  of ; by definition this will be a complex reductive linear algebraic group, of type dual to the Lie
type of . See e.g., <cit.>In the previous section we
used flag varieties associated to complex semisimple Lie groups.
If G is any reductive group with radical (G), then G/(G) is semisimple, 
and the flag varieties for G and G/(G) are the same; 
see e.g., <cit.>. We will tacitly use these facts in
this section..  
Let F be a non archimedean local field, with ring of integers ,
uniformizer ϖ∈, and residue field _q'. Examples
are finite extensions of the field of p-adic numbers, or of the
field of Laurent series over _p. Since  is defined over
, we may consider (F), the group of the F-points of
, with maximal torus (F) and Borel subgroup
(F)=(F) (F). Let I be an Iwahori 
subgroup, i.e., the
inverse image of (_q') under the evaluation map
()→(_q').  To simplify formulas,
we let α, β denote the coroots of . Let R^+ and
R^+∨ denote the positive roots and coroots, respectively.

Let =_c[I\(F)/I] be the Iwahori Hecke
algebra, consisting of compactly supported functions on (F)
which are bi-invariant under I. As a vector space,
=Θ⊗_ H_W(q'), where Θ is a commutative
subalgebra isomorphic to the coordinate ring [] of the
complex dual torus =^*⊗ X^*(), and where
H_W(q') is the finite Hecke (sub)algebra with parameter q'
associated to the (finite) Weyl group W. The finite Hecke algebra
H_W(q') is also a subalgebra of , and it is generated by
elements T_w (w∈ W) such that the following relations hold: T_u
T_v = T_uv if ℓ(uv) = ℓ(u) + ℓ(v), and
(T_s_i+1)(T_s_i -q') = 0 for a simple reflection s_i in W.
 
For every character τ of , and α a coroot define
e^α by e^α(τ)=τ(h_α(ϖ)), where
h_α:F^×→(F) is the one parameter
subgroup. There is a pairing

    ⟨· , ·⟩: (F)/()×→^*
 
given by ⟨ a,z⊗λ⟩=z^(λ(a)).  This induces an isomorphism
between (F)/() and the group X^*() of
rational characters of . It also induces an identification
between  and unramified characters of (F), i.e.,
characters which are trivial on ().

Following Reeder <cit.>,
from now on we take τ to be an unramified
character of (F) such that e^α(τ) ≠ 1 for all
coroots α, and for which the stabilizer W_τ = 1. The principal series representation is the induced representation
I(τ):=_(F)^(F) (τ). As a -vector
space, I(τ) consists of locally constant functions f on
(F) such that f(bg)=τ(b)δ^1/2(b)f(g) for
every b∈(F), where
δ(b):=∏_α>0|α^∨(b)|_F is the modulus
function on the Borel subgroup. The Hecke algebra  acts through
convolution from the right on the Iwahori invariant
subspace I(τ)^I, so that the restriction of this action to
H_W(q') is a regular representation. One can pass back and forth
between left and right -modules by using the standard
anti-involution ι on  given by ι(h)(x) =
h(x^-1). If T_w denote the standard generators of the Hecke
algebra H_W(q'), then ι(T_w)=T_w^-1 and ι(q')=q',
see <cit.>. This is of course
consistent with the left -action on I(τ) described by
Reeder in <cit.>.

We are interested in the Iwahori invariants I(τ)^I of the
principal series representation for an unramified character. One
reason to study the invariants is that as a (F)-module, the
principal series representation I(τ) is generated by I(τ)^I;
cf. <cit.>. As a vector
space, _ I(τ)^I = |W|, the order of the Weyl group
W. We will study the transition between two bases of
I(τ)^I. From the decomposition (F) = _w ∈ W(F) w I one obtains the basis of the characteristic
  functions on the orbits, denoted by {φ_w| w∈
W}[Our φ_w is equal to ϕ_w^-1
  in <cit.>.]. For w∈ W, the element
φ_w is characterized by the following two
conditions <cit.>:


  * φ_w is supported on (F)wI;

  * φ_w(bwg)=τ(b)δ^1/2(b) for every b∈(F) and g∈ I.

The (left) action of  on I(τ)^I, denoted by π, was
calculated e.g., by Casselman
in <cit.>. With the conventions
from Reeder <cit.>, for every simple 
coroot α_i:

    π(T_s_i)(φ_w)=
    {[ q'φ_ws_i+(q'-1)φ_w        if  ws_i<w;;             φ_ws_i        if  ws_i>w. ].


The second basis, called Casselman's basis, and denoted by
{f_w| w∈ W}, was defined by
Casselman <cit.> by duality using certain
intertwiner operators. We recall the relevant definitions, following
again <cit.>. For every character τ and x∈
W, define xτ∈ X^*() by the formula
xτ(a):=τ(x^-1ax) for every a∈.  Since τ is
unramified and it has trivial stabilizer under the Weyl group action,
the space _(F)(I(τ), I(x^-1τ)) is known to be
one dimensional, spanned by an operator[The intertwiner
  _x is related to M_x
  from <cit.> by the formula
  _x=M_x^-1.] _x=_x^τ defined by

    _x(φ)(g):=∫_N_xφ(ẋng)dn,

where ẋ is a representative of x∈ W, N_x=N(F)∩ẋ^-1N^-(F)ẋ where N^- is the unipotent radical of
the opposite Borel subgroup ^-; the measure on N_x is
normalized by the condition that (N_x∩())=1 <cit.>. If x, y∈ W satisfy
ℓ(x)+ℓ(y)=ℓ(xy), then
^x^-1τ_y^τ_x=_xy^τ. Then there exist
unique functions f_w ∈ I(τ)^I such that

    _x^τ (f_w)(1)=δ_x,w.
 
(Under our conventions f_w equals the element denoted
f_w^-1 in <cit.>.) For the longest element w_0 in the
Weyl group, Casselman showed
in <cit.> that

    φ_w_0=f_w_0.
 
Reeder <cit.> calculated the action of  on the
functions f_w: he showed in <cit.> that
the functions f_w are Θ-eigenvectors, and he calculated
in <cit.> the action of
H_W(q'). To describe the latter, let

    c_α=1-q'^-1e^α(τ)/1-e^α(τ).


For every simple coroot α_i and w∈ W, write

    J_i,w=
    {[ c_w(α_i)c_-w(α_i)       if  ws_i>w;;                 1       if  ws_i<w. ].

Then, we have 

    π(T_s_i)(f_w)=q'(1-c_w(α_i))f_w+q'J_i,wf_ws_i.




 §.§ A conjecture of Bump, Nakasuji, and Naruse

 
In this section we state a conjecture of Bump, Nakasuji, and Naruse
regarding a factorization of certain coefficients of the transition
matrix between the bases {φ_w } and { f_w }. Let

    ϕ_u:=∑_u≤ wφ_w ∈ I(τ)^I,
 
and consider the expansion in terms of the Casselman's basis: 

    ϕ_u=∑_wm̃_u,wf_w.

Then by the definition of f_w,
m̃_u,w=_w(ϕ_u)(1). It is also easy to see that
m̃_u,w=0 unless u≤ w, see <cit.>. We
shall see below that m̃_u,w equals the evaluation at τ
of the coefficient m_u,w from (<ref>), defined
for the Langlands dual flag variety. For every u≤ w∈ W, recall
the definition S(u,w):={β∈ R^+, ∨ | u≤ s_β w <
w} (cf. (<ref>)).  Recall that  is the complex Langlands 
dual group, with the corresponding Borel subgroup  and the 
maximal torus . The goal is to prove the following statement.


For every u≤ w∈ W, 

    m̃_u,w=∏_α∈
      S(u,w)1-q'^-1e^α(τ)/1-e^α(τ),

if and only if the opposite Schubert variety
Y(u):= u / in the (dual, complex) flag
manifold / is smooth at the torus fixed point e_w.
 

This is the representation-theoretic counterpart of
Theorem <ref>; its proof will be given in the next
subsection.

We provide further historical context.
Casselman <cit.> asked for an expression of
the basis f_w as a linear combination of the standard basis
φ_w. Bump and Nakasuji found that the basis ϕ_w is better
behaved for this question. Of course the original Casselman's basis
can be obtained from the Möbius inversion

    φ_u=∑_w≥ u(-1)^ℓ(u)-ℓ(w)ϕ_w.
 

The case u=1 of the Bump-Nakasuji-Naruse conjecture is well
known. In this case ϕ_1 is the spherical vector in I(τ),
i.e., the vector fixed by the maximal compact subgroup G(), and

    _w(ϕ_1)(1)=m̃_1,w = ∏_α∈
      S(1,w)1-q'^-1e^α(τ)/1-e^α(τ).

This is the Gindikin–Karpelevich formula, which in the
non-archimedean setting was actually proved by Langlands
<cit.> after Gindikin and Karpelevich proved a similar
statement for real groups. Casselman obtained another proof using his
basis f_w, and this plays a crucial role in his computation of the
Macdonald formula and the spherical Whittaker functions, see
<cit.>. See also
<cit.> for an approach using the stable basis and the
equivariant -theory of the cotangent bundle T^*(/).
We will recover (<ref>) below, as a consequence of
Theorem <ref>.  Other special cases of the conjecture follow
from work of Reeder <cit.>.



 §.§ Casselman's problem and motivic Chern classes

In this section, we construct the promised isomorphism between the
H_W(q')-module of the Iwahori invariants of the principal series
representation of  and the equivariant  group of the flag
variety for the dual group , regarded as an H_W(-y)-module
via the action of the operators ^∨_i.  This construction,
together with the cohomological properties of the motivic Chern
classes from <ref>, will be used to
prove Theorem <ref>.

For now we assume that the unramified character τ is in the open
set in  such that 1-q' e^α(τ)≠ 0, for every
(positive or negative) coroot α. Regard the representation ring
K_() as a subring of [] and let _τ
denote the one dimensional K_()-module induced by
evaluation at τ. Recall that the operators ^∨_i from
Definition <ref> satisfy

    (^∨_i+1)(^∨_i+y)=0,

and the braid relations (Proposition <ref>). 
Hence, they induce an action of the Hecke
algebra H_W(-y) with parameter -y on the -theory ring
K_(/)[y,y^-1] by sending T_w 
to ^∨_w (here the parameter -y
corresponds to the parameter q in <cit.>). We use the
symbol π to denote this action.

As in (<ref>), define the element
b̃_w ∈ K_(/)_[y^-1] by the
formula

    b̃_w:=(-1)^/ - ℓ(w)∏_α>0, wα>0y^-1 +e^-wα/1-e^wαι_w⊗ 1  .
 
Equivalently, b̃_w = b_w ⊗ 1, with b_w from equation
(<ref>).


We now state the main comparison theorem.
We are thankful to a referee for suggesting the current formulation.
 
  
There exists a unique isomorphism of left H_W(q')-modules (assuming the identification q'=-y)

    Ψ:K_(/)[y,y^-1]
    ⊗_K_()[y,y^-1]_τ 
    I(τ)^I ,
 
such that

(a) Ψ(_y^∨(Y(w)^∘) ⊗ 1)= φ_w and

(b) Ψ (b̃_w) = f_w. 

 
There is an analogue of this theorem for the equivariant -theory
of T^*(G/B)
proved in <cit.>. This is also studied by
Lusztig and Braverman–Kazhdan in <cit.> from different points of view.

 
The uniqueness is obvious. Next we define the map Ψ 
by property (a), and prove the remaining claims. The fact that Ψ is a 
map of H_W(q')-modules follows from
comparing Proposition <ref>(a) to equation
(<ref>); these describe the Hecke actions on the basis
of dual motivic Chern classes _y^∨(Y(w)^∘) and on the basis
of characteristic functions φ_w.

To prove property (b), we argue by descending induction on
ℓ(w). Recall that f_w_0 = φ_w_0 and b_w_0 =
ι_w_0 = _y^∨(Y(w_0)) (from Definition <ref>); 
therefore
Ψ(b_w_0)= f_w_0. Now take any w<w_0 and assume
that Ψ(b_z) = f_z for all b̃_z with
ℓ(z)>ℓ(w). Pick a simple root α_i such that
ws_i>w. Then by induction, Ψ (b̃_ws_i) = f_ws_i.
Since Ψ is a homomorphism of Hecke modules,

    Ψ(^∨_i (b_ws_i)) = 
    π(T_i)(Ψ(b_ws_i)) = 
    π(T_i)(f_ws_i).
 
On the one hand, Lemma <ref> gives 

    Ψ(^∨_i (b_ws_i))=
    Ψ(1+y/e^wα_i(τ)-1b̃_ws_i-yb̃_w)
    =1-q'/e^wα_i(τ)-1f_ws_i
    -yΨ(b̃_w).
 
Here we have used that {α > 0 : w(α) > 0 } = {α >
0 : ws_i(α) > 0 }∪{α_i }. On the other hand,
equation (<ref>) gives

    π(T_i)(f_ws_i)=1-q'/e^wα_i(τ)-1
    f_ws_i-yf_w.

Therefore, Ψ(b_w) = f_w. By induction, this
finishes the proof of property (b).



The coefficients m̃_u,w are represented by the
meromorphic functions m_u,w on , defined in equation
(<ref>), for the Langlands dual complex flag variety
/. More precisely, let τ∈ be any regular
unramified character, i.e., with trivial stabilizer W_τ.
Then

    m̃_u,w = m_u,w(τ) .



Observe that Theorem <ref>, together with the
definitions of m̃_u,w and m_u,w imply the equality
m̃_u,w = m_u,w(τ) for all regular unramified
characters τ satisfying 1-q' e^α(τ)≠ 0, for every
coroot α. However, it is known that the intertwiners _x
depend holomorphically on regular characters τ∈; see
e.g., <cit.> 
or <cit.>. Then one can extend meromorphically 
the equality m̃_u,w = m_u,w(τ) to any regular unramified
character τ.


Combining Corollary <ref> with Proposition
<ref> above gives a formula for m̃_u,w in
terms of localizations of motivic Chern classes, and in particular it
recovers the Langlands-Gindikin-Karpelevich formula from
(<ref>). 
Also, Theorem <ref> follows now from 
Theorem <ref>(c) and the main theorem 
from <ref>.
 
This follows from Theorem <ref> above together
with the equality m̃_u,w = m_u,w(τ) for all regular
unramified characters τ.




 §.§ Analytic properties of transition coefficients
 
In this section we prove a conjecture of Bump and
Nakasuji <cit.> about analytic
properties for the transition coefficients m̃_u,w and the
set of coefficients r̃_u,w defined as follows
(cf. <cit.>).  If f=f(q') is a
function, let f̅(q'):=f(q'^-1). 
Define

    r̃_u,w := ∑_u ≤ x ≤ w (-1)^ℓ(x) - ℓ(u)m̃_x,w.
 
Since we are interested only in analytic properties, by the comparison
Theorem <ref>(c) we can replace the coefficients
m̃_u,w by the `geometric' ones m_u,w. We accordingly
let r_u,w:= ∑_u ≤ x ≤ w (-1)^ℓ(x) - ℓ(u)m_x,w be the corresponding coefficients, where
f̅(y):=f(y^-1).  Therefore, r̃_u,w =
r_u,w(τ)|_y=-q' for any regular unramified character τ.
With this notation, we can prove the following statement,
cf. <cit.>.

 
Let u ≤ w be two Weyl group elements. Then the functions

    ∏_α∈ S(u,w)(1-e^α)r_u,w   , 
    ∏_α∈ S(u,w)(1-e^α)m_u,w

are holomorphic on the dual torus .


 
As observed by Bump and Nakasuji in loc. cit., the conjecture
for r_u,w implies the conjecture for m_u,w. Further, using the
formula for m_u,w from Proposition <ref>, 

    (r_u,w)^∨    = ∑_u ≤ x ≤ w (-1)^ℓ(x) -
        ℓ(u) (m_x,w)^∨ = 1/_y(Y(w)^∘)|_w∑_u ≤ x ≤ w (-1)^ℓ(x) - ℓ(u)_y (Y(x))|_w 
        = 1/_y(Y(w)^∘)|_w∑_u ≤ x
      (-1)^ℓ(x) - ℓ(u)_y (Y(x))|_w =
      _y(Y(u)^∘)|_w/_y(Y(w)^∘)|_w;
 
here the third equality holds because _y(Y(x))|_w = 0 for x ≰
w, as e_w ∉ Y(x), and the last equality follows by Möbius
inversion on the Bruhat poset W. It follows that

    ( ∏_α∈
      S(u,w)(1-e^α)r_u,w)^∨ = ∏_α∈
      S(u,w)(1-e^-α)
    _y^-1(Y(u)^∘)|_w/_y^-1 (Y(w)^∘)|_w,
 
therefore it suffices to show that the right-hand side is holomorphic
in y.  Using Proposition <ref>, the definition of
S(u,w) (<ref>), and the description of _y(Y(w)^∘)|_w
from Theorem <ref> (and its proof), we obtain

    ∏_α∈ S(u,w)(1-e^-α)
    _y^-1(Y(u)^∘)|_w/_y^-1 (Y(w)^∘)|_w   
    = _y^-1(Y(u)^∘)|_w/λ_y^-1 (T^*_w
      Y(w))×∏_α∈ S(u,w)
      (1-e^-α)/∏_α > 0, ws_α < w (1 -
      e^wα)
        = _y^-1(Y(u)^∘)|_w/λ_y^-1 (T^*_w
      Y(w))×∏_α > 0, u ≤ w s_α <
        w(1-e^wα)/∏_α > 0, ws_α < w (1 -
      e^wα)
        = _y^-1(Y(u)^∘)|_w/λ_y^-1 (T^*_w
      Y(w)) ·∏_α > 0, u ≰ ws_α < w (1 -
      e^wα).
 
The last expression is a holomorphic function by
Theorem <ref> and we are done.


We also record the following result, obtained in the proof of
Theorem <ref>.

The coefficients r_u,w are obtained from the expansion: 

    _y^∨(Y(u)^∘) = ∑r_u,w b_w
 
or, equivalently 

    (r_u,w)^∨ =  _y(Y(u)^∘)|_w/_y(Y(w)^∘)|_w ∈(K_()[y^± 1] )    .
 


 
The coefficients r_u,w satisfy other remarkable properties, such
as a certain duality and orthogonality; see <cit.>. We will study these properties using motivic
Chern classes in an upcoming note.  






§ APPENDIX: PROOF OF LEMMA <REF>

Recall Lemma <ref>. 

 
(a) Let u,w ∈ W. Under the left Weyl group multiplication, 

    w. _, T^1/2, (u) 
    = _w , w T^1/2, w.(wu) .
 
In particular, if both the polarization T^1/2 and the line bundle
 are G-equivariant, then

    w. _, T^1/2, (u) 
    = _w , T^1/2, (wu) .


(b) The duality automorphism acts by sending q ↦ q^-1 and 

    (_,T^1/2, (w))^∨
    =q^- G/B/2_,T^1/2_, ^-1(w),

where T^1/2_:=q^-1(T^1/2)^∨
is the opposite polarization; see <cit.>, i.e., this duality changes the polarization
and slope parameters to the opposite ones, while keeping the chamber
parameter invariant.

(c) Fix integral weights λ, μ∈ X^*(T) and the equivariant
line bundles _λ= G ×^B _λ and
_μ. Let a ∈ be a rational number. Then

    _, T^1/2,a _λ⊗_μ(w) = e^- w(μ)_μ⊗_, T^1/2,a _λ(w) ,
 
as elements in K_T×^*(T^*(G/B)).


 
Part (a) can be proved directly by checking that the left-hand side 
satisfies the defining properties of the stable basis on the right-hand side.
Part (b) is <cit.> with
ħ^- X/2 replaced by q^- G/B/2 
In <cit.> the variety X is the symplectic
  resolution, and it corresponds to our T^*(G/B), so
  ħ^- X/2 should be q^- G/B; the missing
  factor of 1/2 is a typo.. Since this equation is not 
  proved in loc. cit., we include a proof in the case when the fixed point set 
  X^T is finite
(e.g., X=T^*(G/B)).[Our T is
  denoted by A in loc. cit. and this is the torus
  preserving the symplectic form of X.] By the uniqueness of stable
envelopes, we need to show q^ X/4(_,
  T^1/2,)^∨ satisfies the defining properties of
_, T_^1/2,^-1. The support
condition is obvious, and the degree condition follows because
_T remains unchanged after multiplication by powers of q^±1/2. We turn to the normalization condition. Denote by F
a component of X^T (a point, in our case), and use the notation
N_+, N_-, N^1/2, etc., for the appropriate normal
subspaces to F, as in the paragraphs preceding
Theorem <ref>. Since
N_–T^1/2=q^-1(T^1/2_>0)^∨
-T^1/2_>0 (see <cit.>), the
normalization is (see <cit.>)

    _, T^1/2,|_F
    =(-1)^ T^1/2_>0( N_-/ T^1/2)^1/2_|_F=(-1)^ T^1/2_>0
    q^- T^1/2_>0/2
    ( T^1/2_>0)^∨_N_+|_F.
 
The last equality holds because the normal bundle of 
at F is spanned by the non-attracting weights at F; this is
the same as the normal bundle of N_+ inside N, therefore
_|_F = _N_+|_F = λ_-1^T
  ×^*(N_-^∨).

Let {γ_j} be the torus weights of T^1/2_>0|_F
and let {β_i } be the torus weights of
T^1/2_<0|_F.  Since
T(X)=T^1/2+q^-1(T^1/2)^∨, the torus weights
of N_-|_F are {β_i } and { q^-1γ_j^-1}. We
abuse notation and write λ for e^λ∈ R(T). Then,

    _, T^1/2,|_F   =(-1)^
      T^1/2_>0q^- T^1/2_>0/2∏_j
    γ_j^-1∏_i(1-β_i^-1)∏_j(1-qγ_j)
       =q^ T^1/2_>0/2∏_i(1-β_i^-1)
    ∏_j(1-q^-1γ_j^-1).

Since T^1/2_=q^-1(T^1/2)^∨,
the torus weights of T^1/2_,>0|_F are 
{q^-1β_i^-1} and the torus weights of
T^1/2_,<0|_F are {
q^-1γ_j^-1}. A similar calculation shows

    _, T^1/2_,^-1|_F
    =q^ T^1/2_<0/2∏_j(1-qγ_j)
    ∏_i(1-β_i).

Taking the dual of (<ref>), we get 

    (_, T^1/2,)^∨|_F
    =q^- T^1/2_>0/2∏_i(1-β_i)
    ∏_j(1-qγ_j).

Therefore, 

    (_, T^1/2,)^∨|_F
    =q^- T^1/2/2_, T^1/2_,^-1|_F
    =q^- X/4_, T^1/2_,
    ^-1|_F.

This proves the normalization condition, whence part (b). Part (c)
follows directly from the uniqueness of the stable envelope.

  



halpha


