Constructing tame supercuspidal representations
Jeffrey Hakim







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Constructing Tame Supercuspidal Representations
    Jeffrey Hakim
    December 30, 2023
===============================================

A new approach to Jiu-Kang Yu's construction of tame supercuspidal representations of p-adic reductive groups is presented.  Connections with the theory of cuspidal  representations of finite groups of Lie type and the theory of distinguished representations are also discussed.


=.13in



§ INTRODUCTION


This paper provides a new approach to the construction of  tame supercuspidal representations  of p-adic reductive groups.  It grew out of a desire to unify the theory of distinguished  tame supercuspidal representations from <cit.>   with  Lusztig's analogous theory for  cuspidal representations   of finite groups of Lie type <cit.>.



Our construction may be viewed as a revision of Yu's construction <cit.> that associates supercuspidal representations to certain representations of compact-mod-center subgroups.  In the applications to distinguished representations, it plays a role analogous to  Deligne-Lusztig “induction” of  cuspidal representations from certain characters of tori <cit.> (but our approach  is not cohomological). 

Yu's tame supercuspidal representations are parametrized by  complicated objects called generic, cuspidal G-data.  (See <cit.>.)
The fact that Yu's representations should be associated to simpler objects (such as characters of tori) is already evident  in Murnaghan's paper <cit.>,  Kaletha's paper <cit.>, and the general theory of the local Langlands correspondence.

In <cit.> and  <cit.>, generic, cuspidal G-data are manufactured from simpler data and then  Yu's construction is applied.
By contrast, our intent is to simplify Yu's construction and make it more amenable to applications, such as the applications to distinguished representations discussed in <ref>.

The main source of simplification is the elimination of  Howe factorizations in  Yu's construction.  (See <cit.> and <cit.>.)  Recall that to construct a tame supercuspidal representation with Yu's construction, one has  to make a noncanonical choice of a factorization.  Removing the need for this choice, as we do, allows one to more easily develop applications, because one no longer needs technical lemmas proving the existence of  application-friendly factorizations.

Another technical notion required in Yu's construction is that of “special isomorphisms.”  Though it is already known that special isomorphisms can be chosen canonically and their choice does not affect the isomorphism class of the constructed representation, special isomorphisms have remained a misunderstood and unpleasant aspect of the theory of tame supercuspidal representations.  (Here, we are really referring to the “relevant” special isomorphisms of <cit.>.  See 3.4, especially Proposition 3.26, <cit.> for more details.  Yu's original definition of “special isomorphism” is in <cit.>.)



Our construction takes as its input a suitable representation ρ (called a “permissible representation”) of a compact-mod-center subgroup of our p-adic group G and then  canonically constructs the character of a representation κ of an open compact-mod-center subgroup such that  κ induces a supercuspidal representation π of G. 
The construction of the character of κ (and hence the equivalence classes of κ and π) is canonical, reasonably simple, and makes no reference to special isomorphisms.  We do, in fact, use special isomorphisms to establish the existence of a representation κ with the given character, but the meaning and role of special isomorphisms in the theory becomes more transparent.


We also observe that, in many cases, there is no need to explicitly impose genericity conditions in  our construction, but, instead, the necessary genericity properties are automatic. 
We are indebted to Tasho Kaletha for explaining to us that Yu's main genericity condition, condition GE1, is built into our construction.
 
 
In general, the influence of Kaletha's ideas on this paper is hard to adequately acknowledge.
After the initial draft of the paper was circulated, the author became aware of Kaletha's work <cit.> on regular supercuspidal representations.  This prompted a major revision of the paper, during which various tameness conditions and other technicalities were removed.

The best evidence of the utility of our construction lies in the proof of Theorem <ref>, a result that is  stated in this paper  but proven   in a companion paper <cit.>.  The proof uses the theory in this paper to establish stronger versions of the author's main results with Fiona Murnaghan, with considerably less effort.

Theorem <ref> links  the representation theories of finite and p-adic groups by providing a uniform formula for the dimensions of the spaces of invariant linear forms for distinguished representations.

For finite groups of Lie type, our formula is essentially a reformulation of the main result of <cit.>.  For p-adic reductive groups, it is a reformulation of  the main result of  <cit.>.
In both cases, these reformulations refine earlier reformulations in <cit.>.




This paper is structured in an unorthodox way to accommodate a variety of readers, each of whom is only interested in select aspects of the theory. 
The construction itself is expeditiously explained (without proofs)
in <ref>.  The main result is Theorem <ref>.
The proofs are  taken up in <ref>.
For the most part, we expect that the connection between our construction and Yu's construction will be self-evident in <ref>, but  <ref> formalizes this connection.  Finally, in <ref>, we describe the connection between our construction and the Deligne-Lusztig construction in applications to the theory of distinguished representations.

We gratefully acknowledge the advice and generous help of Jeffrey Adler and Joshua Lansky throughout the course of this project. 



§ THE CONSTRUCTION




 §.§ The inducing data


Let F be a finite extension of a field ℚ_p of p-adic numbers for some odd prime p, 

 and let F be an algebraic closure of F.
 
 Let  be a connected reductive F-group.
 We are interested in constructing supercuspidal representations π of G = (F).
 
The input for our construction is a suitable representation (ρ , V_ρ) of a suitable compact-mod-center subgroup H_x of G, and the map from ρ to π has some formal similarities to Deligne-Lusztig induction of cuspidal representations of finite groups of Lie type.

Let us now specify H_x and ρ more precisely.
 
 Fix an F-subgroup  of  that is a Levi subgroup of  over some tamely ramified finite extension of F.
 We  require that the quotient _/_ of the centers of  and  is F-anisotropic.  

Fix  a vertex x in the reduced building ℬ_ red(,F).  Let H =  (F) and let H_x be the stabilizer of x in H.  Let H_x,0 be the corresponding (maximal) parahoric subgroup in H and let H_x,0+ be its prounipotent radical.
 
Note that the reduced building ℬ_ red(,F) of  is, by definition, identical to the extended building ℬ(_ der , F) of the derived group _ der of .

Let _ sc→_ der be the universal cover of _ der.
We identify the  buildings ℬ (_ sc,F) and ℬ(_ der,F)   (as in <ref>) and let H_ der,x,0+^♭ denote the image of H_ sc,x,0+ in _ der.
 
  A representation of H_x is permissible if it is a smooth, irreducible, complex representation (ρ ,V_ρ) of H_x such that:


  (1) ρ induces an irreducible (and hence supercuspidal) representation of H,

  (2) the restriction of ρ to H_x,0+ is a multiple of some character ϕ of H_x,0+, 

  (3) ϕ is trivial on H_ der,x,0+^♭,

  (4)  the  dual cosets (ϕ |Z^i,i+1_r_i)^* (defined in <ref>) contain elements that satisfy Yu's condition GE2  (stated below in <ref>).







When the order of the fundamental group π_1(_ der) of _ der is not divisible by p, then we show in Lemma <ref> that H_ der,x,0+^♭=H_ der,x,0+.  In this case, we show in Lemma <ref> that every permissible representation
ρ may be expressed as ρ_0 ⊗ (χ|H_x), where ρ_0 has depth zero and χ is a quasicharacter of H that extends ϕ.  (So our representations ρ_0 are essentially the same as the representations denoted by ρ in <cit.>.)  This allows one to appeal to the depth zero theory of Moy and Prasad <cit.> to explicitly describe the representations ρ.  Even when p divides the order of π_1 (_ der), one can apply depth zero Moy-Prasad theory by lifting to a z-extension, as described in <ref>.  In this case, one obtains a factorization  over the z-extension, but the factors may not correspond to representations of H_x.



In light of <cit.>, condition (4) is only required when 
 p is a torsion prime for the dual of the root datum of  (in the sense of  <cit.> and <cit.>). 
 Using Steinberg's results (Lemma 2.5 and Corollary 1.13 of <cit.>), we can make this more explicit as follows.  Since we assume p 2, condition (4) is  needed precisely when 


  * p=3 and _ der has a factor of type E_6, E_7, E_8 or F_4,

  * p=5 and _ der has a factor of type E_8, or 

  * _ der has a factor of type A and
p is a torsion prime for X/Φ, where Φ is the root lattice and X is the character lattice relative to some maximal torus of .

  In particular, condition (4) is unnecessary for  _n, but is required for _n.



A given permissible representation can correspond to supercuspidal representations for two different groups.  For example, suppose Ψ = (,y,ρ_0,ϕ⃗) is a generic cuspidal G-datum with d>0 and ϕ_d =1, with notations as in <cit.>.  Let Ψ' be the datum obtained from Ψ by deleting the last components of  and ϕ⃗.  Then the tame supercuspidal representations of G and G^d-1 associated to Ψ and Ψ', respectively, are both associated to the same permissible representation ρ_0⊗ (∏_i=0^d-1(ϕ_i|G^0_[y])).




 §.§ Basic notations


We use boldface letters for F-groups and non-boldface for the groups of F-rational points.  Gothic letters are used for  Lie algebras, and  the subscript “der” is used for derived groups of algebraic groups and their Lie algebras.  For example, G =  (F),   g =  Lie(), 𝔤 = g(F), and _ der is the derived group of .  We let G_ der denote _ der(F), but caution that this is not necessarily the derived group of G.

In general, our notations tend to follow <cit.>, which, in turn, mostly follows <cit.>.   This applies, in particular, to groups G_y,f associated to concave functions, such as  Moy-Prasad groups G_y,s and groups G⃗_y,s⃗ associated to twisted Levi sequences  and admissible sequences s⃗.  Many of these notations are recalled in  <ref>, as well as conventions regarding variants of exponential maps (such as Moy-Prasad isomorphisms).
We use colons in our notations for quotients, for example,
 G_y,s:r = G_y,s/G_y,r.  Similar notations apply to the Lie algebra filtrations and to F-groups other than . 

Moy-Prasad filtrations  over F are defined with respect to the standard valuation v_F on F.  Moy-Prasad filtrations over extensions of F are also defined with respect to v_F.   Thus we have intersection formulas such as (E)_y,r∩ G = G_y,r.

Let  
    E_r =E∩ v_F^-1([r,∞])
 and, for r>0,

    E_r^× = 1+ E_r.



Given a point y in the reduced building ℬ_ red( , F), we let 𝖦_y denote the associated reductive group defined over the residue field f of F.  Thus 𝖦_y(F) = G(F^ un)_y,0:0+, where F^ un is the maximal unramified extension of F in F, and F is the residue field of F^ un (which is an algebraic closure of f).



 §.§ The torus  and the subtori _𝒪

The construction requires the choice of a maximal, elliptic F-torus  in  with the   properties:


  * The splitting field E in F of  over F is a tamely ramified extension of F.  (Note that E/F is automatically a finite Galois extension.)

  * The point x is the unique fixed point of Gal(E/F) in the apartment 𝒜_ red(,, E).


In choosing , we are appealing to some known facts:

 
  * ℬ_ red( , F)= ℬ_ red(,E)^ Gal(E/F), since E/F is tamely ramified,  according to a result of Rousseau <cit.>. (See also <cit.>.)
 
  * 𝒜_ red (,,F) consists of a single point, since   is elliptic. 

  * Not only do tori  of the above kind exist, but DeBacker <cit.> gives a construction of a special class of such tori:

  * Choose an elliptic maximal f-torus 𝖳 in 𝖧_x.(See Lemma 2.4.1 <cit.>.)  
  * Choose a maximal F^ un-split torus  in  such that


  *  is defined over F, 

  * x∈𝒜_ red(,,F^ un), and 

  * 𝖳(F) is the image of (F^ un)∩ (F^ un)_x,0 in 𝖧_x(F).

(See Lemma 2.3.1 <cit.>.)
 
  * Take  to be the centralizer  of  in .  

Such tori  are called “maximally unramified elliptic maximal tori” in 3.4.1 <cit.>, and  various reformulations of the definition are provided there.  For example, an elliptic maximal F-torus  is maximally unramified  if it is  contained in a Borel subgroup of  over F^ un.  Lemma 3.4.4 <cit.> says that any two maximally unramified elliptic maximal tori in  corresponding to the same point in ℬ_ red(,F) must be H_x,0+-conjugate.
Lemma 3.4.18 <cit.> says every regular depth zero supercuspidal representation of H comes from a regular depth zero character of the F-rational points of a maximally unramified elliptic maximal torus of .



 It is necessarily the case that T⊂ H_x since T preserves both 𝒜_ red(,,E) and ℬ_ red (,F), and the intersection of the latter spaces is { x}.


 When H_ der is compact, every F-torus in  is elliptic and ℬ_ red (,F) is a point x and so H_x =H.  In this case, every maximal F-torus  in  is elliptic and x is (trivially) the unique Gal(E/F)-fixed point in 𝒜_ red(,,E).  This example illustrates, in the extreme, that the point x does not  determine the torus .














Let Φ = Φ (,) and Γ =  Gal(F/F).  Given a Γ-orbit 𝒪 in Φ, let _𝒪 be the torus generated by the tori _a = image(ǎ) as a varies over 𝒪.

For positive depths, the Moy-Prasad filtration of T_𝒪 = _𝒪(F) is given using the norm map
 
    N_E/F: _𝒪(E)   →    T_𝒪
    
    t   ↦   ∏_γ∈ Gal(E/F)γ (t)

as follows.  If a∈𝒪 and r>0 then, according to Lemma <ref>,  
    T_𝒪,r =N_E/F(ǎ (E_r^×))= N_E/F(_a(E)_r)= N_E/F(_𝒪(E)_r),
 for all a∈𝒪.


Though the tori _𝒪 are not explicitly considered in <cit.>, the norm map on _a(E) plays a prominent role there, and our use of the norm map has  been influenced by the theory in <cit.>.  (See <cit.>.)






 §.§ Recovering r⃗ and 




A key  step in our construction is to recursively construct from ϕ  a sequence 
    = (^0,… , ^d)
of subgroups

    = ^0⊊⋯⊊^d=

and a sequence

    r⃗ = (r_0,… ,r_d)

of real numbers

    0≤ r_0 <⋯ < r_d-1≤ r_d.

 

We construct the groups recursively in the order ^d,… ,^0 (and similarly for r_d,… , r_0), so d should be treated initially as an unknown nonnegative integer whose value only becomes evident at the end of the recursion.
This indexing is compatible with <cit.>, but it is the opposite of what is done in <cit.>.

To begin the recursion, we take  ^d = and 

    r_d = depth(ρ) = depth(ϕ).


When =, we declare that d=0.

Now assume ⊊.  Suppose i∈{ 0,… , d-1} and ^i+1 has been defined and strictly contains .

In general, given ^i+1, we take

    ^i = ^i+1_ der∩,    ^i = ^i+1_ der∩.

Note that ^i is the
torus generated by the tori _𝒪 associated to Γ-orbits 𝒪 in Φ (^i+1,).  
(See <cit.>.)


To say  is elliptic in  means that /_ is F-anisotropic or, equivalently, _ der∩ is F-anisotropic.  Since we assume _/_ is F-anisotropic,  must be elliptic in .  So ^d-1, and hence ^i, must be F-anisotropic.
It follows that x determines unique points (that we also denote x) in ℬ_ red(^i+1,F) and ℬ_ red (^i,F).  


Our conventions regarding embeddings of buildings are similar to those in <cit.>.  See, in particular, Remark 3.3 <cit.>, regarding the embedding of ℬ_ red(^i,F) in ℬ_ red(^i+1,F).
It should be understood that the images of x in the various buildings  depends on the choice of .  As in <cit.>, it will be easy to see that varying  does not change the isomorphism class of the resulting supercuspidal representation.  (See the discussion after Lemma 1.3 <cit.>.)


At this point, we encounter a technical issue involving the residual characteristic p.

Suppose first that p does not divide the order of the fundamental group π_1(_ der).  
Then we define

    r_i = depth(ϕ|H^i_x,0+) = depth(ϕ |T^i_0+).

The latter identity of depths is a consequence of Lemma <ref>.

If p divides the order π_1(_ der), we can simply replace  by a z-extension ^♯ of the type considered in <ref>.  In other words, we  identify G with G^♯ /N, where  is the kernel of the z-extension, and  apply our construction to ^♯ instead of .

Though it is somewhat tedious, the latter case can also be translated into terms that are intrinsic to G.  (In other words, one does not need to refer to the z-extension, but one does need to refer to the universal cover of _ der.)
To appreciate the source of complications, consider, for example, the character ϕ of H_x,0+.  According to Lemma 3.5.3 <cit.>, we have a surjection H^♯_x,0+→ H_x,0+, and thus the pullback ϕ^♯ of ϕ to H^♯_x,0+ captures all of the information carried by ϕ.  
But we may not have similar surjections associated to the groups H^i_x,0+ and T^i_0+, and this creates difficulties.  The appropriate depths r_i for the construction may no longer be the depths of the restrictions of ϕ to H^i_x,0+ and T^i_0+.  Therefore, we cannot simply define the r_i's as above.

In light of the previous remarks, we generally assume p does not divide the order of π_1(_ der) when we describe our construction, though we do consider more general p when considering the complete family of representations that the construction captures.  (See Lemma <ref>.)

Continuing with the construction, we now take ^i to be the (unique) maximal subgroup of  such that:


  * ^i is defined over F.

  * ^i contains .

  * ^i is a Levi subgroup of  over E.

  * ϕ |H^i-1_x,r_i=1.


(It follows from Lemma <ref> that the condition ϕ |H^i-1_x,r_i=1 is equivalent to the condition ϕ |T^i-1_r_i=1.)

As soon as ^i =, the recursion ends.  The value of i for which ^i= is declared to be i=0.  The values of d and the other indices also become evident at this point.

  There is an alternate way to construct r⃗ and  that is direct (as opposed to being recursive) and clarifies the meaning of G^i and r_i, as well as the fact that these objects are well defined.  It exploits the choice of  and is inspired by 3.7 <cit.>.

One can define 
    r_0<⋯ < r_d-1
 to be the sequence of positive numbers that occur as depths of the various characters ϕ |T_𝒪,0+ for 𝒪∈ΓΦ.  (This does not preclude the possibility that ϕ |T_𝒪,0+=1 for some orbits 𝒪.)
Next, when i<d, we take 
    Φ^i = ⋃_𝒪∈ΓΦ, ϕ|T_𝒪,r_i=1𝒪.
  One can define ^i to be the unique E-Levi subgroup of  that contains  and is such that Φ (^i,) = Φ^i.



 To connect our narrative with Yu's construction, it might be helpful to think in terms of Howe's theory of factorizations of admissible quasicharacters (and its descendants), even though our approach avoids factorization theory.  In the present context,  a factorization of ϕ would consist of quasicharacters ϕ_i : G^i →^× such that ϕ = ∏_i=0^d (ϕ_i|H_x,0+).  But the factors ϕ_i+1,… , ϕ_d are trivial on [G^i+1,G^i+1]∩ H_x,0+ and the factors ϕ_0,… , ϕ_i-1 are trivial on H_x,r_i.  Therefore, on [G^i+1,G^i+1] ∩ H_x,r_i, the character ϕ coincides with the quasicharacter ϕ_i.  Lemma <ref> (with  replaced by ^i+1) implies  that [G^i+1,G^i+1] ∩ H_x,r_i = H^i_x,r_i, so long as p does not divide the order of the fundamental group of ^i+1_ der.  
Hence, ϕ|H^i_x,r_i = ϕ_i|H^i_x,r_i.  Our present approach to constructing supercuspidal representations is partly motivated by the heuristic that the essential information carried by the factor ϕ_i is contained in the restriction ϕ_i|H^i_x,r_i and, since this restriction agrees with ϕ |H^i_x,r_i, there  is no need to use factorizations.  (See Lemma <ref> for more details.)




Sequences such as = (^0,… ,^d) are called“tamely ramified twisted Levi sequences” (Definition 2.43 <cit.>), and one may view ^i+1 as being constructed from ^i by adding unipotent elements.  By contrast, if ^d = and = (^0,… ,^d) then each ^i+1 is obtained from ^i by adding semisimple elements.




 §.§ The inducing subgroup K


The objective of our construction is to associate to ρ an equivalence class of supercuspidal representations of G.  These representations are induced from a certain compact-mod-center subgroup K of G that is defined in this section.  Later, we define an equivalence class of representations κ of K by canonically defining the (common) character of these representations κ.  The representation π of G induced from such a κ will lie in the desired equivalence class of supercuspidal representations π associated to ρ.

For each i∈{ 0,… , d-1}, let 
    s_i= r_i/2

and

    J^i+1   =    (G^i,G^i+1)_x,(r_i,s_i)
    
    J^i+1_+    =    (G^i,G^i+1)_x,(r_i,s_i+)
    
    J^i+1_++   =    (G^i,G^i+1)_x,(r_i+,s_i+).

(The notations on the right hand side follow <cit.>.
See also the discussion in <ref> below regarding the groups _x,t⃗ associated to the twisted Levi sequence = (^i,^i+1)  and admissible sequences of numbers.)


Our desired inducing subgroup K is 
given by 

    K= H_xJ,

where

    J = J^1⋯ J^d.




 §.§ The subgroup K_+


The (equivalent) representations κ of K mentioned above will turn out to restrict to a multiple of a certain canonical character ϕ̂ of a certain subgroup K_+.
 The subgroup K_+ is 
given by 

    K_+ = H_x,0+L_+,

where

    L_+   =    (ϕ)L^1_+⋯ L^d_+,
    
     L^i+1_+   =    G^i+1_ der∩ J^i+1_++.


 The character ϕ̂ is the inflation 
    ϕ̂=  inf_H_x,0+^K_+-.3em (ϕ )
 of ϕ  to K_+, that is, ϕ̂(hℓ) = ϕ (h), when h∈ H_x,0+ and ℓ∈ L_+.   More details regarding the definition of ϕ̂ are provided in <ref>.

 
 
 The inflation of characters from H_x,0+ to K_+ just considered should not be confused with a different inflation procedure considered in <cit.>, where we are using the decomposition K_+ = H_x,0+J^1_+⋯ J^d_+.
 
 




 §.§ The dual cosets



Fix i∈{ 0,… , d-1} and define ^i,i+1 to be the 
F-torus

    ^i,i+1 = (^i+1_ der∩^i)^∘,
 where ^i is the center of ^i.




Fix a character ψ of F that is trivial on the maximal ideal 𝔓_F but nontrivial on the ring of integers 𝔒_F.

 The dual coset (ϕ |Z^i,i+1_r_i)^* of ϕ |Z^i,i+1_r_i is the coset in z^i,i+1,*_-r_i:(-r_i)+ consisting of elements
 X^*_i in z^i,i+1,*_-r_i that represent ϕ |Z^i,i+1_r_i in the sense that

    ϕ ( exp (Y+z^i,i+1_r_i+)) = ψ (X^*_i(Y)),  ∀ Y∈z^i,i+1_r_i.



(The notation “exp” here is for the Moy-Prasad isomorphism on z^i,i+1_r_i:r_i+.  See <ref> for more details.)

Let z^i,i+1 be the Lie algebra of Z^i,i+1, and let z^i,i+1,* be the dual of z^i,i+1.
The decomposition 
    g^i = z^i⊕g^i_ der
 restricts to a decomposition 
    h^i = z^i,i+1⊕h^i-1.


Using this, we associate to each coset in z^i,i+1,*_-r_i:(-r_i)+ 
a character of H^i_x,r_i:r_i+ that is trivial on H^i-1_x,r_i:r_i+.  The character of H^i_x,r_i:r_i+ associated to the dual coset (ϕ |Z^i,i+1)^* is just the restriction of ϕ since ϕ |H^i-1_x,r_i=1.

In other words, if X∈z^i,i+1_r_i and Y∈h^i-1_x,r_i then

    ϕ (exp (X+Y+ h^i_x,r_i+)) = ψ (X^*_i(X)),
 where X^*_i is any element of (ϕ |Z^i,i+1_r_i)^*  and 
    exp: h^i_x,r_i:r_i+≅ H^i_x,r_i:r_i+
 is the isomorphism defined in <ref>.

Lemma <ref> establishes that every  element X^*_i of the dual coset  (ϕ |Z^i,i+1_r_i)^* is ^i+1-generic of depth -r_i.  (Here, we are using condition (4) in Definition <ref>, as well as Lemma 8.1 <cit.>.)

 The decomposition h^i = z^i,i+1⊕h^i-1
restricts to

    t^i = z^i,i+1⊕t^i-1,

where t^i and  t^i-1 are generated by the t_𝒪's associated to Φ (^i+1,) and Φ (^i,), respectively.  We caution that z^i,i+1 is usually not generated by t_𝒪's and, in fact, may not contain any t_𝒪's.





 §.§ The construction


For each i∈{ 0,… ,d-1}, the 
elements X^*_i of the dual coset (ϕ |Z^i,i+1_r_i)^* also represent a character of J^i+1_+ that we denote by ζ_i.  By construction, ζ_i is the inflation of a ^i+1-generic character of G^i_x,r_i of depth r_i.

Since G^i_x,r_i:r_i+ is isomorphic to a finite vector space over a finite field of characteristic p, every character of G^i_x,r_i:r_i+, in particular ζ_i, must take values in the group μ_p of complex p-th roots of unity.













The space V of our inducing representation κ will be a tensor product

    V = V_ρ⊗ V_0⊗⋯⊗ V_d-1

of V_ρ with spaces V_i, such that each V_i is the space of a Heisenberg representation τ_i attached to the character ζ_i.








































The next step is to define these Heisenberg representations.  

Let W_i be the quotient J^i+1/J^i+1_+ viewed as an _p-vector space with nondegenerate (multiplicative) symplectic form

    ⟨ uJ^i+1_+ ,vJ^i+1_+⟩ = ζ_i(uvu^-1v^-1).

(The fact that this is a nondegenerate symplectic form is shown in <cit.>.)


Let
ℋ_i be the (multiplicative) Heisenberg p-group  such that

    ℋ_i = W_i×μ_p

with multiplication given by

    (w_1,z_1)(w_2,z_2) = (w_1w_2, z_1z_2 ⟨ w_1,w_2⟩^(p+1)/2).

Let (τ_i,V_i) be a Heisenberg representation of ℋ_i whose central character is the identity map on μ_p.  (Up to isomorphism, τ_i is unique.)

Let 𝒮_i be the symplectic group Sp(W_i).  We let 𝒮_i act on ℋ_i via its natural action on the first factor of W_i×μ_p.  The semidirect product 𝒮_i⋉ℋ_i is the Cartesian product 𝒮_i×ℋ_i with multiplication

    (s_1,h_1)(s_2,h_2)= (s_1s_2, (s_2^-1h_1)h_2).

Except for the case in which p=3 and __p(W_i)=2, there is a unique extension τ̂_i of τ_i to a representation 

    τ̂_i :𝒮_i⋉ℋ_i→ (V_i).

When p=3 and __p(W_i)=2, we also have a canonical lift τ̂_i that is specified in <cit.>.
Given h∈ H_x, let 
    ω_i(h) = τ̂_i( Int(h),1),
 where Int(h) comes from the conjugation action of h on J^i+1.


























































We now state our main result:

  Suppose ρ is a permissible representation of H_x.  
Up to isomorphism, there is a unique representation κ = κ (ρ) of K such that:


  (1)  The character of κ has support in H_xK_+.

  (2)  κ |K_+ is a multiple of ϕ̂.

  (3)  κ | H_x = ρ⊗ω_0⊗⋯⊗ω_d-1.

Then π (ρ)=  ind_K^G(κ (ρ)) is an irreducible, supercuspidal representation  of G whose isomorphism class is canonically associated to ρ.  The isomorphism class of every tame supercuspidal representation of G constructed by Yu in <cit.> contains a representation π (ρ).



The fact that the equivalence class of κ (ρ) is well-defined and canonically associated to ρ is proven in Lemma <ref>.
The proof of the latter result also establishes that κ (ρ) is determined by the properties listed above.
The fact that π (ρ) is irreducible  is proven in Lemma <ref>.  This implies that π (ρ) is also supercuspidal, as explained on page 12 of <cit.>.
Lemma <ref> establishes the connection between our construction and Yu's construction.


If ρ is permissible and Θ_π is the smooth function on the regular set of G that represents the character of  π = π(ρ) then Θ_π is given  by the Frobenius formula

    Θ_π (g) =∑_C∈ K G/K ∑_h∈ Cχ̇_κ (h gh^-1),

where χ̇_κ is the function on G defined by

    χ̇_κ (jk) = ϕ̂(k)· tr(ρ (j))· tr (ω_0(j))⋯ tr (ω_d-1(j)),
 
for j∈ H_x and k∈ K_+, and χ̇_κ≡ 0 on G-H_xK_+.
In particular, Θ_π (g) =0 when g does not lie in the G-invariant set generated by H_xK_+.


We refer to <cit.> for details regarding the Frobenius formula.



§ TECHNICAL DETAILS


This chapter is essentially a collection of technical appendices to the previous chapter.



 §.§ z-extensions


A z-extension of  (over F) 
consists of a connected reductive F-group ^♯ with simply connected derived group together with an associated exact sequence

    1→→^♯→→ 1
 of F-groups
such that  is an induced torus that embeds in the center of ^♯.

To say that  is an induced torus means that it is a product of tori of the form Res_L/F(𝔾_m), where each L is a finite extension of F.  
This implies that the Galois cohomology H^1(F,) is trivial and hence
the cohomology sequence associated to the above exact sequence yields
an exact sequence

    1→ N→ G^♯→ G→ 1

of F-points.
Therefore every representation of G may be viewed as a representation of G^♯ that factors through G.

Not only do z-extensions always exist, but we can always choose a z-extension ^♯ of  such that  is a product of factors Res_E/F(𝔾_m), where, as usual, E is the splitting field of  over F.
In this case, the preimage ^♯ of  in ^♯ is an E-split maximal torus in ^♯.  
Assume, at this point, that we have fixed such a z-extension.

The terminology “z-extension” and our definition are derived from  <cit.>, but Kottwitz refers to Langlands's article
 <cit.> as the source of the notion.  Existence results for z-extensions can be found in <cit.> and <cit.>.
 

The Philosophy of z-extensions.  When studying the representation theory of general connected reductive F-groups, it suffices to consider the representation theory of groups with simply connected derived group.  For groups of the latter type, technicalities involving bad primes (such as those related to the Kneser-Tits problem) are minimized (or eliminated).

Here, we are using the terminology “bad primes” to informally refer to any technical issues that might cause a general theory of representations of connected reductive F-groups to break down for a finite number of residual characteristics p of the ground field F.


Let ^♯ be the preimage of  in ^♯.
Note that we have a natural identification of Φ (,) with Φ (^♯,^♯).  Then the center Z() is the intersection of the kernels of the roots in Φ (,).  It is easy to see that the center of ^♯ must be the intersection of the kernels of the roots in Φ (^♯,^♯) that correspond to elements of Φ ( ,).
Since ^♯ is the centralizer in ^♯ of the torus Z(^♯)^∘, it must be the case that ^♯ is a Levi subgroup of ^♯ (over E).  It follows that the fundamental group of ^♯_ der embeds in the (trivial) fundamental group of ^♯_ der.
(This is observed in <cit.> and is easy to prove directly.)
It also follows
that our z-extension of  restricts to a z-extension

    1→→^♯→→ 1
 of .

Next, we observe that we have a natural Gal(E/F)-equivariant identification of reduced buildings

    ℬ_ red(^♯ ,E)≅ℬ_ red(,E)
 as simplicial complexes.  If we identify the groups 
    ^♯ (E) / (E) = (^♯ /)(E) ≅ (E)
 then (E) has the same action on ℬ_ red(^♯ ,E) and ℬ_ red(,E).

(Recall, from <cit.>, that associated to each Chevalley basis of the Lie algebra h_ der of _ der is a point in ℬ_ red(,E).  But h_ der is naturally identified with the Lie algebra of ^♯_ der.
So a Chevalley basis determines a point in both ℬ_ red(,E)
and ℬ_ red(^♯ ,E).  Identifying these points determines our identification of reduced buildings.)


Given x∈ℬ_ red(,E), we have an exact sequence

    1→(E)→^♯(E)_x→ (E)_x→ 1
 of stabilizers of x, as well as exact sequences

    1→(E)_r→^♯ (E)_x,r→(E)_x,r→ 1
 for all r≥ 0.
There are  similar exact sequences 

    1→ N→ H^♯_x→ H_x→ 1
 and 
    1→ N_r→ H^♯_x,r→ H_x,r→ 1
 
for F-rational points.
(See <cit.>.)



Suppose ρ is a representation of H_x and ρ^♯ is its pullback to a representation of H^♯_x.  Then ρ is permissible if and only if ρ^♯ is.  The representation π^♯ of G^♯ associated to ρ^♯ is precisely the pullback of the representation π of G associated to ρ.  The correspondence π^♯↔π gives a bijection between the representations of G that come from applying our construction to G^♯ and the representations that come from applying our construction directly to G.


To end this section, we note a complication that occurs for the group _n.  Even though _n is simply connected, its dual PGL_n is not, and this introduces the issues discussed above in Remark <ref>.  Thus taking z-extensions does not remove all problems related to the residual characteristic.



 §.§ Commutators


In this section, we collect a few facts about commutators.

Let (E)^+ be the (normal) subgroup of (E) generated by the E-rational elements of the unipotent radicals of parabolic subgroups of  that are defined over E. 

We recall now  some basic facts about (E)^+.  (See  <cit.> and <cit.>.)
Since  E is a perfect field,   (E)^+ may also be described as the group generated by the E-rational unipotent elements in (E).  
Since E is a field with at least 4 elements,  any subgroup of (E) that is normalized by (E)^+ either contains (E)^+ or is central.  (This  follows from the main theorem of <cit.>.)  
In particular, (E)^+⊂ [ (E),(E)].  But since  is E-split, (E)/(E)^+ must be abelian and hence we deduce that

    (E)^+ = [ (E),(E)].


If x∈ℬ_ red(,F) then we have inclusions

    H^♭_ der,x,0+   ⊂    [H_ der,H_ der]∩ H_x,0+⊂ [H,H]∩ H_x,0+
       ⊂    [(E),(E)]∩ H_x,0+
    = (E)^+ ∩ H_x,0+
       ⊂   H_ der,x,0+.

If the order of the fundamental group π_1(_ der) is not divisible by p then all of these inclusions are equalities.



The first step is to show  H^♭_ der,x,0+⊂ [H_ der,H_ der]∩ H_x,0+.
We observe that
the universal cover _ sc of _ der decomposes as a direct product

    _ sc = ∏_i _i
 of F-simple factors _i, and,
for each i, we have _i =  Res_F_i/F'_i, where '_i is an absolutely simple, simply connected F_i-group and F_i is a finite extension of F.  
(See <cit.>.)
To prove the desired inclusion, it suffices to show that for each i we have 
    ξ (H_i,x,0+)⊂ [ξ(H_i),ξ(H_i)],

where ξ is the covering map _ sc→_ der.

Suppose first that _i is F-isotropic or, equivalently, '_i is F_i-isotropic.  We have

    H_i^+⊂ [H_i,H_i]⊂ H_i= H_i^+,
 where the first inclusion follows from  <cit.> and the equality follows from <cit.>.  Therefore,

    ξ (H_i,x,0+) ⊂ξ([H_i,H_i]) ⊂ [ξ(H_i),ξ(H_i)].


Now suppose _i is F-anisotropic.   Then H_i = _1(D_i), where D_i is a central division algebra over F_i.  According to <cit.>, we have

    [H_i,H_i]= H_i ∩ (1+𝔓_D_i) = H_i,x,0+,
 where 𝔓_D_i is the maximal ideal in D_i.  Again, we obtain
   ξ (H_i,x,0+)⊂ [ξ(H_i),ξ(H_i)] and hence H^♭_ der,x,0+⊂ [H_ der,H_ der]∩ H_x,0+.
   
The remaining inclusions follow from the inclusions

    [H_ der,H_ der]⊂ [H,H]⊂ [(E),(E)]
    = (E)^+ ⊂_ der(E).


Now suppose the order of the fundamental group π_1(_ der) is not divisible by p.  The fact that H^♭_ der,x,0+=  H_ der, x,0+ is discussed in the proof of <cit.> and it follows from <cit.>.
 
 
The previous proof uses standard methods that can also be found in <cit.> and in the proof of <cit.>.

There are several uses for the latter result.  
The fact that one often has H^♭_ der,x,0+=  H_ der, x,0+ allows one to replace H^♭_ der,x,0+ 
in the definition of “permissible representation” with a simpler object.   
The groups involving commutators are included partly to allow one to link 
our theory with Yu's theory, where quasicharacters of H play more of a role.  In this regard, we note that a character of H_x,0+ extends to a quasicharacter of H precisely when it is trivial on [H,H]∩ H_x,0+.  On the other hand,
the group [H_ der,H_ der]∩ H_x,0+ is relevant to Kaletha's theory and, in particular, 
<cit.>.
Another application is the following:


Suppose ρ is a permissible representation of H_x and χ is a quasicharacter of G.  Then ρ⊗ (χ|H_x) is also permissible.  The representation of G associated to ρ⊗ (χ|H_x) is equivalent to π⊗χ, where π is the representation of G associated to ρ.


 
Showing that ρ⊗ (χ|H_x) is  permissible amounts to showing that
χ | H^♭_ der,x,0+ is trivial.  But this results from the inclusions

    H^♭_ der,x,0+⊂ [H,H]⊂ [G,G],
 which are implied by Lemma <ref>.  The second assertion follows directly from the definition of our construction.
 


Next, we discuss commutators of groups associated to concave functions (as in the previous section) and, in particular, groups associated to admissible sequences.

Given two concave functions f_1 and f_2, a  function f_1∨ f_2 is defined in <cit.> and it is shown in Lemma 5.22 
<cit.> that, under modest restrictions, that [G_x,f_1, G_x,f_2]⊂ G_x,f_1∨ f_2. (The latter result is a refinement of  <cit.>.)  


For groups associated to admissible sequences, Corollary 5.18 <cit.> gives the following  simpler statement of the result in Lemma 5.22 <cit.>.  (The notion of “admissible sequence” is more general in <cit.> than in  <cit.>.)


Given admissible sequences a⃗ = (a_0,… , a_d) and b⃗ = (b_0,… , b_d), 
then [G⃗_x,a⃗ , G⃗_x,b⃗]⊂G⃗_x,c⃗, where 
c⃗ = (c_0,… , c_d), with 

    c_j = 
    min{ a_0+b_0,… , a_d+b_d},   if j=0,
    min{ a_j+m_b,m_a+b_j, a_j+1+b_j+1,… , a_d+b_d},   if j>0,

and m_a = min{ a_0,… , a_d} and m_b = min{ b_0,… , b_d}.
 


We frequently will use the previous lemma (explicitly and implicitly) in combination with the following observation: to show that one subgroup H_1 of G normalizes another subgroup H_2  is equivalent to showing [H_1,H_2]⊂ H_2.  




 §.§ Twists of depth zero representations


The purpose of this section is to examine conditions under which a permissible representation ρ of H_x can be expressed as a tensor product ρ_0⊗χ, where ρ_0 has depth zero and χ is a quasicharacter of H_x.  Such a decomposition is useful in applications because the depth zero representations ρ_0 have a simple description, according to the work of Moy and Prasad <cit.>.  A secondary goal is to contrast our construction of supercuspidal representations with Yu's construction  in the special case in which = or, equivalently, d=0.

We start with a definition:


A quasicharacter χ of H_x is H-normal if it
satisfies one of the following two equivalent conditions:
 

  * χ is trivial on [H,H_x]∩ H_x.

  * The intertwining space 
 I_h (χ) =
 Hom_
hH_x h^-1∩ H_x 
(^hχ ,χ) 
is nonzero for all h∈ H.




The next result and its corollary have obvious proofs:


If ρ is an irreducible, smooth representation of H_x and χ is  an H-normal quasicharacter  of H_x then there is a set-theoretic identity 
    I_h(ρ) = I_h(ρ⊗χ),
 for all h∈ H, where

    I_h(ρ)= Hom_hH_x h^-1∩ H_x(^hρ ,ρ).




If ρ is an irreducible, smooth representation of H_x and χ is  an H-normal quasicharacter  of H_x then  the representation ind_H_x^H(ρ) is irreducible if and only if ind_H_x^H(ρ⊗χ) is irreducible.


The latter result raises the question of when, given a permissible representation ρ of H_x, one can twist ρ by an H-regular quasicharacter to obtain a depth zero representation, that is, a representation of H_x whose restriction to H_x,0+ is a multiple of the trivial representation.
This is answered in the following:


Suppose ρ is a permissible representation of H_x that restricts on H_x,0+ to the character ϕ.  Then the following are equivalent:


  * There exists an H-normal quasicharacter χ of H_x and a depth zero representation ρ_0 of H_x such that ρ = ρ_0⊗χ.

  * ϕ extends to an H-normal quasicharacter χ of H_x.

  * ϕ |([H,H_x]∩ H_x,0+)=1.



For Yu's tame supercuspidal representations, the conditions in the previous lemma can be replaced by the following more restrictive conditions:


  * There exists a  quasicharacter χ of H and a depth zero representation ρ_0 of H_x such that ρ = ρ_0⊗ (χ |H_x).

  * ϕ extends to a  quasicharacter χ of H.

  * ϕ |([H,H]∩ H_x,0+)=1.



If p does not divide the order of π_1(_ der) then every permissible representation ρ of H_x may be expressed as ρ_0⊗ (χ|H_x), where ρ_0 is a depth zero representation of H_x and χ is a quasicharacter of H.



Under the condition on p, Lemma <ref> implies that H^♭_ der,x,0+ = [H,H]∩ H_x,0+.  Therefore the character ϕ of H_x,0+ associated to ρ extends to a quasicharacter χ of H.  The representation ρ_0 = ρ⊗ (χ^-1|H_x) has depth zero.



If p divides the order of π_1(_ der), then the possibility that H^♭_ der,x,0+ is strictly contained in [H,H]∩ H_x,0+
opens up the possibility of new supercuspidal representations  not captured by Yu's construction.



We now compare our theory with Yu's theory in the case of =.  In this case, both theories minimally extend the Moy-Prasad theory of depth zero supercuspidal representations.

Let 𝔣 be the residue field of F, viewed as a subfield of the residue field 𝔉 of the maximal unramified extension of F^ un of F contained in F.
Let 𝖦_x be the 𝔉-group with 𝖦_x (𝔉) = (F^ un)_x,0:0+ and the natural Galois action.  The group 𝖦_x (𝔣) is identified with G_x,0:0+.


Recall from <cit.> that if ρ^∘ is a representation of G_x,0 then
 (G_x,0,ρ^∘) is called a depth zero minimal K-type 
 if it is the inflation of an irreducible cuspidal representation  of G_x,0:0+ = 𝖦_x (𝔣).
 
 According to <cit.> and <cit.>, every irreducible, depth zero, supercuspidal representation π of G has the form ind_G_x^G (ρ_0), where ρ_0 is a smooth, irreducible representation of G_x whose restriction to G_x,0+ is a multiple of the trivial representation and whose restriction to G_x,0 contains a depth zero minimal K-type.
 
 In fact,  every smooth, irreducible representation of G_x that induces π must contain a depth zero minimal K-type, since any two representations of G_x that induce π must be conjugate by an element of G that normalizes G_x.

 
Yu's tame supercuspidal representations π in the special case of =  are constructed as follows.    One starts with a pair (ρ_0  ,χ), where ρ_0|G_x,0+ is 1-isotypic,  ρ_0 induces an irreducible representation of G, and χ is any quasicharacter of G and then one puts 
    π = _G_x^G (ρ_0⊗ (χ |G_x)).

 Consider now the representation ρ =ρ_0 ⊗ (χ|G_x).  In order for  ρ to be  permissible, it must satisfy the following:


  (1) ρ induces an irreducible representation of G,

  (2) ρ|G_x,0+ is a multiple of some character ϕ, 

  (3) ϕ| H^♭_ der,x,0+=1.


Condition (2) is satisfied with ϕ = χ|G_x,0+. 
Let χ^♯ be the pullback of χ to a quasicharacter of a z-extension G^♯ of G.  Then Lemma 3.5.1 <cit.> implies that χ^♯ is trivial on G^♯_ der,x,0+.  Condition  (3) follows.  But the restriction of χ to G_x is G-normal and hence condition (1) is satisfied by Corollary <ref>.  So ρ must be permissible.

It is now evident  that when =  our framework captures all of Yu's supercuspidal representations for which d=0, and perhaps more in view of Remarks <ref> and <ref>.  


At first glance, it appears that  conditions (2) and (3) in the definition of “permissible representation” (Definition <ref>) should be eliminated since  they are either unnecessary or they reduce the number of supercuspidal representations constructed.  On the other hand, using condition (1) alone yields a theory with no content.
When  is a proper subgroup of , it will be readily apparent how all three conditions are needed to construct supercuspidal representations.



 §.§ Root space decompositions and exponential maps


In this section, with F as usual,  can be taken as any connected reductive group that splits over E.  We also fix a point x in the reduced building ℬ_ red(,F).

Let ℳ consist of the following data:


  *  a maximal F-torus that splits over E and contains x in its apartment 𝒜_ red(,,E),

  * a -basis χ_1,… , χ_n for the character group X^*(),

  * a linear ordering of Φ = Φ (,).





The theory of mock exponential maps was developed by Adler <cit.> and our approach is adapted from his, together with theory from <cit.> and <cit.>.

For positive r in 
    = ∪{ s+  :  s∈}∪{∞},

the mock exponential map on t(E)_r,

    ℳ-exp: t (E)_r→(E)_r,
 is defined  by the condition

    χ_i (ℳ-exp (X)) = 1+dχ_i (X),
 for all X∈t(E)_r and  1≤ i≤ n.

Next, for each root a∈Φ, we have an exponential map  

    exp: g_a (E)_x,r→_a(E)_x,r.



Recall that _a (E)_x,r is the group (denoted X_α in  <cit.>) that Tits associates to the affine function α that has  gradient a and satisfies α (x) =r.  Thus _a(E)_x,r = exp(E_α (*)X_a), where * is the point in 𝒜_ red(,,E) that Tits associates to some Chevalley basis that contains the root vector X_a∈g_a.  Since  α (*)  = r-⟨ x-*,a⟩, we have 
    _a (E)_x,r  = exp (g_a (E)_x,r),

where 
    g_a (E)_x,r = E_r-⟨ x-*,a⟩ X_a.



Now fix a function

    f: Φ∪{ 0}→ (0,∞ )
 that is concave in the sense that

    f( ∑_i a_i) ≤∑_i f(a_i),
 where we are summing over a set of elements of Φ∪{0} whose sum is also in Φ∪{ 0}.

Then Bruhat-Tits consider the multiplication map

    μ : (E)_f(0)×∏_a∈Φ_a(E)_x,f(a)→ (E)_x,f,

where the product is taken according some fixed (but arbitrary) ordering of Φ, and (E)_x,f is the group generated by (E)_f(0) and the _a(E)_x,f(a)'s.
It is shown in <cit.> that μ is bijective.  (See also <cit.>.)  We will consider the map μ associated to the ordering of Φ coming from ℳ.


We refer the reader to <cit.> for more details on our application of <cit.>.  Yu explains that the choice of a valuation of the root datum, in the sense of <cit.>, is equivalent to the choice of a point in the reduced building.  In particular, it determines filtrations of the various root groups.  So it should be understood that when we apply the Bruhat-Tits result, we are choosing the valuation of root data corresponding to our given point x.


The corresponding Lie algebra map 

    ν : t(E)_f(0)×∏_a∈Φg_a(E)_x,f(a)→g (E)_x,f,

is obviously bijective and, in fact, is an E-linear isomorphism.

The mock exponential map

    ℳ-exp: g (E)_x,f→(E)_x,f

is defined as the composite

    g (E)_x,f[r]^ℳ-exp[d]_ν^-1   (E)_x,f
    t(E)_f(0)×∏_a∈Φg_a(E)_x,f(a)[r]   (E)_f(0)×∏_a∈Φ_a(E)_x,f(a)[u]_μ,

where the bottom map is obtained via the (mock) exponential maps discussed above on the factors.


Once ℳ and x are fixed, the various mock exponential maps we get for different f are all compatible in the sense that they are all restrictions of the map associated to the constant 0+.


The functions f of most interest to us are the functions

    f(a) = 
    
    r_0,   if a∈Φ (^0,)∪{0},
    
    r_i,   if a∈Φ (^i+1,)- Φ (^i,) and i>0,
 associated to tamely ramified twisted Levi sequences = (^0,… ,^d) and admissible sequences r⃗ = (r_0,…, r_d), as in <cit.>.  In this case, (E)_x,f is denoted  (E)_x,r⃗.  When r⃗ and s⃗ are admissible sequences such that

    0<s_i≤ r_i≤min (s_i,… , s_d)+min (s_0,…, s_d),
 for all i∈{ 0,… , d}, then the mock exponential determines an isomorphism

    exp : g⃗(E)_x,s⃗:r⃗→(E)_x,s⃗:r⃗
 of abelian groups that is independent of ℳ.  (See <cit.>.)  The latter isomorphism is Gal(E/F)-equivariant and yields an isomorphism
g⃗_x,s⃗:r⃗≅G⃗_x,s⃗:r⃗ on F-points.  (See <cit.>.)


The canonical reference for facts about groups associated to concave functions has historically been <cit.>.  More concise and modern treatments of this theory can be found in <cit.> and in the work of Yu.  Of particular importance to us are the results on commutators that are discussed in the next section.



Suppose χ is a character of G_x,0+ that is trivial on (E)^+∩ G_x,0+.  Then the depth of χ is identical to the depth of χ |T_0+.



Suppose r>0.  Then we have a commutative diagram

    g (E)_x,r:r+[r]^ℳ-exp[d]_ν^-1   (E)_x,r:r+
    t(E)_r:r+×∏_a∈Φg_a(E)_x,r:r+[r]   (E)_r:r+×∏_a∈Φ_a(E)_x,r:r+[u]_μ

of isomorphisms.
We are especially interested in μ, but its properties may be deduced from the properties of the other maps, which are studied in <cit.>.
It is easy to see that μ^-1 maps G_x,r:r+ to a set of the form T_r:r+× S, where S is a subgroup of ∏_a∈Φ_a(E)_x,r:r+ such that μ (S) is the image in G_x,r:r+ of a subset of (E)^+∩ G_x,r.

Suppose χ is a character of G_x,0+ that is trivial on G_x,r+.  Then χ |G_x,r determines a character of G_x,r:r+ that factors through μ^-1  to a character of T_r:r+× S that is the product of χ |T_r:r+ with the trivial character of S.  It follows that if χ has depth r then its restriction to T_0+ also has depth r.

Similarly, if χ is a character of G_x,0+ that is trivial on T_r+ and (E)^+∩ G_x,0+ then χ  determines a character of G_x,r:r+ and a corresponding character of T_r:r+× S that is trivial on S.
It follows that if χ|T_0+ has depth r then χ also has depth r.





 §.§ The norm map and the filtrations of the _𝒪's


Fix   an orbit 𝒪 of Γ =  Gal (F/F) in Φ = Φ (,),  and recall that we have let _𝒪 denote the F-torus generated by the E-tori
_a = image(ǎ) for a∈𝒪.


In this section, we study the Moy-Prasad filtration of the T_𝒪 using 
the norm map 
    N_E/F: _𝒪(E)   →    T_𝒪
    
    t   ↦   ∏_γ∈ Gal(E/F)γ (t).


























Our main result is:


If a∈𝒪 and r>0 then 
    T_𝒪,r =N_E/F(ǎ (E_r^×))= N_E/F(_a(E)_r)= N_E/F(_𝒪(E)_r)
 for all a∈𝒪.


 
Given a surjective homomorphism

    →
 of F-tori that split over E
the associated homomorphism

    A_r→ B_r

is surjective for all r>0, so long as the kernel of the original map is connected or finite with order not divisible by p.  (See Lemma 3.1.1 <cit.>.)

We can apply this to the norm map

    N_E/F:R_E/F(_𝒪)→_𝒪,
 since it is surjective and its kernel is a torus.  (See the remarks below, following the proof.)
We deduce that

    N_E/F(_𝒪(E)_r) = T_𝒪,r


Similarly, the surjection

    ǎ :R_E/F(𝔾_m)→_a
 yields surjections

    _a (E)_r =ǎ(E_r^× )

for all r>0.
(If the kernel is nontrivial then it must have order two, in which case we invoke our assumption that p 2.)

Since _𝒪(E)_r is generated by the groups _b (E)_r with b∈𝒪, it follows that 
    N_E/F(_a (E)_r) = N_E/F(_𝒪(E)_r).

Our assertions now follow.



The latter result asserts that, in a certain sense, the norm map “preserves depth,”  but it does not  say that N_E/F preserves the depths of individual elements.
This is similar to the fact that, since E/F is tamely ramified,  
_E/F(E_r) = F_r for all real numbers r, even though the trace clearly does not preserve depths of individual elements.



It is not true in general that _a(E) = ǎ (E^×) for all roots a that are defined over E.
Consider, for example, =  PGL_2 and the subgroup  consisting of cosets g, where g∈_2 and  is the center of _2.  Then ǎ(E^×) consists of cosets 

    [    u    0;    0 u^-1 ] = [ u^2   0;   0   1 ],
 with u∈ E^×, while _a(E) consists of cosets 
    [ u 0; 0 1 ],
 with u∈ E^×.



Let  be an F-torus.  If L is a field containing E then the norm map

    N_E/F:R_E/F() →
 over L is equivalent to the multiplication map

    ^n = ×⋯×→.
  Accordingly, the norm map must be surjective and its kernel must be a torus.  If  is E-split then the choice of an E-diagonalization of  yields a realization of  as a product of R_E/F (𝔾_m) factors. Thus the norm gives an explicit realization of every F-torus as a quotient of an induced torus.  Similarly, the inclusion of  in R_E/F() explicitly shows that every F-torus is contained in an induced torus.
(See Propositions 2.1 and 2.2 <cit.>.)










































 §.§ Yu's genericity conditions


This section is a recapitulation of facts from <cit.> that are, implicitly or explicitly, based on results from  <cit.>.

Fix i∈{ 0,… , d-1} and suppose X^*_i lies in the dual coset (ϕ |Z^i,i+1_r_i)^*∈z^i,i+1,*_-r_i: (-r_i)+.

For each root a∈Φ^i+1 = Φ (^i+1,), let H_a = dǎ(1)  be the associated coroot in the Lie algebra g^i+1.

In Lemma <ref>, we will show that X^*_i necessarily satisfies Yu's condition (<cit.>):

GE1.   v_F(X^*_i(H_a))=-r_i,  ∀ a∈Φ^i+1-Φ^i.

For the rest of this section, we assume GE1 is satisfied.

Now let F be the residue field of the algebraic closure F of F or, equivalently, the algebraic closure of the residue field f.

Choose ϖ_r_i∈F of depth r_i and note that ϖ_r_iX^*_i has depth 0.

Given χ∈ X^*(), we have an associated linear form dχ :t→𝔾_a, and the map χ↦ dχ extends to an isomorphism 
    X^*()⊗_F≅t^*⊗F.

Under the latter isomorphism, the element ϖ_r_iX^*_i corresponds to an element of X^*()⊗_O_F.

Let X^*_i be the residue class of ϖ_r_iX^*_i modulo
X^*()⊗_P_F.
Thus 
    X^*_i∈ X^*()⊗_F .

Though X^*_i is only well-defined up to multiplication by a scalar in F^× (depending on the choice of ϖ_r_i), it is easy to see that this scalar does not affect our discussion.  (In other words, one could treat X^*_i as a projective point.)

Let 
    Φ^i+1_X_i^*= { a∈Φ^i+1 : X_i(H_a)=0}.



Φ^i+1_X_i^*= Φ^i.



GE1 implies that X^*_i (H_a)  0 for all a∈Φ^i+1-Φ^i.  Since X^*_i∈z^i,* and H_a∈g_ der^i for a∈Φ^i, we must have X^*_i (H_a)=0 for all a∈Φ^i.  


We now observe that the Weyl group 

    W(Φ^i+1) = ⟨ r_a : a∈Φ^i+1⟩
 acts on
X^*()⊗_F and we let Z_W(Φ^i+1)(X^*_i) denote the stabilizer of X^*_i.

Yu's second genericity condition (<cit.>) is:

GE2.   Z_W(Φ^i+1)(X^*_i) =W (Φ^i) = W(Φ^i+1_X_i).


W(Φ^i) = W(Φ^i+1_X_i) is identical to the subgroup of
Z_W(Φ^i+1)(X^*_i) generated by the reflections in Z_W(Φ^i+1)(X^*_i).



If a∈Φ^i+1 then

    r_a(X^*_i ) = X^*_i - X^*_i (H_a)  a,

 and thus r_a ∈ Z_W(Φ^i+1)(X^*_i) precisely when a∈Φ^i+1_X^*_i = Φ^i.
Since every reflection in W(Φ^i+1) has the form r_a for some a∈Φ^i+1, our claim follows.




W(Φ^i) is a normal subgroup of Z_W(Φ^i+1)(X^*_i) and the quotient group
Z_W(Φ^i+1)(X^*_i)/W(Φ^i) is isomorphic to a subgroup of the torsion component of X^*()/Φ^i+1, where Φ^i+1 is the lattice generated by Φ^i+1. 



The desired result follows directly from Theorem 4.2(a)  in <cit.> upon  equating Steinberg's objects

    (H,A,L,^AL,Σ^*,W, Z_W(H))

with the objects 

    (X^*_i, F, X^*(), X^*()⊗_F,Φ^i+1, W(Φ^i+1),Z_W (Φ^i+1) (X^*_i)),

respectively.
Note that Z_W(H)^0 in <cit.> denotes the subgroup of Z_W(H) generated by the reflections in Z_W(H).

 (<cit.>)
If p is not a torsion prime for the dual of the root datum of ^i+1, that is, (X_*(),Φ̌^i+1, X^*(),Φ^i+1) then   GE1 implies GE2.  






 §.§ Genericity of the dual coset


This section involves the dual cosets defined in Definition <ref>.
The  result we prove is inspired by Lemma 3.7.5 <cit.>.


If  i∈{ 0,… , d-1} then every  element X^*_i in the dual coset (ϕ |Z^i,i+1_r_i)^* consists of ^i+1-generic elements of depth -r_i.


Given a∈Φ^i+1-Φ^i, let H_a= dǎ (1).
Let 𝒪 be the Γ-orbit of a.
With the obvious notations, we have the Lie algebra analogue
of Lemma <ref>:

    t_𝒪,r_i   =   _E/F(dǎ (E_r_i)) = _E/F(E_r_iH_a)
       =   _E/F(t_a(E)_r_i)= _E/F(t_𝒪(E)_r_i).

(The proof is analogous to that of Lemma <ref>.)

For u∈ E_r_i, we therefore have 
_E/F(uH_a) ∈t_𝒪,r_i⊂t^i_r_i⊂h^i_x, r_i and

    ψ(_E/F (u X^*_i(H_a)))
       =   ψ(X^*_i(_E/F (uH_a)))
       =   ϕ (exp (_E/F(uH_a))) 
       =   ϕ (N_E/F(exp (uH_a))) 
       =   ϕ (N_E/F(exp (dǎ(u)))) 
       =   ϕ (N_E/F(ǎ(1+u))) .

Here, exp is the Moy-Prasad isomorphism h^i_x, r_i:r_i+≅ H^i_x, r_i:r_i+, and we observe that exp restricts to the Moy-Prasad isomorphism t^i_r_i:r_i+≅ T^i_r_i:r_i+.

We now observe  that ϕ |T_𝒪,r_i 1 for all 𝒪∈ΓΦ^i+1 and, according to Lemma <ref>, we also know that if 𝒪 is the orbit of a then N_E/F(ǎ (E^×_r_i)) = T_𝒪,r_i.  Therefore, there exists u∈ E_r_i such that
ψ(_E/F (u X^*_i(H_a))) 1.  
This implies v_F (X^*_i (H_a))=-r_i, which shows that condition GE1 of <cit.> is satisfied.

Now we invoke Lemma 8.1 <cit.> or condition (4) in Definition <ref> to conclude that Yu's condition GE2 is also satisfied.







 §.§ Heisenberg p-groups and finite Weil representations


Fix i∈{ 0,… ,d-1} and let μ_p be the group of complex p-th roots of unity.
We have a surjective homomorphism 
    ζ_i : J^i+1_+ →μ_p
 which we have used to define a symplectic form on the space

    W_i= J^i+1/J^i+1_+

by 
    ⟨ uJ^i+1_+,vJ^i+1_+⟩ = ζ_i (uvu^-1v^-1).

 We have also defined 
a (multiplicative) Heisenberg p-group structure on 

    ℋ_i = W_i×μ_p

using the multiplication rule

    (w_1,z_1)(w_2,z_2) = (w_1w_2, z_1z_2 ⟨ w_1,w_2⟩^(p+1)/2).

The center of ℋ_i is  
    𝒵_i = 1×μ_p
  and we use z↦ (1,z) to identify μ_p with 𝒵_i.

We let H_x act on ℋ_i by

    h· (jJ_+^i+1,z) = (hjh^-1J_+^i+1,z).


We are interested in (central) group extensions of W_i by μ_p that are isomorphic to ℋ_i (as group extensions).  To say that an extension

    1→μ_p→W_i→ W_i→ 1

of W_i by μ_p is isomorphic to ℋ_i as a group extension means that there exists an isomorphism ι  :W_i→ℋ_i such that the diagram 

    1@=[d][r]   μ_p[r]@=[d]   W_i[r][d]^ι   W_i[r]@=[d]   1@=[d]
    
    1[r]   𝒵_i[r]   ℋ_i[r]   W_i[r]   1

commutes.  Such an isomorphism ι is called a special isomorphism.

(Note that, by the Five Lemma, if we merely assume ι is a homomorphism making the latter diagram commute then it is automatically an isomorphism.)


Recall that the cohomology classes in H^2(W_i,μ_p) correspond to group extensions 
W_i
of W_i by μ_p.
Given W_i, we get a 2-cocycle  
    c: W_i× W_i→μ_p
 by choosing a section s:W_i→W_i and defining c by

    s(w_1)s(w_2) = c(w_1,w_2)  s(w_1w_2).

To say that c is a 2-cocycle means that

    c(w_1,w_2) c(w_1w_2,w_3) = (s(w_1)c(w_2,w_3)s(w_1)^-1)  c(w_1,w_2w_3).

Indeed, both sides of the latter equation equal

    s(w_1)s(w_2)s(w_3)s(w_1w_2w_3)^-1.


Changing the section s has the effect of modifying c by a 2-coboundary.
Indeed, any other section s' must be related to s by 
    s' (w) = f(w)s(w)
 where f is a function f:W_i→μ_p, and then c' = c· df, where

    (df)(w_1,w_2) =f(w_1) f(w_2)f(w_1w_2)^-1.


On the other hand, given a 2-cocycle c we get an extension  W_i of W_i by μ_p by taking W_i = W_i×μ_p with multiplication defined by 

    (w_1,z_1)(w_2,z_2) = (w_1w_2, c (w_1,w_2)   z_1z_2)
 for some coycle
c : W_i× W_i→μ_p.  
Associativity of multiplication is equivalent to the cocycle condition.

The cocycle

    c(w_1,w_2) = ⟨ w_1,w_2⟩^(p+1)/2

yields the Heisenberg p-group ℋ_i and we are only interested in cocycles that are cohomologous to this cocycle.
In fact, we are only interested in special isomorphisms that come from  homomorphisms ν_i :J^i+1→ℋ_i.


A special homomorphism is an H_x-equivariant homomorphism ν_i : J^i+1→ℋ_i  that factors to a special isomorphism J^i+1/ζ_i →ℋ_i.


We observe that when ν_i is a  special homomorphism then
ν_i = ζ_i and
the diagram 

    1@=[d][r]   J^i+1_+/ζ_i[r][d]^ζ_i   J^i+1/ζ_i[r][d]^ν_i   W_i[r]@=[d]   1@=[d]
    
    1[r]   𝒵_i[r]   ℋ_i[r]   W_i[r]   1

commutes.


Our notion of special homomorphism is a variant of the notion of “relevant special isomorphism” in <cit.> which, in turn, is derived from Yu's definition of “special isomorphism”  <cit.>.  The existence of special homomorphisms follows from the existence of relevant special isomorphisms.  (See <cit.>.)



Both ℋ_i and J^i+1/ζ_i are Heisenberg p-groups that are canonically associated to ρ.   Special homomorphisms yield isomorphisms between these two Heisenberg p-groups, but the existence of automorphisms of the Heisenberg p-groups leads to the lack of uniqueness of special isomorphisms.  One can choose canonical special homomorphisms, however, there is no one canonical choice that is most convenient for all applications.  (These matters are discussed in <cit.>.)


We have chosen a Heisenberg representation (τ_i ,V_i) of ℋ_i and we have pulled back τ_i to a representation (τ_ν_i,V_i) of J^i+1.  Thus τ_ν_i =τ_i∘ν_i.
We have extended τ_i to a representation τ̂_i of 𝒮_i⋉ℋ_i on V_i and we have let ω_i (h) = τ̂_i ( Int(h),1), for all h∈ H_x.

The next lemma follows routinely from the definitions, but we include a proof to make evident where the H_x-equivariance of ν_i is used.



    τ_ν_i(h^-1jh) = ω_i(h)^-1τ_ν_i(j) ω_i (h),
 whenever j∈ J^i+1 and h∈ H_x.


 Given j∈ J^i+1 and h∈ H_x, our assertion follows from the computation

    τ_ν_i(h^-1jh)
       =   τ̂_i (1,ν_i (h^-1jh))
       =   τ̂_i (1, Int(h)^-1ν_i(j))
       =   ω_i(h)^-1τ_ν_i(j) ω_i(h).


































 §.§ Verifying that ϕ̂ is well defined


Recall from <ref> the following definitions:

    L^i+1_+   =    G^i+1_ der∩ J^i+1_++,
    
    L_+   =    (ϕ)L^1_+⋯ L^d_+,
    
    K_+   =    H_x,0+L_+,
    ^i   =   ^i+1_ der∩.

We have also stated a provisional definition for the character ϕ̂ of K_+, namely, ϕ̂=  inf_H_x,0+^K_+-.3em (ϕ )  or, equivalently, ϕ̂(hℓ)= ϕ( h), when h∈ H_x,0+ and ℓ∈ L_+.  

The  purpose of this section is to verify  that the definition of ϕ̂ make sense.





  (1) L_+ is a group that is normalized by H_x,0+.

  (2) K_+ = G⃗_x,(0+,s_0+,…, s_d-1+)= H_x,0+J_+.

  (3) H ∩ L_+ = ϕ.

  (4) ϕ̂ is a well-defined character of K_+.

  (5) ϕ̂(h) =ζ_i (h), when h∈ G^i+1_ der∩ J^i+1_+ and i∈{ 0,… , d-1}.




Our definition of K_+ is different than the definition  in <cit.> and <cit.>, however, Lemma <ref> shows that the different definitions coincide.  The notation G⃗_x,(0+,s_0+,…, s_d-1+) is a special case of  the notation from <ref> for groups associated to admissible sequences.



By definition,  L_+ =(ϕ)L^1_+⋯ L^d_+ and since H_x,0+ normalizes each factor L^j_+, so does ϕ.
Moreover, L^j_1_+ normalizes L^j_2_+ whenever j_1≤ j_2.  (See Remark <ref>.)
It follows from induction on i that the sets 
    (ϕ)L^1_+⋯ L^i_+
 are groups, for i = 1,… ,d.
This yields (1).

Now suppose i∈{ 0,… , d-1} and let 
    ^i,i+1(E)_x,s_i+=∏_a∈Φ^i+1-Φ^i U_a (E)_x,s_i+,
 where U_a is the root group associated to the root a. In the latter definition, we assume we have fixed an ordering of the set Φ^i+1 -Φ^i and we use this ordering to determine the order of multiplication in the product.  
The resulting set depends on this ordering.





We observe that

    ^i,i+1(E)_x,s_i+⊆^i+1_ der(E)∩ (^i,^i+1)(E)_x,(r_i+,s_i+)⊆^i+1(E)_x,s_i+

and, according to <cit.>,

    (E)_x,(0+,s_0+,…, s_d-1+)
    =  (E)_x,0+ (E)^1_x,s_0+⋯ (E)^d_x,s_d-1+,
 
The latter identity may be sharpened as

    (E)_x,(0+,s_0+,…, s_d-1+)
    =  (E)_x,0+ (E)^0,1_x,s_0+⋯ (E)^d-1,d_x,s_d-1+,
 where we are using <cit.>
and 
 <cit.>.
 
 More precisely, the Bruhat-Tits result can be used to show that one can rearrange the various products so that the contributions of the individual root groups occur in any  specified order.  In particular, one can first group together the roots from Φ^0, then the roots from Φ^1 - Φ^0, and so forth.  

 
 It follows that
 
    (E)_x,(0+,s_0+,…, s_d-1+)
    = (E)_x,0+∏_i=0^d-1(^i+1_ der(E)∩ (^i,^i+1)(E)_x,(r_i+,s_i+)) .


Using the Bruhat-Tits result or an argument as in the proof of <cit.>, we have:

    G⃗_x,(0+,s_0+,…, s_d-1+)   =    H_x,0+∏_i=0^d-1 L^i+1_+
       =    H_x,0+L_+ = K_+.

The identity 
    G⃗_x,(0+,s_0+,…, s_d-1+)= H_x,0+J_+
 is established in <cit.>.  Thus we have proven (2).

Assertion (3) follows from:

    H∩ L^i+1_+ = G^i+1_ der∩ H_x,r_i+ = H^i_x,r_i+⊂ H^i_x,r_i+1⊂ϕ .


We have defined ϕ̂ by ϕ̂(hℓ) = ϕ (h), for h∈ H_x,0+ and ℓ∈ L_+.  The fact  that this is a well-defined function on K_+ is a consequence of the fact that H_x,0+∩ L_+ =ϕ.
We now use (1) and the computation

    ϕ̂(h_1ℓ_1h_2ℓ_2)   =   ϕ̂(h_1h_2(h_2^-1ℓ_1 h_2)ℓ_2)
       =   ϕ(h_1h_2)
       =   ϕ̂(h_1ℓ_1)ϕ̂(h_2ℓ_2),

for h_1,h_2∈ H_x,0+ and ℓ_1,ℓ_2∈ L_+
to deduce that ϕ̂ is a character and prove (4).

Finally, we prove (5).  The methods used earlier in this proof can be used (with essentially no modification) to show that 

    G^i+1_ der∩ J^i+1_+ = H^i_x,r_i L^1_+⋯ L^i+1_+.

We will show that ϕ̂ and ζ_i agree on each of the factors on the right hand side of the latter identity.

On the factor H^i_x,r_i, the characters ϕ̂ and ζ_i coincide since both are represented by any element X^*_i in the dual coset (ϕ |Z^i,i+1_r_i)^* defined in <ref>.

Now consider a factor L^j+1_+, with j∈{ 0,… , i-1}.  By definition,  ϕ̂ is trivial on such a factor.  The fact that ζ_i is also trivial on L^j+1_+ follows from the fact that the dual coset (ϕ |Z^i,i+1_r_i)^* is contained in (𝔷^i,*)_-r_i which is orthogonal to 𝔤^i_ der, x,r_i.  Here, we are using the fact that L^j+1_+ ⊂ G^i_ der∩ G^i_x,r_i.  We also know that both ϕ̂ and ζ_i are trivial on the factor L^i+1_+.  This completes our proof.
























































































































 §.§ Intertwining theory for Heisenberg p-groups


The following section was prompted by an email message from Loren Spice and it was developed in the course of conversations with him.  It addresses an error in the proofs of Proposition 14.1 and Theorem 14.2 of <cit.> that can be fixed using ingredients already present in <cit.>.  

Roughly speaking, our point of view on intertwining is that two representations of overlapping groups intertwine when they can be glued together to form a larger representation of a larger group.  This approach reveals more of the inherent geometric structure present in Yu's theory.

As usual,   will be a connected reductive F-group, but now ' will be an F-subgroup  of  that is a Levi subgroup over E.  In other words, (',) is a twisted Levi sequence in .  Let x be a point in ℬ_ red(',F).

As in <cit.>, we put J= (G',G)_x,(r,s), J_+= (G',G)_x,(r,s+)
and we let ϕ be a -generic character of G'_x,r of depth r.  We let ζ denote the  character   of J_+ associated to ϕ.  (In other words, ζ is the character denoted by ϕ̂|J_+ in <cit.>.)

We remark that the quotient W= J/J_+ is canonically isomorphic to the corresponding Lie algebra quotient and thus our discussion could be carried out on the Lie algebra.

For subquotients S of G, let ^g S denote the subquotient obtained by conjugating by g.
For a representation π of a subquotient S of G, let ^gπ be the representation of ^g S given by ^g π (s) = π (g^-1sg).

Let Φ = Φ (,), Φ' = Φ (',) and 
let Φ_0 be the set of roots a∈Φ such that a(gx-x)=0.

We define groups J_0 and J_0,+ by imitating the definitions of J and J_+ except that we only use roots that lie in Φ_0.

More precisely, let J_0 (E) be the group generated by (E)_r and the groups _a(E)_x,r, with a∈Φ'∩Φ_0, and the groups _a(E)_x,s, with a∈ (Φ -Φ')∩Φ_0.  Define J(E)_0,+ similarly, but replace s by s+.  Now let J_0 = J(E)_0∩ G and J_0,+= J(E)_0,+∩ G.

Let μ_p be the group of complex p-th roots of unity.  Note that all of our symplectic forms are multiplicative and take values in μ_p.  We let I_p denote the identity map on μ_p viewed as a (nontrivial) character of μ_p.

Define W_0 to be the space J_0/J_0,+, viewed as a (multiplicative) _p-vector space with μ_p-valued symplectic form given by 

    ⟨ uJ_0,+,vJ_0,+⟩ = ζ (uvu^-1v^-1 )= ^g ζ (uvu^-1v^-1 ).

(The latter equality follows from <cit.>.)

The embedding J_0↪ J yields an embedding W_0↪ W such that the symplectic form on W, given by

    ⟨ uJ_+,vJ_+⟩ = ζ (uvu^-1v^-1 ),

 restricts to the symplectic form on W_0.
 
Similarly, the embedding ^g J_0↪^g J yields an embedding W_0↪^gW such that the symplectic form on ^gW, given by

    ⟨ u ^gJ_+,v ^g J_+⟩ = ^gζ (uvu^-1v^-1 ),

 restricts to the symplectic form on W_0.
 
 We therefore have a fibered sum (a.k.a., pushout) diagram 

    W_0[r][d]   ^gW[d]
    
    W[r]   W^⋆ .

The fibered sum 
    W^⋆ = W ⊔_W_0^gW
 is
the quotient of W×^g W by the subgroup of elements (w_0,w_0^-1), where w_0 varies over W_0.

Let W_1 denote the image of ^gJ_+∩ J in W.  Then, according to <cit.>, the space W_1 is totally isotropic and its  orthogonal complement W_1^⊥ in W is identical to the image of ^gJ∩ J in W.  We have W_1^⊥ = W_1⊕ W_0.

As with W_0, Yu (canonically) defines a subgroup J_1 such that W_1 = J_1J_+/J_+, except that he uses the roots a such that a(x-gx)<0.
He also defines a subgroup J_3 by using the roots a such that a(x-gx)>0.  Taking W_3 = J_3J_+/J_+, gives a vector space such that we have an orthogonal direct sum decomposition W = W_13⊕ W_0, where the nondegenerate space W_13 has polarization W_1⊕ W_3.

We also have canonical identifications 
    W= W_13⊕ W_0
  and 
    ^gW = W_0⊕^g W_13.
   (See Lemmas 12.8 and 13.6 <cit.>.)
Since the symplectic forms that W_0 inherits from W and ^gW coincide, the space W^⋆ inherits a nondegenerate symplectic structure such that 

    W^⋆ = W_13⊕ W_0⊕^gW_13
 is an orthogonal direct sum of nondegenerate symplectic spaces.





















Let 
    ℋ^⋆ = W^⋆×μ_p
 with the multiplication

    (w_1,z_1)(w_2,z_2) = (w_1w_2, z_1z_2⟨ w_1,w_2⟩^(p-1)/2).

Let (τ^⋆ , V_τ^⋆) be a Heisenberg representation of ℋ^⋆ whose central character is the identity map I_p on μ_p.

For concreteness, one can realize τ^⋆ as follows.
Fix a polarization 
    W_0 = W_2⊕ W_4
 of W_0 and define a polarization 
    W^⋆ = W^⋆ +⊕ W^⋆ -
 by taking

    W^⋆ + = W_1⊕ W_2 ⊕^gW_1
 and

    W^⋆ - = W_3⊕ W_4 ⊕^gW_3.

Now take 
    τ^⋆ =  Ind_W^⋆ +×μ_p^ℋ^⋆(1× I_p ).

Restriction of functions from ℋ^⋆ to W^⋆ - = W^⋆ -× 0 identifies the space V_τ^⋆ of τ^⋆ with the space [W^⋆ -] of complex-valued functions on W^⋆ -.

Let 𝒮^⋆ =  Sp(W^⋆) be the symplectic group of W^⋆ and let 
    τ̂^⋆ :𝒮^⋆⋉ℋ^⋆→ (V_τ^⋆)
  be the unique extension of τ^⋆.  We call this the “Weil-Heisenberg representation” of 𝒮^⋆⋉ℋ^⋆.  Its restriction

    ω^⋆ :𝒮^⋆→ (V_τ^⋆)

to 𝒮^⋆ is called the “Weil representation of 𝒮^⋆.”











Let ℋ = W×μ_p be the Heisenberg group associated to W, and embed ℋ in ℋ^⋆ in the obvious way.

In our above model for the Heisenberg representation τ^⋆, the space 
    V_τ = V_τ^⋆^^gW_1
 of ^g W_1-fixed vectors corresponds to the space [W^- ] of functions in [W^⋆ -] that are supported in the totally isotropic space 
    W^-  = W_3⊕ W_4.

It is an irreducible summand in the decomposition of τ^⋆ |ℋ, and restriction of functions from ℋ^⋆ to ℋ identifies V_τ with the space of the Heisenberg representation

    τ =  Ind_W^+×μ_p^ℋ(1× I_p),
 where 
    W^+ = W_1⊕ W_2.


Similarly, we take 

    V_^g τ = V_τ^⋆^W_1
 and observe that this corresponds to the space [^gW^-].  Note that

    ^g W^- = W_4⊕^g W_3
 since ^g W_4 and W_4 are identified in W^⋆.
Restriction of functions from ℋ^⋆ to ^gℋ identifies V_^gτ with the space of the Heisenberg representation

    ^gτ =  Ind_^gW^+×μ_p^^gℋ(1× I_p),
 where 
    ^g W^+ = ^g W_1⊕ W_2.






Let ℋ_0 be the subgroup 
    ℋ_0 =ℋ∩^gℋ= W_0×μ_p
 of ℋ^⋆.  This is the Heisenberg group associated to W_0.
One may view ℋ^⋆ as a fibered sum 
    ℋ^⋆ = ℋ⊔_ℋ_0^gℋ.


Let 
    V_τ_0 = V_τ∩ V_^gτ= V_τ^⋆^W_1×^gW_1.

This is a common irreducible summand in the decompositions of τ^⋆ |ℋ_0, τ |ℋ_0 and ^gτ|ℋ_0 into irreducibles.
Restriction of functions from ℋ^⋆ to ℋ_0 identifies V_τ_0 with the space of 

    τ_0 =  Ind_W_2×μ_p^ℋ_0(1× I_p).

Restricting functions to W_4 = W_4× 0 identifies V_τ_0 with [W_4].

At this point, we have embedded the representations τ and ^gτ within the representation τ^⋆ and shown that they are identical on the overlap of their domains ℋ_0 =ℋ∩^gℋ.
The explicitly exhibits that τ and ^gτ intertwine or, in other words, that the space
Hom_ℋ_0(^gτ ,τ) is nonzero.  In fact, Yu shows that the latter space has dimension one, and hence τ_0 occurs uniquely in τ and in ^gτ.

We turn now to the intertwining of the Heisenberg-Weil and Weil representations.

The symplectic group 𝒮 =  Sp(W) embeds in 𝒮^⋆ in an obvious way.  The Weil-Heisenberg representation τ̂^⋆ restricts to the Weil-Heisenberg representation 
    τ̂:𝒮⋉ℋ→( V_τ)
 that extends τ.  Let ω denote the resulting Weil representation of 𝒮.

Similar remarks apply with the roles of W and ^gW interchanged and we use the natural notations in this context.

The symplectic group 𝒮_0 =  Sp(W_0) has a natural embedding in 𝒮^⋆ as  a proper subgroup of 𝒮∩^g𝒮.
We obtain the Weil-Heisenberg representation 
    τ̂_0 :𝒮_0 ⋉ℋ_0→(V_τ_0)
 by restricting any of the Weil-Heisenberg representations τ̂^⋆, τ̂ or ^gτ̂.
Let ω_0 be the associated Weil representation of 𝒮_0.

At this point, it is easy to deduce that ω and ^gω intertwine, however, we caution that it is not enough to simply observe 
 that ω and ^gω both restrict to ω_0 on V_τ_0, since 𝒮_0 does not coincide with 𝒮∩^g𝒮.
 
 Instead, we simply observe that
the group 𝒮∩^g𝒮 stabilizes V_τ_0 and thus
 V_τ_0 may be viewed as a common (𝒮∩^g𝒮)⋉ℋ_0-submodule of τ̂ and ^gτ̂.
 We have now shown:
 
 
 The intertwining space Hom_ℋ_0(^gτ,τ) is 1-dimensional and identical to
 Hom_(^g 𝒮∩𝒮)⋉ℋ_0(^g τ̂,τ̂).  One can associate to each complex number c an intertwining operator ℐ_c: V_^gτ→ V_τ  by taking
 

  * ℐ_c(v) = cv for all  v∈ V_τ_0,

  * ℐ_c ≡ 0 on all  irreducible ℋ_0-modules other  than V_τ_0 occurring  V_^gτ.

 



Proposition 14.1 and Theorem 14.2 of <cit.> are valid.



Our claim follows from Lemma <ref> upon pulling back our representations via a special homomorphism and observing that the image of ^gK∩ K in 𝒮^⋆ lies in ^g𝒮∩𝒮.



Note that there is a  representation of 𝒮∩^g𝒮 on V_τ_0 defined by using the natural projection 𝒮∩^g𝒮→𝒮_0 and then composing with ω_0.
Let ω_0^♯ denote this representation.
Let ω̂_0 be the representation of 𝒮∩^g𝒮 on V_τ_0 given by restricting ω^⋆ (or ω or ^gω).  There must be a character χ of 𝒮∩^g𝒮 such that ω̂_0 = ω_0^♯⊗χ, but this character χ need not be trivial.  In the proof of Lemma 14.6 <cit.>, it is incorrectly stated that “N_0 is contained in the commutator subgroup of P_0.”  This leads to problems in the statements of Lemma 14.6 and Proposition 14.7 in <cit.>.  These problems involve the fact that χ need not be trivial, however, as we have indicated, the nontriviality of χ is ultimately is irrelevant for the required intertwining results.





 §.§ The construction of κ


Fix a permissible representation ρ of H_x.  Theorem <ref>  states, in part,  that, 
up to isomorphism, there is a unique representation κ = κ (ρ) of K such that:


  (1)  The character of κ has support in H_xK_+.

  (2)  κ |K_+ is a multiple of ϕ̂.

  (3)  κ | H_x = ρ⊗ω_0⊗⋯⊗ω_d-1.


In this section, we construct such a representation κ.  Since Conditions (1) – (3) completely determine the character of κ, once we have proven the necessary existence result, the uniqueness, up to isomorphism, will follow.


Since K = H_xJ^1⋯ J^d, and since κ is determined on H_x by Condition (3), it suffices to define κ |J^i+1, for all i∈{ 0,… , d-1}.

Now suppose i∈{ 0,… , d-1}.    Fix a  special homomorphism  ν_i :J^i+1→ℋ_i, in the sense of Definition <ref>, and use it to pull back τ_i to a representation τ_ν_i of J^i+1.









 


















Let

    J^i+1_♭   =    G^i+1_ der∩ J^i+1
    
    J^i+1_♭ ,+   =    G^i+1_ der∩ J^i+1_+.

Since [J^i+1,J^i+1]⊂ J^i+1_♭, the quotient J^i+1/J^i+1_♭ is a compact abelian group with subgroup J^i+1_+/J^i+1_♭ ,+.

Choose a character χ_i of J^i+1 that occurs as an irreducible component of the induced representation

    Ind_J^i+1_+/J^i+1_♭,+^J^i+1/J^i+1_♭(  ζ_i^-1(ϕ̂|J^i+1_+)).

Here, we are using Lemma <ref>(5).

Note that Frobenius Reciprocity implies

    ϕ̂|J^i+1_+ =  (χ_i |J^i+1_+)·ζ_i.


Define κ on J^i+1 by

    κ (j) = χ_i(j) (1_ρ⊗ 1_0⊗⋯⊗ 1_i-1⊗τ_ν_i (j)⊗ 1_i+1⊗⋯⊗ 1_d-1),

where 1_ℓ denotes the identity map on V_ℓ.



The definitions of κ |H_x, κ |J^1, …, κ |J^d are compatible and define a representation κ : K→ (V) such that κ |K_+ is a multiple of ϕ̂.



There are three things to prove:


  (i) The given definitions of κ on  H_x, J^1, …, J^d are compatible and hence yield a well-defined mapping κ : K→ (V).

  (ii) The  mapping κ is a homomorphism.

  (iii) κ |K_+ is a multiple of ϕ̂.


It is convenient to start by showing that the definitions of κ on  H_x, J^1, …, J^d are compatible with (iii).

First, we show that the restriction of κ |H_x to H_x∩ K_+ =H_x,0+ is a multiple of ϕ̂|H_x,0+ = ϕ.
By definition, 
    κ | H_x = ρ⊗ω_0⊗⋯⊗ω_d-1
 and ρ | H_x,0+ is a multiple of ϕ.
So it suffices to show that each factor ω_i when restricted to H_x,0+ is a multiple of the trivial representation.  
The latter statement follows directly from the fact that H_x,0+ acts trivially by conjugation on W_i = J^i+1/J^i+1_+.  

(In fact, Lemma 4.2 <cit.> implies that [H_x,0+,J^i+1]⊂ J^i+1_++⊂ J^i+1_+ which, in turn, implies that H_x,0+ acts trivially by conjugation on W_i = J^i+1/J^i+1_+.  See also Remark <ref> below for more general information about commutators.)

Now fix i∈{ 0,… , d-1}.  The restriction of κ |J^i+1 to J^i+1∩ K_+ =J^i+1_+ is a multiple of (χ_i |J^i+1_+)·ζ_i.   
But since ϕ̂|J^i+1_+ =  (χ_i |J^i+1_+)·ζ_i, we see that the restriction of κ |J^i+1 to J^i+1_+ is compatible with (iii).

Next, we observe that

    K = H_x,0+J^1_+⋯ J^d_+ = (H_x∩ K_+)(J^1∩ K_+)⋯ (J^d∩ K_+).
  Therefore, if (i) and (ii) hold then (iii) must also hold, according to the previoous discussion about compatibility with (iii) of the various restrictions of κ.

It remains to prove (i) and (ii).  We prove both things using a single induction.

Suppose we are given i∈{ 0,… , d-1} such that the definitions of κ on H_x,J^1,⋯ , J^i are compatible and yield a representation κ |K^i  of K^i = H_xJ^1⋯ J^i.  (When i=0, this amounts to the trivial assumption that κ |H_x is well defined.)

To show that we have a well-defined representation κ : K→ (V), it suffices to show that the definitions of κ|K^i and κ |J^i+1 are compatible, and that the resulting function κ |K^i+1 is a homomorphism K^i+1→ (V).

Compatibility is equivalent to the condition that κ |K^i and κ|J^i+1 agree on K^i∩ J^i+1 = G^i_x,r_i.
Suppose i≥ 1.  Then since G^i_x,r_i = J^i∩ J^i+1, compatibility reduces to the condition that κ |J^i and κ |J^i+1 agree on G^i_x,r_i.
The latter condition indeed holds since both κ |J^i and κ |J^i+1 restrict to a multiple of ϕ̂|G^i_x,r_i.  A  similar argument may be used when i=0.

It follows that we have a well-defined mapping 
    κ |K^i+1: K^i+1→(V).
 We now verify that this mapping is a homomorphism.

Suppose k,k'∈ K^i and j,j'∈ J^i+1.  It is easy to see that the condition

    κ (kj)κ (k'j') = κ (kjk'j')
 is equivalent to the condition

    κ (k'^-1jk') = κ (k')^-1κ (j)κ (k').

Now write 
    k' = hj_1⋯ j_i,
 with h∈ H_x, j_1∈ J^1, …, j_i∈ J^i and substitute for κ (k') the product
κ (h)κ(j_1)⋯κ (j_i).  Then we obtain the condition

    χ_i (k'^-1jk')  τ_ν_i(h^-1jh) = χ_i (j) τ_ν_i(j).

In light of Lemma <ref>, this reduces to

    χ_i (k'^-1jk')   = χ_i (j)
 or, equivalently,

    χ_i |[K^i,J^i+1]=1.

But, by construction, χ_i is trivial on J^i+1_♭ = G^i+1_ der∩ J^i+1.
Thus our claim  follows from the fact that [K^i,J^i+1]⊂ J^i+1_♭.





The character of κ has support in H_xK_+.
Consequently, up to isomorphism, the  representation κ is independent of the choices of the ν_i's and χ_i's.  



The second assertion follows from the first one, 
since conditions (2) and (3) in Theorem <ref> are independent of the choices of the ν_i's and χ_i's.

So  it suffices to prove that the character χ_κ of κ has support in H_xK_+.

By definition,

    κ (hj_1⋯ j_d) = (∏_i=0^d-1χ_i(j_i+1)) ρ(h)⊗( ⊗_i=0^d-1ω_i(h) τ_ν_i(j_i+1)),

where h∈ H_x, j_1∈ J^1,… , j_d∈ J^d.
Taking traces of the latter finite-dimensional operators gives

    χ_κ (hj_1⋯ j_d) = (∏_i=0^d-1χ_i(j_i+1)) ρ(h)⊗( ⊗_i=0^d-1(ω_i(h) τ_ν_i(j_i+1))).

The desired fact about the support of χ_κ now follows from Howe's computation of the support of the finite Heisenberg/Weil representations. (See <cit.> and <cit.>.)



The previous proof is similar to the proof of Proposition 4.24 in <cit.>.



Recall that κ |J^i+1 is the product of a (noncanonical) character χ_i of J^i+1 with the pullback τ_ν_i of a Heisenberg representation τ_i via a special homomorphism ν_i : J^i+1→ℋ_i.  The character χ_i is chosen in such a way that κ |J^i+1_+ is a multiple of ϕ̂|J^i+1_+.  
  A consequence of Lemma <ref> is that the equivalence classes of the representations κ |J^i+1 are invariants that are canonically associated to ρ.   A priori, knowing the equivalence classes of κ|H_x and the κ |J^i+1's is not enough to determine the equivalence class of κ.

 
 

 §.§ Irreducibility of π

  
 We begin by recapitulating a basic fact from <cit.>.
 
 
 Suppose 𝒢 is a totally disconnected group with center 𝒵, and suppose 𝒦 is an open subgroup of 𝒢 such that 𝒦 contains 𝒵 and the quotient 𝒦/𝒵 is compact.
 
 Assume (μ,V_μ) is an irreducible, smooth, complex representation of 𝒦.  When we write ind_𝒦^𝒢(μ), we are referring to the space of functions 
    f:𝒢→ V_μ
 such that:


  * f(kg) = μ (k)  f(g), for all k∈𝒦, g∈𝒢,

  * f has compact support modulo 𝒵 or, equivalently, the support of f has finite image in 𝒦𝒢,

  * f is fixed by right translations by some open subgroup 𝒦_f of 𝒢.
 
 We view ind_𝒦^𝒢(μ) as a 𝒢-module, where 𝒢 acts on functions by right translations.
 
 A well known and fundamental fact, due to Mackey, is that the representation ind_𝒦^𝒢(μ)
 is irreducible precisely when 
I_g(μ) 0 implies g∈𝒦, where

    I_g(μ) =  Hom_g 𝒦g^-1∩𝒦(^gμ,μ),

for g∈𝒢.
We use this  repeatedly in this section.

 
Now suppose κ is a representation of K as in the statement of Theorem <ref> and let π =  ind_K^G(κ).


The representation π is irreducible.




We may as well assume d>0, since there is nothing to prove if d=0.

Assume g_d∈ G and I_g_d(κ) 0.
It suffices  to show g_d∈ K.



We start the proof with a recursive procedure.

Assume that i∈{ 0,… , d-1} and  we are given g_i+1∈ G^i+1 such that I_g_i+1(κ ) 0.  We will show that there exists

    g_i∈ G^i ∩ J^i+1g_i+1J^i+1.

For such g_i, we also show that it is necessarily the case that I_g_i(κ) 0.

Since κ |K_+ is a multiple of ϕ̂, the assumption I_g_i+1(κ) 0 implies that  I_g_i+1(ϕ̂|K^i+1_+) 0
or, equivalently,

    ϕ̂|([g_i+1^-1,K^i+1_+]∩ K^i+1_+)=1.

Here, K^i+1_+ is the group

    K^i+1_+
       =   H_x,0+J^1_+⋯ J^i+1_+
       =   H_x,0+L^1_+⋯ L^i+1_+
       =   (G^0,…, G^i+1)_x,(0+,s_0+,… ,s_i+).


We have 

    ζ_i |([g_i+1^-1,J^i+1_+]∩ J^i+1_+)= ϕ̂|([g_i+1^-1,J^i+1_+]∩ J^i+1_+)=1,

since, according to Lemma <ref>(5), the characters ζ_i and ϕ̂ agree on G^i+1_ der∩ J^i+1_+.

According to  <cit.>, there exist elements g_i∈ G^i and j_i+1,j'_i+1∈ J^i+1 such that 
    g_i = j_i+1g_i+1j'_i+1.

Then 
    Λ↦κ (j_i+1)∘Λ∘κ (j'_i+1)
 determines a bijection

    I_g_i+1(κ)≅ I_g_i(κ).

Consequently, I_g_i(κ) 0.

Thus we may continue the recursion until we have produced a sequence g_d,… , g_0, where g_i∈ G^i and I_g_i(κ) 0.

In particular, we obtain g_0∈ H such that 
    g_d ∈ J^d⋯ J^1 g_0 J^1⋯ J^d⊂ Kg_0 K.
  So to show g_d lies in K, it suffices to show g_0 lies in H_x = K∩ H.

Hence, it suffices to show  that when h∈ H and I_h(κ ) 0  then it must be the case that h∈ H_x.

So suppose h∈ H and I_h (κ ) 0.  The intertwining space I_h (κ )
 consists of linear endomorphisms Λ of the space V of κ such that

    Λ (^h κ (k)v) =κ (k) Λ(v),  ∀ k∈ hK h^-1∩ K .


Given such Λ∈ I_h (κ ), there is an associated

    λ∈ Hom_hKh^-1∩ K (^h κ⊗κ̃,1),
 where hKh^-1∩ K is embedded diagonally in K × K and (κ̃, V) is the contragredient of (κ,V).  It is defined on elementary tensors in V ⊗V by

    λ (v⊗ṽ) = ⟨Λ (v),ṽ⟩.

The map Λ↦λ gives a linear isomorphism

    I_h (κ) ≅ Hom_hKh^-1∩ K (^hκ⊗κ̃,1).

The latter remarks apply very generally to intertwining, not just to κ, and we will use them for various representations.


We observe that for each i∈{ 0,… , d-1} the intertwining space  I_h (τ_ν_i) has dimension one, according to   <cit.>.  For each such i, fix a nonzero element Λ_i∈ I_h (τ_ν_i) and let 
λ_i be the corresponding element of Hom_hJ^i+1h^-1∩ J^i+1(^h τ_ν_i⊗τ̃_ν_i ,1).

Suppose

    v= v_ρ⊗ v_0⊗⋯⊗ v_d-1∈ V_ρ⊗ V_0⊗⋯⊗ V_d-1 = V

and

    ṽ=   ṽ_ρ⊗ṽ_0⊗⋯⊗ṽ_d-1∈V_ρ⊗V_0⊗⋯⊗V_d-1 = V.


Suppose i∈{ 0,… ,d-1}.
Fix all of the components of v and ṽ except for v_i and ṽ_i and define a linear form λ'_i :V_i⊗V_i→ by

    λ'_i (v_i⊗ṽ_i) = λ (v⊗ṽ).

Then for j∈ hJ^i+1h^-1∩ J^i+1 we have

    λ'_i (v_i⊗ṽ_i)
       =   λ  (κ (h^-1jh)v⊗κ̃(j)ṽ)
       =   χ_i (h^-1jhj^-1) λ'_i (τ_ν_i (h^-1jh)v_i⊗τ̃_ν_i (j)ṽ_i)
       =   λ'_i (τ_ν_i (h^-1jh)v_i⊗τ̃_ν_i (j)ṽ_i),

where the triviality of χ_i (h^-1jhj^-1) follows from the fact that χ_i is trivial on J^i+1_♭ = G^i+1_ der∩ J^i+1.

Therefore, λ'_i lies in 
    Hom_hJ^i+1h^-1∩ J^i+1(^h τ_ν_i⊗τ̃_ν_i ,1) = λ_i.




Using a “multiplicity one” argument  as in the proof of Lemma 5.24 <cit.>, one can show that 
for every linear form 
    λ∈ Hom_hKh^-1∩ K (^hκ⊗κ̃,1)
 there exists a unique linear form

    λ_ρ∈ Hom(V_ρ⊗V_ρ ,)
 such that

    λ (v⊗ṽ) = λ_ρ (v_ρ⊗ṽ_ρ)·∏_i=0^d-1λ_i (v_i⊗ṽ_i).


Roughly speaking, the argument goes as follows.  Fix v_d-1 and ṽ_d-1 so that λ_d-1(v_d-1⊗ṽ_d-1) is nonzero.
Define 

    ∂ v = v_ρ⊗ v_0⊗⋯⊗ v_d-2
 in 
    ∂ V = V_ρ⊗ V_0⊗⋯⊗ V_d-2

and

    ∂ṽ =ṽ_ρ⊗ṽ_0⊗⋯⊗ṽ_d-2

in 
    ∂V = V_ρ⊗V_0⊗⋯⊗V_d-2.
 
Then there is a linear form on ∂ V⊗∂V given on elementary tensors by 

    ∂λ 
    (∂ v⊗∂ṽ) = λ (v⊗ṽ)/λ_d-1(v_d-1⊗ṽ_d-1).


Repeating this procedure one coordinate at a time, one gets a sequence ∂^jλ of linear forms on 
∂^j V⊗∂^j V 
defined on elementary tensors by 

    ∂^jλ 
    (∂^j v⊗∂^jṽ) 
    = λ (v⊗ṽ)/∏_i=1^jλ_d-i(v_d-i⊗ṽ_d-i).

This ultimately yields the desired linear form λ_ρ.



































We claim that

    λ_ρ∈ Hom_hH_xh^-1∩ H_x(^hρ⊗ρ̃,1).
  Indeed, if k∈ hH_xh^-1∩ H_x then, according to  Corollary <ref>, we have

    λ  ( v ⊗ṽ)   =   λ  (κ  (h^-1kh)v ⊗κ̃(k)ṽ)
       =   λ_ρ (ρ(h^-1kh)v_ρ⊗ρ̃(k)ṽ_ρ)∏_i=0^d-1λ_i (ω_i (h^-1kh)v_i ⊗ω̃_i (k)ṽ_i)
       =   λ_ρ (ρ(h^-1kh)v_ρ⊗ρ̃(k)ṽ_ρ)∏_i=0^d-1λ_i ( v_i ⊗ṽ_i).




Since λ_ρ is necessarily nonzero, there must be a corresponding nonzero element Λ_ρ∈  I_h (ρ).    Since I_h(ρ) is nonzero and  ρ induces an irreducible representation of H, we deduce that h lies in H_x, which completes the proof.



In Proposition 4.6 <cit.>, it is shown that  cuspidal G-data that satisfy conditions SC1_i, SC2_i, and SC3_i of  <cit.> yield irreducible (hence supercuspidal) representations of G.  In Theorem 15.1 in <cit.>, it is shown that generic, cuspidal G-data satisfy the SC conditions.  Theorem 9.4, Theorem 11.5, and Theorem 14.2 of <cit.>, respectively, are the results that specifically show that generic data satisfy the conditions SC1_i, SC2_i, and SC3_i, respectively.  Our proof of Lemma <ref> uses  Theorem 9.4  <cit.> and Theorem 14.2 of <cit.> (with the revised proof in Corollary <ref> above).



As just indicated, our proof of Lemma <ref> is based partly on Yu's proof of  his Proposition 4.6.   But  in the latter part of his proof, Yu uses an inductive argument due to Bushnell-Kutzko <cit.>.  
By contrast, we have adapted the proof of Lemma 5.24 <cit.>.  
(The latter result was not intended to handle intertwining issues, but it is interesting  that it can be used for this purpose.)




§ THE CONNECTION WITH YU'S CONSTRUCTION


The theory of generic cuspidal G-data was introduced by Yu in <cit.>, but in discussing this theory we follow the notations and terminology of <cit.>.

Fix a generic cuspidal G-datum Ψ = (, y,ρ_0 ,ϕ⃗) and a torus  as in <cit.>.
Then Yu's construction associates to Ψ a representation κ (Ψ) of an open compact-mod-center subgroup K(Ψ) of G.   The representation κ (Ψ) induces an irreducible, supercuspidal representation π (Ψ) of G.   (See <cit.>.)

Let = ^d, = ^0,  and y come from the given datum Ψ, and let 

    ρ = ρ(Ψ)= ρ_0 ⊗∏_i=0^d (ϕ_i |H_x ),
 where x is the vertex in ℬ_ red (,F) associated to y.

Let r⃗ = (r_0,… , r_d) be the sequence of depths associated to Ψ (as in <cit.>).  Let 
    ϕ = ∏_i=0^d (ϕ_i|H_x,0+).

Then  ϕ and ϕ_i  agree on [G^i+1,G^i+1]∩ H_x,r_i-1+.

For each i∈{ 0,… , d-1}, we have a symplectic space W_i associated to Ψ as in <cit.>.
Associated to W_i, we can define a Heisenberg group structure on 
ℋ_i = W_i×μ_p, as in <ref>.

Define another Heisenberg group W_i^♯ by letting 
    W_i^♯ = W_i × (J^i+1_+/ζ_i)
 with multiplication
defined as in <cit.>.
Then (w,z)↦ (w,ζ_i(z)) defines an isomorphism W^♯_i≅ℋ_i of Heisenberg groups.

Fix a Heisenberg representation (τ_i ,V_i) of ℋ_i.  Let (τ_i^♯ , V_i) be the Heisenberg representation of W_i^♯ obtained via the given isomorphism W^♯_i≅ℋ_i.

Extending τ_i as in <ref>, we obtain a Weil representation 𝒮_i→(V_i).  This is the same as the Weil representation that comes from τ_i^♯ in <cit.>.

We have a map H_x→𝒮_i given by conjugation.  Pulling back the Weil representation via the latter map yields a representation 
    ω_i : H_x→(V_i).


In Yu's construction and ours, we must arbitrarily choose  for each i∈{ 0,… ,d-1} a Heisenberg representation within a prescribed isomorphism class.  
Implicit in the statement of the next result is the assumption that we choose the same family of Heisenberg representations when we construct κ (Ψ) and κ (ρ).
More precisely, we first choose a family of Heisenberg representations τ_i^♯ as in <cit.>, then, after it is established that (Ψ) = (ρ) and r⃗(Ψ) = r⃗(ρ), we use in the construction of κ (ρ) the Heisenberg representations τ_i associated to the τ_i^♯'s as above.



Suppose Ψ is a generic cuspidal G-datum and  ρ is the corresponding representation of H_x.  Then:


  * ρ is a permissible representation.

  * The given sequences = (^0,…, ^d) and r⃗ = (r_0,…, r_d) associated to Ψ are identical to the corresponding sequences associated to ρ.

  * The representations κ (ρ) and κ (Ψ) are  defined on the same group K acting on the same space V and the two representations are identical on the subgroups H_x, J^1_+, …, J^d_+.

  * The characters of both representations have support in H_xJ^1_+⋯ J^d_+.

Consequently, the characters of κ (Ψ) and κ (ρ) are identical and thus these representations of K (and the associated tame supercuspidal representations of G) are equivalent.



Fix Ψ = (, y,ρ_0 ,ϕ⃗) and  , as above.
We may as well replace y in the datum with the point x=[y], since only x is relevant to Yu's construction.  As noted in <cit.>, if ^♯ is z-extension of  as in <ref>, then Ψ pulls back to a generic, cuspidal G^♯-datum Ψ^♯ = (^♯, x,ρ_0^♯, ϕ⃗^♯) such that Yu's representation π (Ψ) of G pulls back to his representation π (Ψ^♯) of G^♯.
Therefore, we may as well assume that _ der is simply connected.

(Note that the complications regarding the definition of the r_i's in <ref> essentially disappear when considering Yu's construction, since Yu imposes more restrictive conditions on his objects.  For example, a given factor ϕ_i always has the same depth as its pullback ϕ_i^♯, since we have surjections G^i,♯_x,r→ G^i_x,r for all r≥ 0, according to Lemma 3.5.3 <cit.>.)



Let

    ρ = ρ(Ψ)= ρ_0 ⊗∏_i=0^d (ϕ_i |H_x ).

In our construction, we assume that there is a character ϕ of H_x,0+ such that ρ|H_x,0+ is a multiple of ϕ.  Such a character ϕ exists in the present situation and, according to <cit.>, it is given by 
    ϕ = ∏_j=0^d(ϕ_j|H_x,0+).



In general, there are various objects, such as the ^i's, associated to Ψ and, in our construction, there are similar objects associated to ϕ.  
We need to show that the objects , r⃗ and ϕ⃗ coming from Ψ can be recovered from
ϕ exactly as in our construction.

Assume i∈{ 1,… , d-1}.
Let ^i denote the center of ^i and let ^i,i+1 = (^i+1_ der∩^i)^∘.
Choose a G^i+1-generic element Z^*_i∈𝔷^i,*_-r_i of depth -r_i that represents ϕ_i |G^i_x,r_i.
Using the decomposition

    𝔷^i,*_-r_i = 𝔷^i+1,*_-r_i⊕z^i,i+1,*_-r_i,

we write Z^*_i = Y^*_i +X^*_i.
Then Y^*_i represents the trivial character of H^i_x,r_i since 𝔷^i+1,* is orthogonal to 𝔤^i+1_ der.  Therefore, Z^*_i and X^*_i represent the same character of H^i_x,r_i, namely, ϕ_i |H^i_x,r_i.

Since we assume _ der is simply connected, it follows from Lemma <ref> (with  replaced by ^i+1) that 

    [G^i+1,G^i+1] ∩ H_x,r_i = H^i_x,r_i

and, consequently, ϕ  |H^i_x,r_i=ϕ_i |H^i_x,r_i.  Therefore, X^*_i must lie in the dual coset (ϕ |Z^i,i+1)^*∈z^i,i+1,*_-r_i:(-r_i)+.





We observe now that, just as in our construction,
 r_i must be the depth of ϕ |H^i_x,0+, and ^i is the unique maximal subgroup of  containing  such that ^i is defined over F and is an E-Levi subgroup of , and ϕ |H^i-1_x,r_i=1.


Thus, we have established  that the sequences  and r⃗ associated to Ψ and ϕ coincide.  It is also follows form the remarks in <ref> that ρ is permissible.
The fact that  and r⃗ coincide for Ψ and ρ implies that the associated subgroups of G, such as K, are  the same for Ψ and ϕ.

It is routine to verify from the definitions that κ (Ψ)|H_x and κ (ρ)|H_x are the identical representation of the group H_x on the same space V.  
We refer the reader to <cit.> for the definition of κ (Ψ) and we note that it is easy to see that the representation ω_i defined just before the statement of the present lemma is given by ω_i (h) = τ̂_i^♯ (f'_i(h)), in the notation of in <cit.>.


We also observe that both κ (Ψ) and κ (ρ) act according to a multiple of the character ϕ̂ on each of the subgroups J^i+1_+.
In the case of κ (Ψ), this is <cit.>.

The fact that the support of the character of κ (Ψ) is contained in H_xJ^1_+⋯ J^d_+ is shown in the proof of <cit.>.  The corresponding fact for κ (ρ) is our Lemma <ref>.






































§ DISTINGUISHED CUSPIDAL REPRE­SEN­TA­TIONS FOR P-ADIC GROUPS AND  FINITE GROUPS OF LIE TYPE


The motivation for this paper was a desire to unify the p-adic and finite field theories of distinguished cuspidal representations.  We now state in a uniform way a theorem that applies both to  p-adic and finite fields.   The proof appears in a companion paper <cit.>.

In this section, we simultaneously address two cases that we refer to as “the p-adic case” and “the finite field case.”
In the p-adic case, F is, as usual, a finite extension of _p with p odd.  In the finite field case, F= 𝔽_q where q is a power of an odd prime p.  Let  be a connected, reductive F-group and let G =  (F).

In the p-adic case, we let ρ be a permissible representation of H_x.  In the finite field case, we let ρ be a character in general position of (𝔽_q), where  is an F-elliptic maximal F-torus.
For the sake of unity, we let L denote H_x in the p-adic case and (𝔽_q) in the finite field case.

Let π (ρ) be the irreducible supercuspidal or cuspidal Deligne-Lusztig representation of G associated to ρ.  Let ℐ be the set of F-automorphisms of  of order two, and let G act on ℐ by

    g·θ =  Int(g)∘θ∘ Int(g)^-1,
 where Int(g) is conjugation by g.  Fix a G-orbit Θ in ℐ.
Given θ∈𝒪, let G_θ be the stabilizer of θ in G.  Let G^θ be the group of fixed points of θ in G.  
When ϑ is an L-orbit in Θ, let 
    m_L (ϑ) = [G_θ :G^θ (G_θ∩ L)],
 for some, hence all, θ∈ϑ.

Let ⟨Θ,ρ⟩_G denote the dimension of the space Hom_G^θ(π (ρ),1)  of -linear forms on the space of π (ρ) that are invariant under the action of G^θ for some, hence all, θ∈Θ.

For each θ such that θ (L) = L, we define a character 
    ε__L,θ :L^θ→{± 1}
 as follows.

In the finite field case,

    ε__L,θ (h) = ( Ad(h)|𝔤^θ).

One can show that this is the same as the character ε
defined in <cit.>.

In the p-adic case,

    ε__L,θ (h) = ∏_i=0^d-1(_ f ( Ad(h)  |  W_i^+)/P_F)_2,

where our notations are as follows.

First, we take

    𝔚_i^+
    =((⊕_a∈Φ^i+1-Φ^ig_a)^ Gal(F/F))^θ_x,s_i:s_i+,
 viewed as a vector space over the residue field f of F.
In other words, 𝔚_i^+ is, roughly speaking,  the space of θ-fixed points in the Lie algebra of W_i= J^i+1/J^i+1_+. 

Next, for u∈f^×, we let (u/P_F)_2 denote the quadratic residue symbol.  This is related to the ordinary Legendre symbol   by

    (u/P_F)_2
    =(N_f/𝔽_p (u)/p) = (N_f/𝔽_p (u))^(p-1)/2= u^(q_F-1)/2.


We remark that in the p-adic case, ε__L,θ is the same as the character η'_θ defined in <cit.>.


In both the finite field and p-adic cases, one can, at least in most cases, compute the determinants arising in the definition of ε__L,θ and provide  elementary expressions for them in terms of Gal(F/F)-orbits of roots.

When ϑ is an L-orbit in Θ, we write ϑ∼ρ when θ (L) = L for some, hence all θ∈ϑ, and when 
the space Hom_L^θ (ρ ,ε__L,θ) is nonzero.
When ϑ∼ρ, we define

    ⟨ϑ ,ρ⟩_L =  Hom_L^θ (ρ ,ε__L,θ),
 where θ is any element of ϑ.  (The choice of θ does not matter.)

With these notations, we may have the following:

 ⟨Θ,ρ⟩_G = ∑_ϑ∼ρ m_L(ϑ) ⟨ϑ , ρ⟩_L.

Note that in  the special case in which
 

  *  is a product _1×_1,

  *  contains the involution θ (x,y) = (y,x), 

  * ρ has the form ρ_1×ρ̃_2,

where ρ̃_2 is the contragredient of ρ_2,
we have

    ⟨Θ,ρ⟩_G =  Hom_G(π (ρ_1),π(ρ_2)).

In the finite field case, our theorem in the latter situation is consistent with  the  Deligne-Lusztig inner product formula <cit.>.  (See <cit.>, for more details.)






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