# Representations of $GL_{n}(D)$ near the identity

Guy Henniart and Marie-France Vignéras

###### Abstract.

Let $p$ be a prime number, $F$ a finite extension of $\mathbb{Q}_{p}$ or of
$\mathbb{F}_{p}((t))$. We consider the group $G=GL_{n}(D)$ for a positive
integer $n$ and a central finite dimensional division $F$-algebra $D$. For an
irreducible smooth complex representation $\pi$ of $G$, inspired by work of R.
Howe when $D=F$, we establish the existence and uniqueness of integers
$c_{\pi}(\lambda)$, for partitions $\lambda$ of $n$, such that for any small
enough compact open subgroup $K$ of $G$ the restriction of $\pi$ to $K$ is the
same as that of the virtual representation $\sum
c_{\pi}(\lambda)\operatorname{Ind}_{P_{\lambda}}^{G}1$, where the sum is over
partitions $\lambda$ of $n$ and $P_{\lambda}$ is a parabolic subgroup of $G$
in the associate class determined by $\lambda$. When $P_{\lambda}$ is minimal
such that $c_{\pi}(\lambda)\neq 0$ we prove that $c_{\pi}(\lambda)$ is
positive, equal .to the dimension of a generalized Whittaker model of $\pi$.
We elucidate the behaviour of $c_{\pi}$ under the Jacquet-Langlands
correspondence $LJ$ of Badulescu from $GL_{dn}(F)$ to $G$. We extend the above
result on $\pi$ near identity to a representation of $G$ over a field $R$ with
characteristic not $p$. For any Moy-Prasad pro-$p$ subgroup $K$ of $G$, we
determine from the $c_{\pi}(\lambda)$ a polynomial $P_{\pi,K}$ with integral
coefficients and degree $d(\pi)$ independent on $K$, such that, for large
enough integers $j$ the dimension of fixed points in $\pi$ under the
congruence subgroup $K_{j}$ of $K$ is $P_{\pi,K}(q^{dj})$ where $q^{d}$ is the
cardinality of the residue field of $D$.

###### Contents

  1. 1 Introduction
  2. 2 Notations
  3. 3 Nilpotent orbits
    1. 3.1
    2. 3.2
  4. 4  Nilpotent orbital integrals
    1. 4.1
    2. 4.2
    3. 4.3
  5. 5 Trace of an admissible representation and parabolic induction
    1. 5.1
    2. 5.2
  6. 6 Complex representations of $G$ near the identity
    1. 6.1
    2. 6.2
  7. 7 Parabolic induction
  8. 8 Whittaker spaces
  9. 9 Jacquet-Langlands correspondence
    1. 9.1
    2. 9.2
    3. 9.3
  10. 10 Coefficient field of characteristic different from $p$
    1. 10.1
    2. 10.2
    3. 10.3
    4. 10.4
    5. 10.5
    6. 10.6
    7. 10.7
    8. 10.8
  11. 11 Invariant vectors by Moy-Prasad subgroups
  12. 12 $G=GL_{2}(D)$
    1. 12.1
    2. 12.2
    3. 12.3
    4. 12.4
    5. 12.5

##  1\. Introduction

Let $p$ be prime number, $F$ a finite extension of $\mathbb{Q}_{p}$ or of
$\mathbb{F}_{p}((T))$. Let $\underline{G}$ a reductive conneccted group over
$F$, and put $G=\underline{G}(F)$. Let $R$ be a field, and $\pi$ a smooth
admissible representation of $G$ on an $R$-vector space $V.$

Our first motivation was in the following question, when $\pi$ is of finite
length: Let $x$ be a point in the Bruhat-Tits building of $G$ and $r$ a
positive real number. For any integer $j\geq 0$, let $d(j)$ be the dimension
of the space of fixed points in $V$ under the Moy-Prasad subgroup $G_{x,r+j}$
of $G$.

######  Question 1.1.

Is there a polynomial $P$ with integer coefficients such that $d(j)=P(p^{j})$
for large enough $j$ ? If so, what can we say about its degree and its leading
coefficient ?

When the characteristic $\operatorname{char}_{R}$ of $R$ is $p$, precise
knowledge of those dimensions for irreducible $\pi$ is available only for
$G=GL(2,\mathbb{Q}_{p})$ (S. Morra, see §12.5). Apart for groups $G$ of
relative rank one those dimensions seem unknown.

Our paper studies the case where $\operatorname{char}_{R}\neq p$. Then a
smooth finite length $R$-representation $\pi$ of G is automatically
admissible, and its restriction to a pro-$p$ subgroup $K$ of G is semisimple,
with finite multiplicities. We write $[\pi]_{K}$ for the image of that
restriction in the Grothendieck group of admissible $R$-representations of
$K$. We ask a more ambitious question:

######  Question 1.2.

Is there an open pro-$p$ subgroup $K$ of G where we can control $[\pi]_{K}$ ?

In the case of $GL_{2}(F)$ an answer to that question was offered by Casselman
[Casselman73]. In this paper we consider the case where $G=GL_{n}(D)$ for a
central division algebra $D$ over $F,$ with finite degree $d^{2}$ over $F$.
For a partition $\lambda=(\lambda_{1},\dots,\lambda_{r})$ of $n$, we let
$P_{\lambda}$ be the upper block triangular subgroup of $G$ with blocks of
size $\lambda_{1},\dots,\lambda_{r}$ down the diagonal, and put
$d_{\lambda}=\sum_{i<j}\lambda_{i}\lambda_{j}$. We have $d_{\lambda}\geq
d_{\mu}$ if $\lambda\leq\mu$ for the classical partial order $\leq$ on
partitions. We let $\pi_{\lambda}$ be the representation of $G$ non-normalized
parabolically induced from the trivial representation of $P_{\lambda}$; it has
finite length.

Let $\pi$ be a finite length smooth representation of $G$ on an $R$-vector
space $V$.

######  Theorem 1.3.

There is a unique function $c_{\pi}$ from partitions of $n$ to $\mathbb{Z}$
and an open pro-p subgroup $K=K_{\pi}$ of $G$ such that
$[\pi]_{K}=\sum_{\lambda}c_{\pi}(\lambda)[\pi_{\lambda}]_{K}$.

If $\lambda$ is minimal in the support of $c_{\pi}$, then $c_{\pi}(\lambda)$
is positive.

Theorem 1.3 has consequences to our first question. We let $q$ be the
cardinality of the residue field of $F$, so the residue field of $D$ has
cardinality $q^{d}$. Let $x$ a point in the Bruhat-Tits building of $G$ and
$r$ a positive real number.

######  Theorem 1.4.

Let $P=P_{\pi,G_{x,r}}$ be the polynomial

(1.1) $P_{\pi,G_{x,r}}(X)=\sum_{\lambda}|P_{\lambda}\backslash
G/G_{x,r}|\,c_{\pi}(\lambda)\,X^{d_{\lambda}}.$

Then $\dim_{R}V^{G_{x,r+j}}=P(q^{dj})$ for large enough integers $j.$ The
degree of $P$ is $d(\pi)=\max(d_{\lambda})$ where the maximum is over
partitions $\lambda$ in the support of $\pi$. The leading coefficient is

(1.2) $a_{\pi,G_{x,r}}=\sum_{\lambda,d_{\lambda}=d(\pi)}|P_{\lambda}\backslash
G/G_{x,r}|\,c_{\pi}(\lambda).$

The function $c_{\pi}$ has good properties with respect to natural operations,
apart from being additive on exact sequences, hence factoring to a function on
the Grothendieck group of finite length smooth representations of $G$. If
$\chi$ is a character of $G$, $c_{\chi\pi}=c_{\pi}$. If $\pi^{\prime}$ the
base change of $\pi$ to an extension $R^{\prime}$ of $R$, then
$c_{\pi^{\prime}}=c_{\pi}$; in particular $c_{\pi}$ is invariant under
automorphisms of $R$. When $p\neq 2,G=GL_{n}(F)$ and
$\operatorname{char}_{F}=0$ the support of $\pi$ contains a single partition
$\lambda$ with $d_{\lambda}=d(\pi)$ [Moeglin-Waldspurger87]. This may be true
for any $p,F$ and $D$.

Parabolic induction Let $P$ be an upper block triangular subgroup of $G$, with
block diagonal Levi subgroup $M$ a product $GL_{n_{1}}(D)\times\ldots\times
GL_{n_{r}}(D)$. For $i=1,\ldots,r$, let $\sigma_{i}$ be a finite length
representation of $GL_{n_{i}}(D)$ and put
$\pi=\operatorname{ind}_{P}^{G}(\sigma_{1}\otimes\ldots\otimes\sigma_{r})$, a
finite length representation of $G$.

Given a partition $\lambda_{i}$ of $n_{i}$ for $i=1,\ldots,r$, we have the
induced partition $\lambda$ of $n$ obtained by gathering all the parts of the
$\lambda_{i}$’s and putting them in decreasing order.

######  Theorem 1.5.

For each partition $\lambda$ of $n$,
$c_{\pi}(\lambda)=\sum\prod_{i=1,\ldots,r}c_{\sigma_{i}}(\lambda_{i})$, where
the sum is over $r-$tuples of partitions $(\lambda_{1},\ldots,\lambda_{r})$
inducing to $\lambda$.

Whittaker models Assume that $R$ contains all the roots of unity of $p$-power
order. We have the notion of Whittaker models, possibly degenerate. Let $U$ be
the upper triangular subgroup of $G$, and $\theta$ a character of $U$. We let
$V_{\theta}$ be the maximal quotient of the space $V$ of $\pi$ on which $U$
acts via $\theta$. Its dimension is finite and depends on $\theta$ only up to
conjugation by the diagonal subgroup $T$ of G. The orbits of $T$ on the
characters of $U$ are parametrized by the compositions of $n$. To each
composition $\lambda$ of $n$ is attached a partition $\lambda^{\dagger}$
obtained by gathering the parts of $\lambda$ in decreasing order. The
Whittaker support of $\pi$ is the set of partitions of $n$ of the form
$\lambda^{\dagger}$ where $\lambda$ is a composition of $n$ such that
$V_{\theta}\neq 0$ for $\theta$ corresponding to the composition $\lambda$.

######  Theorem 1.6.

The minimal elements in the support of $c_{\pi}$ and in the Whittaker support
of $\pi$ are the same. If $\mu$ is such a minimal partition, $\lambda$ is a
composition of $n$ with $\lambda^{\dagger}=\mu$ and $\theta$ a character of
$U$ corresponding to $\lambda$, then $c_{\pi}(\mu)=\dim_{R}V_{\theta}$.

Jacquet-Langlands correspondence I.Badulescu has extended the classical
Jacquet-Langlands correspondence to a morphism $LJ_{\mathbb{C}}$ from the
Grothendieck group of smooth finite length complex representations of
$GL_{dn}(F)$ to that of $G$. For an algebraic closure $\mathbb{Q}_{\ell}^{ac}$
of $\mathbb{Q}_{\ell}$, with a chosen square root of $q$, A.Minguez and
V.Sécherre have transported $LJ_{\mathbb{C}}$ to
$\mathbb{Q}_{\ell}^{ac}$-representations, and showed that it descends to a map
$LJ_{\mathbb{F}_{\ell}^{ac}}$ of $\mathbb{F}_{\ell}^{ac}$-representations,
where $\mathbb{F}_{\ell}^{ac}$ is the residue field of
$\mathbb{Q}_{\ell}^{ac}$ and $\ell\neq p$. We define $LJ_{R}$ for our field
$R$, provided it be algebraically closed, and get:

######  Theorem 1.7.

Assume $R$ to be algebraically closed. Let $\tau$ be a finite length smooth
$R$-representation of $GL_{dn}(F)$ and $\pi=LJ_{R}(\tau)$. For any partition
$\lambda$ of $n$, we have
$(-1)^{n}c_{\pi}(\lambda)=(-1)^{dn}c_{\tau}(d\lambda)$.

For $R=\mathbb{C}$ and a discrete series $\pi$, the result is due to D. Prasad
[P00].

We show in §11 how to get Theorem 1.4 from Theorem 1.3; this amounts to
computing the dimensions of fixed points for the $\pi_{\lambda}$’s. Our method
establishes the other results first for $R=\mathbb{C}$, and then extends them
to $R$. Let us hasten to mention that when $R=\mathbb{C}$ part of the results
were known. Indeed when $D=F$ and $\operatorname{char}_{F}=0$, the first part
of Theorem 1.3 is due to [Howe74]. We actually adapt Howe’s arguments to our
setting. Similarly when $\operatorname{char}_{F}=0$ one can obtain Theorems
1.5, 1.6 (and the second part of Theorem 1.3) from the much more general
results of [Moeglin-Waldspurger87], and we get inspiration from their proofs.
111While we were writing our results, the preprint [Suzuki22] reached us. When
$R=\mathbb{C},D=F$ and $\operatorname{char}_{F}=0$, Suzuki establishes Theorem
1.4 for the congruence subgroups $K_{j}=1+M_{n}(P_{F}^{j})$ of $GL_{n}(F)$. He
also gets a result equivalent to Theorem 1.5 and Theorem 1.7 for square
integrable $\tau$. His methods are similar to ours.

We now give more detail on our method of proof. First we take $R=\mathbb{C}$.
In that case, knowing $[\pi]_{K}$ for an open compact subgroup $K$ of $G$ is
equivalent to knowing the character $\operatorname{trace}(\pi)$ on smooth
functions on $G$ supported in $K$. An expression of
$\operatorname{trace}(\pi)$ on small enough $K$ as a linear combination of
finitely many easier distributions is usually called a germ expansion for
$\pi$. When $\operatorname{char}_{F}=0$, the theory of germ expansions has a
long history. For a reductive group $G$ and $\pi$ irreducible Harish-Chandra
established a germ expansion of $\operatorname{trace}(\pi)$ as a linear
combination of Fourier transforms of nilpotent orbital integrals on the Lie
algebra $\mathfrak{g}$ of $G$, with coefficients a priori only complex numbers
[Harish-Chandra70]. To get from functions on $G$ to functions on
$\mathfrak{g}$, he used the exponential map, which is not available to us when
$\operatorname{char}_{F}>0$. The interest of our group $G=GL_{n}(D)$ is that
$\mathfrak{g}=M_{n}(D)$, so that nilpotent orbits of $G$ in $\mathfrak{g}$ are
parametrized by partitions of $n$, and that one can use the map $e:x\mapsto
1+x$ from $\mathfrak{g}$ to $G$ as a substitute for the exponential. When
$D=F$, Howe proved using $e$ that the Fourier transform of the nilpotent
orbital integral corresponding to a partition $\lambda$ is proportional to
$\operatorname{trace}(\pi_{\lambda})$, and got a germ expansion
$\operatorname{trace}(\pi)=\sum_{\lambda}c_{\pi}(\lambda)\operatorname{trace}(\pi_{\lambda})$
on the congruence subgroup $K_{i}$ for $i$ large enough. He showed that the
$c_{\pi}(\lambda)$ are integers by constructing for any $i>0$ a character
$\xi_{\lambda}$ of $K_{i}$ which appears with multiplicity $1$ in
$\pi_{\lambda}$ and multiplicity 0 in $\pi_{\mu}$ unless $\lambda\geq\mu$
[Howe74]. We show the existence of such characters for $D$ in Lemma 6.2. For
our group $G$ and $\pi$ irreducible, Lemaire proved the local integrability of
the distribution $\operatorname{trace}(\pi)$ (that was new when
$\operatorname{char}_{F}=p$) and adapted Howe’s arguments to get a germ
expansion as a linear combination of Fourier transforms of nilpotent integrals
[Lemaire04], which by our Proposition 5.5 translates into a germ expansion as
in Theorem 1.3. Our characters $\xi_{\lambda}$ then yields the integrality
statement and the positivity statement.

Theorem 1.5 follows from the known behaviour of traces with respect to
parabolic induction. In §7, we give a treatment valid whatever
$\operatorname{char}_{F}$ is.

As already said, when $\operatorname{char}_{F}=0$, Theorem 4 can be obtained
from results of C.Moeglin and J.-L.Waldspurger for a reductive group $G$ and
$\pi$ irreducible. They attach to a nilpotent orbit $\mathfrak{O}$ of $G$ in
$\mathfrak{g}$ a number of generalized Whittaker spaces of $\pi$. They
consider the Harish-Chandra germ expansion of $\pi$ as a linear combination
$\sum c_{\pi}(\mathfrak{O})D_{\mathfrak{O}}$ where $D_{\mathfrak{O}}$ is the
Fourier transform of the orbital integral along $\mathfrak{O}$. They show that
if $\mathfrak{O}$ is maximal in the support of $c_{\pi}$ then the dimension of
any Whittaker space attached to $\mathfrak{O}$ is $c_{\pi}(\mathfrak{O})$. The
nilpotent orbits with that maximality property go by the name of wave front
set of $\pi$ and there is a large literature on that subject. In our more
restricted setting, but allowing $\operatorname{char}_{F}=p$, we get Theorem
1.6 by adapting arguments of [Rodier74]222Rodier assumed
$\operatorname{char}_{F}=0,G$ split and the support of $c_{\pi}$ contains the
maximal nilpotent orbit and [Moeglin-Waldspurger87].

Still with $R=\mathbb{C}$, to prove Theorem 1.7 in §9 we use that the Jacquet-
Langlands correspondence LJ is expressed by character identities, where the
characters are considered as locally $L^{1}$ functions on regular semisimple
elements (by the result of B.Lemaire alluded to above).

In §10 we pass from $R=\mathbb{C}$ to the general case. To transfer the
results from a field $R$ to an isomorphic field $R^{\prime}$ we use that the
theory of smooth representations is essentially algebraic. That gives the case
of $\mathbb{Q}_{\ell}^{ac}$ which is isomorphic to $\mathbb{C}$. We then get
the case of $\mathbb{F}_{\ell}^{ac}$ by reduction, using the results of
[Minguez-Sécherre14]. To transfer the results from an algebraically closed
field $R$ to an algebraically closed extension $R^{\prime}$, we use the fact
that for a cuspidal $R^{\prime}$-representation $\pi$ of $G$, there is a
character $\chi$ of G into $R^{\prime*}$ such that $\chi\pi$ comes by base
change from an $R$-cuspidal representation of $G$. To get the result for any
$R$ we show that Theorem 1.3 over an algebraically closed extension $R^{ac}$
of $R$ implies Theorem 1.3 over $R$ essentially because base change preserves
finite length.

When $n=2$ and $D=F$, we compute in §12 the two coefficients
$c_{\pi}(\lambda)$ for all irreducible $\pi$. When $n=3$ or $4$,
$D=F,\operatorname{char}_{F}=0,R=\mathbb{C}$, F.Murnaghan computes the three
coefficients $c_{\pi}(\lambda)$ for cuspidal representations $\pi$ of G
induced from $F^{*}K_{0}$[Murnaghan91]. For any split reductive group $G$ over
$F$, R.Meyer and M.Solleveld using the Bruhat-Tits building of $G$, give an
upper bound on $\dim_{R}V^{C_{r}}$ for some special cases $C_{r},$ of Moy-
Prasad subgroups ([Meyer-Solleveld12]Theorem 8.5). Their result is far less
precise than ours.

Acknowlegments. We thank I. Badulescu, D. Bernardi, P. Broussous, B. Lemaire,
G. McNinch, A. Minguez, S. Morra, V. Sécherre for a number of conversations
about the topic of the paper. The second author has talked about this work at
numerous conferences in 2022 (Stockholm, Singapore, Grenoble, Heidelberg,
Bangalore), and thanks the organizers for their invitation. A part of this
work was done at the Institute for Mathematical Sciences of the National
University of Singapore. The final version of that paper was written when the
first-named author was enjoying the hospitability of the Graduate School of
Mathematical Sciences of the University of Tokyo.

##  2\. Notations

Let $p$ be a prime number, and $F$ a local non archimedean field of residual
characteristic $p$. We denote by $O_{F}$ the ring of integers of $F$, $P_{F}$
the maximal ideal of $O_{F}$, $p_{F}$ a generator of $P_{F}$,
$k_{F}=O_{F}/P_{F}$ the residue field of cardinal $q=p^{f}$ where
$f=[k_{F}:\mathbb{F}_{p}]$ is the residual degree, and $F^{ac}$ an algebraic
closure of $F$. Let $|\ |$ denote the absolute value of $F^{ac}$ such that for
$x\in F^{ac}$ non-zero, and $N_{E/F}$ the norm of a finite extension $E$ of
$F$ containing $x$, we have $|x|^{[E:F]}=|N_{E/F}(x)|=|O_{F}/N_{E/F}(x)O_{F}|$
([Cassels67] 10.Theorem). In particular $|p_{F}|=q^{-1}$.

Let $D$ be a central division algebra of finite dimension $d^{2}$ over $F$. We
denote by $O_{D}$ the maximal order of $D$, $P_{D}$ the maximal ideal of
$O_{D}$, $p_{D}$ a generator of $P_{D}$, $k_{D}=O_{D}/P_{D}$ the residue field
of cardinal $q^{d}$; we have $p_{F}O_{D}=P_{D}^{d}$ [Reiner75].

Let $n$ be a positive integer and $G=GL_{n}(D)$. Let $Z\simeq F^{*}$ denote
the center and $\mathfrak{g}=M_{n}(D)$ the Lie algebra of $G$. Let
$\operatorname{trd},\operatorname{nrd}:M_{n}(D)\to F$ be the reduced trace,
the reduced norm. The symmetric $G$-invariant bilinear form
$(X,Y)\mapsto\operatorname{trd}(XY):M_{n}(D)\times M_{n}(D)\to F$ is not
degenerate and $G=\\{Z\in M_{n}(D)\ |\ \operatorname{nrd}(Z)\neq 0\\}$.

If $P$ is a parabolic subgroup of $G$, its unipotent radical is usually
written $N$, and $M$ is used for a Levi subgroup so that $P=MN$. We write
$\mathfrak{p},\mathfrak{m},\mathfrak{n}$ for their Lie algebras.

A composition $\lambda=(\lambda_{i})$ of $n=\lambda_{1}+\ldots+\lambda_{r},\
\lambda_{i}\in\mathbb{N}_{>0},$ is called a partition of $n$ when the sequence
$(\lambda_{i})$ is decreasing. To a composition $\lambda$ of $n$ is associated
a parabolic subgroup $P_{\lambda}$ of $G=GL_{n}(D)$ with Levi subgroup
$M_{\lambda}$ block-diagonal with blocks of size
$\lambda_{1},\ldots,\lambda_{r}$ down the diagonal, and unipotent radical
$N_{\lambda}$ contained in the the upper triangular subgroup $B$. We let
$P_{\lambda}^{-}=M_{\lambda}N^{-}_{\lambda}$ the parabolic subgroup opposite
to $P_{\lambda}$ with respect to $M_{\lambda}$. We have $G=P_{(n)}$ and
$P_{(1,\ldots,1)}=B$. We denote by $T$ and $U$ the group $M_{(1,\ldots,1)}$ of
diagonal matrices with entries in $D^{*}$ and the strictly upper triangular
group $N_{(1,\ldots,1)}$. A parabolic subgroup $P$ of $G$ is conjugate to
$P_{\lambda}$ for a unique composition $\lambda$ of $n$ and is associated to
$P_{\lambda^{\dagger}}$ for the unique partition $\lambda^{\dagger}$ of $n$
deduced from $\lambda$ by re-ordering its elements. Let $\mathfrak{P}(n)$
denote the set of partitions of $n$. For
$\lambda=(\lambda_{1},\ldots,\lambda_{r})\in\mathfrak{P}(n)$,
$d\lambda=(d\lambda_{1},\ldots,d\lambda_{r})\in\mathfrak{P}(dn)$.

Let $R$ be a field. We denote by $\operatorname{char}_{R}$ the characteristic
of $R$, and by $C_{c}^{\infty}(G;R)$ or $C_{c}^{\infty}(\mathfrak{g};R)$ the
$R$-module of locally constant functions on $G$ or on $\mathfrak{g}$ with
compact support and values in $R$. The map

$\varphi\mapsto f(1+X)=\varphi(X):C_{c}^{\infty}(M_{n}(P_{D});R)\to
C_{c}^{\infty}(K_{1};R)\ \ \text{where }\ K_{1}=1+M_{n}(P_{D})$

is a $G$-equivariant isomorphism. The extension by $0$ embeds
$C_{c}^{\infty}(M_{n}(P_{D});R)$ in $C_{c}^{\infty}(\mathfrak{g};R)$ and
$C_{c}^{\infty}(K_{1};R)$ in $C_{c}^{\infty}(G;R)$. An $R$-distribution on $G$
or on $\mathfrak{g}$ is a linear form on $C_{c}^{\infty}(G;R)$ or
$C_{c}^{\infty}(\mathfrak{g};R)$. The group $G$ acts on $G$ and on
$\mathfrak{g}$ by conjugation, and on $C_{c}^{\infty}(G;R)$ and
$C_{c}^{\infty}(\mathfrak{g};R)$ and on the distributions. A $G$-invariant
distribution is called invariant.

For $R=\mathbb{C}$, $dg$ will denote the Haar measure on $G$ such that $dg$
gives the volume $1$ to the maximal open compact subgroup
$K_{0}=GL_{n}(O_{D})$ of $G$, et $dZ$ the Haar measure on $\mathfrak{g}$
giving the volume $[K_{0}:K_{1}]^{-1}=|GL_{n}(k_{D})|^{-1}$ to $M_{n}(P_{D})$,
hence the volume $a=q^{dn^{2}}|GL_{n}(k_{D})|^{-1}$ to $M_{n}(O_{D})$. The
Haar measures $dZ$ and $dg=a|\operatorname{nrd}Z|_{F}^{-n}dZ$ ([Weil67] X, §1
Lemma 1) are compatible with the map $x\mapsto 1+x:M_{n}(P_{D})\to
K_{1}=1+M_{n}(P_{D})$. The module of $P$ is
$\delta_{P}(p)=|\det(\operatorname{Ad}p)_{\mathfrak{n}}|_{F}$ ([Vigneras96]
I.2.8). Let $dk$ denote the restriction of $dg$ to $K_{0}$, $dp$ the left Haar
measure on $P$ such that $dg=\delta_{P}(p)dkdp$, $dn^{-}$ the Haar measure on
$N^{-}$ such that $dn^{-}\,dp$ is the restriction of $dg$ to $N^{-}P$ (open in
$G$), $dn$ the Haar measure on $N$ giving the same volume to $N\cap K_{0}$ as
the volume of $N^{-}\cap K_{0}$ for $dn^{-}$, and $dm$ the Haar measure on $M$
such that $dp=dm\,dn$. For each $f\in C_{c}^{\infty}(G;\mathbb{C})$,

$\int_{G}f(g)dg=\int_{K_{0}\times P}f(p^{-1}k)\,dk\,dp=\int_{K_{0}\times
P}f(kp)\,\delta_{P}(p)\,dk\,dp$ $=\int_{K_{0}\times M\times
N}f(kmn)\,\delta_{P}(m)\,dk\,dm\,dn.$

Let $dW,dY^{-},dY$ be the Haar measures on
$\mathfrak{h}=\mathfrak{p},\mathfrak{n}^{-},\mathfrak{n}$ such that $dp$ and
$dW$, $dn^{-}$ and $dY^{-}$, $dn$ and $dY$ are compatible with the map
$x\mapsto 1+x$ for $x\in\mathfrak{h}(P_{D})=\mathfrak{h}\cap M_{n}(P_{D})$. We
have $dZ=dWdY^{-}$.

Let $\pi$ be a smooth representation of $G$ on an $R$-vector space $V$. Each
vector is fixed by some open compact subgroup $K$ of $G$,

(2.1) $V=\cup_{K}V^{K}\ \ \ \text{where}\ V^{K}=\\{\text{vectors of }V\
\text{fixed by }K\\}.$

$\pi$ is called admissible when the dimension $\dim_{R}V^{K}$ of $V^{K}$ is
finite for any open compact subgroup $K$. The categories
$\operatorname{Rep}_{R}^{\infty}(G)$ of smooth $R$-representations of $G$,
$\operatorname{Rep}_{R}^{\infty,f}(G)$ of finite length smooth representations
are abelian. When $\operatorname{char}_{R}\neq p$, the category of admissible
$R$-representations of $G$ is abelian and contains
$\operatorname{Rep}_{R}^{\infty,f}(G)$ (this is not true when
$\operatorname{char}_{R}=p$). We denote by $\operatorname{Gr}_{R}^{\infty}(G)$
the Grothendieck group of $\operatorname{Rep}_{R}^{\infty,f}(G)$, and

$\pi\mapsto[\pi]:\operatorname{Rep}_{R}^{\infty}(G)\to\operatorname{Gr}_{R}^{\infty}(G)$

the natural homomorphism. The map $\chi\mapsto\chi\circ\operatorname{nrd}$ is
a bijection from the smooth characters $F^{*}\to R^{*}$ onto the smooth
characters $G\to R^{*}$.

For a set $X$ and a function $f$ on $X$ with value in $\mathbb{Z}$ or in $R$,
the support $\operatorname{Supp}f$ of $f$ is the set of $x\in X$ with
$f(x)\neq 0$ and $1_{X}$ will denote the characteristic function of $X$.

##  3\. Nilpotent orbits

### 3.1.

An element $X\in\mathfrak{g}$ is nilpotent if and only if $X^{r}=0$ for some
$r\in\mathbb{N}$. The set $\mathfrak{N}$ of nilpotent elements in
$\mathfrak{g}$ is stable by $G$-conjugation. A $G$-orbit in $\mathfrak{N}$ is
called a nilpotent orbit of $G$. The set $G\backslash\mathfrak{N}$ of
nilpotent orbits of $G$ is finite, in bijection with the set $\mathfrak{P}(n)$
of partitions of $n$ ([Bushnell-Henniart-Lemaire10] §2.4-2.6).

### 3.2.

Let $V$ be the right $D$-vector space $D^{n}$. The group $G$ identifies with
$\operatorname{Aut}_{D}(V)$ and its Lie algebra $\mathfrak{g}$ with
$\operatorname{End}_{D}V$. Let $X\in\operatorname{End}_{D}V$ be nilpotent. The
composition $\lambda=(\lambda_{1},\ldots)$ of $n$,

(3.1)
$\displaystyle\lambda_{i}=\dim_{D}\operatorname{Ker}X^{i}-\dim_{D}\operatorname{Ker}X^{i-1}\
\ \ {\text{f}or\ }i\geq 1,$

is a partition because the multiplication by $X$ induces an injection from
$\operatorname{Ker}X^{i}/\operatorname{Ker}X^{i+1}$ to
$\operatorname{Ker}X^{i-1}/\operatorname{Ker}X^{i}$. We get a canonical map
$\mathfrak{N}\to\mathfrak{P}(n)$ sending $0$ to $(n)$. The map is bijective.
Let $\mathfrak{O}_{\lambda}$ denote the nilpotent orbit of $G$ containing $X$.
The dual partition of $\lambda$ is $\hat{\lambda}=(\hat{\lambda}_{i}=|\\{j\ |\
\lambda_{j}\geq i\\}|)$. There is a partial order on $\mathfrak{P}(n)$

$\mu\leq\lambda\ \Leftrightarrow\ \hat{\lambda}\leq\hat{\mu}\ \Leftrightarrow\
\sum_{i=1}^{j}\mu_{i}\leq\sum_{i=1}^{j}\lambda_{i}\text{ for all $j$}.$

There is also a partial order on $G\backslash\mathfrak{N}$

$\mathfrak{O}^{\prime}\leq\mathfrak{O}\
\Leftrightarrow\mathfrak{O}\subset\overline{\mathfrak{O}}\text{ where
$\overline{\mathfrak{O}}$ is the closure of $\mathfrak{O}$, }$

The bijection reverses the partial order.

(3.2)
$\overline{\mathfrak{O}}_{\lambda}=\cup_{\hat{\mu}\leq\hat{\lambda}}\mathfrak{O}_{\mu}.$

The unique maximal partition $(n)$ corresponds the null orbit
$\\{0\\}=\mathfrak{O}_{(n)}$. The unique minimal partition $(1,\ldots,1)$
corresponds to the unique maximal nilpotent orbit
$\mathfrak{O}_{(1,\ldots,1)}$, called regular, of closure
$\overline{\mathfrak{O}}_{(1,\ldots,1)}=\cup_{\lambda\in\mathfrak{P}(n)}\mathfrak{O}_{\lambda}$.
The parabolic subgroup $P$ of $\operatorname{Aut}_{D}(V)$ preserving the flag
$(\operatorname{Ker}X^{i})_{i}$ of the iterated kernels of $X$, is associated
to $P_{\lambda}$. The intersection
$\mathfrak{O}_{\lambda}\cap\mathfrak{n}_{\lambda}$ is open dense in
$\mathfrak{n}_{\lambda}$. The dimension of $\mathfrak{O}_{\lambda}$ as an
$F$-variety is even and equal to

(3.3) $\displaystyle\dim_{F}\mathfrak{O}_{\lambda}$
$\displaystyle=2\dim_{F}\mathfrak{n}_{\lambda}=2d^{2}\dim_{D}\mathfrak{n}_{\lambda},$
(3.4) $\displaystyle\dim_{D}\mathfrak{n}_{\lambda}$
$\displaystyle=\sum_{i<j}\lambda_{i}\lambda_{j}.$

We denote $d_{\lambda}=\sum_{i<j}\lambda_{i}\lambda_{j}$ and
$d(\mathfrak{P}(n))=\\{d_{\lambda}\ |\ \lambda\in\mathfrak{P}(n)\\}$,

(3.5)
$d(\mathfrak{P}(n))=\\{d_{(n)}=0<d_{(n-1,1)}=n-1<\ldots<d_{(1,\ldots,1)}=n(n-1)/2\\}.$

The map $\lambda\mapsto d_{\lambda}:\mathfrak{P}(n)\to\mathbb{N}$ is injective
only when $n\leq 5$.

$d(\mathfrak{P}(2))=\\{0<1\\}$.

$d(\mathfrak{P}(3))=\\{0<2=d_{(2,1)}<3\\}$.

$d(\mathfrak{P}(4))=\\{0<3=d_{(3,1)}<4=d_{(2,2)}<5=d_{(2,1,1)}<6\\}$.

$d(\mathfrak{P}(5))=\\{0<4=d_{(4,1)}<6=d_{(3,2)}<7=d_{(3,1,1)}<8=d_{(2,2,1)}<9=d_{(2,1,1,1)}<10\\}$

$d(\mathfrak{P}(6))=\\{0<5=d_{(5,1)}<8=d_{(4,2)}<9=d_{(4,1,1)}=d_{(3,3)}<\ldots<15\\}$.

##  4\.  Nilpotent orbital integrals

The nilpotent orbital integral of the zero nilpotent orbit $\\{0\\}$ is the
value at $0$,

$\mu_{\\{0\\}}(\varphi)=\varphi(0)\ \ \ (\varphi\in
C_{c}^{\infty}(\mathfrak{g};\mathbb{C})).$

Let $\mathfrak{O}$ be a non-zero nilpotent orbit of $G$ and
$\lambda\in\mathfrak{P}(n)\setminus\\{(n)\\}$ such that
$\mathfrak{O}=\mathfrak{O}_{\lambda}$. The nilpotent orbital integral of
$\mathfrak{O}$ is a linear form sending $\varphi\in
C_{c}^{\infty}(\mathfrak{g};\mathbb{C})$ to

(4.1) $\displaystyle\mu_{\mathfrak{O}_{\lambda}}(\varphi)$
$\displaystyle=\int_{\mathfrak{n}_{\lambda}}\varphi_{K_{0}}(Y)\,dY$ (4.2)
$\displaystyle\varphi_{K_{0}}(Z)$
$\displaystyle=\int_{K_{0}}\varphi(kZk^{-1})dk\ \ \text{for}\
Z\in\mathfrak{g},$

$dY$ and $dk$ are the Haar measures on $\mathfrak{n}_{\lambda}$ and $K_{0}$
given in §2. For (4.1), see [Howe74] when $D=F$, [Lemaire04] for $D$ general).

### 4.1.

Homogeneity. For $t\in F^{*},\varphi\in C_{c}^{\infty}(\mathfrak{g})$, write
$\varphi_{t}(Z)=\varphi(t^{-1}Z)$ for $Z\in\mathfrak{g}$.

######  Proposition 4.1.

The nilpotent integral orbital of a nilpotent orbital integral $\mathfrak{O}$
of $G$ satisfies the homogeneity relation:

$\mu_{\mathfrak{O}}(\varphi_{t})=|t|_{F}^{d(\mathfrak{O})}\,\mu_{\mathfrak{O}}(\varphi),\
\ \ \dim_{F}(\mathfrak{O})=2d(\mathfrak{O}).$

###### Proof.

For $\lambda\in\mathfrak{P}(n)\setminus\\{(n)\\}$, we have
$d(\mathfrak{O}_{\lambda})=\dim_{F}\mathfrak{n}_{\lambda}$ by (3.4) and by
(4.1)

$\mu_{\mathfrak{O}_{\lambda}}(\varphi)=\int_{\mathfrak{n}_{\lambda}}\varphi_{K_{0}}(Y)\,dY=|t|_{F}^{\dim_{F}\mathfrak{n}_{\lambda}}\int_{\mathfrak{n}_{\lambda}}(\varphi_{K_{0}}(tY)\,dY=|t|_{F}^{\dim_{F}\mathfrak{n}_{\lambda}}\mu_{\mathfrak{O}_{\lambda}}(\varphi_{t^{-1}}).$

∎

For a nilpotent orbit $\mathfrak{O}$ of $G$ and a lattice $\mathfrak{L}$ in
$\mathfrak{g}$, we denote by $\mu_{\mathfrak{O},\mathfrak{L}}$ the restriction
of $\mu_{\mathfrak{O}}$ to
$C_{c}^{\infty}(\mathfrak{g}/\mathfrak{L};\mathbb{C})$ (identified to the
functions on $\mathfrak{g}$ invariant by translation by $\mathfrak{L}$). The
homogeneity implies ([Harish-Chandra78] Lemma 14 when the characteristic of
$F$ is $0$):

######  Corollary 4.2.

For any lattice $\mathfrak{L}$ in $\mathfrak{g}$, the linear forms
$\mu_{\mathfrak{O},\mathfrak{L}}$ of
$C_{c}^{\infty}(\mathfrak{g}/\mathfrak{L};\mathbb{C})$ for the nilpotent
orbits $\mathfrak{O}$ of $G$ are linearly independent.

###### Proof.

For any $d\in\mathbb{N}$, let $G\backslash\mathfrak{N}_{d}$ denote the set of
nilpotent orbits of dimension $\leq d$. Any nilpotent orbit $\mathfrak{O}$ of
dimension $d>1$ is open in $G\backslash\mathfrak{N}_{d}$ and $\mathfrak{O}\cup
G\backslash\mathfrak{N}_{d-1}$ is closed. We choose:

a) $\varphi_{\mathfrak{O}}\in
C_{\mathbb{C}}^{\infty}(\mathfrak{g};\mathbb{C})$ such that

$\mu_{\mathfrak{O}}(\varphi_{\mathfrak{O}^{\prime}})=\begin{cases}1\
\text{if}\ \mathfrak{O}=\mathfrak{O}^{\prime}\\\ 0\ \text{if}\
\mathfrak{O}\neq\mathfrak{O}^{\prime}\end{cases},$

b) a lattice $\mathfrak{L}_{0}$ in $\mathfrak{g}$ such that
$\varphi_{\mathfrak{O}}\in
C_{c}^{\infty}(\mathfrak{g}/\mathfrak{L}_{0};\mathbb{C})$ for each
$\mathfrak{O}\in G\backslash\mathfrak{N}$,

c) $t\in F^{*}$ such that $\mathfrak{L}\subset t\mathfrak{L}_{0}$.

Then, $(\varphi_{\mathfrak{O}})_{t}$ belongs to
$C_{c}^{\infty}(\mathfrak{g}/\mathfrak{L};\mathbb{C})$ and by homogeneity.

$\mu_{\mathfrak{O}}((\varphi_{\mathfrak{O}^{\prime}})_{t})=|t|^{d(\mathfrak{O})}\mu_{\mathfrak{O}}(\varphi_{\mathfrak{O}^{\prime}})=\begin{cases}|t|^{d(\mathfrak{O})}\
&\text{if}\ \mathfrak{O}=\mathfrak{O}^{\prime}\\\ 0\ &\text{if}\
\mathfrak{O}\neq\mathfrak{O}^{\prime}\end{cases}.$

∎

### 4.2.

Fourier transform. The bilinear map
$(Z,Y)\mapsto\operatorname{trd}(ZY):\mathfrak{g}\times\mathfrak{g}\to F$ is
non degenerate. Let $\psi:F\to\mathbb{C}^{*}$ be a non-trivial additive
character on $F$. The Fourier transform in
$C_{c}^{\infty}(\mathfrak{g};\mathbb{C})$ with respect to $\psi$ and the Haar
measure $dZ$ (fixed in §1) is the endomorphism of
$C_{c}^{\infty}(\mathfrak{g};\mathbb{C})$:

(4.3)
$\varphi\mapsto\hat{\varphi}(Y)=\int_{\mathfrak{g}}\varphi(Z)\,\psi(\operatorname{trd}(ZY))\,dZ\
\ \ (Y\in\mathfrak{g},\ \varphi\in C_{c}^{\infty}(\mathfrak{g};\mathbb{C}))$

There exists a positive real number $c_{\psi}>0$ such that
$\hat{\hat{\varphi}}(Z)=c_{\psi}\varphi(-Z)$ for $Z\in\mathfrak{g}$ 333 The
non-trivial additive characters $F\to\mathbb{C}^{*}$ are
$\psi^{a}(x)=\psi(ax),x\in F$ for $a\in F^{*}$. As
$d(aZ)=|a|_{F}^{d^{2}n^{2}}dZ$, we have
$c_{\psi^{a}}=|a|_{F}^{-n^{2}d^{2}}c_{\psi}$. In particular

(4.4) $\int_{\mathfrak{g}}\hat{\varphi}(Y)dY=c_{\psi}\varphi(0).$

For an $O_{F}$-lattice $\mathfrak{L}$ in $\mathfrak{g}$, the Fourier transform
of $1_{\mathfrak{L}}$ is
$\operatorname{vol}(\mathfrak{L},dZ)\,1_{\mathfrak{L}^{*}_{\psi}}$ where

$\mathfrak{L}^{*}_{\psi}=\\{Z\in M_{n}(D)\ |\
\psi(\operatorname{trd}(Z\mathfrak{L})))=1\\}=\\{Z\in M_{n}(D)\ |\
\operatorname{trd}(Z\mathfrak{L})))\subset\operatorname{Ker}(\psi)\\}.$

######  Example 4.3.

When $\psi$ is trivial on $P_{F}$ and not on $O_{F}$,
$M_{n}(O_{D})_{\psi}^{*}=M_{n}(P_{D})$ ([Weil67] X, §2, Proposition 5).

For an open subset $\mathfrak{C}$ of $\mathfrak{g}$, the extension by zero
embeds $C_{c}^{\infty}(\mathfrak{C};\mathbb{C})$ into
$C_{c}^{\infty}(\mathfrak{g};\mathbb{C})$.

######  Proposition 4.4.

Let $\mathfrak{C}$ be an open neighborhood of zero in $\mathfrak{g}$. The
linear forms

$\varphi\mapsto\mu_{\mathfrak{O}}(\hat{\varphi}):C_{c}^{\infty}(\mathfrak{C};\mathbb{C})\to\mathbb{C}$

for $\mathfrak{O}\in G\backslash\mathfrak{N}$, are linearly independent.

###### Proof.

This follows from the linear independence of the
$\mu_{\mathfrak{O},\mathfrak{L}}$ for any lattice $\mathfrak{L}$ (Corollary
4.2) ([Harish-Chandra78] corollary of Lemma 14). ∎

### 4.3.

Let $\mathfrak{O}$ be a nilpotent orbit of $G$ and $\psi$ a non-trivial smooth
character of $F$. We compute the nilpotent orbital integral
$\mu_{\mathfrak{O}}(\hat{\varphi})$ (4.1) of the Fourier transform
$\hat{\varphi}$ with respect to $\psi$ of $\varphi\in
C_{c}^{\infty}(\mathfrak{g};\mathbb{C})$. Let $\lambda$ be the partition of
$n$ such that $\mathfrak{O}=\mathfrak{O}_{\lambda}$. Write $(P,M,N)$ for
$(P_{\lambda},M_{\lambda},N_{\lambda})$. The bilinear map
$(Y,Y^{-})\mapsto\operatorname{trd}(YY^{-}):\mathfrak{n}\times\mathfrak{n}^{-}\to\mathbb{C}$
is non degenerate because $\operatorname{trd}(YW)=0$ for
$Y\in\mathfrak{n},W\in\mathfrak{p}$. The corresponding Fourier transform with
respect to $\psi$ is the linear map :

$\varphi_{2}\mapsto\hat{\varphi}_{2}(Y)=\int_{\mathfrak{n}^{-}}\varphi_{2}(Y^{-})\psi(\operatorname{trd}(YY^{-}))\,dY^{-}:C_{c}(\mathfrak{n}^{-};\mathbb{C})\to
C_{c}(\mathfrak{n};\mathbb{C}).$

There exists a positive real number $c_{\psi,\mathfrak{n}}$ such that

$\int_{\mathfrak{n}\times\mathfrak{n}^{-}}\varphi_{2}(Y^{-})\varphi_{2}(\operatorname{trd}(YY^{-}))\,dY^{-}\,dY=\int_{\mathfrak{n}}\hat{\varphi}_{2}(Y)dY=c_{\psi,\mathfrak{n}}\,\varphi_{2}(0).$

For $\varphi\in C_{c}^{\infty}(\mathfrak{g};\mathbb{C})$ of Fourier transform
$\hat{\varphi}$ with respect to $\psi$, put

(4.5)
${\bf\hat{\mu}_{\mathfrak{O}}(\varphi)=\mu_{\mathfrak{O}}(c_{\psi,\mathfrak{n}}^{-1}\,\hat{\varphi})}.$

######  Proposition 4.5.

We have
$\hat{\mu}_{\mathfrak{O}}(\varphi)=\int_{\mathfrak{p}}\varphi_{K_{0}}(W)\,dW$
.

This was proved only “for some Haar measures” when $D=F$ and the
characteristic of $F$ is $0$ [Howe74]. The proposition follows from the next
three lemmas where $\varphi\in C_{c}^{\infty}(\mathfrak{g};\mathbb{C})$.

######  Lemma 4.6.

$\int_{\mathfrak{n}\times\mathfrak{p}\times\mathfrak{n}^{-}}\varphi((Y^{-}+W))\psi(\operatorname{trd}(YY^{-}))\,dY^{-}\,dY\,dW=c_{\psi,\mathfrak{n}}\,\int_{\mathfrak{p}}\varphi(W)\
dW.$

###### Proof.

We have $C_{c}(\mathfrak{g};\mathbb{C})=C_{c}(\mathfrak{p};\mathbb{C})\otimes
C_{c}(\mathfrak{n}^{-};\mathbb{C})$. For $\varphi_{1}\in
C_{c}(\mathfrak{p};\mathbb{C}),\varphi_{2}\in
C_{c}(\mathfrak{n}^{-};\mathbb{C})$ and $\varphi\in
C_{c}(\mathfrak{g};\mathbb{C})$ such that
$\varphi(Y^{-}+W)=\varphi_{1}(W)\varphi_{2}(Y^{-})$ for
$Y^{-}\in\mathfrak{n},W\in\mathfrak{p}$, we have

$\int_{\mathfrak{n}\times\mathfrak{p}\times\mathfrak{n}^{-}}\varphi((Y^{-}+W))\psi(trd(YY^{-}))\,dY^{-}\,dY\,dW=c_{\psi,\mathfrak{n}}\,\int_{\mathfrak{p}}\varphi_{1}(W)\varphi_{2}(0)\,dW=c_{\psi,\mathfrak{n}}\,\int_{\mathfrak{p}}\varphi(W)\,dW.$

∎

######  Lemma 4.7.

The integration over $\mathfrak{n}$ of the Fourier transform is integration
over $\mathfrak{p}$:

(4.6)
$\displaystyle\int_{\mathfrak{n}}\hat{\varphi}(Y)\,dY=c_{\psi,\mathfrak{n}}\,\int_{\mathfrak{p}}\varphi(W)\,dW.$

###### Proof.

The left hand side of (4.6) is

$\displaystyle\int_{\mathfrak{n}\times\mathfrak{g}}\varphi(Z)\psi(\operatorname{trd}(YZ))\,dZ\,dY=\int_{\mathfrak{n}\times\mathfrak{p}\times\mathfrak{n}^{-}}\varphi(Y^{-}+W)\psi(trd(Y(Y^{-}+W))\,dY^{-}\,dW\,dY$
$\displaystyle=\int_{\mathfrak{n}\times\mathfrak{p}\times\mathfrak{n}^{-}}\varphi(Y^{-}+W)\psi(trd(YY^{-}))\,dY^{-}\,dW\,dY=c_{\psi,\mathfrak{n}}\,\int_{\mathfrak{p}}\varphi(W)\
dW$

because $trd(YW)=0$ for $Y\in\mathfrak{n},W\in\mathfrak{p}$, and by Lemma 4.6.
∎

######  Lemma 4.8.

The Fourier transform of $\varphi_{K_{0}}$ is $(\hat{\varphi})_{K_{0}}$ for
$\varphi\in C_{c}^{\infty}(\mathfrak{g};\mathbb{C})$.

###### Proof.

Write $K=K_{0}$. Then
$(\hat{\varphi})_{K}(Y)=\int_{K}\hat{\varphi}(kYk^{-1})\,dk$ for
$Y\in\mathfrak{g}$ is equal to

$\int_{K}\int_{\mathfrak{g}}\varphi(Z)\psi(\operatorname{trd}(kYk^{-1}Z))\,dZ\,dk=\int_{K\times\mathfrak{g}}\varphi(kZk^{-1})\psi(\operatorname{trd}(kYk^{-1}\,kZk^{-1}))\,dZ,\,dk$

because $dZ$ is $K$-invariant. This is

$\displaystyle\int_{K\times\mathfrak{g}}\varphi(kZk^{-1})\psi(\operatorname{trd}(kYZk^{-1}))\,dZ\,dk=\int_{K\times\mathfrak{g}}\varphi(kZk^{-1})\psi(\operatorname{trd}(YZ))\,dZ\,dk$
$\displaystyle=\int_{\mathfrak{g}}\varphi_{K}(Z)\psi(\operatorname{trd}(YZ))\,dZ.$

∎

##  5\. Trace of an admissible representation and parabolic induction

### 5.1.

Let $R$ be a field of characteristic $\operatorname{char}_{R}\neq p$ and $dg$
a Haar measure on $G$ with values in $R$. Let
$\pi\in\operatorname{Rep}_{R}^{\infty}(G)$ be an admissible representation of
$G$ on an $R$-vector space $V$. The linear endomorphism of $V$

(5.1) $\pi(f(g))=\int_{G}f(g)\pi(g)dg$

has a finite rank. Its trace is an invariant $R$-distribution on $G$

$\operatorname{trace}(\pi):f\mapsto\operatorname{trace}(\pi(f(g)),\ \ f\in
C_{c}^{\infty}(G;R),$

called the character of $\pi$.

The characters of the irreducible smooth complex representations of $G$ are
linearly independent ([Vigneras96] I.6.13 where $c=0$ should be $0$).

For any exact sequence $0\to\pi_{1}\to\pi\to\pi_{2}\to 0$ of admissible
$R$-representations of $G$,
$\operatorname{trace}(\pi)=\operatorname{trace}(\pi_{1})+\operatorname{trace}(\pi_{2})$.
By the universal property of Grothendieck groups, the character induces a
linear map from the Grothendieck group $\operatorname{Gr}_{R}^{\infty}(G)$ to
the space of invariant $R$-distributions on $G$.

For any open compact subgroup $K$ of $G$, the restriction to $K$ induces a
linear map

(5.2)
$\nu\mapsto\nu|_{K}:\operatorname{Gr}_{R}^{\infty}(G)\to\operatorname{Gr}_{R}^{\infty}(K)$

from the Grothendieck group $\operatorname{Gr}_{R}^{\infty}(G)$ of
$\operatorname{Rep}_{R}^{\infty,f}(G)$ to the Grothendieck group
$\operatorname{Gr}_{R}^{\infty}(K)$ of admissible smooth $R$-representations
of $K$ (any finite length smooth $R$-representation of $G$, is admissible).
When $K$ is a pro-$p$ group, the category $\operatorname{Rep}_{R}^{\infty}(K)$
is semi-simple.

### 5.2.

Parabolic induction. Let $R$ be a field and $P$ a parabolic subgroup of $G$ of
Levi subgroup $M$ and unipotent radical $N$. The parabolic induction
$\operatorname{ind}_{P}^{G}:\operatorname{Rep}_{R}^{\infty}(M)\to\operatorname{Rep}_{R}^{\infty}(G)$
sends $(\sigma,W)\in\operatorname{Rep}_{R}^{\infty}(M)$ to
$(\operatorname{ind}_{P}^{G}(\sigma),V)\in\operatorname{Rep}_{R}^{\infty}(G)$
where $V$ is the space of functions $f:G\to W$ right invariant by some open
subgroup of $G$ and satisfying $f(pg)=\tilde{\sigma}(p)f(g)$ for $(p,g)\in
P\times G$ and $\tilde{\sigma}$ is the inflation to $P$ of $\sigma$. It is an
exact functor respecting admissibility and finite length.

Replacing $P$ by a $G$-conjugate does not change the isomorphism class of
$\operatorname{ind}_{P}^{G}(\sigma)$ and a $G$-conjugate of $P$ contains $B$.

We suppose in this section that $B\subset P$. This implies
$G=K_{0}P=PK_{0}=K_{0}P^{-}=P^{-}K_{0}$ where $K_{0}=GL_{n}(O_{D})$ and
$P^{-}=MN^{-}$ the opposite parabolic subgroup with respect to $M$.

The parabolic induction of the trivial $R$-character of $M$

$\pi_{P}=\operatorname{ind}_{M}^{G}1$

will play an important role. As our parabolic induction is not normalized,
$[\pi_{P}]\in\operatorname{Gr}_{R}^{\infty}(G)$ depends on the choice of $P$
of Levi $M$, but we note that for two arbitray associated parabolic subgroups
$P,P^{\prime}$ of $G$ (having isomorphic Levi subgroups), the restrictions of
$[\pi_{P}]$ and $[\pi_{P}^{\prime}]$ to $K_{0}$ are equal.

######  Lemma 5.1.

Assume $\operatorname{char}_{R}\neq p$ and let $P^{\prime}$ be a parabolic
subgroup of $G$ associated to $P$. The representation
$\operatorname{ind}_{P}^{G}1$ has the same restriction to $K_{0}$ as
$\operatorname{ind}_{P^{\prime}}^{G}1$.

###### Proof.

When $\operatorname{char}_{R}\neq p$, by ([Minguez-Sécherre14] see §10.6 for
the notations), the trivial representation of a block of $M$ appears as a Speh
representation $Z(\Delta)$ where $\Delta$ is a segment supported on unramified
characters of $D^{*}$, and $\operatorname{ind}_{P}^{G}1$ which is the
normalized parabolic induction of an unramified character of $M$ is a product
of $Z(\Delta_{i})$ where the segments $\Delta_{i}$ are segments supported on
unramified characters. Twisting $1$ by a unramified smooth character $\chi$
amounts to twisting each segment by an unramified character $\chi_{i}$. When
$R$ is not finite, a generic choice of the $\chi_{i}$, the twisted segments
are not linked, so the product representation is irreducible and we can
interchange the segments to put them in any order. If $P^{\prime}$ is
associated to $P$, then $P^{\prime}$ is conjugate to a parabolic subgroup
$P^{\prime\prime}=M^{\prime\prime}N^{\prime\prime}$ containing $B$ and the
blocks of $M^{\prime\prime}$ are those of $M$ in a different order. So
$\operatorname{ind}_{P}^{G}\chi\simeq\operatorname{ind}_{P^{\prime\prime}}^{G}\chi^{\prime\prime}$
for some unramified smooth character $\chi^{\prime\prime}$ of
$M^{\prime\prime}$. Therefore $\operatorname{ind}_{P}^{G}1$ and
$\operatorname{ind}_{P^{\prime\prime}}^{G}1\simeq\operatorname{ind}_{P^{\prime}}^{G}1$
have the same restriction to $K_{0}$. When $R$ is finite,
$\pi_{P^{\prime}}=\operatorname{ind}_{P^{\prime}}^{G}1$ has the same
restriction to $K_{0}$ as $\pi_{P}=\operatorname{ind}_{P}^{G}1$ on a finite
extension $R^{\prime}/R$ where there is enough place for the $\chi_{i}$.
Taking the restriction to $R$ of the extension to $R^{\prime}$,
$\oplus^{r}\pi_{P^{\prime}}\simeq\oplus^{r}\pi_{P}$ as a representation of
$K_{0}$, where $r=[R:R^{\prime}]$. For any $j\geq 1$, taking the invariants by
$K_{j}$, the finite dimensional representations
$\oplus^{r}(\pi_{P^{\prime}})^{K_{j}}$ and $\oplus^{r}(\pi_{P})^{K_{j}}$ of
the finite group $K_{0}/K_{j}$ are isomorphic. By Krull-Remak-Schmidt,
$(\pi_{P})^{K_{j}}\simeq(\pi_{P})^{K_{j}}$. As this is true for any $j$, we
have $\pi_{P^{\prime}}\simeq\pi_{P}$. ∎

#### 5.2.1.

When $R=\mathbb{C}$ and $\sigma\in\operatorname{Rep}_{\mathbb{C}}^{\infty}(M)$
is admissible, we compute the character of
$\operatorname{ind}_{P}^{G}(\sigma)$ in terms of the character of $\sigma$.

######  Lemma 5.2.

For $f\in C^{\infty}_{c}(G,\mathbb{C})$, the function
$Sf(m)=\int_{N}\int_{K_{0}}f(kmnk^{-1})dkdn$ on $M$ belongs to
$C^{\infty}_{c}(M,\mathbb{C})$.

###### Proof.

The normal open compact subgroups $K$ of $K_{0}$ form a fundamental system of
neighborhoods of $1$ in $G$ and for $g\in G$ the open compact sets $KgK$ form
a fundamental system of neighborhoods of $g$ in $G$. For $g\in G$ and $m\in
M$, $m^{-1}KgK\cap N$ is open in $N$. The set of $m\in M$ such that
$m^{-1}KgK\cap N\neq\emptyset$ is open compact in $M$, $S1_{KgK}$ is $0$
outside of this set and $S1_{KgK}(m)=\operatorname{vol}(m^{-1}KgK\cap N,dn)\ \
\ \text{for}\ m^{-1}KgK\cap N\neq\emptyset.$ ∎

######  Remark 5.3.

For a normal open compact subgroup $K$ of $K_{0}$ such that $K\cap P=(K\cap
M)(K\cap N)$, $S1_{K}=\operatorname{vol}(K\cap N,dn)1_{M\cap K}$. For $f\in
C^{\infty}_{c}(G,\mathbb{C})$ with $\operatorname{Supp}f\subset K$, then
$\operatorname{Supp}Sf\subset K\cap M$.

######  Proposition 5.4.

We have $\operatorname{trace}(\pi(f(g)dg))=trace(\sigma(Sf(m)dm))$ for
$\sigma\in\operatorname{Rep}_{\mathbb{C}}^{\infty}(M)$ admissible,
$\pi=\operatorname{ind}_{P}^{G}(\sigma)$, and $(f,Sf)$ as in Lemma 5.2.

###### Proof.

a) Preliminaries. As $G=PK_{0}$, a function in the space $V$ of $\pi$ is
determined by its restriction to $K_{0}$, and
$\pi|_{K_{0}}\simeq\operatorname{ind}_{P\cap K_{0}}^{K_{0}}(\sigma|_{M\cap
K_{0}})$. Denote $V|_{K_{0}}$ the restrictions to $K_{0}$ of the functions in
$V$. Let $W$ denote the space of $\sigma$ and $\rho$ the action of $K_{0}$ on
$C^{\infty}(K_{0};W)$ by right translation. We identify $C^{\infty}(K_{0};W)$
and $C^{\infty}(K_{0};R)\otimes_{R}W$. Then $(\operatorname{ind}_{P\cap
K_{0}}^{K_{0}}(\sigma|_{M\cap K_{0}}),V|_{K_{0}})$ is a subrepresentation of
$(\rho,C^{\infty}(K_{0};R)\otimes_{R}W)$. Let $dx$ denote the restriction to
$P\cap K_{0}$ of $dp$ (equal to the restriction of $dk$). The map
$B:(\rho,C^{\infty}(K_{0};R)\otimes_{R}W)\to(\operatorname{ind}_{P\cap
K_{0}}^{K_{0}}(\sigma|_{M\cap K_{0}},V|_{K_{0}})$

$B(h\otimes w)(k)=\operatorname{vol}(P\cap K_{0},dx)^{-1}\int_{P\cap
K_{0}}h(x^{-1}k)\tilde{\sigma}(x)(w)dx\ \ \ \ (h\in C^{\infty}(K_{0};R),w\in
W,k\in K_{0}),$

is a $K_{0}$-equivariant projection. The function $B(h\otimes w)$ on $K_{0}$
extends to a function $F_{h,w}\in V$

$F_{h,w}(pk)=\operatorname{vol}(P\cap K_{0},dx)^{-1}\int_{P\cap
K_{0}}h(x^{-1}k)\tilde{\sigma}(px)(w)dx\ \ \ ((p,k)\in P\times K_{0}).$

b) Choose a normal open pro-$p$ subgroup $K$ of $K_{0}$ such that $f$ is
binvariant by $K$. The endomorphism $\pi(f(g)dg)$ of $V$ restricted to $V^{K}$
is an endomorphism $A$ of $V^{K}$ of trace
$\operatorname{trace}(A)=\operatorname{trace}(\pi(f(g)dg)$. Choose a disjoint
decomposition $K_{0}=\sqcup_{i}y_{i}K$. The $1_{y_{i}K}$ form a basis of
$C^{\infty}(K_{0};R)^{K}$, the support of $B(1_{y_{i}K}\otimes w)$ is in
$y_{i}K$, and $\operatorname{trace}(A)$ is the trace of the endomorphism
$w\mapsto\sum_{i}B(F_{1_{y_{i}K},w})(y_{i})$ of $W$. For $y\in K_{0}$,
$B(F_{1_{yK},w})(y)$ is equal to

$\displaystyle\int_{G}f(g)F_{1_{yK},w}(yg)dg=\int_{G}f(y^{-1}g)F_{1_{yK},w}(g)dg=\int_{K_{0}\times
P}f(y^{-1}p^{-1}k)F_{1_{yK},w}(p^{-1}k)dk\,dp$
$\displaystyle=\operatorname{vol}(P\cap K_{0},dx)^{-1}\int_{K_{0}\times
P\times P\cap
K_{0}}f(y^{-1}p^{-1}k)h_{y}(x^{-1}k)\sigma(p^{-1}x)(w)\,dk\,dp\,dx$
$\displaystyle=\int_{K_{0}\times
P}f(y^{-1}p^{-1}k)h_{y}(k)\sigma(p^{-1})(w)\,dk\,dp=\int_{K_{0}\times
P}f(y^{-1}pk)1_{yK^{\prime}}(k)\tilde{\sigma}(p)(w)\,dk\,dp$
$\displaystyle=\operatorname{vol}(K^{\prime},dk)\int_{P}f(y^{-1}py)\tilde{\sigma}(p)(w)\,dp.$

Therefore

$\displaystyle\sum_{i}B(F_{1_{y_{i}K},w})(y_{i})=\operatorname{vol}(K,dk)\int_{P}\sum_{i}f(y_{i}^{-1}py_{i})\tilde{\sigma}(p)(w)\,dp=\int_{K_{0}\times
P}f(k^{-1}pk)\tilde{\sigma}(p)(w)dk\,dp$ $\displaystyle=\int_{K_{0}\times
M\times N}f(k^{-1}mnk)\sigma(m)(w)dk\,dm\,dn=\sigma(Sf(m)dm)(w).$

We deduce that the trace of $\pi(f(g)dg)$ is the trace of $\sigma(Sf(m)dm)$. ∎

The set $\\{P_{\lambda}\ |\ \lambda\in\mathfrak{P}(n)\\}$ represents the
parabolic subgroups of $G$ modulo association.

######  Proposition 5.5.

When $P$ is a parabolic subgroup of $G$ associated to $P_{\lambda}$, we have

$\operatorname{trace}(\pi_{P}(f(g)\,dg))=\hat{\mu}_{\mathfrak{O}_{\lambda}}(\varphi).$

for $f\in C_{c}^{\infty}(K_{1};\mathbb{C})$ and $\varphi\in
C_{c}^{\infty}(M_{n}(P_{D});\mathbb{C})$ such that $f(1+X)=\varphi(X)$ for
$X\in M_{n}(P_{D})$, and $\hat{\mu}_{\mathfrak{O}_{\lambda}}$ as in (4.5).

###### Proof.

For $(f,\varphi)$ as in the proposition, the functions

(5.3) $\displaystyle f_{K_{0}}(g)=\int_{K_{0}}f(kgk^{-1})\,dk\ \ (g\in G),\ \
\ \varphi_{K_{0}}(X)=\int_{K_{0}}\varphi(kXk^{-1})\,dk\ \ (X\in M_{n}(D)),$

belong also to
$C_{c}^{\infty}(K_{1};\mathbb{C}),C_{c}^{\infty}(M_{n}(P_{D});\mathbb{C})$ and
$f_{K_{0}}(1+X)=\varphi_{K_{0}}(X)$ for $X\in M_{n}(P_{D})$,

$\int_{P}f(p)\,dp=\int_{\mathfrak{p}}\varphi(W)\,dW,$

$\operatorname{trace}(\pi_{P}(f(g)\,dg))=\operatorname{trace}(\pi_{P_{\lambda}}(f(g)\,dg))$
as $\pi_{P}=\pi_{P_{\lambda}}$ on $K_{0}$(Lemma 5.1), and

$\displaystyle\operatorname{trace}(\pi_{P}(f(g)\,dg))=\int_{M}Sf(m)\,dm=\int_{P}f_{K_{0}}(p)\,dp=\int_{\mathfrak{p}}\varphi_{K_{0}}(W)\,dW=\hat{\mu}_{\mathfrak{O}}(\varphi).$

for $P=P_{\lambda},\mathfrak{O}=\mathfrak{O}_{\lambda}$, by Propositions 5.2
and 4.5. ∎

######  Corollary 5.6.

For any non zero map $c:\mathfrak{P}(n)\to\mathbb{Z}$, the restriction of

$\sum_{\lambda\in\mathfrak{P}(n)}c(\lambda)\,[\pi_{P_{\lambda}}]\in\operatorname{Gr}_{\mathbb{C}}^{\infty}(G)$

to an arbitrary open compact subgroup $K$ of $G$ is not $0$.

###### Proof.

By Propositions 5.5 and 4.4, the characters of $\pi_{P_{\lambda}}$ are
linearly independant on any neighborhood of $1$, because their values on $f\in
C_{c}^{\infty}(K_{1};\mathbb{C})$ are the Fourier transforms of the nilpotent
orbital integrals of $\mathfrak{O}_{\lambda}$ on $\varphi\in
C_{c}^{\infty}(M_{n}(P_{D});\mathbb{C})$ when $f(1+X)=\varphi(X)$. ∎

##  6\. Complex representations of $G$ near the identity

### 6.1.

By [Harish-Chandra78] when $\operatorname{char}_{F}=0$ (for any reductive
$p$-adic group) and ([Lemaire04] Proposition 4.3 with $E=F$), any non-zero
representation $\pi\in\operatorname{Rep}_{\mathbb{C}}^{\infty,f}G$ non-zero
$\pi$ has a germ expansion of map $c_{\pi}$ on $K_{\pi}$, meaning that:

There exists a map $c_{\pi}:G\backslash\mathfrak{N}\to\mathbb{C}$ (the
coefficient map) and an open subgroup $K_{\pi}$ of $K_{1}=1+M_{n}(P_{D})$ such
that

(6.1) $\displaystyle\operatorname{trace}(\pi(fdg))=\sum_{\mathfrak{O}\in
G\backslash\mathfrak{N}}c_{\pi}(\mathfrak{O})\,\hat{\mu}_{\mathfrak{O}}(\varphi)$

for $f\in C_{c}^{\infty}(K_{\pi};\mathbb{C}),\varphi\in
C_{c}^{\infty}(M_{n}(P_{D});\mathbb{C})$ such that $f(1+X)=\varphi(X)$ for
$X\in M_{n}(P_{D})$ and $f(1+X)=\varphi(X)$ for $X\in
M_{n}(P_{D});\mathbb{C})$.

It is convenient to see $c_{\pi}$ as a map on the set $\mathfrak{P}(n)$ of
partitions of $n$, or on the set of parabolic subgroups $P$ of $G$,

(6.2) $c_{\pi}(\lambda)=c_{\pi}(\mathfrak{O}_{\lambda})=c_{\pi}(P)\ \text{for
$\lambda\in\mathfrak{P}(n)$ and $P$ associated to $P_{\lambda}$}.$

For example, $c_{\pi}((n))=c_{\pi}(\\{0\\})=c_{\pi}(G)$. By Proposition 5.5,
we have for $f\in C_{c}^{\infty}(K_{\pi};\mathbb{C})$,

(6.3)
$\displaystyle\operatorname{trace}(\pi(fdg))=\sum_{\lambda\in\mathfrak{P}(n)}c_{\pi}(\lambda)\,\operatorname{trace}(\pi_{P_{\lambda}}(fdg))=\sum_{P}c_{\pi}(P)\,\operatorname{trace}(\pi_{P}(fdg)).$

the last sum is over a system of representatives $P$ of the parabolic
subgroups of $G$ modulo association. We list some properties of the map
$c_{\pi}$ for $\pi\in\operatorname{Rep}_{\mathbb{C}}^{\infty,f}(G)$.

  * •

The map $c_{\pi}$ is unique by Corollary 5.6 and is not $0$ because

(6.4)
$\dim_{\mathbb{C}}\pi^{K}=trace(\pi(1_{K}\operatorname{vol}(K,dg)^{-1}\,dg)\neq
0\ \ \text{for any open subgroup $K$of $K_{\pi}$.}$

  * •

Two representations
$\pi,\pi^{\prime}\in\operatorname{Rep}_{\mathbb{C}}^{\infty,f}(G)$ have the
same coefficient map if and only if their restrictions to some open compact
subgroup of $G$ are isomorphic, because the linear forms
$\hat{\mu}_{\mathfrak{O}}$ restricted to
$C_{c}^{\infty}(-1+K_{\pi});\mathfrak{C})$ are linearly independent
(Proposition 4.4).

  * •

In particular,

(6.5) $c_{\pi}=c_{\pi\otimes\chi}$

for any smooth character $\chi$ of $G$, because $\chi$ is trivial on some open
compact subgroup.

  * •

The map $c_{\pi}$ depends only on the image $[\pi]$ of $\pi$ in the
Grothendieck group $\operatorname{Gr}_{\mathbb{C}}^{\infty}(G)$. It passes to
a linear map $\nu\mapsto c_{\nu}$ on the Grothendieck group
$\operatorname{Gr}_{\mathbb{C}}^{\infty}(G)$ such that $c_{\pi}=c_{[\pi]}$ for
$\pi\in\operatorname{Rep}_{\mathbb{C}}^{\infty(,f}G)$. But $c_{\nu}=0$ does
not imply $\nu=0$. For example, $c_{\nu}=0$ but
$\nu=\pi_{P_{\lambda}}-\pi_{P_{\lambda^{\prime}}}\neq 0$, for two distinct
composition $\lambda,\lambda^{\prime}$ of $n$ giving the same partition by
reordering its terms.

  * •

When $\pi$ is finite dimensional, it is trivial on some $K_{\pi}\subset K_{1}$
hence

(6.6) $c_{\pi}((n))=\dim_{\mathbb{C}}\pi,\ \ \ c_{\pi}(\lambda)=0\ \
\text{for}\ \lambda\neq(n).$

Conversely, if $c_{\pi}(\lambda)=0$ for $\lambda\neq(n)$ then

(6.7)
$\operatorname{trace}(\pi(fdg))=c_{\pi}(\\{0\\})\,\hat{\mu}_{\\{0\\}}(\varphi)=c_{\pi}((n))\,\int_{G}f(g)dg$

for $(f,\varphi)$ as in (6.1). Hence $\dim_{\mathbb{C}}\pi^{K}=c_{\pi}((n))$
for any open subgroup $K$ of $K_{\pi}$, so $\pi$ is finite dimensional.

  * •

When $D\neq F$, a finite dimensional irreducible smooth representation of
$D^{*}$ may have dimension $>1$, but:

######  Lemma 6.1.

When $R$ an algebraically closed field, $D=F$ or $n>1$, then a finite
dimensional irreducible $R$-representation of $G$ is of the form
$\pi=\chi\circ nrd$ for some $R$-character $\chi$ of $F^{*}$.

###### Proof.

This clear when $G=F^{*}$ because $F^{*}$ is commutative and the Schur’s lemma
is valid for $G$. When $n>1$, then $\operatorname{Ker}(\pi)$ is an open
subgroup of $G$, and in particular contains an open subgroup of $U$. But
$\operatorname{Ker}(\pi)$ is also normal in $G$, so it contains $U$, and all
the conjugates of $U$. Those conjugates generate
$\operatorname{Ker}(\operatorname{nrd})$, so $\pi$ factors through
$\operatorname{nrd}$ implying the lemma. ∎

### 6.2.

We revert to $R=\mathbb{C}$ and show that the values of $c_{\pi}$ are integers
(proved in [Howe74] when $D=F$ has characteristic $0$ and $\pi$ is irreducible
supercuspidal). The key of the proof is the next lemma 6.2 inspired by Howe
([Howe74] Lemma 6).

For a partition $\lambda=(\lambda_{1},\ldots,\lambda_{r})$ of $n$, let
$A_{\lambda}$ be the matrix of the endomorphism of the right $D$-vector space
$D^{n}$ operating on the canonical basis $e_{1},\ldots,e_{n}$ by sending
$e_{1},\ldots,e_{\lambda_{1}}$ to $0$,
$e_{\lambda_{1}+1},\ldots,e_{\lambda_{1}+\lambda_{2}}$ to
$e_{1},\ldots,e_{\lambda_{2}}$, and $e_{\lambda_{1}+\ldots+\lambda_{i}+j}$ to
$e_{\lambda_{1}+\ldots+\lambda_{i-1}+j}$ for
$i=2,\ldots,r-1,j=1,\ldots,\lambda_{i+1}$. Then,
$\operatorname{Ker}A_{\lambda}^{i}$ is the $D$-subspace generated by
$e_{1},\ldots,e_{\lambda_{1}+\ldots+\lambda_{i}}$. The parabolic subgroup of
$G$ stabilizing the flag $(\operatorname{Ker}A_{\lambda}^{i})_{i}$ is
$P_{\lambda}$, and $A_{\lambda}\in\mathfrak{n}_{\lambda}$. Fixing a character
$\psi$ of $F$ trivial on $P_{F}$ and not on $O_{F}$, for an integer $j\geq 1$,
let $\xi_{\lambda}$ be the character of $K_{j}=1+M_{n}(P_{D}^{j})$ trivial on
$K_{2j}$ defined by

(6.8)
$\xi_{\lambda}(1+x)=\psi\circ\operatorname{trd}(A_{\lambda}\,p_{D}^{1-2j}x)\ \
\text{ for}\ x\in M_{n}(P_{D}^{j}).$

######  Lemma 6.2.

For $\mu\in\mathfrak{P}(n)$, the multiplicity $m(\xi_{\lambda},\pi_{P_{\mu}})$
of $\xi_{\lambda}$ in $\pi_{P_{\mu}}$ is $0$ unless $\lambda\geq\mu$. We have
$m(\xi_{\lambda},\pi_{P_{\lambda}})=1$.

###### Proof.

For $\mu\in\mathfrak{P}(n)$, $m(\xi_{\lambda},\pi_{P_{\mu}})$ is the
cardinality of

$(P_{\mu}\cap GL_{n}(O_{D}))\backslash\\{k\in GL_{n}(O_{D})\ |\
\xi_{\lambda}(k^{-1}(P_{\mu}\cap K_{j})k)=1\\}/K_{j}.$

Let $k\in K_{0}=GL_{n}(O_{D})$. We have $\xi_{\lambda}(k^{-1}(P_{\mu}\cap
K_{j})k)=1$ if and only if

(6.9) $\xi_{\lambda}(k^{-1}(1+\mathfrak{p}_{\mu}(P_{D}^{j}))k)=1,$

where $\mathfrak{p}_{\mu}(P_{D}^{j})=\mathfrak{p}_{\mu}\cap M_{n}(P_{D}^{j})$.
The weaker condition
$\xi_{\lambda}(k^{-1}(1+\mathfrak{p}_{\mu}(P_{D}^{2j-1}))k)=1$ already implies
$m(\xi_{\lambda},\pi_{P_{\mu}})=0$ unless $\lambda\geq\mu$. Indeed, it reads
$\psi\circ\operatorname{trd}(A_{\lambda}\,k^{-1}\mathfrak{p}_{\mu}(O_{D})k)=1$.
It depends on the images $\overline{k},\overline{A}_{\lambda}$ of
$k,A_{\lambda}$ in $GL_{n}(k_{D})$ and says that
$trd(\overline{k}\,\overline{A}_{\lambda}\,\overline{k}^{-1}\,\mathfrak{p}_{\mu}(k_{D}))=0$,
that is,
$\overline{k}\,\overline{A}_{\lambda}\,\overline{k}^{-1}\in\mathfrak{n}_{\mu}(k_{D})$.
Let $0\subset W_{1}\subset\ldots$ be the flag of $k_{D}^{n}$ whose stabilizer
is $P_{\mu}(k_{D})$. Then
$\overline{k}\,\overline{A}_{\lambda}\,\overline{k}^{-1}\in\mathfrak{n}_{\mu}(k_{D})$
means $\overline{k}\,\overline{A}_{\lambda}\,\overline{k}^{-1}(W_{i})\subset
W_{i-1}$ for $i\geq 1$, and in particular that
$\operatorname{Ker}(\overline{k}\,(\overline{A}_{\lambda})^{i}\,\overline{k}^{-1})=\overline{k}(\operatorname{Ker}(\overline{A}_{\lambda})^{i})$
contains $W_{i}$. As $\dim_{D}W_{i+1}-\dim_{D}W_{i}=\mu_{i}$, one obtains
$\lambda_{1}+\ldots+\lambda_{i}\geq\mu_{1}+\ldots+\mu_{i}$ for each $i$, that
is $\lambda\geq\mu$.

Suppose now $\mu=\lambda$. We prove that (6.9) is equivalent to $k\in
P_{\lambda}(O_{D})K_{j}$. By its definition $\xi_{\lambda}$ is trivial on
$1+\mathfrak{p}_{\lambda}(P_{D}^{j})$ because
$A_{\lambda}\in\mathfrak{n}_{\lambda}$ hence
$\operatorname{trd}(A_{\lambda}\mathfrak{p}_{\lambda})=0$, so
$P_{\lambda}(O_{D})K_{j}$ does satisfy (6.9). Conversely,
$B_{\lambda}=A_{\lambda}p_{D}^{1-2j}\in\mathfrak{n}_{\lambda}(P_{D}^{1-2j})$.
The condition (6.9) means that
$\operatorname{trd}(B_{\lambda}k^{-1}\mathfrak{p}_{\lambda}(P_{D}^{j})k)\in
P_{F}$ and implies

$B_{\lambda}=k^{-1}Xk+Y,\ \text{where}\ X\in\mathfrak{n}_{\lambda},Y\in
M_{n}(P_{D}^{1-j}).$

Indeed, writing $kB_{\lambda}k^{-1}=X+Y$ with
$X\in\mathfrak{n}_{\lambda},Y\in\mathfrak{p}_{\lambda}^{-}$, we have:

$\displaystyle\operatorname{trd}(B_{\lambda}k^{-1}\mathfrak{p}_{\lambda}(P_{D}^{j})k)=\operatorname{trd}(kB_{\lambda}k^{-1}\mathfrak{p}_{\lambda}(P_{D}^{j}))=\operatorname{trd}(Y\mathfrak{p}_{\lambda}(P_{D}^{j}))=\operatorname{trd}(YM_{n}(P_{D}^{j})),$
$\displaystyle\operatorname{trd}(YM_{n}(P_{D}^{j}))\in
P_{F}\Leftrightarrow\operatorname{trd}(P_{D}^{j-d}YM_{n}(O_{D}))\in
O_{F}\Leftrightarrow Y\in M_{n}(P_{D}^{1-j}).$

See Example 4.3 for the last equivalence. One gets
$B_{\lambda}k^{-1}=k^{-1}X+Y_{1}$ with $Y_{1}\in M_{n}(P_{D}^{1-j})$. Note
that $B_{\lambda}\in M_{n}(P_{D}^{1-2j})$ hence also $X$. We get
$B_{\lambda}^{2}k^{-1}=B_{\lambda}k^{-1}X+B_{\lambda}Y_{1}=k^{-1}X^{2}+Y_{1}X+B_{\lambda}Y_{1}=k^{-1}X^{2}+Y_{2}$
with $Y_{2}\in M_{n}(P_{D}^{j+2(1-2j)})$. By induction
$B_{\lambda}^{i}k^{-1}=k^{-1}X^{i}+Y_{i}$ with $Y_{i}\in
M_{n}(P_{D}^{j+i(1-2j)})$ for $1\leq i\leq r$. For a basis vector
$e\in\operatorname{Ker}A_{\lambda}^{i}$, we have $X^{i}e=0$ because
$X\in\mathfrak{n}_{\lambda}$, and
$B_{\lambda}^{i}k^{-1}e=k^{-1}X^{i}e+Y^{i}e=Y^{i}e$. As
$(A_{\lambda}p_{D}^{1-2j})^{i}k^{-1}e\in
M_{n}(P_{D}^{j+i(1-2j})e\Leftrightarrow A_{\lambda}^{i}k^{-1}e\in
M_{n}(P_{D}^{j})e$, the coefficients of $k^{-1}e$ on the basis vectors which
are not in $\operatorname{Ker}A_{\lambda}^{i}$ are in $P_{D}^{j}$. This means
$k^{-1}\in K_{j}P_{\lambda}(O_{D})$, what we wanted. ∎

We shall need more properties of $\xi_{\lambda}$ in the section on Whittaker
spaces.

######  Lemma 6.3.

The normalizer of $\xi_{\lambda}$ in $K_{0}=GL_{n}(O_{D})$ is
$P_{\lambda}(O_{D})K_{j}$.

###### Proof.

For $k\in K_{0}$, the property $\xi_{\lambda}(1+x)=\xi_{\lambda}(1+kxk^{-1})$
for all $x\in M_{n}(P_{D}^{j})$ means $k^{-1}B_{\lambda}k-B_{\lambda}\in
M_{n}(P_{D}^{1-j})$. As in the proof of Lemma 6.2 one deduces
$B^{i}k-kB^{i}\in M_{n}(P_{D}^{j-i(1-2j)})$ for $i\geq 1$ and one sees that
$k\in P(O_{D})K_{j}$. ∎

######  Remark 6.4.

There is a function in $\pi_{P_{\lambda}}$ with support $P_{\lambda}K_{j}$ and
restriction $\xi_{\lambda}$ to $K_{j}$ since $\xi_{\lambda}$ is trivial on
$1+\mathfrak{p}_{\lambda}(P_{D}^{j})$. That function is a basis of the line of
vectors in $\pi_{P_{\lambda}}$ transforming according to $\xi_{\lambda}$ under
the action of $K_{j}$.

We prove now that the $c_{\pi}(\lambda)$ are integers. By (6.1), when
$K_{j}=1+M_{n}(P_{D}^{j})\subset K_{\pi}$ and
$\delta\in\operatorname{Rep}_{\mathbb{C}}^{\infty}(K_{j})$ irreducible, the
multiplicity $m(\delta,\pi)$ of $\delta$ in
$\pi\in\operatorname{Rep}_{\mathbb{C}}^{\infty}(G)$ satisfies

(6.10) $\displaystyle
m(\delta,\pi)=\sum_{\mu\in\mathfrak{P}(n)}c_{\pi}(\mu)\,m(\delta,\pi_{P_{\mu}}).$

Lemma 6.2 and (6.10) imply:

(6.11) $\displaystyle c_{\pi}(\lambda)$
$\displaystyle=m(\xi_{\lambda},\pi)-\sum_{\mu\in\mathfrak{P}(n),\mu<\lambda}c_{\pi}(\mu)\,m(\xi_{\lambda},\pi_{P_{\mu}}).$

In particular when $\lambda$ is minimal in $\operatorname{Supp}c_{\pi}$,
$c_{\pi}(\lambda)=m(\xi_{\lambda},\pi)$ is positive and independent of $j\geq
j_{0}$. By upwards induction on $\mathfrak{P}(n)$ (downwards induction on the
nilpotent orbits), we obtain that the $c_{\pi}(\lambda)$ are integers.

As the values of the map $c_{\pi}$ are integers, we get more properties::

  * •

$c_{\pi}=c_{\sigma(\pi)}$ when $\sigma$ is an automorphism of $\mathbb{C}$.

  * •

For $\nu\in\operatorname{Gr}_{R}^{\infty,f}(G)$, there exists a map
$c_{\nu}:\mathfrak{P}(n)\to\mathbb{Z}$ and an open subgroup $K_{\nu}$ of $G$
such that $\nu$ and
$\sum_{\lambda\in\mathfrak{P}(n)}c_{\nu}(\lambda)\,[\pi_{P_{\lambda}}]\in\operatorname{Gr}_{R}^{\infty,f}(G)$
have isomorphic restrictions to $K_{\nu}$.

When $R=\mathbb{C}$, the first part of Theorem 1.3 is a version of the germ
expansion. For any $R$, when $\pi$ satisfies the first part of Theorem 1.3 we
say sometimes that $\pi$ has a germ expansion with map $c_{\pi}$ on $K_{\pi}$.

##  7\. Parabolic induction

In this section $R$ is a field and $\operatorname{char}_{R}\neq p$. We prove
now that the first part of Theorem 1.3 implies Theorem 1.5. Let
$P,M,(n_{i}),\sigma_{i},r,\pi$ as in Theorem 1.5. Given partitions
$\lambda_{i}$ of $n_{i}$ for $1\leq i\leq r$, we have the parabolic subgroup
$P_{(\lambda_{i})}$ of $M$ corresponding to the parabolic subgroups
$P_{\lambda_{i}}$ of $GL_{n_{i}}(D)$. Given functions
$c_{i}:\mathfrak{P}(n_{i})\to\mathbb{Z}$ for $1\leq i\leq r$, the function
$c:\mathfrak{P}(n)\to\mathbb{Z}$ defined by

$c(\lambda)=\sum\prod_{i=1,\ldots,r}c_{i}(\lambda_{i}),$

where the sum is over $r-$tuples of partitions
$(\lambda_{1},\ldots,\lambda_{r})$ inducing to $\lambda$ before Theorem 1.5,
is called induced by $(c_{1},\ldots,c_{r})$.

######  Theorem 7.1.

Assume that for $i=1,\ldots,r$, there exists a function
$c_{\sigma_{i}}:\mathfrak{P}(n_{i})\to\mathbb{Z}$ and an open compact subgroup
$K_{\sigma_{i}}$ of $GL_{n_{i}}(D)$ such that
$\sigma_{i}=\sum_{\lambda_{i}\in\mathfrak{P}(n_{i})}c_{\sigma_{i}}(\lambda_{i})\operatorname{ind}_{P_{\lambda_{i}}}^{GL_{n_{i}}(D)}1$
on $K_{\sigma_{i}}$. Then
$\pi=\sum_{\lambda\in\mathfrak{P}(n)}c_{\pi}(\lambda)\operatorname{ind}_{P_{\lambda}}^{G}1$
on $K_{\pi}$ where $c_{\pi}:\mathfrak{P}(n)\to\mathbb{Z}$ is the function
induced by the $c_{\sigma_{i}}$ and $K_{\pi}$ is a open compact subgroup of
$G$ such that $\cap_{g\in P\backslash G/K_{\pi}}\,M\cap gK_{\pi}g^{-1}$ is
contained in $K_{\sigma_{1}}\times\ldots\times K_{\sigma_{r}}$.

###### Proof.

The theorem follows from the fact that for any field $R$,
$\operatorname{ind}_{P}^{G}(\operatorname{ind}_{P_{(\lambda_{i})}}^{M}1)$ has
the same restriction to $K_{0}$ than $\operatorname{ind}_{P_{\lambda}}^{G}1$
by Lemma 5.1, and for given a open compact subgroup $C_{M}$ of $M$, there
exists an open compact subgroup $C$ of $G$ such that

(7.1) $\cap_{g\in P\backslash G/C}\,M\cap gCg^{-1}\ \subset\ C_{M}.$

The existence of $K_{\pi}$ follows from (7.1) applied to
$C_{M}=K_{\sigma_{1}}\times\ldots\times K_{\sigma_{r}}$.

The restriction of a smooth $R$-representation $\sigma$ of $M$ to $C_{M}$
determines the restriction of $\operatorname{ind}_{P}^{G}\sigma$ to $C$, equal
to

(7.2) $\oplus_{g\in P\backslash
G/C}\operatorname{ind}_{P}^{PgC}\sigma\simeq\oplus_{g\in P\backslash
G/C}\operatorname{ind}_{K\cap g^{-1}Pg}^{C}(\sigma^{g})$

where $\sigma^{g}(k)=\sigma(gkg^{-1})$ for $g\in G,k\in g^{-1}Pg\cap C$, and
$\sigma^{g}$ depends only on the restriction of $\sigma$ to $M\cap gCg^{-1}$.
If $\sigma^{\prime}\in\operatorname{Rep}_{R}^{\infty,f}(M)$ is isomorphic to
$\sigma$ on $C_{M}$, then $\operatorname{ind}_{P}^{G}\sigma^{\prime}$ and
$\operatorname{ind}_{P}^{G}\sigma$ are isomorphic on $C$. The same holds true
for virtual representations $\nu,\nu^{\prime}$ of $M$. Take
$\nu=\sigma_{1}\otimes\ldots\otimes\sigma_{r}$ and
$\nu^{\prime}=\nu^{\prime}_{1}\otimes\ldots\otimes\nu^{\prime}_{r}$ with
$\nu^{\prime}_{i}=\sum_{\lambda_{i}\in\mathfrak{P}(n_{i})}c_{\sigma_{i}}(\lambda_{i})\operatorname{ind}_{P_{\lambda_{i}}}^{GL_{n_{i}}(D)}1$.
∎

######  Corollary 7.2.

(Variant of the theorem) Assume that $GL_{n_{i}}(D)$ satisfies the first part
of Theorem 1.3 for $i=1,\ldots,r$. Then for
$\sigma\in\operatorname{Rep}_{R}^{\infty,f}(M)$, there exists an open compact
subgroup $K_{\sigma}$ of $M$ and a unique map
$c_{\sigma}:\mathfrak{P}(n_{1})\times\ldots P(n_{r})\to\mathbb{Z}$ such that
$\sigma=\sum_{(\lambda_{i})\in(\mathfrak{P}(n_{i}))}c_{\sigma}((\lambda)_{i})\pi_{P_{(\lambda_{i})}}$
on $K_{\sigma}$, and $\pi=\operatorname{ind}_{P}^{G}\sigma$ is equal to
$\sum_{\lambda}c_{\pi}(\lambda)\pi_{P_{\lambda}}$ on any open compact subgroup
$K_{\pi}$ of $G$ such that $K_{\sigma}\subset\cap_{g\in P\backslash
G/K_{\pi}}\,M\cap gK_{\pi}g^{-1}$ and $c_{\pi}:\mathfrak{P}(n)\to\mathbb{Z}$
is induced by $c_{\sigma}$.

######  Remark 7.3.

When $G=GL_{n}(F)$, given partitions $\lambda_{i}$ of $n_{i}$ for
$i=1,\ldots,r$, and $\lambda\in\mathfrak{P}(n)$ induced by the $\lambda_{i}$,
the nilpotent orbit $\mathfrak{O}_{\lambda}$ is the nilpotent orbit induced by
the nilpotent orbit $\mathfrak{O}_{(\lambda_{i})}$ of $M$ corresponding to the
$\lambda_{i}$, in the sense of [Lusztig-Spaltenstein79] (see [Jantzen04]). If
$R=\mathbb{C}$, $\operatorname{char}_{F}=0$, $p\neq 2$, $D=F$, the formula for
$c_{\pi}$ follows from ([Moeglin-Waldspurger87] §II.1.3 where $G$ is a
classical group).

##  8\. Whittaker spaces

Our purpose in this section is to relate the coefficient map $c_{\pi}$ to the
dimensions of the different Whittaker spaces of $\pi$ when
$\pi\in\operatorname{Rep}_{\mathbb{C}}^{\infty}(G)$ is irreducible. We first
introduce those subspaces.

The commutator subgroup of the group $U$ of upper unipotent matrices is the
group $U^{\prime}$ of upper unipotent matrices with coefficients $u_{i,i+1}=0$
for $i=1,\ldots,n-1$ (use the identities $E_{a,b}E_{c,d}=E_{a,d}$ if $b=c$ and
$0$ otherwise). The map sending $(u_{i,j})\in U$ to
$(u_{1,2},\ldots,u_{n-1,n})$ induces an isomorphism from $U/U^{\prime}$ to the
additive group $D^{n-1}$. The action of the group $T\simeq(D^{*})^{n}$ of
diagonal matrices by conjugation on $U$ and on $U^{\prime}$ induces an action
on $D^{n-1}$, the diagonal matrix $\operatorname{diag}(a_{1},\ldots,a_{n})\in
T$ sends $(d_{1},\ldots,d_{n-1})\in D^{n-1}$ to
$(a_{1}d_{1}a_{2}^{-1},\ldots,a_{n-1}d_{n-1}a_{n}^{-1})$.

Let us fix a non-trivial smooth character $\psi$ of $F$. Then
$\psi_{D}=\psi\circ\operatorname{trd}$ is a non-trivial character of $D$.
Sending $y\in D$ to the character $\psi_{D}^{y}(x)=\psi_{D}(yx)$ for $x\in D$,
is an isomorphism from the additive group $D$ to its group of smooth
characters. Sending $y=(y_{1}\ldots y_{n-1})\in D^{n-1}$ to
$(\psi_{D}^{y_{1}},\ldots,\psi_{D}^{y_{n-1}})$, is an isomorphism from
$D^{n-1}$ to its group of smooth characters. The above action of $T$ on
$D^{n-1}$ induces an action on its groups of characters, the diagonal matrix
$\operatorname{diag}(a_{1},\ldots,a_{n})$ sends $y=(y_{1},\ldots,y_{n-1})\in
D^{n-1}$ to $(a_{2}^{-1}y_{1}a_{1},\ldots,a_{n}^{-1}y_{n-1}a_{n-1})$.

Let $y=(y_{1},\ldots,y_{n-1})\in D^{n-1}$, $r$ be the number of indices $i$
where $y_{i}=0$, and

(8.1) $I=I(y)=\begin{cases}\emptyset\ \text{ if }r=0,\\\
\\{i_{1}<\ldots<i_{r}\\}\ \text{ the set of indices $i$ where $y_{i}=0$ if
$r\neq 0$}.\end{cases}$

The smooth character of $U$ corresponding to $y$ is

$\theta(u)=\psi\circ\operatorname{trd}(X_{y}v)\ \ \ u=1+v\in U,$

where $X_{y}\in M_{n}(D)$ is the nilpotent matrix with
$(y_{1},\ldots,y_{n-1})$ just below the diagonal and $0$ elsewhere. The
character $\theta$ is called non-degenerate if $I(y)=\emptyset$, and
degenerate otherwise. The character $\theta$ is trivial if and only if
$I(y)=\\{1,\ldots,n\\}$. The group $B=TU$ is its own normalizer in $G$, so the
$G$-normalizer of $\theta$ is of the form $T_{\theta}U$ where $T_{\theta}$ is
the $T$-normalizer of $\theta$. It is the intersection of $B$ with the
commutant of $X_{y}$.

The element $y$ is conjugate under $T$ to the element $\delta_{I}\in D^{n-1}$
with coefficient $0$ in $I$ and $1$ elsewhere. The nilpotent matrix
$X_{\delta_{I}}$ is in Jordan form. The sizes of the Jordan blocks of
$X_{\delta_{I}}$ form a composition $\lambda_{I}$ of $n$,

(8.2) $\lambda_{I}=\begin{cases}(n)\ \text{ when }\ I=\emptyset,\\\
(i_{1},i_{2}-i_{1},\ldots,n-i_{r})\ \text{ when }\ I\neq\emptyset.\end{cases}$

Any composition $\lambda$ of $n$ is equal to $\lambda_{I}$ for a unique subset
$I$ of $\\{1,\ldots,n\\}$. We denote by $X_{\lambda}\in M_{n}(D)$ is the
nilpotent matrix with $(y_{1},\ldots,y_{n-1})$ just below the diagonal and $0$
elsewhere, by

(8.3) $\theta_{\lambda}(u)=\psi\circ\operatorname{trd}(X_{\lambda}v)\ \ \
u=1+v\in U,$

the character $\theta_{\delta_{I}}$ of $U$ and by $T_{\lambda}$ its
$T$-normalizer. The group $T_{\lambda}$ contains the group
$T_{(n)}=\\{\operatorname{diag}(d,\ldots,d)\ |\ d\in D^{*}\\}$ isomorphic to
$D^{*}$.

We fix a representation $\pi\in\operatorname{Rep}_{\mathbb{C}}^{\infty}(G)$ of
space $V$. Given a smooth character $\theta$ of $U$, we look at the space
$V_{\theta}$ of $\theta$-coinvariants of $U$ in $V$, or at its dual, the
(Whittaker) space of linear forms $\Lambda$ on $V$ such that
$\Lambda(uv)=\theta(u)\Lambda(v)$ for $u\in U,v\in V$. It is customary to say
that $\pi$ has a Whittaker model with respect to $\theta$ if $V_{\theta}\neq
0$. Indeed any choice of non-zero linear form $\Lambda$ on $V_{\theta}$ gives
a non-zero intertwining from $\pi$ to $\operatorname{Ind}_{U}^{G}(\theta)$ by
sending $v\in V$ to the function taking value $\Lambda(gv)$ at $g\in G$; that
intertwining is an embedding if $\pi$ is irreducible, hence the name “model”.
We put say that $\pi$ has a non-degenerate Whittaker model, or that $\pi$ is
generic if $V_{\theta}\neq 0$ for some (equivalently all) non-degenerate
characters $\theta$ of $U$. We say that $\pi$ has a Whittaker model it it has
a Whittaker model with respect to some choice of $\theta$.

Using the action of $T$ on $U$ by conjugation, we see that to analyse the
$V_{\theta}$ for all choices of $\theta$, it is enough to consider the
$\theta_{\lambda}$ associated to the compositions $\lambda$ of $n$.

######  Remark 8.1.

1) It is known that if $\pi$ is irreducible then $V_{\theta}$ is finite
dimensional (when $\theta$ is not degenerate [Bushnell-Henniart02], in general
[Aizenbud-BS87]; these papers treat the case of a general reductive group
$G$). The group $T_{\theta}$ acts on $V_{\theta}$; since $T_{\theta}$ is not
commutative if $D\neq F$, we cannot expect $V_{\theta}$ to have always
dimension $0$ or $1$ (as when $D=F$ and $\theta$ not degenerate).

2) Moeglin and Waldspurger [Moeglin-Waldspurger87] consider more general
Whittaker spaces, but ours are enough for our purpose (Theorem 8.2 below).
Also they use the exponential map, which is not available when $F$ has
positive characteristic. Instead we use the map $X\mapsto 1+X:M_{n}(P_{D})\to
1+M_{n}(P_{D})$, as in [Howe74] and [Rodier74] when $D=F$.

3) If $\pi$ is irreducible cuspidal, $\pi$ can only have non-degenerate
Whittaker models because $\theta_{I}$ is trivial on the unipotent radical
$N_{\lambda_{I}}$ of the parabolic group $P_{\lambda_{I}}$. Hence
$\pi_{\theta_{I}}$ is a quotient of the $N_{I}$-coinvariant space
$\pi_{N_{\lambda_{I}}}$ of $\pi$. If $\pi_{N_{\lambda_{I}}}=0$ then
$\pi_{\theta_{I}}=0$, and $N_{\lambda_{I}}$ is trivial if and only if
$I=\emptyset$.

4) It is possible to extend to $GL_{n}(D)$ the theory of [Bernstein-Zelevinski
77] 5.1 to 5.15 to show that a non-zero $\pi$ has a Whittaker model (see [Abe-
Herzig23] 3.4). But that is a consequence of our theorem below (Corollary
8.3).

We now prove Theorem 1.6 (for $R=\mathbb{C}$). We can assume that $\pi$ is
irreducible. We want to relate the coefficient map
$c_{\pi}:\mathfrak{P}(n)\to\mathbb{Z}$ of the germ expansion of $\pi$ with the
dimensions of the spaces $V_{\theta_{\lambda}}$ for the compositions $\lambda$
of $n$, following [Moeglin-Waldspurger87]. We define the Whittaker support of
$\pi$ as the set of partitions $\mu$ of $n$ such that
$V_{\theta_{\lambda}}\neq 0$ for some composition $\lambda$ of $n$ with
associated partition $\hat{\mu}$ (the partition dual to $\mu$).

######  Theorem 8.2.

The minimal elements in $\operatorname{Supp}c_{\pi}$ and in the Whittaker
support of $\pi$ are the same.

Let $\mu$ be a partition of $n$ minimal in $\operatorname{Supp}c_{\pi}$ and
let $\lambda$ be a composition of $n$ with associate partition $\hat{\mu}$.
Then $c_{\pi}(\mu)=\dim_{\mathbb{C}}V_{\theta_{\lambda}}$.

Since $\pi$ has a non-zero germ expansion, the theorem implies:

######  Corollary 8.3.

Any irreducible smooth complex representation $\pi$ of $G$ has a Whittaker
model.

######  Remark 8.4.

1) By the theorem $(1,\ldots,1)$ is minimal in $\operatorname{Supp}c_{\pi}$ if
and only if $V$ has a non-degenerate Whittaker model. This was proved when
$D=F$ [Rodier74].

2) $(n)$ is minimal in $\operatorname{Supp}c_{\pi}$ if and only if
$\dim_{\mathbb{C}}(V)$ is finite. By the theorem that happens if and only if
$V$ has only the trivial Whittaker model.

3) In part 2 of the theorem, $\dim(V_{\theta_{\lambda}})$ does not depend on
the choice of the composition $\lambda$ with associated partition $\hat{\mu}$.
It is the multiplicity in $\pi$ of the character $\xi_{\mu}$ of $K_{j}$
defined in (6.8), if $j$ is large enough.

We turn back to the proof of the theorem. As said at the beginning of this
section, our proof is based on the method of [Moeglin-Waldspurger87],
replacing the exponential by $x\to 1+x$. The starting idea is already in
[Rodier74], but that paper is restricted to the non-degenerate Whittaker
models, and $D=F$. Compared to those works, we work with the germ expansion of
$\pi$ in terms of the $\pi_{P_{\lambda}}$ rather than with Fourier transforms
of nilpotent orbits. We find that it simplifies matter a bit, and it is
coherent with our approach.

###### Proof.

We fix a composition $\lambda=(\lambda_{1},...,\lambda_{r})$ of $n$. We write
$\theta$ for the character $\theta_{\lambda}$ of $U$ and $X$ for the lower
triangular nilpotent matrix in Jordan blocks of size
$\lambda_{1},...,\lambda_{r}$ down the diagonal (if $I$ is the subset of
$\\{1,\ldots,n\\}$ such that $\lambda=\lambda_{I}$, then $X=X_{\delta_{I}}$).
For each positive integer $j$ we define a character $\psi_{j}$ of
$K_{j}=1+M_{n}(P_{D}^{j})$ trivial on $K_{2j}$,

(8.4) $\psi_{j}(1+x)=\psi\circ trd(Xp_{D}^{1-2j}x),\ \ \ x\in
M_{n}(P_{D}^{j}),$

where $\psi$ is a character of $F$ trivial on $P_{F}$ but not on $O_{F}.$ In
other words, $\psi_{j}$ is obtained, in the formula (6.8) for $\xi_{\lambda}$
by replacing the matrix $A$ there with the matrix $X$. We let
$\lambda^{\prime}$ the partition of $n$ obtained from $\lambda$ by putting its
parts in decreasing order, and $B$ the matrix associated as in Lemma 6.2 to
the partition $\lambda^{\prime}\,\hat{}$.

######  Lemma 8.5.

The matrices $B$ and $X$ are conjugate by permutation matrices (corresponding
to permutations of the canonical basis of $D^{n}$).

###### Proof.

A suitable permutation of the canonical basis puts the blocks of $X$ in
decreasing size order, and we get the matrix $X^{\prime}$ analogous to $X$ but
corresponding to $\lambda^{\prime}$. Let us describe a permutation of the
basis which conjugates $X^{\prime}$ to $B$. Let $d$ be the size of the largest
blocks of $X^{\prime}$. Put at the end the first vectors of the blocks of
$X^{\prime}$ of size $d$. Before them, put a bunch of vectors: the images
under $X^{\prime}$ of the previous ones, completed with the first vectors of
the blocks of size $d-1$ of $X^{\prime}$, if any. Once you have the vectors
corresponding to size $i$, put before them the images under $X^{\prime}$ of
the already chosen vectors, completed with the first vectors of the blocks of
size $i-1$. Reaching $i=1$ completes the process. ∎

######  Remark 8.6.

By this lemma, we can apply Lemma 6.2 to $\psi_{j}$. Hence, For any positive
integer $j$, one has $m(\psi_{j},\pi_{P_{\lambda^{\prime}\,\hat{}}})=1$ and
$m(\psi_{j},\pi_{P_{\mu}})=0$ unless $\lambda^{\prime}\,\hat{}\geq\mu$. If
$\lambda^{\prime}\,\hat{}$ is minimal in $\operatorname{Supp}c_{\pi}$, then
for $j\geq j_{0}$ ($j_{0}$ is a positive integer such that the germ expansion
of $\pi$ is valid on $K_{j_{0}}$), we have
$c_{\pi}(\lambda^{\prime}\,\hat{})=m(\psi_{j},\pi)$ is independent of $j\geq
j_{0}$. We recall that $j_{0}$ is a positive integer such that the germ
expansion of $\pi$ is valid on $K_{j_{0}}$.

We now turn to the Whittaker quotient $V_{\theta}$, approaching it (following
Rodier’s initial idea) by a suitable conjugate $\psi^{\prime}_{j}$ of
$\psi_{j}$ and letting $j$ go to infinity.

The diagonal matrix $t=\operatorname{diag}(1,p_{D},...,p_{D}^{n-1})$ acts by
conjugation on $M_{n}(D)$, multiplying the $(a,b)$-coefficient $x$ of a matrix
by $p_{D}^{a}xp_{D}^{-b}$. Conjugating $\psi_{j}$ yields a character
$\psi^{\prime}_{j}$ of the group $K^{\prime}_{j}=t^{2j-1}K_{j}t^{-2j+1}$ which
satisfies also Remark 8.6. The group $U$ is the increasing union of $U\cap
K^{\prime}_{j}$ over $j$, whereas the decreasing subgroups $B^{-}\cap
K^{\prime}_{j}$ have trivial intersection. The restriction of
$\psi^{\prime}_{j}$ to $K^{\prime}_{j}\cap U$ is equal to that of $\theta$,
whereas its restriction to $K^{\prime}_{j}\cap B^{-}$ is trivial. The
multiplication induces a bijection (an Iwahori decomposition):

$(K^{\prime}_{j}\cap U)\times(K^{\prime}_{j}\cap B^{-})\to K^{\prime}_{j}$

The projector $e^{\prime}_{j}:V\to V(\psi^{\prime}_{j})$ of $V$ onto its
$\psi^{\prime}_{j}$-isotypic space $V(\psi^{\prime}_{j})$ (which has dimension
$m(\psi^{\prime}_{j},\pi)=m(\psi_{j},\pi)$) can be obtained by first
projecting onto vectors fixed by $K^{\prime}_{j}\cap B^{-}$, and then applying
the projector $f_{j}$

$f_{j}(v)=\int_{K^{\prime}_{j}\cap U}\theta(u)^{-1}\,uv\,du,\ \ \ v\in V,$

with respect to the Haar measure $du$ giving measure $1$ to
$K^{\prime}_{j}\cap U$.

We write $p:V\to V_{\theta}$ for the projection of $V$ onto $V_{\theta}$ and
$p_{j}:V(\psi^{\prime}_{j})\to V_{\theta}$ for its restriction to
$V(\psi^{\prime}_{j})$.

######  Lemma 8.7.

The map $p_{j}:V(\psi^{\prime}_{j})\to V_{\theta}$ is surjective for large
$j$.

###### Proof.

Let $v\in V$. For large enough $j$, $v\in V^{K^{\prime}_{j}\cap B^{-}}$ hence
$e^{\prime}_{j}(v)=f_{j}(v)$ and $p(e^{\prime}_{j}(v))=p(v)$. Lifting in that
way a basis of the finite-dimensional space $V_{\theta}$ gives the result. ∎

######  Lemma 8.8.

Il $V_{\theta}\neq 0$, then there is a partition $\mu$ in
$\operatorname{Supp}c_{\pi}$ with $\mu\leq\lambda^{\prime}\,\hat{}$.

###### Proof.

Il $V_{\theta}\neq 0$ is not 0, then by Lemma 8.7, $V(\psi^{\prime}_{j})\neq
0$ for large $j$, so $\operatorname{tr}(\pi(e^{\prime}_{j}))\neq 0$. Applying
the germ expansion of $\pi$ to $e^{\prime}_{j}$ there is a minimal partition
$\mu$ of $n$ in $\operatorname{Supp}c_{\pi}$. By Remark 8.6,
$c_{\pi}(\mu)=m(\psi_{j},\pi_{P_{\mu}})$ and $\mu\leq\hat{\lambda^{\prime}}$.
∎

######  Lemma 8.9.

Let $j^{\prime}_{0}=j_{0}+2n-2$. If $\lambda^{\prime}\,\hat{}$ is minimal in
$\operatorname{Supp}c_{\pi}$ and $j\geq j^{\prime}_{0}$, then the endomorphism
$v\to e^{\prime}_{j}e^{\prime}_{j+1}v$ of $V(\psi^{\prime}_{j})$ is a non-zero
homothety.

In [Moeglin-Waldspurger87], that Lemma is given for unspecified large j by
their Lemmas I.13 and I.15. They are rather more involved than Lemme 4 in
[Rodier74], which however applies only to non-degenerate Whittaker models and
$D=F$. The proof of Lemma 8.9 will be given later.

######  Proposition 8.10.

If $\lambda^{\prime}\,\hat{}$ is minimal in $\operatorname{Supp}c_{\pi}$ and
$j\geq j^{\prime}_{0}$, then $p_{j}$ is an isomorphism, so that
$\dim_{\mathbb{C}}(V_{\theta})=\dim_{\mathbb{C}}V(\psi^{\prime}_{j})$.

###### Proof.

We already know by Lemma 8.7 that $p_{j}$ is surjective for $j$ large. We also
know by Remark 8.6 that
$\dim_{\mathbb{C}}V(\psi^{\prime}_{j})=m(\psi^{\prime}_{j},\pi)$ is constant
for$j\geq j_{0}$. The main point is Lemma 8.9 which implies that for $j\geq
j^{\prime}_{0}$, the linear map $q_{j}:V(\psi^{\prime}_{j})\to
V(\psi^{\prime}_{j+1}),v\to v_{1}=e^{\prime}_{j+1}v$ is injective, hence is an
isomorphism because the two spaces have the same dimension. Moreover a vector
$v\in V(\psi^{\prime}_{j})$ is already invariant under $K^{\prime}_{j+1}\cap
B$ so what was said before Lemme 7.7 we have $e^{\prime}_{j+1}v=f_{j+1}v$, and
$v_{1}=e^{\prime}_{j+1}v$ has the same image in $V_{\lambda}$ as $v$.
Iterating the process we get for positive integers $k$, vectors
$v_{k}=e^{\prime}_{j+k}v_{k-1}=f_{j+k}v_{k-1}$. By definition of the projector
$f_{j}$, we have $f_{j+k}f_{j+k-1}=f_{j+k}$ and consequently $v_{k}=f_{j+k}v$.
But $p(v)=0$ if and only if $f_{j+k}v=0$ for large $k$ (Bernstein-Zelevinsky
xyz). As $v_{k}=0$ implies $v_{k-1}=0$ by the injectivity already established,
we get $\operatorname{Ker}(p_{j})=0$. But for large $j$, $p_{j}$ is surjective
so is an isomorphism, and
$\dim_{\mathbb{C}}(V(\psi^{\prime}_{j})=\dim_{\mathbb{C}}(V_{\theta})$. But
for $j\geq j^{\prime}_{0}$, the dimension of $V(\psi^{\prime}_{j})$ is
constant so $p_{j}$ is an isomorphism and the Proposition follows. ∎

Proposition 8.10 implies Part 2 of Theorem 8.2 and that a partition of $n$
which is minimal in $\operatorname{Supp}c_{\pi}$ belongs to the Whittaker
support of $\pi$. Conversely, let $\mu\in\mathfrak{P}(n)$ minimal in the
Whittaker support of $\pi$. Then by Lemma 8.8, there is a partition
$\mu^{\prime}$ in $\operatorname{Supp}c_{\pi}$ with $\mu^{\prime}\leq\mu$, and
we may assume that $\mu^{\prime}$ is minimal in $\operatorname{Supp}c_{\pi}$.
But by Proposition 8.10, that implies that $\mu^{\prime}$ belongs to the
Whittaker support of $\pi$, so $\mu^{\prime}=\mu$. Assuming Lemma 8.9, Theorem
8.2 is proved. ∎

It remains to prove Lemma 8.9. We can conjugate by $t^{1-2j}$ to transform
$\psi^{\prime}_{j}$ back to $\psi_{j}$, and even further conjugate (Lemma 8.5)
by a permutation matrix $\sigma$ to transform $\psi_{j}$ into the character
$\xi_{j}$ attached to the matrix $B$. We need to prove that the endomorphism
of $eV$ sending $v$ to $efv$ is a non-zero homothety, where $e$ is the
$K_{j}$-projector onto the one dimensional space $eV=V(\xi_{j})$ and $f$ is
integration on the group $J=\sigma(t^{2})(K_{j}\cap U)(\sigma(t^{2})^{-1}$
against its character
$(1+x)\mapsto\psi\circ\operatorname{trd}(-B.(p_{D})^{-1-2j}x)$. Clearly $efe$
is an element of $eHe$ where $H$ is the full Hecke algebra of $G$, so we may
restrict the mentioned integration to elements in the support of the Hecke
algebra $eHe$. Also if $j\geq 2n-2$, the group $J$ is contained in
$K_{j-2n+2}$ so it normalizes $K_{j}$, and the support of $efe$ is contained
in the normalizer of $\xi_{j}$ in $K_{j-2n+2}$.

By Lemma 6.3, the normalizer of $\xi_{\lambda}$ in $K_{0}=GL_{n}(O_{D})$ is
$P_{\lambda^{\prime}}(O_{D})K_{j}$. Take $j-2n+2\geq j_{0}$ and $g=1+x$ be in
the support of $efe$. The trace of $ege$ in $eV$ can be computed using the
germ expansion of $\pi$ as the sum over $\mu\in\mathfrak{P}(n)$ of
$c_{\pi}(\mu)$ times the trace of $efe$ in $\pi_{P_{\mu}}$. By our choice
$\lambda^{\prime}\,\hat{}$ is minimal in $\operatorname{Supp}c_{\pi}$, so the
only contribution is $c(\lambda^{\prime}\,\hat{})$. Applying that to any $ege$
in the support of $efe$ gives Lemma 8.9, and even that the homothety is via a
positive integer.

##  9\. Jacquet-Langlands correspondence

The Jacquet-Langlands correspondence extended by Badulescu ([Badulescu07]
Théorème 3.1), is a surjective morphism $LJ$ with a section $JL$

$LJ:\operatorname{Gr}_{\mathbb{C}}^{\infty}(GL_{dn}(F))\to\operatorname{Gr}_{\mathbb{C}}^{\infty}(G),\
\
JL:\operatorname{Gr}_{\mathbb{C}}^{\infty}(G)\to\operatorname{Gr}_{\mathbb{C}}^{\infty}(GL_{dn}(F))$

which is an injective morphism of $\mathbb{Z}$-modules extending the classical
Jacquet-Langlands correspondence between essentially square integrable
representations.

######  Theorem 9.1.

For $\nu\in\operatorname{Gr}_{\mathbb{C}}^{\infty}(GL_{dn}(F))$ and
$\lambda\in\mathfrak{P}(n)$, we have
$(-1)^{n}c_{LJ(\nu)}(\lambda)=(-1)^{dn}c_{\nu}(d\lambda)$.

######  Corollary 9.2.

For $\nu\in\operatorname{Gr}_{\mathbb{C}}^{\infty}(G)$ and
$\lambda\in\mathfrak{P}(n)$, we have
$(-1)^{n}c_{\nu}(\lambda)=(-1)^{dn}c_{JL(\nu)}(d\lambda)$.

The remainer of this section gives the proof of the theorem.

### 9.1.

Badulescu-Jacquet-Langlands correspondence.

####  9.1.1. Preliminaries

Let $\operatorname{Irr}^{2}_{\mathbb{C}}(G)$ denote the set of isomorphism
classes of essentially square integrable irreducible smooth complex
representations of $G$. Any irreducible smooth complex representation of
$D^{*}$ is essentially square integrable.

As in §1, $P_{\lambda}=M_{\lambda}N_{\lambda}$ is a parabolic subgroup of $G$
for $\lambda\in\mathfrak{P}(n)$. For $\mu\in\mathfrak{P}(dn)$, we denote now
by $P_{\mu}=P_{\mu}N_{\mu}$ a parabolic subgroup of $GL_{dn}(F)$ (and not of
$GL_{dn}(D)$).

A basis of the Grothendieck group $\operatorname{Gr}_{\mathbb{C}}^{\infty}(G)$
is

$\mathfrak{B}_{G}=\\{[n.\operatorname{ind}_{P_{\lambda}}^{G}\sigma]\ |\
\sigma\in\operatorname{Irr}^{2}_{\mathbb{C}}(M_{\lambda}),\lambda\in\mathfrak{P}(n)\\}$

where $n.\operatorname{ind}_{P_{\lambda}}^{G}$ is the normalized parabolic
induction ([Badulescu07] Proposition 2.2). As
$\operatorname{Irr}^{2}_{\mathbb{C}}(G)$ is stable by the twist by a smooth
character of $G$,

$\mathfrak{B}^{\prime}_{G}=\\{[\operatorname{ind}_{P_{\lambda}}^{G}\sigma]\ |\
\sigma\in\operatorname{Irr}^{2}_{\mathbb{C}}(M_{\lambda}),\lambda\in\mathfrak{P}(n)\\}$

is also a basis of $\operatorname{Gr}_{\mathbb{C}}^{\infty}(G)$. Let $C_{d}$
be the submodule of $\operatorname{Gr}_{\mathbb{C}}^{\infty}(GL_{dn}(F))$ of
basis

$\mathfrak{B}_{d}^{\prime}=\\{[\operatorname{ind}_{P_{\mu}}^{GL_{dn}(F))}\sigma]\
|\
\sigma\in\operatorname{Irr}^{2}_{\mathbb{C}}(M_{\mu}),\mu\in\mathfrak{P}(dn)\
\text{but}\ \mu\not\in d\mathfrak{P}(n)\\}.$

The Aubert involution $\iota$ of $\operatorname{Gr}_{\mathbb{C}}^{\infty}(G)$
sends an irreducible representation $\pi$ to an irreducible representation
modulo a sign:

(9.1) $\iota(\pi)=(-1)^{n-r}|\iota(\pi)|$

where $|\iota(\pi)|$ is irreducible and $r$ is the number of elements of the
cuspidal support of $\pi$, meaning that
$\pi\subset\operatorname{ind}_{P_{\lambda}}^{G}\sigma$ for
$\lambda=(\lambda_{1},\ldots,\lambda_{r})\in\mathfrak{P}(n)$ and
$\sigma\in\operatorname{Irr}^{2}(M_{\lambda})$ cuspidal ([Badulescu07] (3.4),
[Tadic-90] §1.)

We identify the space $S$ of unitary polynomials in $F[T]$ of degree $dn$ with
$F^{dn}$ by taking the non-dominant coeffficients. The map sending $X\in
M_{n}(D)$ to its reduced characteristic polynomial $P_{X}$ which belongs to
$S$, is continuous ([BourbakiA-8] §17 Définition 1, [Reiner75] §9a).

An element $g\in G$ is called regular semi-simple when the roots of $P_{g}$ in
an algebraic closure $F^{ac}$ of $F$ have multiplicity $1$. The set $G^{rs}$
of regular semi-simple elements of $G$ is open dense in $G$. The conjugacy
class of $g\in G^{rs}$ is the set of elements $g^{\prime}\in G$ with
$P_{g^{\prime}}=P_{g}$. Note that $g=1+p_{F}^{j}X\in G^{rs}$ is conjugate to
an element of $1+p_{F}^{j}M_{n}(O_{D})$ if and only if $X$ is conjugate to an
element of $M_{n}(O_{D})$ if and only the coefficients of
$P_{X}(T)=p_{F}^{-jdn}P_{g}(Tp_{F}^{j}+1)$ belong to $O_{F}$. The set
$\\{P_{g}\ |\ g\in G^{rs}\\}$ consists of the monic polynomials in $F[T]$ of
degree $dn$ without multiple roots in $F^{ac}$, with a non-zero constant term
and with all irreducible factors of degree divisible by $d$. Let
$GL_{dn}(F)^{rs,d}$ be the set of $h\in GL_{dn}(F)^{rs}$ such that
$P_{h}\in\\{P_{g}\ |\ g\in G^{rs}\\}$. We say that $g\in G^{rs}$ and $h\in
GL_{dn}(F)^{rs,d}$ correspond and we write $g\leftrightarrow h$ when
$P_{g}=P_{h}$.

Let $g\in G^{rs}$. The $G$-centralizer $T_{g}$ of $g$ is a maximal torus,
isomorphic to the group of units of $F[T]/(P_{g})$. We put on $G/T_{g}$ the
quotient measure $dx^{*}$ of the Haar measure on $G$ (§1) and on the Haar
measure on $T_{g}$ giving the value $1$ to the maximal torus. The orbital
integral of $f\in C_{c}^{\infty}(G;\mathbb{C})$ at $g$ is

(9.2) $\Phi(f,g)=\int_{G/T_{g}}f(xgx^{-1})\,dx^{*}.$

Let $C_{c}^{\infty}(GL_{dn}(F)^{rs};\mathbb{C})^{(d)}$ be the set of
$\varphi\in C_{c}^{\infty}(GL_{nd}(F)^{rs};\mathbb{C})$ with
$\Phi(\varphi,h)=0$ when $h$ is not in $GL_{nd}(F)^{rs,d}$. We say that $f\in
C_{c}^{\infty}(G^{rs};\mathbb{C})$ and $\varphi\in
C_{c}^{\infty}(GL_{nd}(F)^{rs};\mathbb{C})^{(d)}$ correspond and we write
$f\leftrightarrow\varphi$ when $\Phi(f,g)=\Phi(\varphi,h)$ if $g\in G^{rs}$
and $h\in GL_{nd}(F)^{rs,d}$ correspond. For $f\in
C_{c}^{\infty}(G^{rs};\mathbb{C})$ there exists $\varphi\in
C_{c}^{\infty}(GL_{nd}(F)^{rs};\mathbb{C})^{(d)}$ such that
$f\leftrightarrow\varphi$, and conversely ([Bad18] Proposition 5.1).

####  9.1.2. Jacquet-Langlands correspondence

The classical Jacquet-Langlands correspondence ([DKV84], [Badulescu02]) is the
unique bijective map

$\displaystyle
JL:\operatorname{Irr}^{2}_{\mathbb{C}}(G))\to\operatorname{Irr}^{2}_{\mathbb{C}}(GL_{dn}(F))\
\ \ \text{ such that for }\ \pi\in\operatorname{Irr}^{2}_{\mathbb{C}}(G),$
$\displaystyle(-1)^{n}\,trace(\pi(f(g)dg))=(-1)^{dn}\,trace(JL(\pi)(\varphi(h)dh))$

when $f\in C_{c}^{\infty}(G);\mathbb{C})^{rs},\ \varphi\in
C_{c}^{\infty}(GL_{dn}(F);\mathbb{C})^{rs,d},\ f\leftrightarrow\varphi$. The
image by $JL$ of the Steinberg representation of $G$ is the Steinberg
representation of $GL_{dn}(F))$. The maps $JL$ extends to

1) a bijective map

$JL:\operatorname{Irr}^{2}_{\mathbb{C}}(M_{\lambda}))\to\operatorname{Irr}^{2}_{\mathbb{C}}(M_{d\lambda})\
\text{for any composition $\lambda$ of $n$.}$

2) an injective map

$JL:\mathfrak{B}_{G}\to\mathfrak{B}_{GL_{dn}(F)}$ (9.3)
$JL([n.\operatorname{ind}_{P_{\lambda}}^{G}\sigma]=[n.\operatorname{ind}_{P_{d\lambda}}^{GL_{dn}(F)}JL(\sigma)]\
\text{for}\
\sigma\in\operatorname{Irr}^{2}_{\mathbb{C}}(M_{\lambda}),\lambda\in\mathfrak{P}(n),$

and by linearity to an injective homomorphism

$JL:\operatorname{Gr}_{\mathbb{C}}^{\infty}(G)\to\operatorname{Gr}_{\mathbb{C}}^{\infty}(GL_{dn}(F)),$

satisfying ([Badulescu07] Théorème 3.1):

(9.4)
$(-1)^{n}\,\operatorname{trace}\nu(f(g)dg)=(-1)^{dn}\,\operatorname{trace}JL(\nu)(\varphi(h)dh)$

when $f\in C_{c}^{\infty}(GL_{n}(D)^{rs};\mathbb{C}),\ \varphi\in
C_{c}^{\infty}(GL_{dn}(F)^{rs};\mathbb{C})^{(d)},\ f\leftrightarrow\varphi$.
We have

$\operatorname{Gr}_{\mathbb{C}}^{\infty}(GL_{dn}(F))=JL(\operatorname{Gr}_{\mathbb{C}}^{\infty}(G))\oplus
C_{d}.$

The homomorphism $JL$ commutes with

a) the twist by smooth characters:

$JL((\chi\circ\operatorname{nrd})\otimes\nu)=(\chi\circ\det)\otimes JL(\nu)$
when $\chi$ is a smooth character of $F^{*}$,

b) the normalized parabolic induction ([Badulescu07] Théorème 3.6):

$JL(\operatorname{ind}_{P_{\lambda}}^{G}(\delta_{\lambda}^{1/2}\nu)=\operatorname{ind}_{P_{d\lambda}}^{GL_{dn}(F)}(\delta_{d\lambda}^{1/2}JL(\nu))$).

3) a surjective homorphism extending the inverse $LJ$ of the classical
Jacquet-Langlands correspondence $JL$ for the Levi subgroups :

$LJ:\mathfrak{B}_{GL_{dn}(F)}\to\mathfrak{B}_{G}$ (9.5)
$LJ([n.\operatorname{ind}_{P_{\mu}}^{{}^{GL_{dn}(F)}}\sigma]=\begin{cases}[n.\operatorname{ind}_{P_{\lambda}}^{G}LJ(\sigma)]&\
\text{for}\ \sigma\in\operatorname{Irr}^{2}_{\mathbb{C}}(M_{\mu}),\
\mu=d\lambda\in d\mathfrak{P}(n),\\\ 0&\ \text{for}\
\sigma\in\operatorname{Irr}^{2}_{\mathbb{C}}(M_{\mu}),\
\mu\in\mathfrak{P}(dn)\ \text{but}\ \mu\not\in d\mathfrak{P}(n)\end{cases}$

giving by linearity a surjective homomorphism (the Badulescu-Jacquet-Langlands
correspondence):

$LJ:\operatorname{Gr}_{\mathbb{C}}^{\infty}(GL_{dn}(F))\to\operatorname{Gr}_{\mathbb{C}}^{\infty}(G)$

of kernel $C_{d}$, section $JL$, satisfying

(9.6)
$(-1)^{dn}\,\operatorname{trace}\nu(f(g)dg)=(-1)^{n}\,\operatorname{trace}LJ(\nu)(\varphi(h)dh)$

when $f\in C_{c}^{\infty}(GL_{dn}(F)^{rs};\mathbb{C}),\ \varphi\in
C_{c}^{\infty}(G^{rs};\mathbb{C})^{(d)},\ f\leftrightarrow\varphi$. The
homorphism $LJ$ commutes with the parabolic induction, is compatible with the
Aubert involution $\iota$ [Aubert-95] up to a sign ([Badulescu07] Proposition
3.16):

(9.7) $(-1)^{n}\iota\circ LJ=LJ\circ(-1)^{dn}\iota.$

As $LJ$ sends the Steinberg representation of $GL_{dn}(F)$ to the Steinberg
representation of $G$, the Aubert involution of the Steinberg representation
is the trivial representation up to a sign, and $LJ$ commutes with the
parabolic induction, we have:

(9.8) $\
(-1)^{nd}\,LJ(\pi_{P_{\mu}})=\begin{cases}(-1)^{n}\,\pi_{P_{\lambda}}&\
\text{if}\ \mu=d\lambda,\\\ 0\ &\ \text{otherwise}.\end{cases}$

### 9.2.

The theorem 9.1 is an easy consequence of (9.6), (9.8), and of the linear
independance of the restrictions to $K\cap GL_{n}(D)^{rs}$ of the characters
of the representations $\pi_{P_{\mu}}$ of $GL_{dn}(F)$ for
$\mu\in\mathfrak{P}(dn)$, for any open compact subgroup $K$ of $GL_{dn}(F)$.

We give the details. Let $P=MN$ be a parabolic subgroup of $G$ of Levi $M$,
$\sigma\in\operatorname{Irr}^{2}_{\mathbb{C}}(M)$,
$\pi=\operatorname{ind}_{P}^{G}\sigma$. Let $c_{\pi},c_{JL(\pi)}$ be the maps
and $K_{\pi},K_{JL(\pi)}$ groups in the germ expansions (6.1) of
$[\pi],JL([\pi])$, such that for any $g\in K_{\pi}\cap G^{rs}$ there exists
$h\in K_{JL(\pi)}\cap GL_{dn}(F)^{rs,d}$ with $g\leftrightarrow h$, as we can
because for $g\in GL_{n}(D)^{rs}$, $h\in GL_{dn}(F)^{rs,d}$ with the same
reduced characteristic polynomial $P(T)$, the coefficients of
$p_{F}^{-jdn}P(Tp_{F}^{j}+1)$ belong to $O_{F}$ if and only if $g$ is
conjugate to an element of $1+p_{F}^{j}M_{n}(O_{D})$ if and only if $h$ is
conjugate to element of $1+p_{F}^{j}M_{dn}(O_{F})$. By (9.6) and the germ
expansion (6.1),

$(-1)^{n}\,\sum_{\lambda\in\mathfrak{P}(n)}c_{LJ(\pi)}(\lambda)\,\operatorname{trace}\pi_{P_{\lambda}}(f(g)dg)=(-1)^{dn}\,\sum_{\mu\in\mathfrak{P}(dn)}c_{\pi}(\mu)\,\operatorname{trace}\pi_{P_{\mu}}(\varphi(g)dg)$
$=(-1)^{n}\,\sum_{\mu\in\mathfrak{P}(dn)}c_{\pi}(\mu)\,\operatorname{trace}LJ(\pi_{P_{\mu}})(f(g)dg)=(-1)^{dn}\,\sum_{\lambda\in\mathfrak{P}(n)}c_{\pi}(d\lambda)\,\operatorname{trace}\pi_{P_{\lambda}}(f(g)dg)$

when $f\in C_{c}^{\infty}(K_{\pi}\cap G^{rs};\mathbb{C}),\ \varphi\in
C_{c}^{\infty}(K_{JL(\pi)}\cap GL_{dn}(F)^{rs};\mathbb{C})^{(d)},\
f\leftrightarrow\varphi$. The linear independence of the characters of
$\pi_{P_{\lambda}}$ on $K_{LJ(\pi)}\cap G^{rs}$ for
$\lambda\in\mathfrak{P}(n)$ imply

$(-1)^{dn}c_{\pi}(\lambda)=(-1)^{n}c_{LJ(\pi)}(d\lambda)\ \ \text{for }\
\lambda\in\mathfrak{P}(n),$

for any $[\pi]$ in the basis $\mathfrak{B}_{G}$ of
$\operatorname{Gr}_{\mathbb{C}}^{\infty,f}(G)$. This ends the proof of the
theorem 9.1.

### 9.3.

Applications to $c_{\pi}((n))$ For
$\pi\in\operatorname{Irr}^{2}_{\mathbb{C}}(G)$ and a division central
$F$-algebra $D_{dn}$ of reduced degree $dn$, there exists a unique
$\pi_{dn}\in\operatorname{Irr}_{\mathbb{C}}(D_{dn}^{*})$ such that their
images by the classical Jacquet-Langlands correspondence in
$\operatorname{Irr}_{\mathbb{C}}^{2}(GL_{dn}(F))$ are equal. The dimension of
$\pi_{dn}$ is finite and by Theorem 9.1)
$(-1)^{n}c_{\pi}(n)=-c_{JL(\pi_{dn})}(dn)=-\dim_{\mathbb{C}}\pi_{dn}$. An
irreducible smooth complex representation $\pi$ of $G$ is tempered if and only
if $\pi=\operatorname{ind}_{P}^{G}\sigma$ for a parabolic subgroups $P=MN$ of
$G$ and $\sigma\in\operatorname{Irr}^{2}_{\mathbb{C}}(M)$ ([Lapid-Minguez-
Tadic16] A.11).

For $\nu\in\operatorname{Gr}_{\mathbb{C}}^{\infty}(GL_{dn}(F))$ and
$\lambda\in\mathfrak{P}(n)$, we have
$(-1)^{n}c_{LJ(\nu)}(\lambda)=(-1)^{dn}c_{\nu}(d\lambda)$.

######  Corollary 9.3.

Let $\pi\in\operatorname{Rep}_{\mathbb{C}}^{\infty}(G)$ irreducible and
tempered. Then

$c_{\pi}((n))=\begin{cases}(-1)^{n-1}\,\dim_{\mathbb{C}}\pi_{dn}&\text{if}\
\pi\in\operatorname{Irr}^{2}_{\mathbb{C}}(G)\\\ 0&\text{if}\
\pi\not\in\operatorname{Irr}^{2}_{\mathbb{C}}(G)\end{cases}.$

##  10\. Coefficient field of characteristic different from $p$

Let $R$ be a field. The Harish-Chandra germ expansion for complex
representations does not extend to $R$-representations. Our goal is to show
that Theorem 1.3 proved using the Harish-Chandra germ expansion remain valid
for $R$-representations when the characteristic of $R$ is not $p$. There are
two simple reasons:

For a parabolic subgroup $P$ of $G$, the representation
$\operatorname{ind}_{P}^{G}1$ is defined over $\mathbb{Z}$.

For a field extension $R^{\prime}/R$, the scalar extension from $R$ to
$R^{\prime}$ respects finite length, and for an irreducible smooth
$R$-representation $\pi$ of a profinite group, $R^{\prime}\otimes_{R}\pi$ is
$\pi$-isotypic.

From now on, $\operatorname{char}_{R}\neq p$. When
$\pi\in\operatorname{Rep}_{R}^{\infty,f}$ is equal to
$\sum_{\lambda\in\mathfrak{P}(n)}c_{\pi}(\lambda)\operatorname{ind}_{P_{\lambda}}^{G}1$
on $K_{\pi}$ as in Theorem 1.3, the map $c_{\pi}$ is unique because:

######  Proposition 10.1 (Corollary 5.6).

Let $K$ be an open pro-$p$ subgroup of $G$. For any non zero map
$c:\mathfrak{P}(n)\to\mathbb{Z}$, the restriction to $K$ of

$\sum_{\lambda\in\mathfrak{P}(n)}c(\lambda)\,[\pi_{P_{\lambda}}]\in\operatorname{Gr}_{R}^{\infty}(G)$

is not $0$.

###### Proof.

We can suppose $R$ algebraically closed. The categories
$\operatorname{Rep}_{R}^{\infty}(K)$ and
$\operatorname{Rep}_{\mathbb{C}}^{\infty}(K)$ are equivalent and the
Grothendieck groups $\operatorname{Gr}_{R}^{\infty}(K)$ and
$\operatorname{Gr}_{\mathbb{C}}^{\infty}(K)$ are isomorphic because $K$ is a
pro-$p$ group and $\operatorname{char}_{R}\neq p$. The proposition is true
when $R=\mathbb{C}$ (Corollary 5.6) and the representations
$\pi_{P_{\lambda}}$ correspond. Hence the proposition is true. ∎

We list other properties which will be used in the proof of the theorem 1.3.

### 10.1.

Twist by a character, image by an automorphism

Assume that $\pi\in\operatorname{Rep}_{R}^{\infty,f}(G)$ has a germ expansion
of map $c_{\pi}$ on $K_{\pi}$ (the first part of Theorem 1.3), $\chi$ is a
smooth $R$-character of $G$ and $\sigma$ is an automorphism of $R$. Then the
representations $\pi\otimes\chi$ and $\sigma(\pi)$ have a germ expansion of
maps $c_{\pi\otimes\chi}=c_{\sigma(\pi)}=c_{\pi}$ on
$K_{\pi\otimes\chi}=K_{\sigma(\pi)}=K_{\pi}$ if $\chi$ is trivial on
$K_{\pi}$. The reason is that
$\sum_{\lambda\in\mathfrak{P}(n)}c_{\pi}(\lambda)\,[\pi_{P_{\lambda}}]$ is
defined over $\mathbb{Z}$.

### 10.2.

Germ expansion on the Grothendieck group Assume that any
$\pi\in\operatorname{Rep}_{R}^{\infty,f}(G)$ has a germ expansion of map
$c_{\pi}$ on some open open compact .subgroup $K_{\pi}$ of $G$. Then, the
linear map $\nu\mapsto
c_{\nu}:\operatorname{Gr}_{R}^{\infty}(G)\to\\{\mathfrak{P}(n)\to\mathbb{Z}\\}$
such that $c_{[\pi]}=c_{\pi}$ for
$\pi\in\operatorname{Rep}_{R}^{\infty,f}(G)$, has the property that the
restrictions to some open compact subgroup $K_{\nu}$ of $G$ of $\nu$ and of
$\sum_{\lambda\in\mathfrak{P}(n)}c_{\nu}(\lambda)\,[\pi_{P_{\lambda}}]$ are
isomorphic.

Parabolic induction: For a parabolic subgroup $P$ of $G$ of Levi $M$, the
parabolic induction $\operatorname{ind}_{P}^{G}$ is exact and respects finite
length and passes to a linear map between the Grothendieck groups:

$\operatorname{ind}_{P}^{G}:\operatorname{Gr}_{R}^{\infty}(M)\to\operatorname{Gr}_{R}^{\infty}(G),\
\ \ \operatorname{ind}_{P}^{G}[\sigma]=[\operatorname{ind}_{P}^{G}\sigma]\
\text{for }\sigma\in\operatorname{Rep}_{R}^{\infty,f}(M).$

When $\nu\in\operatorname{Gr}_{R}^{\infty}(M)$ has a germ expansion of map
$c_{\nu}$, then $\operatorname{ind}_{P}^{G}\nu$ has a germ expansion of map
induced by $c_{\nu}$ (Corollary 7.2).

### 10.3.

For $j\in\mathbb{N}_{>0}$ and $\lambda$ is a composition of $n$, the values of
the character $\xi_{\lambda}$ of $K_{j}=1+M_{n}(P_{D}^{j})$ defined by (6.8)
and of the character $\theta_{\lambda}$ of $U$ defined by (8.3) are roots of
$1$ of order $p^{m}$ for any $m\in\mathbb{N}$. Assume that the field $R$
contains roots of unity of any $p$-power order,

$\text{we write}\ \mu_{p^{\infty}}\subset R,\ \text{ implying}\
\operatorname{char}_{R}\neq p.$

We can define $\xi_{\lambda}$ and $\theta_{\lambda}$ over $R$ as before, and
the Whittaker support of an irreducible smooth $R$-representation of $G$ as
before Theorem 8.2.

Let $\pi\in\operatorname{Rep}_{R}^{\infty,f}(G)$ having a germ expansion of
map $c_{\pi}:\mathfrak{P}(n)\mapsto\mathbb{Z}$: for a positive integer $j_{0}$
the restriction of $\pi$ and of
$\sum_{\lambda\in\mathfrak{P}(n)}c_{\pi}(\lambda)\operatorname{ind}_{P_{\lambda}}^{G}1$
to $K_{j_{0}}$ are equal. With the same proofs as for $R=\mathbb{C}$, we have:

######  Theorem 10.2.

1) For any integer $j\geq j_{0}$ and any $\lambda$ partition of $n$, we have

(10.1) $\displaystyle c_{\pi}(\lambda)$
$\displaystyle=m(\xi_{\lambda},\pi)-\sum_{\mu\in\mathfrak{P}(n),\mu<\lambda}c_{\pi}(\mu)\,m(\xi_{\lambda},\pi_{P_{\mu}}).$

In particular when $\lambda$ is minimal in the support of $c_{\pi}$,
$c_{\pi}(\lambda)=m(\xi_{\lambda},\pi)$ is positive and independent of $j\geq
j_{0}$.

2) Theorem 8.2 is valid.

An algebraically closed field $R$ with $\operatorname{char}_{R}\neq p$
contains $\mu_{p}^{\infty}$. To prove Theorem 1.3 for $R$ algebraically
closed, by Theorem 10.2 and Proposition 10.1, we have only to prove that any
$\pi\in\operatorname{Rep}_{R}^{\infty,f}(G)$ has a germ expansion: there
exists a map $c_{\pi}:\mathfrak{P}(n)\to\mathbb{Z}$ such that $\pi$ and
$\sum_{\lambda}c_{\pi}(\lambda)\operatorname{ind}_{P_{\lambda}}^{G}1$ have
equal on some open compact subgroup $K_{\pi}$ of $G$.

We prove now Theorem 1.3 going from $R=\mathbb{C}$ to
$R=\mathbb{Q}_{\ell}^{ac}$ to $R=\mathbb{F}_{\ell}^{ac}$, $\ell\neq p$, to an
algebraically closed field $R$, and finally to a not necessarily algebraically
closed field $R$.

### 10.4.

$R\simeq R^{\prime}$. For any prime number $\ell$, the fields $\mathbb{C}$ and
$\mathbb{Q}_{\ell}^{ac}$ are isomorphic. It is easy to see that if Theorem 1.3
is true for a field $R$, it is also true for an isomorphic field $R^{\prime}$.
Indeed, a field isomorphism $j:R\to R^{\prime}$ induces isomorphisms of
categories
$j_{G}:\operatorname{Rep}^{\infty}_{R}(G)\to\operatorname{Rep}^{\infty}_{R^{\prime}}(G)$
and
$j_{K}:\operatorname{Rep}^{\infty}_{R}(K)\to\operatorname{Rep}^{\infty}_{R^{\prime}}(K)$
for any open compact subgroup $K$ of $G$. The isomorphisms commute with the
restriction to $K$ and the parabolic induction $\operatorname{ind}_{P}^{G}$.
For $\pi\in\operatorname{Rep}^{\infty}_{R}(G)$ and
$\sigma\in\operatorname{Rep}^{\infty}_{R}(M)$,

$j_{K}(\pi|_{K})=j_{G}(\pi)|_{K},\ \
\operatorname{ind}_{P}^{G}(j_{M}(\sigma))=j_{G}(\operatorname{ind}_{P}^{G}\sigma).$

When the theorems are true for $R$ they are also true for $R^{\prime}$. For
$\pi\in\operatorname{Rep}^{\infty,f}_{R}(G)$, then $c_{\pi}=c_{j_{G}(\pi)}$
and we can take $K_{j_{G}(\pi)}=K_{\pi}$.

### 10.5.

$R\simeq\mathbb{F}_{\ell}^{ac}$ for $\ell\neq p$.

The theorems over $\mathbb{Q}_{\ell}^{ac}$ imply the theorem over
$\mathbb{F}_{\ell}^{ac}$ by reduction modulo $\ell$ for $\ell\neq p$. We
denote by $\mathbb{Z}_{\ell}^{ac}$ the ring of integers of
$\mathbb{Q}_{\ell}^{ac}$. A lattice in a $\mathbb{Q}_{\ell}^{ac}$-vector space
$V$ is a free $\mathbb{Z}_{\ell}^{ac}$-module of generated by a
$\mathbb{Q}_{\ell}^{ac}$-basis of $V$.

Let $\pi\in\operatorname{Rep}_{\mathbb{Q}_{\ell}^{ac}}^{\infty,f}(G)$. One
says that $\pi$ is integral when the space of $\pi$ contains a $G$-stable
lattice $\mathfrak{L}_{\pi}$. Then, the reduction modulo $\ell$ of
$\mathfrak{L}_{\pi}$ equal to
$\mathbb{F}_{\ell}^{ac}\otimes_{\mathbb{Z}_{\ell}^{ac}}\mathfrak{L}_{\pi}$
belongs to $\operatorname{Rep}_{\mathbb{F}_{\ell}^{ac}}^{\infty,f}(G)$ and its
image in the Grothendieck group
$\operatorname{Gr}_{\mathbb{F}_{\ell}^{ac}}^{\infty}(G)$ does not depend on
the choice of $\mathfrak{L}_{\pi}$. It is called the reduction modulo $\ell$
of $\pi$, and denoted by $r_{\ell}(\pi)$. The subcategory of integral
representations
$\operatorname{Rep}_{\mathbb{Q}_{\ell}^{ac}}^{\infty,f,int}(G)$ in
$\operatorname{Rep}_{\mathbb{Q}_{\ell}^{ac}}^{\infty,f}(G)$ is abelian; let
$\operatorname{Gr}_{\mathbb{Q}_{\ell}^{ac}}^{\infty,int}(G)$ be its
Grothendieck group. The reduction modulo $\ell$ passes to a surjective (not
injective) map between the Grothendieck groups:

$r_{\ell}:\operatorname{Gr}_{\mathbb{Q}_{\ell}^{ac}}^{\infty,int}(G)\to\operatorname{Gr}_{\mathbb{F}_{\ell}^{ac}}^{\infty}(G),$

and there is an explicit subset $E(G)$ of
$\operatorname{Rep}_{\mathbb{Q}_{\ell}^{ac}}^{\infty,f,int}(G)$ such that the
set $\\{r_{\ell}(\pi)\ |\ \pi\in E(G)\\}$ is a basis of the Grothendieck group
$\operatorname{Gr}_{\mathbb{F}_{\ell}^{ac}}^{\infty}(G)$ ([Minguez-Sécherre14]
Théorème 9.35).

For a parabolic subgroup $P$ of $G$ with Levi $M$, the parabolic induction
$\operatorname{ind}_{P}^{G}:\operatorname{Rep}_{\mathbb{Q}_{\ell}^{ac}}^{\infty}(M)\to\operatorname{Rep}_{\mathbb{Q}_{\ell}^{ac}}^{\infty}(G)$
is exact, respects finite length and integrality hence passes to the
Grothendieck groups and
$r_{\ell}\circ\operatorname{ind}_{P}^{G}=\operatorname{ind}_{P}^{G}\circ
r_{\ell}$ on $\operatorname{Rep}_{\mathbb{Q}_{\ell}^{ac}}^{\infty,f,int}(M)$.

The representation $\pi_{P}$ over $\mathbb{Q}_{\ell}^{ac}$ are integral, with
a canonical integral structure (the functions with values in
$\mathbb{Z}_{\ell}^{ac}$: $\pi_{P}$ over $\mathbb{Z}_{\ell}^{ac}$) of
reduction modulo $\ell$ the representation $\pi_{P}$ over
$\mathbb{F}_{\ell}^{ac}$

If $\pi\in\operatorname{Rep}_{\mathbb{Q}_{\ell}^{ac}}^{\infty,f,int}(G)$ has a
germ expansion of map $c_{\pi}$ on $K_{\pi}$, then
$r_{\ell}(\pi)\in\operatorname{Gr}_{\mathbb{F}_{\ell}^{ac}}^{\infty}(G)$ has a
germ expansion of map $c_{\pi}$ on $K_{\pi}$.

######  Lemma 10.3.

Let
$\pi,\pi^{\prime}\in\operatorname{Rep}_{\mathbb{Q}_{\ell}^{ac}}^{\infty,f,int}(G)$
with $r_{\ell}(\pi)=r_{\ell}(\pi^{\prime})$. Then $c_{\pi}=c_{\pi^{\prime}}.$

###### Proof.

When $j$ is large, we have (10.1) for $\pi$ and $\pi^{\prime}$. As $K_{j}$ is
a pro-$p$ group,
$m(\xi_{\lambda},\pi)=m(r_{\ell}(\xi_{\lambda}),r_{\ell}(\pi)).$ Therefore
$r_{\ell}(\pi)=r_{\ell}(\pi^{\prime})$ implies
$m(\xi_{\lambda},\pi)=m(\xi_{\lambda},\pi^{\prime})$. By induction on
$\lambda$ we deduce $c_{\pi}=c_{\pi^{\prime}}$. ∎

As the $r_{\ell}(\pi)$ for $\pi\in E(G)$ generate
$\operatorname{Gr}_{\mathbb{F}_{\ell}^{ac}}^{\infty}(G)$, Lemma 10.3 gives the
existence of a linear map

$c:\operatorname{Gr}_{\mathbb{F}_{\ell}^{ac}}^{\infty}(G)\to\\{\mathfrak{P}(n)\to\mathbb{Z}\\}\
\text{ defined by }c_{r_{\ell}(\pi)}=c_{\pi}\ \text{ for
$\pi\in\operatorname{Rep}_{\mathbb{Q}_{\ell}^{ac}}^{\infty,f,int}(G)$}.$

For $\pi\in\operatorname{Rep}_{\mathbb{F}_{\ell}^{ac}}^{\infty,f}(G)$, the
restrictions of $\pi$ and of
$\sum_{\lambda\in\mathfrak{P}(n)}c_{\pi}(\lambda)\,r_{\ell}(\pi_{P_{\lambda}})$
to some open pro-$p$ group $K_{\pi}$ of $G$ are isomorphic. Theorem 1.3 when
$R=\mathbb{F}_{\ell}^{ac}$ is proved.

### 10.6.

$R^{\prime}/R$ algebraically closed fields Given an extension $R^{\prime}/R$
of algebraically closed fields of characteristic different from $p$, we prove
that the germ expansion over $R$ for all $n\geq 1$ is equivalent to the germ
expansion over $R^{\prime}$ for all $n\geq 1$. Therefore we get Theorem 1.3
over any algebraically closed field $R$, because we already proved for
$R=\mathbb{C}$ and $R=\mathbb{F}_{\ell}^{ac}$ when $\ell\neq p$.

The proof relies on properties, that we now recall, of the scalar extension
$\pi\mapsto
R^{\prime}\otimes_{R}\pi:\operatorname{Rep}_{R}^{\infty}(G)\to\operatorname{Rep}_{R^{\prime}}^{\infty}(G)$
from $R$ to $R^{\prime}$ and of the representations of $G$ parabolically
induced from Speh representations of the Levi subgroups of $G$. Fix the same
square root of $q=p^{f}$ in $R$ and in $R^{\prime}$, when $f$ is odd.

The scalar extension from $R$ to $R^{\prime}$ respects irreducible
representations, is exact and passes to an injective linear map $\nu\mapsto
R^{\prime}\otimes_{R}\nu:\operatorname{Gr}_{R}^{\infty}(G)\to\operatorname{Gr}_{R^{\prime}}^{\infty}(G)$
between the Grothendieck groups, commutes with the parabolic induction and for
any open pro-$p$ subgroup $K$ of $G$ is an isomorphism of categories
$\delta\mapsto
R^{\prime}\otimes_{R}\delta:\operatorname{Rep}_{R}^{\infty}(K)\to\operatorname{Rep}_{R^{\prime}}^{\infty}(K)$.
When $\pi\in\operatorname{Rep}_{R}^{\infty,f}(G)$ the multiplicity
$m(\delta,\pi)$ in $\pi$ of $\delta\in\operatorname{Rep}_{R}^{\infty}(K)$
irreducible is equal to
$m(R^{\prime}\otimes_{R}\delta,R^{\prime}\otimes_{R}\pi)$.

Any irreducible cuspidal $R^{\prime}$-representation $\rho^{\prime}$ of $G$ is
the twist by an unramified smooth $R^{\prime}$-character $\chi$ of $G$ of an
irreducible cuspidal $R$-representation $\rho$ of $G$,
$\rho^{\prime}=\chi\otimes(R^{\prime}\otimes_{R}\rho)=\chi\otimes_{R}\rho$
[Vigneras96]. By Lemma 10.4 below, this is also true Speh representations.

Let $m$ be a divisor of $n$ and $\rho$ an irreducible cuspidal
$R$-representation of $GL_{m}(D))$. To $(\rho,n)$ are attached

  * •

an unramified smooth $R$-character $\chi_{\rho}$ of $GL_{m}(D)$,

  * •

a cuspidal $R$-segment
$\Delta_{\rho,n}=(\rho,\chi_{\rho}\otimes\rho,\ldots,\chi_{\rho}^{-1+n/m}\otimes\rho)$
of length $n/m$,

  * •

an irreducible representation
$Z(\Delta_{\rho,n})\in\operatorname{Rep}_{R}^{\infty}(GL_{n}(D))$ (a Speh
representation) equal to the socle of the normalized parabolic induction
$\rho\times\ldots\times(\chi_{\rho}^{-1+n/m}\otimes\rho)$ of
$\rho\otimes\ldots\otimes(\chi_{\rho}^{-1+n/m}\otimes\rho)\in\operatorname{Rep}_{R}^{\infty}M_{\lambda}$
for $\lambda=(m,\ldots,m)\in\mathfrak{P}(n)$.

Any smooth $R^{\prime}$-character of $G=GL_{n}(D)$ is of the form
$\chi_{n}=\chi\otimes\operatorname{nrd}$ for a smooth $R^{\prime}$-character
of $F^{*}$.

######  Lemma 10.4.

For a smooth $R^{\prime}$-character $\chi_{n}$ of $G$ we have
$\chi_{n}\otimes_{R}Z(\Delta_{\rho,n})\simeq
Z(\Delta_{\chi_{m}\otimes_{R}\rho,n})$.

###### Proof.

It is easy to see on the definition that the map $\rho\mapsto
Z(\Delta_{\rho},n)$ for $\rho\in\operatorname{Rep}_{R}(GL_{m}(D)$ irreducible
cuspidal, commutes with the scalar extension from $R$ to $R^{\prime}$. The
$R^{\prime}$-representation $\rho^{\prime}=R^{\prime}\otimes_{R}\rho$ of
$GL_{m}(D)$ is irreducible cuspidal, $\chi_{\rho}=\chi_{\rho^{\prime}}$, and
$R^{\prime}\otimes_{R}Z(\Delta_{\rho,n})=Z(\Delta_{\rho^{\prime},n})$. The
twist by a character commutes with parabolic induction and the twist of a
representation $\pi$ of $G$ by a character gives an isomorphism of the lattice
of the subrepresentations of $\pi$ onto the lattice of subrepresentations of
the twist $\pi$. The twist of
$\rho^{\prime}\times\ldots\times(\chi_{\rho}^{-1+n/m}\otimes\rho^{\prime})$ by
$\chi_{n}$ is
$(\chi_{m}\otimes\rho^{\prime})\times\ldots\times(\chi_{m}\chi_{\rho}^{-1+n/m}\otimes\rho^{\prime})$.
As $Z(\Delta_{\rho^{\prime},n)}$ is the socle of
$\rho^{\prime}\times\ldots\times(\chi_{\rho}^{-1+n/m}\otimes\rho^{\prime})$ we
deduce that $\chi_{n}\otimes
Z(\Delta_{\rho^{\prime},n})=Z(\Delta_{\chi_{m}\otimes\rho^{\prime},n})$. ∎

To a composition $(n_{1},\ldots,n_{r})$ of $n$, a divisor $m_{i}$ of $n_{i}$
and an irreducible cuspidal $R$-representation $\rho_{i}$ of $GL_{m_{i}}(D))$
for $1\leq i\leq r$, are associated

  * •

a cuspidal $R$-multisegment
$\mathfrak{M}=(\Delta_{\rho_{1},n_{1}},\ldots,\Delta_{\rho_{r},n_{r}})$,

  * •

a Speh $R$-representation
$Z(\mathfrak{M})=Z(\Delta_{\rho_{1},n_{1}})\otimes\ldots
Z(\Delta_{\rho_{r},n_{r}})$ of $M=M_{(n_{1},\ldots,n_{r})}$,

  * •

the normalized parabolic induction
$n.I(\mathfrak{M})=\operatorname{ind}_{P}^{G}(Z(\mathfrak{M})\delta_{P}^{1/2})$
of $Z(\mathfrak{M})$ where $P=P_{(n_{1},\ldots,n_{r})}$ and $\delta_{P}$ is
the module of $P$.

The Grothendieck group $\operatorname{Gr}_{R}^{\infty}(G)$ is generated by the
$[n.I(\mathfrak{M})]$ for the cuspidal $R$-multisegments $\mathfrak{M}$ of
$GL_{n}(D)$ ([Minguez-Sécherre14] proof of Lemma 9.36 with Proposition 9.29).

But $Z(\mathfrak{M})\delta_{P}^{1/2}$ is also a Speh representation
$Z(\mathfrak{M}^{\prime})=Z(\Delta_{\rho^{\prime}_{1},n_{1}})\otimes\ldots
Z(\Delta_{\rho^{\prime}_{r},n_{r}})$ where $\rho^{\prime}_{i}$ is the twist of
$\rho_{i}$ by an unramified character. Therefore
$\operatorname{Gr}_{R}^{\infty}(G)$ is also generated by the images of the
parabolic induction
$I(\mathfrak{M})=\operatorname{ind}_{P}^{G}(Z(\mathfrak{M}))$ for the cuspidal
$R$-multisegments $\mathfrak{M}$. If the Speh $R$-representations
$Z(\mathfrak{M})$ of $G$ have a germ expansion then the $I(\mathfrak{M})$ have
a germ expansion (Corollary 7.2) and any
$\pi\in\operatorname{Rep}_{R}^{\infty,f}(G)$ has a germ expansion.

We are now ready to prove that the existence of a germ expansion over $R$ is
equivalent to the existence of a germ expansion over $R^{\prime}$. Let
$\mathfrak{M}^{\prime}=(\Delta_{\rho^{\prime}_{1},n_{1}},\ldots,\Delta_{\rho^{\prime}_{r},n_{r}})$
be a cuspidal $R^{\prime}$-multisegment of $GL_{n}(D)$. For $i=1\ldots,r$,
$\rho^{\prime}_{i}$ is an irreducible smooth cuspidal
$R^{\prime}$-representation of $GL_{m_{i}}(D)$ for a divisor $m_{i}$ of
$n_{i}$; there exists an unramified smooth $R^{\prime}$-character
$\chi^{\prime}_{i}$ and an irreducible smooth cuspidal
$R^{\prime}$-representation of $GL_{m_{i}}(D)$ such that
$\rho^{\prime}_{i}=\rho_{i}\chi_{i}$ and
$Z(\Delta_{\rho^{\prime}_{i},n_{i}})=\chi^{\prime}_{i}Z(\Delta_{\rho_{i},n_{i}})$.
Let $\mathfrak{M}=(\Delta_{\rho_{1},n_{1}},\ldots,\Delta_{\rho_{r},n_{r}})$
and $\chi^{\prime}$ the unramified $R^{\prime}$-character of
$M_{n_{1},\ldots,n_{r}}$ corresponding to the $\chi^{\prime}_{i}$. Then
$Z(\mathfrak{M}^{\prime})=\chi^{\prime}Z(\mathfrak{M})$. The Speh
$R^{\prime}$-representation $Z(\mathfrak{M}^{\prime})$ has a germ expansion if
and only if the Speh $R$-representation $Z(\mathfrak{M})$ has a germ
expansion.

### 10.7.

$R$ not necessarily algebraically closed Let $R$ be a field of characteristic
different from $p$. We prove that there is a germ expansion over $R$ when
there is a germ expansion over an algebraic closure $R^{ac}$ of $R$, using the
properties of the scalar extension from $R$ to $R^{ac}$ [Henniart-Vignéras19].

For $\pi\in\operatorname{Rep}_{R}^{\infty}(G)$ irreducible, the
$R^{ac}$-representaton $R^{ac}\otimes_{R}\pi$ has finite length because $\pi$
is admissible as the characteristic of $R$ is different from $p$. Assume that
there is a map $c:\mathfrak{P}(n)\to\mathbb{Z}$ such that
$R^{ac}\otimes_{R}\pi=R^{ac}\otimes_{R}(\sum_{\lambda}c(\lambda)\pi_{P_{\lambda}}$
on an open compact subgroup $K$ of $G$. The scalar extension
$\operatorname{Gr}_{R}^{\infty}(K)\to\operatorname{Gr}_{R^{ac}}^{\infty}(K)$
from $R$ to $R^{ac}$ is injective, by the properties of the scalar extension
from $R$ to $R^{ac}$ [Henniart-Vignéras19]. Therefore
$\pi=\sum_{\lambda}c(\lambda)\pi_{P_{\lambda}}$ on $K$. The representation
$\pi$ has a germ expansion with the same map
$c_{\pi}=c_{R^{ac}\otimes_{R}\pi}=c$. The properties of the scalar extension
from $R$ to $R^{ac}$ imply:

For any irreducible subquotient $\pi^{\prime}$ of $R^{ac}\otimes_{R}\pi$, we
have

(10.2) $c_{\pi}=\ell_{\pi}c_{\pi^{\prime}}\ \ \text{where $\ell_{\pi}$ is the
length of }R^{ac}\otimes_{R}\pi.$

Therefore $c_{\pi}$ and $c_{\pi^{\prime}}$ have the same support. As
$c_{\pi^{\prime}}(\lambda)>0$ when $\lambda$ is minimal in the support of
$c_{\pi^{\prime}}$ (Theorem 10.2). This ends the proof of Theorem 1.3.

### 10.8.

The Jacquet-Langlands correspondence

The classical Jacquet-Langlands correspondence $JL$ between essentially square
integrable representations on both sides, is compatible with character twists
and equivariant under the action of $\operatorname{Aut}(\mathbb{C})$.
Transported to $\mathbb{Q}_{\ell}^{ac}$ 444 (for the root of $q$ in
$\mathbb{Q}_{\ell}^{ac}$ image of $\sqrt{q}\in\mathbb{C}$ via the
isomorphism),

$JL:\operatorname{Irr}^{2}_{\mathbb{Q}_{\ell}^{ac}}(G)\to\operatorname{Irr}^{2}_{\mathbb{Q}_{\ell}^{ac}}(GL_{dn}(F))$

preserves the property of being integral, and two integrals representations of
$G$ are congruent modulo $\ell$ if and only if their images under $JL$ are
congruent modulo $\ell$ ([Minguez-Sécherre17] Theorem 1.1). Once a square root
of $q=p^{f}$ in $\mathbb{Q}_{\ell}^{ac}$ has been chosen when $f$ is odd to
normalize parabolic induction, the Jacquet-Langlands correspondence $LJ$
transported to the Grothendieck groups of
$\mathbb{Q}_{\ell}^{ac}$-representations does reduce modulo $\ell$ thus
yielding a map for $\mathbb{F}_{\ell}^{ac}$-representations ([Minguez-
Sécherre17] Theorem 1.16)

$LJ:\operatorname{Gr}_{\mathbb{F}_{\ell}^{ac}}^{\infty}(GL_{dn}(F))\to\operatorname{Gr}_{\mathbb{F}_{\ell}^{ac}}^{\infty}(G).$

By our argument of reduction modulo $\ell$ in §10.5 we see that Theorem 9.1 is
valid for $\mathbb{F}_{\ell}^{ac}$-representations. When $R$ is an
algebraically closed field of characteristic different from $p$, the reasoning
of §10.6 then gives a map

$LJ:\operatorname{Gr}_{R}^{\infty}(GL_{dn}(F))\to\operatorname{Gr}_{R}^{\infty}(G)$

satisfying Theorem 9.1 for $R$-representations.

######  Theorem 10.5.

(Theorem 9.1). When $R$ is an algebraically closed field of characteristic
different from $p$, for $\nu\in\operatorname{Gr}_{R}^{\infty}(GL_{dn}(F))$ and
$\lambda\in\mathfrak{P}(n)$, we have
$(-1)^{n}c_{LJ(\nu)}(\lambda)=(-1)^{dn}c_{\nu}(d\lambda)$.

######  Remark 10.6.

When $D\neq F$, there are cuspidal complex representations of $GL_{n}(D)$ that
are rational over the field $\mathbb{R}$ of real numbers but not defined over
$\mathbb{R}$. So the Jacquet-Langlands correspondence does not descend to an
arbitrary fied $R$.

A counter-example occurs already for $D^{*}$ and $D$ is a quaternion field
over $F$ with $q\equiv 3\mod 4$. Take a regular complex character $\chi$ of
$k_{D}^{*}$ of order $4$, seen as a character of $O_{D}^{*}$ and extended by
$-1$ on a uniformizer $p_{F}$ of $F$. The induced representation
$\operatorname{ind}_{F^{*}O_{D}^{*}}^{D^{*}}\chi$ has dimension $2$ and its
image is the quaternion group of order $8$ which is not defined over
$\mathbb{R}$. The irreducible representation
$\pi^{0}=JL(\operatorname{ind}_{F^{*}O_{D}^{*}}^{D^{*}}\chi)$ of $GL_{2}(F)$
is cuspidal of level $0$ and can be explicited. For example for
$F=\mathbb{Q}_{3}$, the irreducible cuspidal representation $\sigma^{0}$ of
$GL_{2}(\mathbb{F}_{3})$ corresponding to $\pi^{0}$ has dimension $2$ and is
defined over $\mathbb{R}$. As the central character of $\pi^{0}$ is trivial on
$O_{F}^{*}$, $\sigma^{0}$ factorizes by $PGL_{2}(\mathbb{F}_{3})=S_{4}$ which
has all its irreducible representations defined over $\mathbb{R}$ and even
over $\mathbb{Q}$.

##  11\. Invariant vectors by Moy-Prasad subgroups

We prove in this section Theorem 1.4. Let $R$ be a field, $P$ a parabolic
subgroup of $G$ of Levi $M$ and $K$ an open compact subgroup of $G$. The
positive integer

$\dim_{R}(\pi_{P})^{K}=|P\backslash G/K|$

depends only on $[\pi_{P}]$, hence only on the conjugacy class of $M$ and of
$K$. We can suppose that $P=P_{\lambda}$ for $\lambda\in\mathfrak{P}(n)$ and
$K\subset K_{0}$. We have $G=P_{\lambda}K_{0}$ and $P_{\lambda}\backslash
G/K\simeq(P_{\lambda}\cap K_{0})\backslash K_{0}/K$.

######  Example 11.1.

We have $(P_{\lambda}\cap K_{0})\backslash K_{0}/1+M_{n}(P_{D})\simeq
P_{\lambda}(k_{D})\backslash GL_{n}(k_{D})$ where $k_{D}=O_{D}/P_{D}$ is the
residue field of $D$, $q_{D}$ its cardinality. We deduce

$|P_{\lambda}\backslash
G/1+M_{n}(P_{D})|=[n!]_{q_{D}}/\prod_{i}[\lambda_{i}!]_{q_{D}},$

where $[n!]_{q}=\prod_{m=1}^{n}[m]_{q},\ [m]_{q}=(q^{m}-1)/(q-1)$ ([Suzuki22]
Lemma 1.13).

######  Proposition 11.2.

Let $G_{x,r}$ denote the a Moy-Prasad subgroup of $G$ fixing an element $x$ of
the building of the adjoint group $\mathcal{BT}$ of $G$, and $r$ is a positive
real number, and $j\in\mathbb{N}$. We have

(11.1) $|P\backslash G/G_{x,r+j/d}|=|P\backslash
G/G_{x,r}|\,q^{d\,d_{\lambda}j}.$

When $K^{\prime}$ is a normal open subgroup of $K$,

$|P\backslash G/K^{\prime}|=\sum_{g\in P\backslash G/K}|P\backslash
PgK/K^{\prime}|,\ \ |P\backslash PgK/K^{\prime}|=\frac{[K:K^{\prime}]}{[(K\cap
g^{-1}Pg):(K^{\prime}\cap g^{-1}Pg)]}.$

The group $G_{x,r+j/d}$ is normal in $G_{x,r}$, and (11.1) follows from :

######  Proposition 11.3.

We have $[G_{x,r}\cap P:G_{x,r+1/d}\cap P]=q^{d\,(n^{2}-d_{\lambda})}$.

Note that the index is the same for all $(x,r)$. The $D$-dimension of the Lie
algebra $\mathfrak{p}$ of $P$ is $n^{2}-d_{\lambda}$ where
$\lambda\in\mathfrak{P}(n)$ is the partition such that $P$ is associated to
$P_{\lambda}$.

######  Example 11.4.

When $P=G$, then $\lambda=(n),d_{(n)}=0$,
$[G_{x,r}:G_{x,r+1/d}]=q^{d\,n^{2}}$.

When $P=B$, then $\lambda=(1,\ldots,1),d_{(1,\ldots,1)}=n(n-1)/2$,
$[G_{x,r}\cap B:G_{x,r+1/d}\cap B]=q^{d\,(n(n+1)/2)}$.

###### Proof.

It is more convenient to use lattice functions rather than points in the
Bruhat-Tits building $\mathcal{BT}$, For that we follow [Broussous-Lemaire22]
denoted here by [BL]. Recall that a lattice function is a map $\Phi$ from
$\mathbb{R}$ to $O_{D}$-lattices in $D^{n}$ satisfying the conditions of ([BL]
Definition 2.1); in particular

(11.2) $\Phi(s+1/d)=P_{D}\,\Phi(s)\ \ \text{for any}\ s\in\mathbb{R}.$

The group $\mathbb{R}$ acts on lattice functions by translations, and to a
lattice function is associated a point in $\mathcal{BT}$. That point is the
same for a translate, and one gets in that way a $G$-equivariant bijection
from the set of lattice functions up to translation onto $\mathcal{BT}$. For
any lattice function $\Phi$ and any $r\in\mathbb{R}$, one defines a lattice in
$M_{n}(D)$

$\mathfrak{g}_{\Phi,r}=\\{A\in M_{n}(D)\ |\ A(\Phi(s))=\Phi(r+s)\ \ \text{for
any}\ s\in\mathbb{R}\\}.$

In their introduction [BL] indicate that
$\mathfrak{g}_{\Phi,r}=\mathfrak{g}_{x,r}$ where $x\in\mathcal{BT}$
corresponds to $\Phi$ and $\mathfrak{g}_{x,r}$ is the lattice in $M_{n}(D)$
defined by Moy and Prasad. They also say that the filtration $G_{x,r}$ for
$r\geq 0$ defined by Moy and Prasad satisfies:

$G_{x,0}=(\mathfrak{g}_{\Phi,0})^{*},\ \ \ G_{x,r}=1+\mathfrak{g}_{\Phi,r}\ \
\text{for any}\ r>0.$

They refer to their Appendix A, written by Lemaire; the relevant comments are
in the lines before their Proposition A.3.6.

An immediate consequence of condition (11.2) is that
$\mathfrak{g}_{\Phi,r+1/d}=P_{D}\,\mathfrak{g}_{\Phi,r}$. That implies in
particular that

$[\mathfrak{g}_{\Phi,r}:\mathfrak{g}_{\Phi,r+1/d}]=q^{d\,n^{2}}\ \ \text{for
any}\ r>0.$

More generally, if $W$ is a sub-$D$-vector space of $M_{n}(D)$,
$\mathfrak{g}_{\Phi,r+1/d}\cap W=P_{D}\,(\mathfrak{g}_{\Phi,r}\cap W)$.
Applying that to $\mathfrak{p}$, we get

$[\mathfrak{g}_{\Phi,r}\cap\mathfrak{p}:\mathfrak{g}_{\Phi,r+1/d}\cap\mathfrak{p}]=q^{d\,dim_{D}(\mathfrak{p})}\
\ \text{for any}\ r>0.$

This proves the proposition because $[G_{x,r}\cap P:G_{x,r+1/d}\cap
P]=[\mathfrak{g}_{\Phi,r}\cap\mathfrak{p}:\mathfrak{g}_{\Phi,r+1/d}\cap\mathfrak{p}]$
and $\dim_{D}(\mathfrak{p})=n^{2}-d_{\lambda}$. ∎

We deduce:

######  Corollary 11.5.

Let $P$ be a parabolic subgroup of $G$ associated to $P_{\lambda}$ for
$\lambda\in\mathfrak{P}(n)$, and $G_{x,r+j/d}$ a Moy-Prasad subgroup for
$x\in\mathcal{BT},r\in\mathbb{R},r>0$ and $j\in\mathbb{N}$. We have for $g\in
G$,

(11.3) $\displaystyle|P\backslash PgG_{x,r}/G_{x,r+1/d}|$
$\displaystyle=\frac{[G_{x,r}:G_{x,r+1/d}]}{[(G_{x,r}\cap P):(G_{x,r+1/d}\cap
P)]}=q^{d\,d_{\lambda}}.$

Clearly, (11.1) follows from (11.3).

######  Example 11.6.

1) For a vertex $x$ of $\mathcal{BT}$, the Moy-Prasad group $G_{x,0}$ is
conjugate to $K_{0}=GL_{n}(O_{D})$ and $G_{x,r}$ is conjugate to
$K_{1}=1+p_{D}M_{n}(O_{D})$ for $0<r\leq 1/d$. Hence

$|P_{\lambda}\backslash G/G_{x,r}|=\begin{cases}|P_{\lambda}\backslash
G/K_{0}|=1\ \text{ if }\ r=0,\\\ |P_{\lambda}\backslash
G/K_{1}|=\frac{[n]_{q^{d}}!}{\prod_{k}[\lambda_{k}]_{q^{d}}!}\ \text{ if }\
0<r\leq 1/d.\end{cases}$

where $[n]_{q}!=\frac{q-1}{q-1}\ldots\frac{q^{n}-1}{q-1}$. Indeed
$|P_{\lambda}\backslash G/K_{0}|=1$ because $G=P_{\lambda}K_{0}$, and
$|P_{\lambda}\backslash
G/K_{1}|=[GL_{n}(\mathbb{F}_{q^{d}}):P_{\lambda}(\mathbb{F}_{q^{d}})]$.

2) For the barycenter $x$ of an alcove, $G_{x,0}$ is conjugate to the Iwahori
group $I$, inverse image in $K_{0}$ of the upper triangular group of
$GL_{n}(k_{D})$, and $G_{x,r}$ is conjugate to the pro-Iwahori group
$I_{1/d}$, inverse image of the strictly upper triangular group of
$GL_{n}(k_{D})$, for $0<r\leq 1/d$. Write $\mathfrak{J}$ for the lattice of
$(x_{i,j})\in M_{n}(O_{D})$ with $x_{i,j}\in P_{D}$ when $i>j$, and
$\mathfrak{J}_{1/d}$ for the lattice of $(x_{i,j})\in M_{n}(O_{D})$ with
$x_{i,j}\in P_{D}$ when $i\geq j$. Then,

$I=\mathfrak{I}^{*},\ \ I_{1/d}=1+\mathfrak{J}_{1/d}\ \ \text{for}\ 0<r\leq
1/d.$

We have $P_{\lambda}\backslash G/I\simeq P_{\lambda}\backslash
G/I_{1/d}\simeq(S_{\lambda_{1}}\times\ldots\times S_{\lambda_{r}})\backslash
S_{n}$ hence

$|P_{\lambda}\backslash G/G_{x,r}|=|P_{\lambda}\backslash
G/I|=|P_{\lambda}\backslash G/I_{1/d}|=\frac{n!}{\prod_{k}\lambda_{k}!}.$

######  Remark 11.7.

Proposition 11.3 reduces the computation of $|P_{\lambda}\backslash
G/G_{x,r}|$ for $r>0$ to the case $0<r<1/d$. For $g\in
G,x\in\mathcal{BT},r\geq 0$, we have $gG_{x,r}g^{-1}=G_{g(x),r}$; this reduces
the computation of $|P_{\lambda}\backslash G/G_{x,r}|$ for $x\in\mathcal{BT}$
to the case where $x$ belongs to the the closed alcove $\mathcal{A}$ of
$\mathcal{BT}$ determined by $B$.

Theorem 1.3 implies for $\pi\in\operatorname{Rep}_{R}^{\infty,f}(G)$,

(11.4)
$\dim_{R}\pi^{G_{x,r+j/d}}=\sum_{\lambda\in\mathfrak{P}(n)}c_{\pi}(\lambda)\,|P_{\lambda}\backslash
G/G_{x,r+j/d}|.$

and the integer $c_{\pi}(\lambda)$ is positive if $d_{\lambda}=d(\pi)$ then
$\lambda$ is minimal in the support of $c_{\pi}$. Applying (11.1), we deduce
Theorem 1.4.

######  Remark 11.8.

  1. (i)

The polynomial $P_{\pi,G_{x,r}}(X)$ is determined by those where $x$ is in a
closed alcove of $\mathcal{B}T$ and $0<r<1/d$ because

$P_{\pi,G_{x,r+j/d}}(X)=P_{\pi,G_{x,r}}(q^{dj}X)\ \ \text{for }\ 0<r<1/d,\
j\in\mathbb{N}.$ $P_{\pi,G_{x,r}}(X)=P_{\pi,G_{g(x),r}}(X)\ \ \text{for }\
0\leq r,\ g\in G.$

  2. (ii)

For $\pi\in\operatorname{Rep}_{R}^{\infty,f}(G)$, and any Moy-Prasad pro-$p$
group $G_{x,r}$ of $G$

$\dim_{R}\pi^{G_{x,r+j/d}}\ \sim\ a_{\pi,G_{x,r}}\,q^{d(\pi)dj}\ \ \text{when
$j\in\mathbb{N}$ goes to infinity}.$

The integer $d(\pi)$ can be called the Gelfand-Kirillov dimension of $\pi$.

##  12\. $G=GL_{2}(D)$

In this section we assume that $G=GL_{2}(D)$, $R$ is a field of characteristic
different from $p$ except in §12.5 where its characteristic is $p$, and we
give more details on the polynomial $P_{\pi,K}(X)$ attached to
$\pi\in\operatorname{Rep}_{R}^{\infty,f}(G)$ and a Moy-Prasad subgroup $K$.

### 12.1.

The Moy-Prasad open compact subgroups of $G$ are conjugate to the open compact
subgroups

$K_{0}\supset I_{0}\supset I_{1/2}\supset K_{1}\supset I_{1}\supset
I_{3/2}\supset K_{2}\supset I_{2}\supset\ldots,$

where $K_{0}=GL_{2}(O_{D}),\