# Gelfand-Kirillov dimension of Representations of $\GL_{n}$ over a non-
archimedean local field

Kenta Suzuki  Department of Mathematics, Massachusetts Institute of
Technology, Cambridge, MA 02142-4307 USA<EMAIL_ADDRESS>
(Date: August 2022)

###### Abstract.

We calculate the asymptotic behavior of the dimension of the fixed vectors of
$\pi$ with respect to compact open subgroups
$1+M_{n}({\mathfrak{p}}^{N})\subset\GL_{n}(F)$ for $\pi$ an admissible
representation of $\GL_{n}(F)$, and $F$ a nonarchimedean local field. Such
dimensions can be calculated by germs of the character of $\pi$. We also make
some observations on how those dimensions behave under instances of Langlands
functoriality, such as the Jacquet-Langlands correspondence and cyclic base
change, where relations between characters are known.

###### Contents

  1. 1 Introduction
  2. 2 Terminology and Conventions
  3. 3 Reduction to supercuspidal representations
    1. 3.1 The growth polynomial of $\langle\Delta\rangle$
    2. 3.2 Growth polynomial of arbitrary admissible representations
  4. 4 More examples
  5. 5 General theory of local character expansions
  6. 6 Under Langlands functoriality
    1. 6.1 Jacquet-Langlands correspondence
    2. 6.2 Base change
  7. A Representations of ${\mathrm{SL}}_{n}(F)$
  8. B Direct calculation of the Gelfand-Kirillov dimension of supercuspidal representations
  9. C Level zero representations of $\GL_{2}$ and $\GL_{3}$

##  1\. Introduction

Smooth representations of $\GL_{n}(F)$, for a nonarchimedean local field $F$
such as ${\mathbb{Q}}_{p}$ or $\mathbb{F}_{p}((t))$, is of great interest in
number theory, for instance due to the local Langlands correspondence.
However, most such representations are infinite-dimensional, so finding a
finite measure of their size is important. One such measure is the Gelfand-
Kirillov dimension:

######  Definition.

Let $\pi$ be a (complex) admissible representation of $\GL_{n}(F)$. The
_Gelfand-Kirillov dimension_ (henceforth, GK-dimension) of $\pi$ is a real
number $r\geq 0$, denoted $\dim_{GK}(\pi)$, such that

$\dim(\pi^{K_{N}})\sim q^{rN},$

where $K_{N}\colonequals 1+M_{n}({\mathfrak{p}}_{F}^{N})$ is a compact open
subgroup of $\GL_{n}(F)$ and $q$ is the order of the residue field
${\mathscr{O}}_{F}/{\mathfrak{p}}_{F}$. Here, for functions $f(N)$ and $g(N)$,
denote $f(N)\sim g(N)$ if there exists real numbers $0<a<b$ such that for
sufficiently large integers $N>0$, the inequality $af(N)<g(N)<bf(N)$ is
satisfied.

In fact, Harish-Chandra and Howe’s local character formula says this function
$\dim(\pi^{K_{N}})$ is a polynomial in $q^{N-1}$ for $N$ large (see Corollary
5.7), which we call the _growth polynomial_ of $\pi$. Now, $\dim_{GK}(\pi)$ is
exactly the degree of the growth polynomial (and, in particular, is an
integer). A more ambitious goal would be to calculate the entire growth
polynomial. The coefficients of the growth polynomial contain important
information about the representation, as they behave well under instances of
Langlands functoriality (see Section 6.)

Irreducible representations of $\GL_{n}(F)$ were classified by Bernstein and
Zelevinsky [21] in terms of multisegments—combinatorial data—of supercuspidal
representations (those corresponding to irreducible representations under the
Langlands correspondence). Roughly, arbitrary representations are obtained by
inducing from parabolic subgroups, and supercuspidal representations are those
not obtained from such a procedure.

For supercuspidal representations $\rho$ of $\GL_{n}(F)$, it is a result of
Howe [12] that $\dim_{GK}(\rho)=\frac{n(n-1)}{2}$, and in fact, the leading
term of the growth polynomial is known (see Proposition 5.8):

$\dim(\rho^{K_{N}})=(1+o(1))[n!]_{q}q^{\frac{n(n-1)}{2}(N-1)},$

where
$[n!]_{q}\colonequals[1]_{q}\cdots[n]_{q}=\frac{q-1}{q-1}\cdots\frac{q^{n}-1}{q-1}$
is the $q$-factorial. This was extended to generic representations by Rodier
[15].

We further extend these results, and calculate the leading term of the growth
polynomial for _arbitrary_ admissible representations of $\GL_{n}(F)$, based
on the combinatorial datum of the Bernstein-Zelevinsky classification [21]:

######  Theorem 1.1.

Let $\pi$ be an arbitrary admissible irreducible representation of
$\GL_{n}(F)$. Then, there exists a multisegment
$a=\\{\Delta_{1},\dots,\Delta_{m}\\}$ such that $\rho=\langle a\rangle$, where
$\Delta_{i}=[\rho_{i},\nu^{r_{i}-1}\rho_{i}]$ for supercuspidal
representations $\rho_{i}$ of $\GL_{n_{i}}(F)$. Then,

$\dim(\pi^{K_{N}})=(1+o(1))\frac{[n!]_{q}}{[r_{1}!]_{q^{n_{1}}}\cdots[r_{m}!]_{q^{n_{m}}}}(q^{N-1})^{\frac{1}{2}(n^{2}-n_{1}r_{1}^{2}-\cdots-
n_{m}r_{m}^{2})},$

so in particular,

$\dim_{GK}(\pi)=\frac{1}{2}(n^{2}-n_{1}r_{1}^{2}-\cdots-n_{m}r_{m}^{2}).$

In fact, we are able to express the growth polynomial of arbitrary admissible
representations of $\GL_{n}(F)$ _exactly_ in terms of growth polynomials for
supercuspidal representations. As a consequence, we deduce that
$\dim(\pi^{K_{N}})$ is an _integer-coefficient_ polynomial in $q^{N-1}$ (see
Corollary 3.8), generalizing a result of Howe [12].

The difficulty is that although every irreducible representation of
$G=\GL_{n}(F)$ appears as a subquotient of some induced representation
$\operatorname{n-Ind}_{P}^{G}(\rho)$ with $\rho$ a supercuspidal
representation of the parabolic subgroup $P\subset G$, such representations
may have _multiple_ irreducible subquotients, and we can a priori only detect
the maximal GK-dimension (see Lemma 3.4). We identify exactly the irreducible
subquotient of $\operatorname{n-Ind}_{P}^{G}(\rho)$ with maximal GK-dimension
(see Lemma 3.9) and for each irreducible representation $\pi$ find a pair
$(P,\rho)$ such that $\pi$ has maximal GK-dimension among the irreducible
subquotients of $\operatorname{n-Ind}_{P}^{G}(\rho)$. Here, $\rho$ may no
longer be supercuspidal, but are products of representation of the form
$\langle\Delta\rangle$ for some segment $\Delta$, whose growth polynomial is
calculated in Subsection 3.1.

The paper is structured as follows: In section 2 we gather the notation used
throughout the paper. In section 3 we prove the main result, the calculation
of the leading term of the growth polynomial for arbitrary irreducible
representations of $\GL_{n}(F)$. In section 4 we calculate some examples,
corresponding to those multisegments with particularly simple combinatorial
structure. In section 5 we look at Howe’s local character formula, and
reproduce a proof that indeed $\dim(\pi^{K_{N}})$ is a polynomial in
$q^{N-1}$. Finally, in section 6 we explore how growth polynomials behave
under some instances of Langlands functoriality, such as Jacquet-Langlands,
base change, and automorphic induction.

We also briefly consider the growth polynomial for representations of
${\mathrm{SL}}_{n}(F)$ in appendix A, which we deduce from the theory for
$\GL_{n}(F)$. In appendix B we provide a more direct proof of
$\dim_{GK}(\rho)=\frac{n(n-1)}{2}$ for $\rho$ supercuspidal, based on Bushnell
and Kutzko’s classification [7]. Finally, in appendix C we provide explicit
calculations for growth polynomials of level zero representations of $\GL_{2}$
and $\GL_{3}$, based on the techniques developed in appendix B.

##  2\. Terminology and Conventions

Throughout the paper, $F$ is a nonarchimedean local field of residual
characteristic $p$. That is, $F$ is a finite extension of ${\mathbb{Q}}_{p}$
or $\mathbb{F}_{p}((T))$. Let ${\mathscr{O}}_{F}$ denote the ring of integers
of $F$, and let ${\mathfrak{p}}_{F}\subset{\mathscr{O}}_{F}$ denote the unique
maximal ideal. Moreover, let $q_{F}$ denote the size of the residue field
$k_{F}\colonequals{\mathscr{O}}_{F}/{\mathfrak{p}}_{F}$. Subscripts are
dropped whenever $F$ is clear from context.

Note that $F$ is assumed to have characteristic zero in Sections 5 and 6.

Fix a nontrivial character $\psi\colon F\to{\mathbb{C}}^{\times}$ of level
zero, i.e., $\psi$ is trivial on ${\mathfrak{p}}_{F}$ but not on
${\mathcal{O}}_{F}$.

By a _representation_ of $\GL_{n}(F)$, we always mean a complex, admissible,
smooth representations. That is, we will only consider representations
$(\pi,V)$ such that $\pi\colon\GL_{n}(F)\to\GL_{\mathbb{C}}(V)$ is continuous
with respect to the discrete topology on $\GL_{\mathbb{C}}(V)$, and such that
for any compact open subgroup $K\subset\GL_{n}(F)$ the space of fixed vectors

$V^{K}\colonequals\\{v\in V:\pi(k)v=v\text{ for any }k\in K\\}$

is finite-dimensional. The notation $\pi$ is used for arbitrary admissible
smooth representations of $\GL_{n}(F)$, while $\rho$ is reserved for
supercuspidal representations. We will denote induction by
$\operatorname{Ind}$, compact induction by $\operatorname{c-Ind}$, and
normalized induction by $\operatorname{n-Ind}$ (i.e.,
$\operatorname{n-Ind}_{H}^{G}(\pi)\colonequals\operatorname{Ind}_{H}^{G}(\delta_{G}^{1/2}\delta_{H}^{-1/2}\otimes\pi)$).

Let $\mathcal{R}$ be the Grothendieck ring of representations of $\GL$, i.e.,
the free abelian group generated by representations of $\GL_{n}$, modulo
$\phi-\pi-\sigma$ for any exact sequence

$0\to\pi\to\phi\to\sigma\to 0$

of representations of $\GL_{n}$. It becomes a graded commutative ring by the
product defined in Section 3 (see also [21, Thm 1.9]).

We define a series of compact open subgroups of $\GL_{n}(F)$, by
$K_{0}\colonequals\GL_{n}({\mathscr{O}}_{F})$, the maximal compact subgroup,
and for $N>0$, by

$K_{N}\colonequals
1+M_{n}({\mathfrak{p}}_{F}^{N})=\ker(K_{0}\to\GL_{n}({\mathscr{O}}_{F}/{\mathfrak{p}}_{F}^{N})).$

The subgroups $\\{K_{N}\\}_{N\geq 0}$ form a fundamental system of
neighborhoods of $1\in\GL_{n}(F)$.

######  Definition.

Let $\pi$ be a representation of $\GL_{n}(F)$. Then, the _growth polynomial_
of $\pi$ is the polynomial $G_{\pi}(X)\in{\mathbb{C}}[X]$ such that for $N\gg
0$,

$\dim(\pi^{K_{N}})=G_{\pi}(q^{N-1}).$

Such a polynomial exists by Corollary 5.7 and Lemma 3.4. Denote by
$\dim_{GK}(\pi)$ the degree of $G_{\pi}$.

Finally, let
$\mathrm{rec}_{F}\colon\mathbf{Irr}(\GL_{n}(F))\to\mathcal{G}_{n}(F)$ be the
local Langlands correspondence, where $\mathbf{Irr}(\GL_{n}(F))$ is the set of
irreducible representations of $\GL_{n}(F)$ and $\mathcal{G}_{n}(F)$ is the
set of semisimple Deligne representations of the Weil group $\mathcal{W}_{F}$
(a close relative to the absolute Galois group
$\operatorname{Gal}(\overline{F}/F)$ of $F$).

##  3\. Reduction to supercuspidal representations

Notation and terminology in this section mostly follows [21], so $\nu\colon
F^{\times}\to{\mathbb{C}}^{\times}$ is the character
$\nu(g)\colonequals|\det(g)|$, and a _segment_ is a set of supercuspidal
representations of the form
$[\rho,\nu^{r-1}\rho]\colonequals\\{\rho,\nu\rho,\dots,\nu^{r-1}\rho\\}$ for
some supercuspidal representation $\rho$ and integer $r>0$. By [21, Thm 6.1]
irreducible representations of $\GL_{n}$ are classified by _multisegments_
—multisets of segments.

A key construction in Bernstein and Zelevinsky’s classification is inducing
from parabolic subgroups $P\subset G$. Conjugacy classes of $P$ correspond to
partitions of $n$. Indeed, a partition $n=n_{1}+\cdots+n_{r}$, denoted
$\lambda$, corresponds to

$P_{\lambda}\colonequals\begin{pmatrix}\GL_{n_{1}}&*&&*\\\ &\GL_{n_{2}}&*\\\
&&\ddots&*\\\ 0&&&\GL_{n_{r}}\end{pmatrix}\subset G.$

Let $\pi_{1},\dots,\pi_{r}$ be admissible representations of
$\GL_{n_{1}},\dots,\GL_{n_{r}}$, respectively. Following the notation in [21],
we denote

$\pi_{1}\times\cdots\times\pi_{r}\colonequals\operatorname{n-Ind}_{P_{\lambda}}^{G}(\pi_{1}\boxtimes\cdots\boxtimes\pi_{r}),$

where $\pi_{1}\boxtimes\cdots\boxtimes\pi_{r}$ is inflated via the surjection
$P_{\lambda}\to\GL_{n_{1}}\times\cdots\times\GL_{n_{r}}$, which quotients out
the unipotent radical $N_{\lambda}\subset P_{\lambda}$.

The following lemma on the size of the double cosets $P_{\lambda}\backslash
G/K_{N}$ is useful:

######  Lemma 3.1.

Let $\lambda$ be the partition $n=n_{1}+\cdots+n_{r}$. Then,

$|P_{\lambda}\backslash G/K_{N}|=\begin{bmatrix}n\\\
n_{1},\cdots,n_{r}\end{bmatrix}_{q}(q^{N-1})^{(n^{2}-n_{1}^{2}-\cdots-
n_{r}^{2})/2}$

for $N>0$, where $\begin{bmatrix}n\\\
n_{1},\dots,n_{r}\end{bmatrix}_{q}\colonequals\frac{[n!]_{q}}{[n_{1}!]_{q}\cdots[n_{r}!]_{q}}$,
and $[n!]_{q}$ is the $q$-analogue of $n!$, defined in Section 1.

###### Proof.

Since $G=P_{\lambda}K_{0}$, we have

$P_{\lambda}\backslash G/K_{N}\cong P_{\lambda}({\mathscr{O}})\backslash
K_{0}/K_{N}\cong
P_{\lambda}({\mathscr{O}}/{\mathfrak{p}}^{N})\backslash\GL_{n}({\mathscr{O}}/{\mathfrak{p}}^{N}),$

the partial flag variety over ${\mathscr{O}}/{\mathfrak{p}}^{N}$ corresponding
to the partition $\lambda$. Now, the subgroup

$\begin{pmatrix}I_{n_{1}}&0&\cdots&0\\\
{\mathfrak{p}}/{\mathfrak{p}}^{N}&I_{n_{2}}&&\vdots\\\ \vdots&&\ddots&0\\\
{\mathfrak{p}}/{\mathfrak{p}}^{N}&\cdots&{\mathfrak{p}}/{\mathfrak{p}}^{N}&I_{n_{r}}\end{pmatrix}\subset\GL_{n}({\mathscr{O}}/{\mathfrak{p}}^{N})$

acts simply on
$P_{\lambda}({\mathscr{O}}/{\mathfrak{p}}^{N})\backslash\GL_{n}({\mathscr{O}}/{\mathfrak{p}}^{N})$
from the right, with quotient $P_{\lambda}(k)\backslash\GL_{n}(k)$, so

$\displaystyle|P_{\lambda}\backslash G/K_{N}|$
$\displaystyle=|P_{\lambda}(k)\backslash\GL_{n}(k)|\cdot(q^{N-1})^{(n^{2}-n_{1}^{2}-\cdots-
n_{r}^{2})/2}$ $\displaystyle=\begin{bmatrix}n\\\
n_{1},\cdots,n_{r}\end{bmatrix}_{q}(q^{N-1})^{(n^{2}-n_{1}^{2}-\cdots-
n_{r}^{2})/2}.$

Here, the last equality is because the quotient
$P_{\lambda}(k)\backslash\GL_{n}(k)$ is the partial flag variety over $k$, and
the projective space ${\mathbb{P}}_{k}^{n-1}$ has size
$[n]_{q}=\frac{q^{n}-1}{q-1}$. ∎

######  Remark.

Here, $\dim(N_{\lambda})=\frac{1}{2}(n^{2}-n_{1}^{2}-\cdots-n_{r}^{2})$, where
$N_{\lambda}\subset P_{\lambda}$ is the nilpotent radical.

The following Mackey-type formula is also useful, and will be used repeatedly
throughout the paper:

######  Lemma 3.2.

Let $G$ be a locally profinite group, $H\subset G$ a closed subgroup, and
$K\subset G$ a compact open subgroup. Let $S\subset G$ be a set of
representatives for $H\backslash G/K$. Then, for a smooth representation
$\lambda$ of $H$,

$\displaystyle(\operatorname{c-Ind}_{H}^{G}\lambda)^{K}$
$\displaystyle\xrightarrow{\sim}\bigoplus_{g\in S}\lambda^{H\cap gKg^{-1}}$
$\displaystyle f$ $\displaystyle\mapsto(f(g))_{g\in S}.$

###### Proof.

We will first show well-definedness. Let
$f\in(\operatorname{c-Ind}_{H}^{G}\lambda)^{K}$. For each $g\in S$ and $h\in
H\cap gKg^{-1}$, we have:

$f(g)=f(h\cdot g\cdot g^{-1}h^{-1}g)=\lambda(h)f(g),$

so $f(g)\in\lambda^{H\cap gKg^{-1}}$. Moreover, there is a subset $T\subset S$
such that $\operatorname{supp}f=\sqcup_{g\in T}HgK$, where each $HgK$ is open
(it is a union of $K$-cosets). Thus,

$H\backslash\operatorname{supp}f=\bigsqcup_{g\in T}H\backslash HgK,$

so since $H\backslash\operatorname{supp}f$ is compact, $T\subset S$ must be
finite. That is, the image lies in the direct sum.

We will construct an inverse. Let $\mathbf{a}=(a_{g})_{g\in
S}\in\bigoplus_{g\in S}\lambda^{H\cap gKg^{-1}}$ be arbitrary. For any $u\in
G=HSK$, there is a $h\in H$, $g\in S$, and $k\in K$ such that $u=hgk$. Let
$f_{\mathbf{a}}(u)\colonequals\lambda(h)a_{g}$. We claim $f_{\mathbf{a}}\colon
G\to\lambda$ is well-defined. Indeed, if $u=hgk=h^{\prime}gk^{\prime}$, then

$\lambda(h)a_{g}=\lambda(h^{\prime})\lambda((h^{\prime})^{-1}h)a_{g}=\lambda(h^{\prime})a_{g},$

since $(h^{\prime})^{-1}h=gk^{\prime}k^{-1}g^{-1}\in H\cap gKg^{-1}$. Now,
$f_{\mathbf{a}}\in(\operatorname{Ind}_{H}^{G}\lambda)^{K}$ is clear. Finally,
$f_{\mathbf{a}}$ is compactly supported modulo $H$, since

$H\backslash\operatorname{supp}f=\bigcup_{g\in T}H\backslash HgK$

where $T=\\{g\in S:a_{g}\neq 0\\}$ is finite, and each $H\backslash HgK$ is
the continuous image of the compact set $K$. ∎

Now, the growth polynomials (defined in Section 2) of induced representations
of the form $\pi_{1}\times\cdots\times\pi_{r}$ are readily calculated:

######  Lemma 3.3.

Let $n=n_{1}+\cdots+n_{r}$ be a partition and let $\pi_{1},\dots,\pi_{r}$ be
admissible representations of $\GL_{n_{1}},\dots,\GL_{n_{r}}$, respectively.
Then,

$G_{\pi_{1}\times\cdots\times\pi_{r}}(X)=\begin{bmatrix}n\\\
n_{1},\cdots,n_{r}\end{bmatrix}_{q}X^{(n^{2}-n_{1}^{2}-\cdots-
n_{r}^{2})/2}\prod_{i=1}^{r}G_{\pi_{i}}(X),$

so in particular,

$\dim_{GK}(\pi_{1}\times\cdots\times\pi_{r})=\frac{1}{2}(n^{2}-n_{1}^{2}-\cdots-
n_{r}^{2})+\sum_{i=1}^{r}\dim_{GK}(\pi_{i}).$

###### Proof.

Let $\pi=\pi_{1}\times\cdots\times\pi_{r}$. The normalization may be ignored
since the character $\delta_{G}^{1/2}\delta_{P}^{-1/2}$ is trivial on a
sufficiently small neighborhood of $P$. By Lemma 3.2 we have

$\rho^{K_{N}}=\bigoplus_{g\in P\backslash
G/K_{N}}(\rho_{1}\boxtimes\cdots\boxtimes\rho_{r})^{P(F)\cap gK_{N}g^{-1}}.$

Now, by Iwasawa decomposition $G=BK_{0}=PK_{0}$, so the representatives $g\in
P\backslash G/K_{N}$ can be chosen so that $g\in K_{0}$. Then,
$gK_{N}g^{-1}=K_{N}$, so

$\displaystyle\pi^{K_{N}}$
$\displaystyle=\big{(}(\pi_{1}\boxtimes\cdots\boxtimes\pi_{r})^{P(F)\cap
K_{N}}\big{)}^{\oplus|P\backslash G/K_{N}|}$
$\displaystyle=\big{(}(\pi_{1}\boxtimes\cdots\boxtimes\pi_{r})^{K_{N}\times\cdots\times
K_{N}}\big{)}^{\oplus|P\backslash G/K_{N}|}$
$\displaystyle=\big{(}\pi_{1}^{K_{N}}\otimes\cdots\otimes\pi_{r}^{K_{N}}\big{)}^{\oplus|P\backslash
G/K_{N}|}.$

Now the result follows from Lemma 3.1. ∎

Moreover, growth polynomials behave well with short exact sequences:

######  Lemma 3.4.

Let

$0\to\pi\to\phi\to\sigma\to 0$

be an exact sequence of admissible representations of $G$. Then,

$G_{\phi}(X)=G_{\pi}(X)+G_{\sigma}(X),$

so in particular,
$\dim_{GK}(\phi)=\max\\{\dim_{GK}(\pi),\dim_{GK}(\sigma)\\}$.

###### Proof.

Follows immediately from [6, Section 2.3] Corollary 1, that $(-)^{K_{N}}$ is
an exact functor. ∎

######  Remark.

A nice formalization of Lemma 3.3 and Lemma 3.4, together with Corollary 5.7
is the following: the map from the Grothendieck ring of representations of
$\GL$

$\displaystyle\mathscr{I}\colon\mathcal{R}$
$\displaystyle\to{\mathbb{C}}[X,X^{-1}]$
$\displaystyle\pi\in\mathrm{Rep}(\GL_{n})$
$\displaystyle\mapsto[n!]_{q}^{-1}X^{-\frac{n^{2}-n}{2}}G_{\pi}(X)$

is a well-defined ring homomorphism. That is, $\mathscr{I}$ normalizes
$\dim(\pi^{K_{N}})$ by the size of the full flag variety over
${\mathscr{O}}/{\mathfrak{p}}^{N}$.

###  3.1. The growth polynomial of $\langle\Delta\rangle$

We calculate the growth polynomial for representations of the form
$\langle\Delta\rangle$ for some segment $\Delta$, which is of particular
importance since they generate the Grothendieck ring $\mathcal{R}$ [21, Cor
7.5]. Thus, growth polynomial for arbitrary representations can be deduced
from that of $\langle\Delta\rangle$, via Lemma 3.3 and Lemma 3.4, as carried
out in the next subsection.

We first review the theory of Bernstein-Zelevinsky derivatives, as defined in
[3, Section 3.5]:

######  Definition.

Let $M_{n}\subset\GL_{n}$ be the _mirabolic subgroup_ , the group of matrices
whose last row is $(0,\cdots,0,1)$. For an admissible representation
$(\tau,U)$ of $M_{n}$ and $0<i\leq n$, define the _$i$ -th derivative_
$\tau^{(i)}$ of $\tau$, which is a representation of $\GL_{n-i}$, as

$\tau^{(k)}\colonequals U/(\theta_{i}(h)v-\pi(h)v:v\in U,h\in N_{i}),$

where $N_{i}$ consists of elements of the form

$h=\left(\begin{array}[]{@{}c|c@{}}\begin{matrix}\begin{matrix}1\\\
&\ddots&\\\ &&1\end{matrix}\\\ \hline\cr\begin{matrix}&\\\ &\\\
&\end{matrix}\end{matrix}&\begin{matrix}*&*&*\\\ \vdots&\vdots&\vdots\\\
a_{n-i}&*&*\\\ 1&a_{n-i+1}&*\\\ &\ddots&a_{n-1}\\\
&&1\end{matrix}\end{array}\right)$

and $\theta_{i}(h)\colonequals\psi(a_{n-i+1}+\cdots+a_{n-1})$, where $\psi$ is
the fixed character of $F$ as defined in Section 2. Here, the top-left block
has size $(n-i)\times(n-i)$, and the bottom-right block has size $i\times i$.
Moreover, for representations of $\GL_{n}$ define the $i$-th derivative as the
$i$-th derivative of the restriction to $M_{n}\subset\GL_{n}$.

######  Definition.

Let $\pi$ be a representation of $\GL_{n}(F)$. Then, let the _total
derivative_ be the formal sum in the Grothendieck ring

$\mathscr{D}(\pi)\colonequals\pi+\pi^{(1)}+\cdots+\pi^{(n)}\in\mathcal{R}.$

Extending $\mathscr{D}$ to $\mathcal{R}$ by linearity defines a ring
homomorphism $\mathcal{R}\to\mathcal{R}$, by [3, Lem 4.5].

Now, the following calculations of total derivatives are of particular
interest to us:

######  Lemma 3.5 (Bernstein-Zelevinsky [3, Cor 4.14 (a)]).

Let $\pi$ be a representation of $\GL_{m}$, and let $\varphi$ be a
representation of the parabolic subgroup $P_{(n-1)+1}\subset\GL_{n}$. Then,
the derivatives of

$\sigma\colonequals\operatorname{n-Ind}_{P_{m+(n-1)+1}}^{P_{(m+n-1)+1}}(\pi\boxtimes\varphi)$

are given by, for $i>0$,

$\sigma^{(i)}=\pi\times\varphi^{(i)}.$

######  Lemma 3.6 (Zelevinsky [21, Thm 3.5]).

Let $\Delta=[\rho,\nu^{r-1}\rho]$ be a segment, where $\rho$ is a
supercuspidal representation. Then,

$\mathscr{D}(\langle\Delta\rangle)=\langle\Delta\rangle+\langle\Delta^{-}\rangle,$

where $\Delta^{-}\colonequals[\rho,\nu^{r-2}\rho]$.

Now, we calculate the growth polynomial for representations of the form
$\langle\Delta\rangle$ for some segment $\Delta$, heavily utilizing the fact
that they remain irreducible when restricted to $M_{n}$ [21, Rmk 3.6]:

######  Proposition 3.7.

Let $\rho$ be a supercuspidal representation of $\GL_{n_{1}}(F)$, and let
$\Delta=[\rho,\nu^{r-1}\rho]$, so $\langle\Delta\rangle$ is a representation
of $\GL_{n}(F)$, with $n=n_{1}r$. Then,

$\displaystyle G_{\langle\Delta\rangle}(X)$
$\displaystyle=[r!]_{q^{n_{1}}}^{-1}\begin{bmatrix}n\\\
n_{1},\dots,n_{1}\end{bmatrix}_{q}X^{\frac{n_{1}(n_{1}-1)r(r-1)}{2}}G_{\rho}(X)^{r}$
$\displaystyle=\frac{[n!]_{q}}{[r!]_{q^{n_{1}}}}X^{\frac{n(n-r)}{2}}+(\text{lower
order terms}),$

so in particular,

$\dim_{GK}\langle\Delta\rangle=\frac{n(n-r)}{2}.$

###### Proof.

Throughout the proof, we abbreviate the parabolic subgroup $P_{\lambda}$ as
the partition $\lambda$.

There is a $M_{n}$-homomorphism

$\displaystyle i\colon\langle\Delta\rangle$
$\displaystyle\hookrightarrow\langle\Delta^{-}\rangle\times\nu^{r-1}\rho=\operatorname{n-Ind}_{(n-n_{1})+n_{1}}^{n}(\langle\Delta^{-}\rangle\boxtimes\nu^{r-1}\rho)$
$\displaystyle\xrightarrow{res}\sigma\colonequals\operatorname{n-Ind}_{(n-n_{1})+(n_{1}-1)+1}^{(n-1)+1}(\langle\Delta^{-}\rangle\boxtimes\nu^{r-1}\rho|_{(n_{1}-1)+1}),$

where the inclusion
$\langle\Delta\rangle\hookrightarrow\langle\Delta^{-}\rangle\times\nu^{r-1}\rho$
is the $G$-homomorphism defined by the characterization of
$\langle\Delta\rangle$ as the unique irreducible submodule of
$\rho\times\cdots\times\nu^{r-1}\rho$ [21, Section 3.1], and $res$ is the
surjection given by restricting functions $f\colon
G\to\langle\Delta^{-}\rangle\boxtimes\nu^{r-1}\rho$ to $P_{(n-1)+1}$. We
calculate their derivatives using Lemma 3.5 and Lemma 3.6:

$\displaystyle\mathscr{D}(\langle\Delta\rangle)$
$\displaystyle=\langle\Delta^{-}\rangle+\langle\Delta\rangle$
$\displaystyle\mathscr{D}(\langle\Delta^{-}\rangle\times\nu^{r-1}\rho)$
$\displaystyle=\langle\Delta^{--}\rangle+\langle\Delta^{-}\rangle+\langle\Delta^{--}\rangle\times\nu^{r-1}\rho+\langle\Delta^{-}\rangle\times\nu^{r-1}\rho$
$\displaystyle\mathscr{D}(\sigma)$
$\displaystyle=\langle\Delta^{-}\rangle+\sigma.$

Here $i$ is nonzero, since both $\langle\Delta\rangle$ and $\sigma$ have
$n_{1}$-th derivative $\langle\Delta^{-}\rangle$, but the $n_{1}$-th
derivative
$\langle\Delta^{-}\rangle+\langle\Delta^{--}\rangle\times\nu^{r-1}\rho$ of
$\langle\Delta^{-}\rangle\times\nu^{r-1}\rho$ only contains the irreducible
representation $\langle\Delta^{-}\rangle$ with multiplicity one. Now, since
$\langle\Delta\rangle$ is an irreducible $M_{n}$-representation by [21, Cor
7.9], the map $i$ must be injective.

Moreover, if $\tau$ is the cokernel of $i$, then its total derivative is

$\displaystyle\mathscr{D}(\tau)$
$\displaystyle=\mathscr{D}(\sigma)-\mathscr{D}(\langle\Delta\rangle)$
$\displaystyle=\sigma-\langle\Delta\rangle$ $\displaystyle=\tau.$

That is, $\tau$ has no derivatives of degree $>0$, so by [21, Section 3.5] we
have $\tau=0$. Thus, $i$ is an isomorphism of $M_{n}$-representations.

The overall strategy is as follows: Use the isomorphism $i$ to calculate the
dimension of $\langle\Delta\rangle^{K_{N}\cap
M_{n}}\supseteq\langle\Delta\rangle^{K_{N}}$. In fact, we can show

$\langle\Delta\rangle^{K_{N}\cap
M_{n}}=\langle\Delta\rangle^{K_{N}\cap\GL_{n-1}}=\langle\Delta\rangle^{K_{N}\cap{{}^{t}}M_{n}},$

so

$\langle\Delta\rangle^{K_{N}}=\langle\Delta\rangle^{K_{N}\cap
M_{n}}\cap\langle\Delta\rangle^{K_{N}\cap{{}^{t}}M_{n}}=\langle\Delta\rangle^{K_{N}\cap
M_{n}},$

which allows us to calculate $\dim(\langle\Delta\rangle^{K_{N}})$.

First, for $N_{1}\geq N$ let

$J_{N_{1}}\colonequals\left(\begin{array}[]{@{}c|c@{}}\begin{matrix}\\\
K_{N}\cap\GL_{n-1}\\\ {}\end{matrix}&\begin{matrix}{\mathfrak{p}}^{N_{1}}\\\
\vdots\\\ {\mathfrak{p}}^{N_{1}}\end{matrix}\\\ \hline\cr 0\quad\cdots\quad
0&1\end{array}\right)\subset M_{n}$

be compact open subgroups which approximate the subgroup $K_{N}\cap\GL_{n-1}$
of $M_{n}$, with $J_{N}=K_{N}\cap M_{n}$. Here,

(3.1) $\langle\Delta\rangle^{K_{N}\cap\GL_{n-1}}=\bigcup_{N_{1}\geq
N}\langle\Delta\rangle^{J_{N_{1}}},$

where $\supseteq$ is obvious and $\subseteq$ is from smoothness of the
representation $\langle\Delta\rangle|_{M_{n}}$. This allows us to use Lemma
3.2 on the groups $J_{N_{1}}$.

For each $N_{1}\geq N$ we have

(3.2) $P_{(n-n_{1})+(n_{1}-1)+1}\backslash P_{(n-1)+1}/J_{N_{1}}\cong
P_{(n-n_{1})+(n_{1}-1)}\backslash\GL_{n-1}/K_{N}\cap\GL_{n-1}$

has size $\begin{bmatrix}n-1\\\ n_{1}-1\end{bmatrix}_{q}q^{n(n_{1}-1)(N-1)}$
by Lemma 3.1, which notably does not depend on the choice of $N_{1}$. Since
$i$ is a $M_{n}$-isomorphism,

$\displaystyle\dim(\langle\Delta\rangle^{J_{N_{1}}})$
$\displaystyle=\dim(\sigma^{J_{N_{1}}})$ (3.3)
$\displaystyle=|P_{(n-n_{1})+(n_{1}-1)+1}\backslash
P_{(n-1)+1}/J_{N_{1}}|\dim\big{(}\langle\Delta^{-}\rangle^{K_{N}}\otimes(\nu^{r-1}\rho)^{K_{N}}\big{)}$
$\displaystyle=\begin{bmatrix}n-1\\\
n_{1}-1\end{bmatrix}_{q}q^{n(n_{1}-1)(N-1)}\dim(\langle\Delta^{-}\rangle^{K_{N}})\dim(\rho^{K_{N}}),$

where the second equality is by a similar argument as in Lemma 3.3. By the
independence on the choice of $N_{1}$, the chain of inclusions

$\langle\Delta\rangle^{J_{N}}\subseteq\langle\Delta\rangle^{J_{N+1}}\subseteq\langle\Delta\rangle^{J_{N+2}}\subseteq\cdots$

must all be equalities, so in particular, (3.1) tells us that
$\langle\Delta\rangle^{K_{N}\cap
M_{N}}=\langle\Delta\rangle^{K_{N}\cap\GL_{n-1}}$.

On the other hand, by [2, Thm 7.3] we have
$\langle\Delta\rangle\circ\iota\cong\langle\Delta\rangle^{\vee}\cong\langle[\nu^{1-r}\rho^{\vee},\rho^{\vee}]\rangle$,
where $\iota(g)\colonequals{{}^{t}}g^{-1}$ is the involution on $\GL_{n}(F)$
corresponding to the nontrivial automorphism of the Dynkin diagram $A_{n-1}$.
Thus, for any subgroup $H\subset M_{n}$ we have

$\langle\Delta\rangle^{{{}^{t}}H}\cong(\langle\Delta\rangle^{\vee})^{H}.$

Symmetrically, all the same arguments, with $\rho$ replaced by
$\nu^{1-r}\rho^{\vee}$, tells us that
$\langle\Delta\rangle^{K_{N}\cap{{}^{t}}M_{N}}=\langle\Delta\rangle^{K_{N}\cap\GL_{n-1}}$.

Since the groups $K_{N}\cap M_{n}$ and $K_{N}\cap{{}^{t}}M_{n}$ generate
$K_{N}$,

$\langle\Delta\rangle^{K_{N}}=\langle\Delta\rangle^{K_{N}\cap
M_{n}}\cap\langle\Delta\rangle^{K_{N}\cap{{}^{t}}M_{n}}=\langle\Delta\rangle^{K_{N}\cap
M_{n}}.$

But this dimension was calculated in equation 3.1, thus giving us a recursive
formula for $\langle\Delta\rangle^{K_{N}}$.

By downward induction on $r$, and observing that

$\begin{bmatrix}n-1\\\
n_{1}-1\end{bmatrix}_{q}=\frac{[n_{1}]_{q}}{[n]_{q}}\cdot\begin{bmatrix}n\\\
n_{1}\end{bmatrix}_{q}=[r]_{q^{n_{1}}}^{-1}\begin{bmatrix}n_{1}r\\\
n_{1}\end{bmatrix}_{q},$

we obtain

$\dim(\langle\Delta\rangle^{K_{N}})=[r!]_{q^{n_{1}}}^{-1}\begin{bmatrix}n\\\
n_{1},\dots,n_{1}\end{bmatrix}_{q}(q^{N-1})^{\frac{n_{1}(n_{1}-1)r(r-1)}{2}}\dim(\rho^{K_{N}})^{r}.$

The final asymptotic is by Proposition 5.8. ∎

######  Remark.

In the notation of the remark after Lemma 3.4, Proposition 3.7 becomes:

$\mathscr{I}_{\langle\Delta\rangle}(X)=[r!]_{q^{n_{1}}}^{-1}X^{-\frac{r-1}{2}n}\mathscr{I}_{\rho}(X)^{r}.$

As a corollary, we generalize a result of Howe [12] that
$G_{\rho}(X)\in{\mathbb{Z}}[X]$, for $\rho$ supercuspidal:

######  Corollary 3.8.

Let $\pi$ be an arbitrary admissible representation of $\GL_{n}(F)$. Then, the
growth polynomial $G_{\pi}(X)$ has integer coefficients.

###### Proof.

The statement holds for $\pi$ supercuspidal, by Corollary 5.7 and [12, Thm 3],
which states $c_{\mathcal{O}}(\pi)\in{\mathbb{Z}}$ for any nilpotent orbit
${\mathcal{O}}$.

The general case follows from Proposition 3.7, Lemma 3.3, and the fact that
the Grothendieck ring $\mathcal{R}$ of representations of $\GL$ is generated
as a ring by representations of the form $\langle\Delta\rangle$ (see [21, Cor
7.5]). That $[r!]_{q^{n_{1}}}^{-1}\begin{bmatrix}n\\\
n_{1},\dots,n_{1}\end{bmatrix}_{q}$ in Proposition 3.7 is an integer is more
readily seen from the recursion (3.1). ∎

######  Remark.

The weaker statement, that $G_{\pi}(X)\in{\mathbb{Q}}[X]$, is immediate since
$\dim(\pi^{K_{N}})$ is always an integer.

###  3.2. Growth polynomial of arbitrary admissible representations

Although Proposition 3.7 allows for the calculation of growth polynomials for
arbitrary representations, explicitly expressing representations in terms of
$\langle\Delta\rangle$ is difficult in general. Nonetheless, Theorem 3.10
calculates the leading term in terms of the Bernstein-Zelevinsky
classification.

The following lemma allows us to relate the leading term of the growth
polynomial of the representation $\langle a\rangle$ to that of the easier-
understood representation $\pi(a)$.

######  Lemma 3.9.

For any multiset $a$ of segments,

$\dim(\langle a\rangle^{K_{N}})=(1+o(1))\dim(\pi(a)^{K_{N}})$

so in particular, $\dim_{GK}(\pi(a))=\dim_{GK}(\langle a\rangle)$.

###### Proof.

First of all, we claim that if $a<b$ (i.e., $a$ can be obtained from $b$ via a
chain of elementary operations) then $\dim_{GK}\pi(a)<\dim_{GK}\pi(b)$.
Indeed, by Lemma 3.3 it suffices to check that for linked segments $\Delta$
and $\Delta^{\prime}$,

$\dim(\langle\Delta\cap\Delta^{\prime}\rangle\times\langle\Delta\cup\Delta^{\prime}\rangle)<\dim(\langle\Delta\rangle\times\langle\Delta^{\prime}\rangle).$

Let $\Delta=[\rho,\nu^{r-1}\rho]$ and
$\Delta^{\prime}=[\nu^{s}\rho,\nu^{s+t-1}\rho]$ for some $0<s\leq r<s+t$ and
$\rho$ a supercuspidal representation of $\GL_{n_{1}}(F)$. We hope to show

$\dim_{GK}(\langle\Delta\cap\Delta^{\prime}\rangle\times\langle\Delta\cup\Delta^{\prime}\rangle)=(t+s)(r-s)n_{1}^{2}+((t+s)^{2}+(r-s)^{2})\frac{n_{1}(n_{1}-1)}{2}.$

is less than

$\dim_{GK}(\langle\Delta\rangle\times\langle\Delta^{\prime}\rangle)=rtn_{1}^{2}+(r^{2}+t^{2})\frac{n_{1}(n_{1}-1)}{2}.$

This reduces to the fact that $(t+s)^{2}+(t-s)^{2}>r^{2}+t^{2}$, i.e., the
convexity of $x^{2}$.

Now by [21, Thm 7.1], in the Grothendieck ring $\mathcal{R}$ of
representations of $\GL$ we have

$\pi(a)=\langle a\rangle+\sum_{b<a}m(b;a)\langle b\rangle$

for some nonnegative integers $m(b;a)$. For $b<a$ we have $\dim_{GK}\langle
b\rangle\leq\dim_{GK}\pi(b)<\dim_{GK}\pi(a)$, so

$\dim(\pi(a)^{K_{N}})=\dim(\langle
a\rangle^{K_{N}})+\sum_{b<a}m(b;a)\dim(\langle
b\rangle^{K_{N}})=(1+o(1))\dim(\langle a\rangle^{K_{N}}).\qed$

We finally arrive at our main theorem:

######  Theorem 3.10.

Let $\pi$ be an arbitrary admissible irreducible representation of
$\GL_{n}(F)$. Then, there exists a multisegment
$a=\\{\Delta_{1},\dots,\Delta_{m}\\}$ such that $\rho=\langle a\rangle$, where
$\Delta_{i}=[\rho_{i},\nu^{r_{i}-1}\rho_{i}]$ for supercuspidal
representations $\rho_{i}$ of $\GL_{n_{i}}(F)$. Then,

$G_{\pi}(X)=\frac{[n!]_{q}}{[r_{1}!]_{q^{n_{1}}}\cdots[r_{m}!]_{q^{n_{m}}}}X^{\frac{1}{2}(n^{2}-n_{1}r_{1}^{2}-\cdots-
n_{m}r_{m}^{2})}+(\text{lower order terms}),$

so in particular, $\dim_{GK}(\pi)=\frac{1}{2}(n^{2}-n_{1}r_{1}^{2}-\cdots-
n_{m}r_{m}^{2})$.

###### Proof.

Combining Lemma 3.3, Lemma 3.9, and Proposition 3.7, we obtain:

$\displaystyle\dim(\pi^{K_{N}})$
$\displaystyle=(1+o(1))\dim((\langle\Delta_{1}\rangle\times\cdots\times\langle\Delta_{m}\rangle)^{K_{N}})$
$\displaystyle=(1+o(1))\begin{bmatrix}n\\\
n_{1}r_{1},\dots,n_{m}r_{m}\end{bmatrix}_{q}(q^{N-1})^{\frac{n^{2}-n_{1}^{2}r_{1}^{2}-\cdots-
n_{m}^{2}r_{m}^{2}}{2}}\prod_{i=1}^{m}\dim(\langle\Delta_{i}\rangle^{K_{N}})$
$\displaystyle=(1+o(1))\frac{[n!]_{q}}{[r_{1}!]_{q^{n_{1}}}\cdots[r_{m}!]_{q^{n_{m}}}}(q^{N-1})^{\frac{1}{2}(n^{2}-n_{1}r_{1}^{2}-\cdots-
n_{m}r_{m}^{2})}.\qed$

######  Example 3.11.

Consider representations with support
$\\{\rho,\nu\rho,\nu\rho,\nu^{2}\rho\\}$, as in [21, Ex 11.4], with $\rho$ a
supercuspidal representation of $\GL_{n_{1}}(F)$. Equivalently, they are
irreducible subquotients of $\rho\times\nu\rho\times\nu\rho\times\nu^{2}\rho$
[21, Thm 7.1]. Then, the Hasse diagram for multisegments supported on it is
(where the partial order is induced from elementary operations):

$\\{\rho,\nu\rho,\nu\rho,\nu^{2}\rho\\}$$\\{[\rho,\nu\rho],\nu\rho,\nu^{2}\rho\\}$$\\{\rho,\nu\rho,[\nu\rho,\nu^{2}\rho]\\}$$\\{[\rho,\nu\rho],[\nu\rho,\nu^{2}\rho]\\}$$\\{[\rho,\nu^{2}\rho],\nu\rho\\}$

and the corresponding GK-dimensions of $\langle a\rangle$ (which, due to Lemma
3.9 and its proof respects the Hasse diagram) are:

$8n_{1}^{2}-2n_{1}$$8n_{1}^{2}-3n_{1}$$8n_{1}^{2}-3n_{1}$$8n_{1}^{2}-4n_{1}$$8n_{1}^{2}-5n_{1}$.

######  Example 3.12 (Generalized Steinberg representations).

For a supercuspidal representation $\rho$ of $\GL_{n_{1}}$ let
$\Delta=[\rho,\nu^{r-1}\rho]$, and consider the generalized Steinberg
representation
$\langle\Delta\rangle^{t}=\langle\rho,\dots,\nu^{r-1}\rho\rangle$ of
$\GL_{n}$. Then, by Theorem 3.10,

$G_{\langle\Delta\rangle^{t}}(X)=[n!]_{q}X^{\frac{1}{2}(n-1)n}+(\text{lower
order terms}).$

Proposition 4.1 gives a more explicit formula.

######  Example 3.13 (Generic representations).

More generally, if $\pi$ is a generic representation of $\GL_{n}(F)$, then

$G_{\pi}(X)=[n!]_{q}X^{\frac{1}{2}(n-1)n}+(\text{lower order terms}).$

so the asymptotic formula is independent of the supercuspidal support. In
fact, generic representations are characterized by being maximal (with respect
to the order induced by elementary operations) in the poset of representations
with a fixed supercuspidal support [21, Thm 9.7]. Thus,
$\dim_{GK}(\pi)=\frac{n^{2}-n}{2}$ if and only if $\pi$ is generic, recovering
Rodier’s result [15].

##  4\. More examples

In this section, we will only consider $\rho$-rigid representations, for some
fixed supercuspidal representation $\rho$:

######  Definition.

An irreducible representation $\pi$ of $\GL_{n}(F)$ is _$\rho$ -rigid_ for
some supercuspidal representation $\rho$ if the supercuspidal support of $\pi$
only contains representations of the form $\nu^{i}\rho$ for some
$i\in{\mathbb{Z}}$.

Only considering such representations suffice, by [21, Prop 8.6] and Lemma
3.3.

An explicit formula for the growth polynomial is available in some cases, when
the structure of the lattice of multisegments is simple:

######  Proposition 4.1.

Let a $\rho$-rigid multiset of segments $a$ with support of size $s$ be such
that any two of its segments have an empty intersection, where $\rho$ is a
supercuspidal representation of $\GL_{n_{1}}(F)$ (so $n=n_{1}s$). Then,

$G_{\langle a\rangle}(X)=\begin{bmatrix}n\\\
n_{1},\dots,n_{1}\end{bmatrix}_{q}X^{\frac{s(s-1)}{2}n_{1}^{2}}G_{\rho}(X)^{s}\sum_{b\leq
a}(-1)^{s-|b|}\prod_{i=1}^{m}[r_{i}!]_{q^{n_{1}}}^{-1}X^{-\frac{r_{i}(r_{i}-1)}{2}n_{1}},$

where $b=\\{\Delta_{1},\dots,\Delta_{m}\\}\leq a$, with $\Delta_{i}$ a segment
of size $r_{i}$.

###### Proof.

By [21, Prop 9.13], in the Grothendieck ring we have the identity

$\langle a\rangle=\sum_{b\leq a}(-1)^{|a|-|b|}\pi(b),$

so by Lemma 3.3 and Lemma 3.4 we have

$\displaystyle G_{\langle a\rangle}(X)$ $\displaystyle=\sum_{b\leq
a}(-1)^{|a|-|b|}G_{\pi(b)}(X)$ $\displaystyle=\sum_{b\leq
a}(-1)^{|a|-|b|}\begin{bmatrix}n\\\
r_{1}n_{1},\dots,r_{m}n_{m}\end{bmatrix}_{q}X^{\frac{s^{2}-r_{1}^{2}-\dots-
r_{m}^{2}}{2}n_{1}^{2}}\prod_{i=1}^{m}G_{\langle\Delta_{i}\rangle}(X).$

Now, applying Proposition 3.7 and simplifying gives the desired result. ∎

######  Example 4.2.

Some example calculations are:

$\displaystyle G_{\langle[\rho,\nu\rho]\rangle^{t}}(X)$
$\displaystyle=\begin{bmatrix}2n_{1}\\\
n_{1}\end{bmatrix}_{q}X^{n_{1}^{2}}G_{\rho}(X)^{2}\\{1-[2!]_{q^{n_{1}}}^{-1}X^{-n_{1}}\\},$
$\displaystyle G_{\langle[\rho,\nu^{2}\rho]\rangle^{t}}(X)$
$\displaystyle=\begin{bmatrix}3n_{1}\\\
n_{1},n_{1},n_{1}\end{bmatrix}_{q}X^{3n_{1}^{2}}G_{\rho}(X)^{3}\\{1-2[2!]_{q^{n_{1}}}^{-1}X^{-n_{1}}+[3!]_{q^{n_{1}}}^{-1}X^{-3n_{1}}\\}.$

######  Example 4.3.

In the notation of Proposition 4.1, when $n_{1}=1$ and $\rho=\chi$ is a
character,

$G_{\langle a\rangle}(X)=\sum_{b\leq a}(-1)^{s-|b|}\begin{bmatrix}s\\\
r_{1},\dots,r_{m}\end{bmatrix}_{q}X^{\frac{1}{2}(s^{2}-\sum_{i=1}^{m}r_{i}^{2})}.$

######  Remark.

More generally, for a $\rho$-rigid multiset of segments $a$ with support of
size $s$,

$G_{\langle a\rangle}(X)=Q(X)\cdot G_{\rho}(X)^{s},$

for some polynomial $Q(X)$ determined by the structure of the poset of
multisegments on the support of $a$ (as in Example 3.11). In Proposition 4.1,
said poset is a Boolean lattice.

##  5\. General theory of local character expansions

Section 3 reduces the calculation of growth polynomials of representations to
that of supercuspidal representations. To calculate the growth polynomials of
supercuspidal representations, we use the _local character expansion_ , which
tells us the germ of a character.

Throughout this section, _assume $F$ has characteristic zero_. Let $G$ be the
group of $F$-rational points of a connected reductive algebraic $F$-group
(e.g., $\GL_{n}(F)$ or $\GL_{m}(D)$, for a division algebra $D$ over $F$.)
Also let ${\mathfrak{g}}$ be its Lie algebra, so there is an exponential map

$\exp_{G}\colon{\mathfrak{g}}_{0}\to G_{1},$

for some neighborhood ${\mathfrak{g}}_{0}$ of $0\in{\mathfrak{g}}$ and
neighborhood $G_{1}$ of $1\in G$. In fact, $\exp_{G}$ is bijective for
sufficiently small neighborhoods ${\mathfrak{g}}_{0}\ni 0$ and $G_{1}\ni 1$.

######  Example 5.1.

For $G=\GL_{n}(F)$ (resp., $\GL_{m}(D)$), we have ${\mathfrak{g}}=M_{n}(F)$
(resp., $M_{m}(D)$), and for ${\mathfrak{g}}_{0}\colonequals
M_{n}(p{\mathscr{O}}_{F})$ (resp., $M_{m}(p{\mathscr{O}}_{D})$) and
$G_{1}\colonequals 1+{\mathfrak{g}}_{0}$, there is an isomorphism of groups

$\displaystyle{\mathfrak{g}}_{0}$ $\displaystyle\to G_{1}$ $\displaystyle X$
$\displaystyle\mapsto 1+X+\frac{X^{2}}{2}+\cdots.$

Unlike in the archimedean case, $\sum_{n\geq 0}\frac{X^{n}}{n!}$ does not
converge on the entire Lie algebra.

Let $Ch_{\pi}$ denote the character associated to an admissible representation
$(\pi,V)$ of $G$. That is, for $f\in C_{c}^{\infty}(G)$ a compactly supported
function on $G$,

$\mathrm{tr}\,\pi(f)=\int_{G}Ch_{\pi}(g)f(g)dg,$

where $\pi(f)=\int_{G}f(g)\pi(g)dg\colon V\to V$ has finite rank so the trace
is well-defined (restrict to a finite-dimensional subspace $\pi(f)(V)\subset
W$ such that $\pi(f)(W)\subset W$). Note that $\pi(g)\colon V\to V$ for $g\in
G$ need not have finite rank so its “literal” trace is ill-defined.

For a compactly supported function $f\in C_{c}^{\infty}({\mathfrak{g}})$,
define the _Fourier transform_ as

$\widehat{f}(X)\colonequals\int_{\mathfrak{g}}f(Y)\cdot\psi(B(X,Y))dY,$

where $B(-,-)$ is the Killing form on ${\mathfrak{g}}$ and $\psi$ is the fixed
character of $F$ as defined in Section 2.

######  Definition.

Let ${\mathcal{N}}_{G}$ be the set of nilpotent $\mathrm{Ad}\ G$-orbits in
${\mathfrak{g}}$, called the nilpotent orbits of $G$. It is partially ordered
with respect to the order ${\mathcal{O}}_{1}\leq{\mathcal{O}}_{2}$ when
$\overline{{\mathcal{O}}}_{1}\subseteq\overline{{\mathcal{O}}}_{2}$, where
$\overline{{\mathcal{O}}}_{1}\subset G$ is the Zariski closure of
$\overline{{\mathcal{O}}}_{1}$.

######  Example 5.2.

For $G=\GL_{n}(F)$, there is a 1-1 correspondence between the partially
ordered sets:

  1. (1)

${\mathcal{N}}_{G}$, the set of nilpotent orbits of $G$;

  2. (2)

equivalence classes of parabolic subgroups $P\subset G$; and

  3. (3)

partitions of $n$,

where the correspondence between (1) and (2) reverses order. Indeed, for
${\mathcal{O}}\in{\mathcal{N}}_{G}$ let $x\in{\mathcal{O}}$ and consider the
parabolic subgroup fixing the flag defined by $x$. Conversely, for any
parabolic $P\subset G$, let $N\subset P$ be the unipotent radical and let
$\mathfrak{n}\subset{\mathfrak{g}}$ be its Lie algebra, so
$N=I_{n}+\mathfrak{n}$. Then, consider the unique nilpotent orbit
${\mathcal{O}}$ such that ${\mathcal{O}}\cap\mathfrak{n}\subset\mathfrak{n}$
is open. For each partition $\lambda$ of $n$, denote by
${\mathcal{O}}_{\lambda}\in{\mathcal{N}}_{G}$ the corresponding nilpotent
orbit. Finally, the correspondence between (2) and (3) is demonstrated in the
beginning of Section 3.There is a similar correspondence for
$G^{\prime}=\GL_{m}(D)$.

Note that when $\lambda$ is the partition $n=n_{1}+\cdots+n_{r}$, then for
$N_{\lambda}\subset P_{\lambda}$ the nilpotent radical,

$\dim({\mathcal{O}}_{\lambda})=2\dim(N_{\lambda})=\sum_{i\neq
j}n_{i}n_{j}=n^{2}-n_{1}^{2}-\cdots-n_{r}^{2}.$

######  Lemma 5.3.

Any nilpotent orbit ${\mathcal{O}}$ in $G$ has a $G$-invariant Radon measure
$\mu_{\mathcal{O}}$. Moreover, letting
${\widehat{\mu}}_{\mathcal{O}}(f)\colonequals\mu_{\mathcal{O}}(\widehat{f})$
for $f\in C_{c}^{\infty}({\mathfrak{g}})$ (where $\widehat{f}$ denotes the
Fourier transform with respect to the self-dual measure), the distribution
$\widehat{\mu}_{\mathcal{O}}$ is represented by a locally integrable function,
again denoted $\widehat{\mu}_{\mathcal{O}}$.

######  Example 5.4.

For $f\in C_{c}^{\infty}({\mathfrak{g}})$ we have $\mu_{\\{0\\}}(f)=f(0)$, so
$\widehat{\mu}_{\\{0\\}}=1$.

Now, recall Howe and Harish-Chandra’s local character expansion [10]:

######  Proposition 5.5.

Let $(\pi,V)$ be an irreducible admissible representation of $G$. Then, there
exists a neighborhood $U$ of $0\in{\mathfrak{g}}$ such that for regular $X\in
U$,

$Ch_{\pi}(\exp(X))=\sum_{{\mathcal{O}}}c_{\mathcal{O}}(\pi)\widehat{\mu}_{\mathcal{O}}(X),$

or, equivalently, for a function $f$ supported on $\exp(U)$,

$\mathrm{tr}\,\pi(f)=\sum_{\mathcal{O}}c_{\mathcal{O}}(\pi)\mu_{\mathcal{O}}(\widehat{f\circ\exp}),$

where the sum runs over nilpotent orbits of ${\mathfrak{g}}$, and the constant
$c_{\mathcal{O}}(\pi)\in{\mathbb{C}}$ only depends on $\pi$ and
${\mathcal{O}}$.

######  Lemma 5.6.

Let ${\mathcal{O}}$ be a nilpotent orbit in $\GL_{n}(F)$ or $\GL_{m}(D)$, and
let $P$ be the corresponding parabolic subgroup. Then, for
$X\in{\mathfrak{g}}$ regular in a sufficiently small neighborhood of $0$,

$\widehat{\mu}_{\mathcal{O}}(X)=Ch_{\operatorname{Ind}_{P}^{G}(1_{P})}(\exp
X).$

In particular, for $f\in C_{c}^{\infty}({\mathfrak{g}})$ supported on a
sufficiently small neighborhood of $0$,

$\mu_{\mathcal{O}}(\widehat{f})=\mathrm{tr}\operatorname{Ind}_{P}^{G}(1_{P})(f).$

###### Proof.

For $\GL_{n}(F)$, this is [12, Lem 5], and the same argument works for
$\GL_{m}(D)$. ∎

Let $(\pi,V)$ be an irreducible admissible representation of $\GL_{n}(F)$.
Then, we may express $\dim(\pi^{K_{N}})$ as a polynomial of $q^{N-1}$, as in
[16]:

######  Corollary 5.7.

Let $\pi$ be an irreducible admissible representation of $\GL_{n}(F)$. Then,

$G_{\pi}(X)=\sum_{{\mathcal{O}}\in{\mathcal{N}}_{G}}c_{\mathcal{O}}(\pi)\begin{bmatrix}n\\\
n_{1},\dots,n_{r}\end{bmatrix}_{q}X^{\frac{1}{2}\dim{\mathcal{O}}},$

where $n=n_{1}+\cdots+n_{r}$ is the partition corresponding to ${\mathcal{O}}$
under the identification in Example 5.2.

###### Proof.

Let $V$ be the underlying vector space for $\pi$. Let $e_{K_{N}}$ be the
normalized characteristic function, i.e.,

$e_{K_{N}}(x)=\begin{cases}\frac{1}{\mathrm{vol}(K_{N})}&\text{if }x\in
K_{N}\\\ 0&\text{if }x\notin K_{N}.\end{cases}$

Then, $\pi(e_{K_{N}})\colon\pi\to\pi$ is a projection to $\pi^{K_{N}}$, so

$\dim(\pi^{K_{N}})=\mathrm{tr}\,\pi(e_{K_{N}}).$

Now by Proposition 5.5 and Lemma 5.6, for any integer $N\gg 0$,

$\displaystyle\mathrm{tr}\,\pi(e_{K_{N}})$
$\displaystyle=\frac{1}{\mathrm{vol}(K_{N})}\int_{K_{N}}Ch_{\pi}(g)dg$
$\displaystyle=\sum_{\mathcal{O}}c_{\mathcal{O}}(\pi)\frac{1}{\mathrm{vol}(K_{N})}\int_{K_{N}}Ch_{\operatorname{Ind}_{P}^{G}(1)}(g)dg$
$\displaystyle=\sum_{\mathcal{O}}c_{\mathcal{O}}\mathrm{tr}\,(\operatorname{Ind}_{P}^{G}(1_{P}))(e_{K_{N}}).$

Now, by Lemma 3.3 and Example 5.2,

$\displaystyle\mathrm{tr}\,(\operatorname{Ind}_{P}^{G}(1_{P}))(e_{K_{N}})$
$\displaystyle=\dim(\operatorname{Ind}_{P}^{G}(1_{P})^{K_{N}})$
$\displaystyle=\begin{bmatrix}n\\\
n_{1},\dots,n_{r}\end{bmatrix}_{q}q^{\frac{1}{2}\dim{\mathcal{O}}(N-1)}.\qed$

######  Remark.

More generally, let $(\pi,V)$ be an admissible irreducible representation of
$\GL_{n}(F)$, let $N\gg 0$, and let $\varphi$ be an arbitrary irreducible
representation of $K_{N}$. Then, by [6, Prop 4.4],

$e_{\varphi}(x)\colonequals\begin{cases}\frac{\dim\varphi}{\mathrm{vol}(K_{N})}\mathrm{tr}\,(\varphi(x^{-1}))&\text{if
}x\in K_{N}\\\ 0&\text{if }x\notin K_{N}\end{cases}$

is such that $\pi(e_{\varphi})\colon V\to V$ is a projection to the
$\varphi$-isotypic component, so a similar argument shows

$\dim(\hom_{K_{N}}(\varphi,\pi|_{K_{N}}))=\sum_{{\mathcal{O}}\in{\mathcal{N}}_{G}}c_{\mathcal{O}}(\pi)\dim(\varphi^{K_{N}\cap
P})\begin{bmatrix}n\\\
n_{1},\dots,n_{r}\end{bmatrix}_{q}q^{\frac{1}{2}\dim{\mathcal{O}}(N-1)}.$

Thus, the coefficients $c_{\mathcal{O}}$ completely determine the structure of
the $K_{N}$-representation $\pi|_{K_{N}}$.

It is known that:

######  Proposition 5.8.

For $\rho$ a supercuspidal representation of $\GL_{n}(F)$,

$G_{\rho}(X)=[n!]_{q}X^{\frac{n(n-1)}{2}}+(\text{lower order terms}).$

###### Proof.

Follows from [12, Thm 3], together with Corollary 5.7. ∎

Moreover, in for certain supercuspidal representations, the entire growth
polynomial is known [13, Prop 5.3]:

######  Proposition 5.9.

Let $\rho$ be the supercuspidal representation corresponding to the generic
character $(E/F,\theta)$ where $\theta=(\chi\circ N_{E/F})\phi$ for some
generic character $\phi$ of $E$ over $F$, where $\phi$ has level $j$ (i.e.,
$j$ is the minimal integer such that $\phi$ is trivial on
$1+{\mathfrak{p}}_{E}^{j}$). Then,

  1. (1)

If $E/F$ is unramified,

$G_{\rho}(X)=\sum_{\mathcal{O}}(-1)^{n+r}n(r-1)!w_{\mathcal{O}}^{-1}q^{(n^{2}-n-\dim{\mathcal{O}})j/2}\begin{bmatrix}n\\\
n_{1},\dots,n_{r}\end{bmatrix}_{q}X^{\frac{1}{2}\dim{\mathcal{O}}}$

  2. (2)

If $E/F$ is totally ramified,

$G_{\rho}(X)=\sum_{\mathcal{O}}(-1)^{n+r}(r-1)!w_{\mathcal{O}}^{-1}\frac{q}{q-1}(r-\sum_{i=1}^{r}q^{-n_{i}})q^{\frac{(n^{2}-n-\dim{\mathcal{O}})j}{2n}}\begin{bmatrix}n\\\
n_{1},\dots,n_{r}\end{bmatrix}_{q}X^{\frac{1}{2}\dim{\mathcal{O}}},$

where $w_{\mathcal{O}}$ is the number of permutations of $r$ letters which fix
the $r$-tuple $(n_{1},\dots,n_{r})$.

###### Proof.

Follows from [13, Prop 5.3], together with Corollary 5.7. ∎

######  Remark.

For $\GL_{\ell}(F)$ with $\ell$ prime, this result suffices to give an
explicit formula for all supercuspidal representations.

##  6\. Under Langlands functoriality

Again, throughout this section, _assume $F$ has characteristic zero_.

###  6.1. Jacquet-Langlands correspondence

Recall the Jacquet-Langlands correspondence [9]:

######  Proposition 6.1.

Let $D$ be a division algebra of dimension $d^{2}$ over $F$, and let
$E^{2}(\GL_{n}(F))$ (resp., $E^{2}(\GL_{m}(D))$) be the set of irreducible
square-integrable representations of $\GL_{n}(F)$ (resp, $\GL_{m}(D)$), where
$n=md$. Then, there exists a bijection

$\displaystyle E^{2}(\GL_{n}(F))$ $\displaystyle\to E^{2}(\GL_{m}(D))$
$\displaystyle\pi$ $\displaystyle\mapsto\pi_{D}$

such that for $g^{\prime}\in\GL_{m}(D)$ a regular element, whose conjugacy
class corresponds to a regular element $g\in\GL_{n}(F)$, then

$(-1)^{m}Ch_{\pi_{D}}(g^{\prime})=(-1)^{n}Ch_{\pi}(g).$

Now, we have:

######  Proposition 6.2.

Let $\pi\in E^{2}(\GL_{n}(F))$ be an irreducible square-integrable
representation, corresponding to $\pi_{D}\in E^{2}(\GL_{m}(D))$ under the
Jacquet-Langlands correspondence. Then

$(-1)^{m}c_{\mathcal{O}}(\pi)=(-1)^{n}c_{{\mathcal{O}}^{\prime}}(\pi_{D}),$

where ${\mathcal{O}}^{\prime}\subseteq{\mathcal{O}}$ are nilpotent orbits of
$G^{\prime}$ and $G$, respectively (via identifying the Lie algebras of
$G^{\prime}$ and $G$, tensored with $\overline{F}$).

###### Proof.

Let $m=m_{1}+\dots+m_{r}$ be an arbitrary partition of $m$, and consider a
block diagonal matrix $g=(g_{1},\dots,g_{r})\in\GL_{n}(F)$ where
$g_{i}\in\GL_{dm_{i}}(F)$ are elliptic elements (i.e., has an irreducible
characteristic polynomial) and the minimal polynomials of $g_{i}$ are coprime.
Then $g$ is regular semisimple, corresponding to the diagonal matrix
$g^{\prime}=(g_{1}^{\prime},\dots,g_{r}^{\prime})\in\GL_{m}(D)$ where each
$g_{i}^{\prime}\in D_{dm_{i}}^{\times}\subset\GL_{m_{i}}(D)$ is similar to
$g_{i}\in\GL_{dm_{i}}(F)$, where $D_{dm_{i}}\supset D$ is a division algebra
of dimension $d^{2}m_{i}^{2}$ over $F$.

Then, by Proposition 5.5 and Lemma 5.6, when $g$ as above is sufficiently
close to $1$,

$Ch_{\pi}(g)=\sum_{{\mathcal{O}}}c_{\mathcal{O}}(\pi)Ch_{\operatorname{Ind}_{P}^{G}(1)}(g).$

Here by [18, Thm 3] the character $Ch_{\operatorname{Ind}_{P}^{G}(1)}(g)$ is
locally supported on the union of the conjugates of $P$, and so
$Ch_{\operatorname{Ind}_{P}^{G}(1)}(g)=0$ for $P$ not corresponding to a
partition of the form $a_{1}d+\cdots+a_{i}d$.

Let ${\mathcal{N}}_{G}^{d}$ be the set of such nilpotent orbits, i.e., those
corresponding to partitions with all parts a multiple of $d$. There is a
bijection ${\mathcal{N}}_{G^{\prime}}\simeq{\mathcal{N}}_{G}^{d}$, sending
${\mathcal{O}}^{\prime}\in{\mathcal{N}}_{G^{\prime}}$ to the unique nilpotent
orbit ${\mathcal{O}}$ of $G$ such that
${\mathcal{O}}^{\prime}\otimes\overline{F}\subseteq{\mathcal{O}}\otimes\overline{F}$
(on the level of partitions, $a_{1}+\cdots+a_{i}\mapsto
a_{1}d+\cdots+a_{i}d$).

Now, Proposition 6.1 gives the equality

(6.1)
$(-1)^{m}\sum_{{\mathcal{O}}\in\mathcal{N}_{G}^{d}}c_{\mathcal{O}}(\pi)Ch_{\operatorname{Ind}_{P}^{G}(1)}(g)=(-1)^{n}\sum_{{\mathcal{O}}^{\prime}\in\mathcal{N}_{G^{\prime}}}c_{{\mathcal{O}}^{\prime}}(\pi_{D})Ch_{\operatorname{Ind}_{P}^{G}(1)}(g),$

since by Lemma 5.6,

$\widehat{\mu}_{{\mathcal{O}}^{\prime}}(g^{\prime})=Ch_{\operatorname{Ind}_{P^{\prime}}^{G^{\prime}}(1_{P^{\prime}})}(g^{\prime})=Ch_{\operatorname{Ind}_{P}^{G}(1_{P})}(g).$

where $P^{\prime}\colonequals(P\otimes\overline{F})\cap G^{\prime}\subset
G^{\prime}$. Moreover, for parabolic subgroups $P_{1}\subsetneq P_{2}$ of $G$
we have
$\operatorname{supp}(Ch_{\operatorname{Ind}_{P_{1}}^{G}(1)})\subsetneq\operatorname{supp}(Ch_{\operatorname{Ind}_{P_{2}}^{G}(1)})$
in a neighborhood of $1\in G$, so the characters
$\\{Ch_{\operatorname{Ind}_{P}^{G}(1)}:P\subset G\text{ parabolic}\\}$ are
linearly independent. Thus, equation 6.1 gives

$(-1)^{m}c_{\mathcal{O}}(\pi)=(-1)^{n}c_{{\mathcal{O}}^{\prime}}(\pi_{D}).\qed$

######  Corollary 6.3.

Let $\pi\in E^{2}(\GL_{n}(F))$ be an irreducible square-integrable
representation, and let $\pi_{D}\in E^{2}(D^{\times})$ be the corresponding
representation under the Jacquet-Langlands correspondence, where $D$ is a
$n^{2}$-dimensional division algebra. Then the growth polynomial $G_{\pi}(X)$
has constant term $(-1)^{n-1}\dim(\pi_{D})$.

###### Proof.

Since $D^{\times}$ is compact modulo center, the representation $\pi_{D}$ is
finite-dimensional. Thus, $c_{0}(\pi_{D})=\dim(\pi_{D})$, and so
$c_{0}(\pi)=(-1)^{n-1}c_{0}(\pi_{D})=(-1)^{n-1}\dim(\pi_{D})$. ∎

The combination of Proposition 5.8 and Corollary 6.3 allows the calculation
for the polynomial $\dim(\pi^{K_{N}})$ for representations $\pi$ of
$\GL_{2}(F)$:

######  Example 6.4.

When $n=2$, [6, Section 56] provides an explicit description of the Jacquet-
Langlands correspondence for minimal supercuspidal representations $\rho$ of
$\GL_{2}$. Only treating these suffices since arbitrary supercuspidal
representations are twists of minimal supercuspidal representations.

  * •

If $\rho$ has level zero, then $\rho_{D}$ is induced from a character of
$E^{\times}U_{D}^{1}$, so

$G_{\rho}(X)=(q+1)X-2.$

Otherwise, there is a simple stratum $({\mathfrak{A}},n,\alpha)$ such that
$\rho$ contains the character $\psi_{\alpha}$ of $U_{\mathfrak{A}}^{[n/2]+1}$,
so $\rho\cong\operatorname{c-Ind}_{J}^{G}\Lambda$ where
$J=E^{\times}U_{\mathfrak{A}}^{[(n+1)/2]}$ and $\Lambda$ is an irreducible
representation on $J$:

  * •

If $e_{\mathfrak{A}}=2$ then $\rho_{D}$ is induced from a character on
$E^{\times}U_{D}^{[n/2]+1}$, so

$G_{\rho}(X)=(q+1)X-|E^{\times}U_{D}^{[n/2]+1}\backslash D^{\times}|.$

  * •

If $e_{\mathfrak{A}}=1$ and $n$ is odd then $\rho_{D}$ is induced from a
$q$-dimensional representation of $E^{\times}U_{D}^{n}$, so

$G_{\rho}(X)=(q+1)X-q|E^{\times}U_{D}^{n}\backslash D^{\times}|,$

  * •

If $e_{\mathfrak{A}}=1$ and $n$ is even then $\rho_{D}$ is induced from a
character of $E^{\times}U_{D}^{n+1}$, so

$G_{\rho}(X)=(q+1)X-|E^{\times}U_{D}^{n+1}\backslash D^{\times}|.$

In each of these cases, we have:

$\displaystyle|E^{\times}U_{D}^{m}\backslash D^{\times}|$
$\displaystyle=|\langle\varpi_{E}\rangle\backslash\langle\varpi_{D}\rangle|\cdot|k_{E}^{\times}\backslash
k_{D}^{\times}|\cdot|U_{E}^{1}U_{D}^{m}\backslash U_{D}^{1}|$
$\displaystyle=\frac{2}{e}\cdot\frac{q^{2}-1}{q^{2/e}-1}|U_{E}^{1}U_{D}^{m}\backslash
U_{D}^{1}|.$

Now, there is an exact sequence

$1\to(U_{E}^{1}\cap U_{D}^{m})\backslash U_{E}^{1}\cong U_{D}^{m}\backslash
U_{E}^{1}U_{D}^{m}\to U_{D}^{m}\backslash U_{D}^{1}\to
U_{E}^{1}U_{D}^{m}\backslash U_{D}^{1}\to 1,$

where $m\geq 1$. Noting that $U_{E}^{1}\cap U_{D}^{m}=U_{E}^{1+[(m-1)e/2]}$,
we obtain

$|U_{E}^{1}U_{D}^{m}\backslash U_{D}^{1}|=\frac{|U_{D}^{m}\backslash
U_{D}^{1}|}{|U_{E}^{1+[(m-1)e/2]}\backslash
U_{E}^{1}|}=q^{2(m-1)-\frac{2}{e}[\frac{e}{2}(m-1)]}.$

Thus, combining everything gives

$G_{\rho}(X)=\begin{cases}(q+1)X-2&\text{if }\rho\text{ has level zero}\\\
(q+1)X-(q+1)q^{[n/2]}&\text{if }e_{\mathfrak{A}}=2\\\ (q+1)X-2q^{n}&\text{if
}e_{\mathfrak{A}}=1.\end{cases}$

###  6.2. Base change

Another instance of Langlands functoriality is base change and automorphic
induction. For a finite extension $E/F$ of degree $d$, base change is given by
the commutative diagram

(6.2)
${\mathbf{Irr}(\GL_{n}(F))}$${\mathcal{G}_{n}(F)}$${\mathbf{Irr}(\GL_{n}(E))}$${\mathcal{G}_{n}(E),}$$\scriptstyle{\mathrm{rec}_{F}}$$\scriptstyle{\mathrm{BC}_{E/F}}$$\scriptstyle{\mathrm{Res}^{\mathcal{W}_{F}}_{\mathcal{W}_{E}}}$$\scriptstyle{\mathrm{rec}_{E}}$

and automorphic induction is given by

${\mathbf{Irr}(\GL_{n}(E))}$${\mathcal{G}_{n}(E)}$${\mathbf{Irr}(\GL_{nd}(F))}$${\mathcal{G}_{nd}(F),}$$\scriptstyle{\mathrm{rec}_{E}}$$\scriptstyle{\mathrm{AI}_{E/F}}$$\scriptstyle{\mathrm{Ind}^{\mathcal{W}_{F}}_{\mathcal{W}_{E}}}$$\scriptstyle{\mathrm{rec}_{F}}$

where $\mathrm{rec}_{F}$ denotes the local Langlands correspondence. When
$E/F$ is a cyclic extension of prime degree, $\mathrm{BC}_{E/F}$ has a
characterization in terms of characters [1, Thm 6.2]:

######  Definition.

Let $E/F$ be a cyclic extension of prime degree, and let
$\tau\in\operatorname{Gal}(E/F)$ be a generator. Moreover, let $(\Pi,V)$ be an
irreducible representation of $\GL_{n}(E)$ such that $\Pi\cong\Pi^{\tau}$,
where $\Pi^{\tau}(g)\colonequals\Pi(\tau(g))$ for $g\in\GL_{n}(E)$. Let
$I_{\tau}\colon\Pi\to\Pi^{\tau}$ be such an isomorphism, appropriately
normalized (require the Whittaker functional to be fixed). Then there exists a
locally constant function $Ch_{\Pi,\tau}$ defined on $\GL_{n}(E)^{\tau-reg}$
such that

$\mathrm{tr}(\pi(f)\circ I_{\tau})=\int_{\GL_{n}(E)^{\tau-
reg}}f(g)Ch_{\Pi,\tau}(g)dg,$

where $\GL_{n}(E)^{\tau-reg}$ is the set of $g\in\GL_{n}(E)$ such that
$g\tau(g)\cdots\tau^{d-1}(g)$ is regular.

######  Proposition 6.5 (Shintani character formula).

Let $E/F$ be a cyclic extension of prime degree $d$, and let
$\tau\in\operatorname{Gal}(E/F)$ be a generator. Then for $\pi$ an irreducible
tempered representation of $\GL_{n}(F)$,

$Ch_{\mathrm{BC}_{E/F}(\pi),\tau}(g)=Ch_{\pi}(\mathcal{N}g),$

for any $g\in\GL_{n}(E)^{\tau-reg}$, where $\mathcal{N}g\in\GL_{n}(F)$ is the
norm of $g$, i.e., the unique (up to conjugacy) element of $\GL_{n}(F)$
conjugate to $g\tau(g)\cdots\tau^{d-1}(g)$ (which is regular by definition).

Shintani’s character formula gives another method to calculate the growth
polynomial:

######  Proposition 6.6.

Let $E/F$ be a cyclic extension of prime degree $d$, and let $\theta$ be a
generic character of $E/F$ (i.e., $\theta\neq\theta^{\tau}$). Then for an
integer $N\gg 0$,

$G_{\mathrm{AI}_{E/F}\theta}(q^{[N/e]-1})=\mathrm{tr}(I_{\tau}|_{V^{K_{N}}})$

where
$V\colonequals\theta\times\theta^{\tau}\times\cdots\times\theta^{\tau^{d-1}}$.

###### Proof.

Let $\pi=\mathrm{AI}_{E/F}(\theta)$ and $\Pi=\mathrm{BC}_{E/F}(\pi)$. Then, by
Mackey’s formula,

$\displaystyle\mathrm{rec}_{E}(\Pi)$
$\displaystyle=(\operatorname{Ind}_{\mathcal{W}_{E}}^{\mathcal{W}_{F}}\mathrm{rec}_{E}\theta)|_{\mathcal{W}_{E}}$
$\displaystyle=\bigoplus_{\sigma\in\mathcal{W}_{E}/\mathcal{W}_{F}}\mathrm{rec}_{E}\theta^{\sigma}$
$\displaystyle=\mathrm{rec}_{E}\theta\oplus\mathrm{rec}_{E}\theta^{\tau}\oplus\cdots\oplus\mathrm{rec}_{E}\theta^{\tau^{d-1}},$

so that $\Pi=\theta\times\cdots\times\theta^{\tau^{d-1}}$. Now, for $N\gg 0$,
by Proposition 6.5 and Lemma 5.6,

$\displaystyle\mathrm{tr}(I_{\tau}|_{\Pi^{K_{N}}})$
$\displaystyle=\frac{1}{\mathrm{vol}(K_{N}(E))}\int_{K_{N}(E)^{\tau-
reg}}Ch_{\Pi,\tau}(g)dg$
$\displaystyle=\frac{1}{\mathrm{vol}(K_{N}(E))}\int_{K_{N}(E)^{\tau-
reg}}Ch_{\pi}(\mathcal{N}g)dg$
$\displaystyle=\sum_{\mathcal{O}}c_{\mathcal{O}}(\pi)\frac{1}{\mathrm{vol}(K_{N}(E))}\int_{K_{N}(E)^{\tau-
reg}}Ch_{\operatorname{Ind}_{P}^{G}(1)}(\mathcal{N}g)dg,$

where the use of Lemma 5.6 is justified because $\mathcal{N}g\in\GL_{n}(F)$ is
regular and lies in a small neighborhood of $1$. We may safely replace
$\operatorname{Ind}_{P}^{G}(1)$ with
$\operatorname{Ind}_{P}^{G}(\chi_{1}\boxtimes\cdots\boxtimes\chi_{r})$ for
unitary characters $\chi_{i}$ such that $\chi_{i}\circ N_{E/F}$ are distinct,
since the germs of their characters match. But then (by following the diagram
(6.2)), the base change is

$\mathrm{BC}_{E/F}(\operatorname{Ind}_{P}^{G}(\chi_{1}\boxtimes\cdots\boxtimes\chi_{r}))=\operatorname{Ind}_{P(E)}^{G(E)}(\chi_{1}\circ
N_{E/F}\boxtimes\cdots\boxtimes\chi_{r}\circ N_{E/F}),$

so by the same argument as above,

$\displaystyle\frac{1}{\mathrm{vol}(K_{N}(E))}\int_{K_{N}(E)^{\tau-
reg}}Ch_{\operatorname{Ind}_{P}^{G}(1)}(\mathcal{N}g)dg$
$\displaystyle=\mathrm{tr}\big{(}I_{\tau}|_{\operatorname{Ind}_{P(E)}^{G(E)}(\chi_{1}\circ
N_{E/F}\boxtimes\cdots\boxtimes\chi_{r}\circ N_{E/F})^{K_{N}}}\big{)}.$

Vectors of the invariant space $\operatorname{Ind}_{P(E)}^{G(E)}(\chi_{1}\circ
N_{E/F}\boxtimes\cdots\boxtimes\chi_{r}\circ N_{E/F})^{K_{N}}$ can be viewed
as functions on $P(E)\backslash G(E)/K_{N}(E)$, and $\tau$ acts on it by the
Galois action, so the trace is the size of the invariant elements

$(P(E)\backslash G(E)/K_{N}(E))^{\tau}=P(F)\backslash G(F)/(K_{N}(E)\cap
G(F))=P(F)\backslash G(F)/K_{[N/e]}(F),$

so altogether, by Corollary 5.7,

$\displaystyle\mathrm{tr}(I_{\tau}|_{\Pi^{K_{N}}})$
$\displaystyle=\sum_{\mathcal{O}}c_{\mathcal{O}}(\pi)|P(F)\backslash
G(F)/K_{[N/e]}(F)|$
$\displaystyle=G_{\mathrm{AI}_{E/F}\theta}(q^{[N/e]-1}).\qed$

We will give an alternative proof of the $n=2$ case of Proposition 5.9 (1), as
well as Example 6.4:

######  Proposition 6.7.

Let $E/F$ be an unramified degree $2$ extension and let $\theta$ be a generic
character of $E/F$, such that $\theta^{\tau}\theta^{-1}$ has level $\ell$
(i.e., $\ell$ is the minimal integer such that $\theta^{\tau}\theta^{-1}$ is
trivial on $1+{\mathfrak{p}}_{E}^{\ell+1}$). Then,

$G_{\mathrm{AI}_{E/F}(\theta)}(X)=(q+1)X-2q^{\ell}.$

###### Proof.

Let $E=F(\sqrt{\alpha})$ with $v_{F}(\alpha)=0$ and let
$\varpi\in{\mathscr{O}}_{F}$ be the generator of the maximal ideal
${\mathfrak{p}}_{F}$. By Proposition 6.6, it suffices to calculate the trace
of $I_{\tau}$ on $(\theta\times\theta^{\tau})^{K_{N}(E)}$. By [4, Prop 4.5.6]
it is given by

$\displaystyle
q^{-\ell}I_{\tau}\colon\operatorname{n-Ind}_{B(E)}^{G(E)}(\theta\boxtimes\theta^{\tau})$
$\displaystyle\to\operatorname{n-Ind}_{B(E)}^{G(E)}(\theta\boxtimes\theta^{\tau})^{\tau}$
$\displaystyle f$ $\displaystyle\mapsto\bigg{(}g\mapsto\lim_{s\to
0^{-}}\int_{E}f\big{(}\begin{pmatrix}&1\\\
1&x\end{pmatrix}\tau(g)\big{)}|x|^{s}dx\bigg{)}.$

For $g_{0}\in\GL_{2}(E)$, define the twisted indicator function

${\mathbf{1}}_{g_{0}}(g)\colonequals\begin{cases}(\theta(\frac{1}{2})\boxtimes\theta^{\tau}(-\frac{1}{2}))(b)&\text{if
}g=bg_{0}u\text{ for some }b\in B(E),u\in K_{N}(E)\\\ 0&\text{if }g\notin
B(E)g_{0}K_{N}(E),\end{cases}$

so that $\\{{\mathbf{1}}_{g_{0}}\\}_{g_{0}\in
B(E)\backslash\GL_{2}(E)/K_{N}(E)}$ is a basis for
$(\theta\times\theta^{\tau})^{K_{N}(E)}$. Note that by Iwasawa decomposition
we may take $g_{0}\in K_{0}(E)$, in which case ${\mathbf{1}}_{g_{0}}$ is well-
defined for $N>\ell$. Thus,

(6.3)
$\mathrm{tr}(q^{-\ell}I_{\tau}|_{(\theta\times\theta^{\tau})^{K_{N}}})=\sum_{g_{0}\in
B(E)\backslash\GL_{2}(E)/K_{N}(E)}\lim_{s\to
0^{-}}\int_{E}{\mathbf{1}}_{g_{0}}\big{(}\begin{pmatrix}&1\\\
1&x\end{pmatrix}\tau(g_{0})\big{)}|x|^{s}dx.$

Moreover, for arbitrary $k\in K_{0}(F)$ we have

${\mathbf{1}}_{g_{0}}\big{(}\begin{pmatrix}&1\\\
1&x\end{pmatrix}\tau(g_{0})\big{)}={\mathbf{1}}_{g_{0}k}\big{(}\begin{pmatrix}&1\\\
1&x\end{pmatrix}\tau(g_{0}k)\big{)},$

since if $\begin{pmatrix}&1\\\ 1&x\end{pmatrix}\tau(g_{0})=bg_{0}u$ then
$\begin{pmatrix}&1\\\ 1&x\end{pmatrix}\tau(g_{0}k)=bg_{0}k(k^{-1}uk)$, where
$k^{-1}uk\in K_{N}(E)$. Thus, the sum in equation (6.3) can be combined by
$K_{0}(F)$-orbits. Those orbits are represented by

$\big{\\{}I_{2},\begin{pmatrix}1\\\
\varpi^{i}\sqrt{\alpha}&1\end{pmatrix}:0\leq i<N\big{\\}},$

where the orbit of $I_{2}$ has size $q^{N-1}(q+1)$ by Lemma 3.1, and the orbit
of $\begin{pmatrix}1\\\ \varpi^{i}\sqrt{\alpha}&1\end{pmatrix}$ has size
$(q^{2}-q)q^{2(N-1)}$ for $i=0$ and $(q^{2}-1)q^{2(N-1)-i}$ for $i>0$. What is
left is to calculate the integral in equation 6.3 for each of these choices of
$g_{0}$.

When $g_{0}=I_{2}$, then $\begin{pmatrix}&1\\\ 1&x\end{pmatrix}\in BK_{N}$
exactly when $v_{E}(x)\leq-N$, in which case $b=\begin{pmatrix}-x^{-1}&*\\\
&x\end{pmatrix}$ and so the desired integral is

$\lim_{s\to 0^{-}}\int_{v(x)\leq-N}\theta(-x^{-1})\theta^{\tau}(x)|x|^{s}dx.$

The change of variables $x\mapsto ux$ for $u\in{\mathscr{O}}_{E}^{\times}$
such that $\theta(\tau(u))\neq\theta(u)$ shows the integral is $0$.

Next, when $g_{0}=\begin{pmatrix}1\\\ \varpi^{i}\sqrt{\alpha}&1\end{pmatrix}$,
we have

$\begin{pmatrix}&1\\\ 1&x\end{pmatrix}\begin{pmatrix}1\\\
-\varpi^{i}\sqrt{\alpha}&1\end{pmatrix}=\begin{pmatrix}*&*\\\
1-x\varpi^{i}\sqrt{\alpha}&x\end{pmatrix}\in Bg_{0}K_{N}$

exactly when
$(1-x\varpi^{i}\alpha)x^{-1}\equiv\varpi^{i}\alpha\pmod{{\mathfrak{p}}_{E}^{N}}$,
i.e., $x\in\frac{1}{2\varpi^{i}\sqrt{\alpha}}(1+{\mathfrak{p}}_{E}^{N-i})$.
Thus, the integral is (using that the measure $\frac{dx}{|x|}$ is a Haar
measure for $E^{\times}$):

$\int_{\frac{1}{2\varpi^{i}\sqrt{\alpha}}(1+{\mathfrak{p}}_{E}^{N-i})}\theta(-x^{-1})\theta^{\tau}(x)\frac{dx}{|x|}=\int_{1+{\mathfrak{p}}_{E}^{N-i}}\theta(x^{-1})\theta^{\tau}(x)\frac{dx}{|x|}.$

If $N-i\geq\ell+1$ then $\theta(x^{-1})\theta^{\tau}(x)=1$ on
$1+{\mathfrak{p}}_{E}^{N-i}$, so the integral is simply
$\int_{1+{\mathfrak{p}}^{N-i}}\frac{dx}{|x|}=q_{E}^{(i-N)+\frac{1}{2}}=q^{2(i-N)+1}$.
On the other hand, if $N-i\leq\ell$ then looking at the change of variables
$x\mapsto ux$ for $u\in 1+{\mathfrak{p}}_{E}^{\ell}$ such that
$\theta(\tau(u))\neq\theta(u)$ shows the integral is $0$. Thus, in conclusion,

$\displaystyle
q^{-\ell}\mathrm{tr}(I_{\tau}|_{(\theta\times\theta^{\tau})^{K_{N}}})$
$\displaystyle=(q^{2}-q)q^{2(N-1)}q^{-2N+1}+\sum_{i=0}^{N-\ell-1}(q^{2}-1)q^{2(N-1)-i}q^{2(i-N)+1}$
$\displaystyle=(q+1)q^{N-\ell-1}-2.\qed$

######  Remark.

More generally, let $\rho\in\mathbf{Irr}(\GL_{n}(E))$ be supercuspidal and
consider $\pi=\mathrm{AI}_{E/F}\rho$. Then, the same arguments show the growth
polynomial $G_{\pi}$ is related to the trace of the intertwining integral
$\rho\times\cdots\times\rho^{\tau^{d-1}}\to(\rho\times\cdots\times\rho^{\tau^{d-1}})^{\tau}$.
Moreover, the trace is given by a sum over the double coset
$P_{n+\cdots+n}(E)\backslash\GL_{nd}(E)/K_{N}(E)K_{0}(F)$, the
$K_{0}(F)$-orbits in the partial flag variety over
${\mathscr{O}}_{E}/{\mathfrak{p}}_{E}^{N}$. Looking more closely at the
combinatorial structure of such double cosets is expected to give new results.

## Acknowledgement

The author thanks Hao Peng for mentoring this project and giving helpful
advice. We thank Zhiyu Zhang for suggesting this project. We thank Prof. Wei
Zhang and Prof. Ju-Lee Kim for giving us some relevant references and pointing
us in the right direction. We thank Dr. Max Gurevich for pointing out an error
in the original argument for Proposition 3.7. We thank Prof. Marie-France
Vignéras and Prof. Guy Henniart for helpful discussions, as well as fixing an
error in Remark 6.4.

Finally, we thank Prof. David Jerison and Prof. Ankur Moitra for organizing
the SPUR program and helping us through insightful discussions.

##  Appendix A Representations of ${\mathrm{SL}}_{n}(F)$

Let $(\varphi,V)$ be an irreducible representation of ${\mathrm{SL}}_{n}(F)$.
We reduce the calculation of the _leading term_ of the growth polynomial of
$\varphi$ to the theory for $\GL_{n}(F)$.

We first extend $\varphi$ to a representation of $Z{\mathrm{SL}}_{n}(F)$,
which is of finite-index in $\GL_{n}(F)$. The central character

$\omega_{\varphi}\colon Z\cap{\mathrm{SL}}_{n}(F)\to{\mathbb{C}}^{\times}$

can be extended to a character $\omega\colon Z\to{\mathbb{C}}^{\times}$, since
the Pontryagin dual of $Z\cap{\mathrm{SL}}_{n}(F)\hookrightarrow Z$ is
surjective. Now, consider an extension of $\varphi$

$\displaystyle\widetilde{\varphi}\colon Z{\mathrm{SL}}_{n}(F)$
$\displaystyle\to\GL(V)$ $\displaystyle zg$
$\displaystyle\mapsto\omega(z)\varphi(g).$

Since $Z{\mathrm{SL}}_{n}\subset\GL_{n}$ has finite index, by [6, Lem 2.7] the
induced representation
$\operatorname{Ind}_{Z{\mathrm{SL}}_{n}}^{\GL_{n}}\widetilde{\rho}$ decomposes
into a direct product of irreducible representations of $\GL_{n}(F)$, say as
$\pi_{1}\oplus\cdots\oplus\pi_{k}$. Moreover, by [17, Thm 1.2] the
restrictions $\pi_{i}|_{Z{\mathrm{SL}}_{n}(F)}$ only contain
$\widetilde{\varphi}$ once, so the $\pi_{i}$ are distinct:

$\hom_{\GL_{n}}(\pi_{i},\operatorname{Ind}_{Z{\mathrm{SL}}_{n}}^{\GL_{n}}\widetilde{\varphi})=\hom_{Z{\mathrm{SL}}_{n}}(\pi_{i}|_{Z{\mathrm{SL}}_{n}},\widetilde{\varphi})$

is one-dimensional. Since there are nonzero homomorphisms

$\pi_{i}|_{Z{\mathrm{SL}}_{n}}\to\widetilde{\varphi}\to\pi_{j}|_{Z{\mathrm{SL}}_{n}},$

by [17, Prop 2.4] there exists a character
$\chi\colon\GL_{n}/Z{\mathrm{SL}}_{n}\to{\mathbb{C}}^{\times}$ such that
$\pi_{i}\cong\chi\pi_{j}$. Conversely, if $\pi_{i}$ is an irreducible
component of
$\operatorname{Ind}_{Z{\mathrm{SL}}_{n}}^{\GL_{n}}\widetilde{\varphi}$ then so
is any twist $\chi\pi_{1}$ by a character
$\chi\colon\GL_{n}/Z{\mathrm{SL}}_{n}\to{\mathbb{C}}^{\times}$. Thus, in fact

$(\operatorname{Ind}_{Z{\mathrm{SL}}_{n}}^{\GL_{n}}\widetilde{\varphi})^{\oplus
d}\cong\bigoplus_{\chi\colon\GL_{n}/Z{\mathrm{SL}}_{n}\to{\mathbb{C}}^{\times}}\chi\pi_{1},$

where $d$ is the size of the subgroup

$\\{\chi\colon\GL_{n}/Z{\mathrm{SL}}_{n}\to{\mathbb{C}}^{\times}:\chi\pi_{1}\cong\pi_{1}\\}.$

Note that $d$ is finite since $\GL_{n}/Z{\mathrm{SL}}_{n}\cong
F^{\times}/(F^{\times})^{n}\cong\mathbb{Z}/n$ is finite. Now, the $K_{N}$
invariants are, by Lemma 3.2,

(A.1) $\bigoplus_{g\in
Z{\mathrm{SL}}_{n}\backslash\GL_{n}/K_{N}}\widetilde{\varphi}^{Z{\mathrm{SL}}_{n}\cap
gK_{N}g^{-1}}=\bigoplus_{g\in
Z{\mathrm{SL}}_{n}\backslash\GL_{n}/K_{N}}\widetilde{\varphi}^{gK_{N}g^{-1}}\cong(\pi_{1}^{K_{N}})^{\oplus
n/d}.$

Since each $\\{gK_{N}g^{-1}\\}$ is a fundamental system of neighborhoods of
${\mathrm{SL}}_{n}$ decreasing at the same rate, by Harish-Chandra’s local
character formula they all behave the same asymptotically:

$\dim(\widetilde{\varphi}^{gK_{N}g^{-1}})=(1+o(1))\dim(\widetilde{\varphi}^{hK_{N}h^{-1}}).$

Thus, equation A.1 gives:

$\frac{n}{d}\dim(\pi_{1}^{K_{N}})=(1+o(1))\cdot|Z{\mathrm{SL}}_{n}\backslash\GL_{n}/K_{N}|\dim(\widetilde{\varphi}^{K_{N}})=(n+o(1))\dim(\varphi^{K_{N}\cap{\mathrm{SL}}_{n}}),$

so that

$\dim(\varphi^{K_{N}\cap{\mathrm{SL}}_{n}})=(\frac{1}{d}+o(1))\dim(\pi_{1}^{K_{N}}).$

To summarize:

######  Proposition A.1.

Let $\varphi$ be an irreducible representation of ${\mathrm{SL}}_{n}$. There
exists an irreducible representation $\pi_{1}$ of $\GL_{n}$ such that
$\varphi\hookrightarrow\pi_{1}|_{{\mathrm{SL}}_{n}}$. Then,

$\dim(\varphi^{K_{N}\cap{\mathrm{SL}}_{n}})=\big{(}\frac{1}{d}+o(1)\big{)}\dim(\pi_{1}^{K_{N}}),$

where $d$ is the number of characters
$\chi\colon\GL_{n}/Z{\mathrm{SL}}_{n}(F)\to{\mathbb{C}}^{\times}$ such that
$\chi\pi_{1}\cong\pi_{1}$ as $\GL_{n}$-representations.

######  Remark.

If $\varphi$ is supercuspidal, then so is $\pi_{1}$ [17]. In the language of
[5], let $\pi_{1}=\operatorname{c-Ind}_{\mathbf{J}}^{G}\Lambda$, where
$(\mathbf{J}=\mathbf{J}(\beta,\mathfrak{a}),\Lambda)$ is an EMST in $G$. Then,
by [5, Lem 3.1.3] and [5, 3.1.4], $d=e(F[\beta]/F)$, the ramification degree,
where $\beta\in\GL_{n}(F)$ is such that $F[\beta]$ is a field.

The correcting factor $d$ can be deduced from the Bernstein-Zelevinsky
classification of the representation $\pi_{1}$:

######  Corollary A.2.

Let $\varphi\hookrightarrow\pi_{1}|_{{\mathrm{SL}}_{n}(F)}$ be an irreducible
representation of ${\mathrm{SL}}_{n}$, where $\pi_{1}=\langle a\rangle$, with
$a=\\{\Delta_{1},\dots,\Delta_{m}\\}$ a multisegment, where
$\Delta_{i}=[\rho_{i},\nu^{r_{i}-1}\rho_{i}]$ for a supercuspidal
representation $\rho_{i}$ of $\GL_{n_{i}}(F)$. Let $d$ be the number of
characters $\chi\colon F^{\times}/(F^{\times})^{n}\to{\mathbb{C}}^{\times}$
such that
$\\{\Delta_{1}\otimes\chi,\dots,\Delta_{m}\otimes\chi\\}=\\{\Delta_{1},\dots,\Delta_{m}\\}$
as multisets, where
$[\rho,\nu^{r-1}\rho]\otimes\chi\colonequals[\rho\otimes\chi\circ\det,\nu^{r-1}\rho\otimes\chi\circ\det]$.
Then,

$\dim(\varphi^{K_{N}\cap{\mathrm{SL}}_{n}})=(1+o(1))\frac{1}{d}\frac{[n!]_{q}}{[r_{1}!]_{q^{n_{1}}}\cdots[r_{m}!]_{q^{n_{m}}}}(q^{N-1})^{\frac{1}{2}(n^{2}-n_{1}r_{1}^{2}-\cdots-
n_{m}r_{m}^{2})},$

###### Proof.

Immediate from Proposition A.1, Theorem 3.10 and the uniqueness of the
Bernstein-Zelevinsky classification [21, Thm 6.1]. ∎

Thus, as opposed to $\GL_{n}$, where the leading coefficient of growth
polynomials only involved the combinatorial structure of the multisegment, the
leading coefficient for the growth polynomial of representations of
${\mathrm{SL}}_{n}$ involves the properties of the supercuspidal
representations $\rho_{i}$.

######  Example A.3.

Let
$\varphi\hookrightarrow(\langle[\rho,\nu\rho]\rangle\times\langle[\chi\rho,\nu\chi\rho]\rangle)|_{{\mathrm{SL}}_{n}}$
where $\chi(x)\colonequals(-1)^{v_{F}(x)}$ is a character of $F^{\times}$, and
$\rho=\operatorname{c-Ind}_{\mathbf{J}}^{\GL_{n_{1}}}\Lambda$ where
$(\mathbf{J}=\mathbf{J}(\beta,\mathfrak{a}),\Lambda)$ is an EMST in
$\GL_{n_{1}}(F)$, with $d_{1}=e(F[\beta]/F)$ odd. Then, $2d_{1}$ characters of
$F^{\times}/(F^{\times})^{n}$ fix the multisegment
$\\{[\rho,\nu\rho],[\chi\rho,\nu\chi\rho]\\}$ (generated by $\chi$ and the
characters fixing $\rho$), so:

$\dim(\varphi^{K_{N}\cap{\mathrm{SL}}_{n}})=(1+o(1))\frac{1}{2d_{1}}\frac{[n!]_{q}}{[2]_{q^{n_{1}}}^{2}}(q^{N-1})^{8n_{1}^{2}-4n_{1}}.$

##  Appendix B Direct calculation of the Gelfand-Kirillov dimension of
supercuspidal representations

We will provide a different proof of a weaker form of Proposition 5.8, based
on Bushnell’s classification of supercuspidal representations of $\GL_{n}(F)$:

######  Theorem B.1 (Bushnell-Kutzko [7]).

If $\rho$ is an irreducible supercuspidal representation of $G=\GL_{n}(F)$,
then

$\rho\cong\operatorname{c-Ind}_{{\mathbf{K}}}^{G}\Lambda,$

for some compact modulo center subgroup ${\mathbf{K}}\supset Z$ and
representation $\Lambda$ on ${\mathbf{K}}$.

Here, since $\Lambda$ is an irreducible representation on a compact modulo
center subgroup ${\mathbf{K}}$, it must be finite-dimensional.

The following lemma on the size of double cosets is useful:

######  Lemma B.2.

For $g\in G$, the size of the double coset $K_{0}\backslash K_{0}gK_{0}/K_{N}$
depends only $g\in K_{0}\backslash G/ZK_{0}$, and when
$g=(\varpi^{a_{1}},\cdots,\varpi^{a_{n}})$ with $a_{1}\geq\cdots\geq a_{n}$
then

$|K_{0}\backslash
K_{0}gK_{0}/K_{N}|=q^{\frac{1}{2}(n_{1}^{2}+\cdots+n_{r}^{2}-n^{2})+\sum_{i<j\leq
n}\min\\{a_{i}-a_{j},N\\}}\begin{bmatrix}n\\\
n_{1},\dots,n_{r}\end{bmatrix}_{q},$

where $n=n_{1}+\cdots+n_{r}$ is a partition such that

$a_{1}=\cdots=a_{n_{1}}>a_{n_{1}+1}=\cdots=a_{n_{1}+n_{2}}>\cdots>a_{n_{1}+\dots+n_{r-1}+1}=\cdots=a_{n},$

and

$\begin{bmatrix}n\\\
n_{1},\dots,n_{r}\end{bmatrix}_{q}\colonequals\frac{[n!]_{q}}{[n_{1}!]_{q}\cdots[n_{r}!]_{q}},$

where $[n!]_{q}\colonequals\prod_{k=1}^{n}\frac{q^{n}-1}{q-1}$ is the
$q$-analogue of $n!$.

###### Proof.

First of all, there is a bijection

$\displaystyle(K_{0}\cap gK_{0}g^{-1})\backslash K_{0}/K_{N}$
$\displaystyle\xrightarrow{\sim}K_{0}\backslash K_{0}gK_{0}/K_{N}$
$\displaystyle x$ $\displaystyle\mapsto gx.$

When $g=(\varpi^{a_{1}},\cdots,\varpi^{a_{n}})$ with $a_{1}\geq\cdots\geq
a_{n}\geq 0$, we have

$K_{0}\cap
gK_{0}g^{-1}=\begin{pmatrix}{\mathscr{O}}^{\times}&{\mathfrak{p}}^{a_{1}-a_{2}}&\cdots\\\
{*}&{\mathscr{O}}^{\times}&{\mathfrak{p}}^{a_{2}-a_{3}}&\\\ {*}&{*}&\ddots\\\
{*}&&&{\mathscr{O}}^{\times}\end{pmatrix}\subset K_{0}.$

Let $\lambda$ be the partition $n=n_{1}+\cdots+n_{r}$ and denote the distinct
values $a_{n_{1}+\dots+n_{i}}$ as $b_{i}$. Then, if ${{}^{t}}P_{\lambda}$ is
the corresponding parabolic subgroup,

${{}^{t}}P_{\lambda}\colonequals\begin{pmatrix}\GL_{n_{1}}&0&\cdots&0\\\
{*}&\GL_{n_{2}}&0&\vdots\\\ &&\ddots&0\\\ {*}&{*}&&\GL_{n_{r}}\end{pmatrix},$

then $(K_{0}\cap
gK_{0}g^{-1})/K_{N}\supset{{}^{t}}P_{\lambda}({\mathscr{O}}/{\mathfrak{p}}^{N})$.
Now, the abelian group

$\frac{1+M_{n}({\mathfrak{p}}/{\mathfrak{p}}^{N})}{(1+M_{n}({\mathfrak{p}}/{\mathfrak{p}}^{N}))\cap
gK_{0}g^{-1}}$

acts simply on the quotient $(K_{0}\cap gK_{0}g^{-1})\backslash K_{0}/K_{N}$
from the right with quotient $P_{\lambda}^{-}(k)\backslash\GL_{n}(k)$, so

$\displaystyle|\frac{\GL_{n}({\mathscr{O}}/{\mathfrak{p}}^{N})}{K_{0}\cap
gK_{0}g^{-1}}|$
$\displaystyle=|\frac{1+M_{n}({\mathfrak{p}}/{\mathfrak{p}}^{N})}{(1+M_{n}({\mathfrak{p}}/{\mathfrak{p}}^{N}))\cap
gK_{0}g^{-1}}|\cdot|\frac{\GL_{n}(k)}{{{}^{t}}P_{\lambda}(k)}|$
$\displaystyle=\prod_{i<j\leq
r}|({\mathfrak{p}}/{\mathfrak{p}}^{N})/({\mathfrak{p}}^{b_{i}-b_{j}}/{\mathfrak{p}}^{N})|^{n_{i}n_{j}}\cdot\begin{bmatrix}n\\\
n_{1},\dots,n_{r}\end{bmatrix}_{q}$ $\displaystyle=q^{\sum_{i<j\leq
r}n_{i}n_{j}(\min\\{b_{i}-b_{j},N\\}-1)}\begin{bmatrix}n\\\
n_{1},\dots,n_{r}\end{bmatrix}_{q}$
$\displaystyle=q^{\frac{1}{2}(n_{1}^{2}+\cdots+n_{r}^{2}-n^{2})+\sum_{i<j\leq
n}\min\\{a_{i}-a_{j},N\\}}\begin{bmatrix}n\\\
n_{1},\dots,n_{r}\end{bmatrix}_{q}.\qed$

######  Proposition B.3.

Let $\rho$ be an irreducible supercuspidal representation of $\GL_{n}(F)$.
Then,

$\dim_{GK}(\rho)=\frac{n(n-1)}{2}.$

###### Proof.

Let $\rho=\operatorname{c-Ind}_{{\mathbf{K}}}^{G}(\lambda)$ for an irreducible
representation $\lambda$ of ${\mathbf{K}}$. The intersection
$K\colonequals{\mathbf{K}}\cap\\{g\in G:|\det g|=1\\}$ must be compact open
(since ${\mathcal{O}}_{F}^{\times}\subset F^{\times}$ is open), so by
conjugating we may assume $K\subset K_{0}$. Then,

$\displaystyle\rho^{K_{N}}$
$\displaystyle=\bigoplus_{g\in{\mathbf{K}}\backslash
G/K_{N}}\Lambda^{{\mathbf{K}}\cap gK_{N}g^{-1}}$
$\displaystyle=\bigoplus_{g\in{\mathbf{K}}\backslash G/K_{N}}\lambda^{K\cap
gK_{N}g^{-1}},$

where $\lambda=\Lambda|_{K}$ is irreducible. Since $\lambda$ is finite-
dimensional

$\dim(\rho^{K_{N}})\sim|{\mathbf{K}}\backslash S/K_{N}|,$

where $S\colonequals\\{g\in G:\lambda^{K\cap gK_{N}g^{-1}}\neq 0\\}$.
Moreover, since ${\mathbf{K}}\supset ZK\subset ZK_{0}$ is a chain of finite-
index inclusions,

$\dim(\rho^{K_{N}})\sim|ZK_{0}\backslash S/K_{N}|.$

Here, $\lambda$ is trivial on some compact subgroup $K_{m}\subset K$.

The general idea of the calculation is to “bound” the set $S$ by double cosets
$ZK_{0}\backslash G/K_{0}$. That is, we find subsets $A,B\subset
ZK_{0}\backslash G/K_{0}$ such that

$ZK_{0}AK_{0}\subset S\subset ZK_{0}BK_{0},$

and calculate the size of the double cosets $ZK_{0}\backslash
ZK_{0}AK_{0}/K_{N}$ and $ZK_{0}\backslash ZK_{0}BK_{0}/K_{N}$. Such a strategy
is fruitful since the double coset $ZK_{0}\backslash G/K_{0}$ has a
particularly simple description, the Cartan decomposition.

Firstly, since if $K_{0}\cap gK_{N}g^{-1}\subset K_{m}$ then $\lambda$ is
trivial on $K\cap gK_{N}g^{-1}\subset K_{m}$, we have the inclusion

$S\supset\\{g\in G:K_{0}\cap gK_{N}g^{-1}\subset K_{m}\\}.$

Now, since $ZK_{0}$ normalizes $K_{N}$ and $K_{m}$, the set on the right is a
$ZK_{0}\backslash G/K_{0}$-double coset. A representative
$(\varpi^{a_{1}},\dots,\varpi^{a_{n}})\in ZK_{0}\backslash G/K_{0}$ is such
that

$K_{0}\cap gK_{N}g^{-1}=1+({\mathfrak{p}}^{\max\\{0,N+a_{i}-a_{j}\\}}),$

which a subset of $K_{m}$ exactly when $a_{i}-a_{j}<N-m$ for each $i$ and $j$.
Thus,

(B.1) $S\supset
K_{0}\\{(\varpi^{a_{1}},\cdots,\varpi^{a_{n}}):a_{1}\geq\cdots\geq
a_{n},a_{i}-a_{i+1}\leq N-m\\}ZK_{0}.$

On the other hand,

(B.2) $S\subset K_{0}\\{(\varpi^{a_{1}},\cdots,\varpi^{a_{n}})\in
G:a_{1}\geq\cdots\geq a_{n},\text{ and for each }i,a_{i}-a_{i+1}\leq
N\\}K_{0}.$

We will show the converse; let

$g=u(\varpi^{a_{1}},\cdots,\varpi^{a_{r}},\varpi^{a_{r+1}},\cdots,\varpi^{a_{n}})v$

where $a_{r}-a_{r+1}>N$ and $u,v\in K_{0}$. We hope to show $g\notin S$. For
any $M>0$, we claim

$(K_{0}\cap gK_{N}g^{-1})\cdot K_{m}=(K_{0}\cap hK_{N}h^{-1})\cdot K_{m}$

so in particular

$\lambda^{K\cap gK_{N}g^{-1}}=\lambda^{K\cap hK_{N}h^{-1}}$

if

$h=u(\varpi^{a_{1}+M},\cdots,\varpi^{a_{r}+M},\varpi^{a_{r+1}},\cdots,\varpi^{a_{n}})v.$

Indeed,

$(K_{0}\cap hK_{N}h^{-1})\cdot K_{m}=(1_{n\times
n}+({\mathfrak{p}}^{a_{ij}}))\cdot K_{m},$

where

$a_{ij}\colonequals\begin{cases}\max\\{N+a_{i}-a_{j},0\\}&\text{if }i,j\leq
r\text{ or }i,j>r,\\\ \max\\{N+a_{i}-a_{j}+M,0\\}&\text{if }i\leq r<j,\\\
\max\\{N+a_{i}-a_{j}-M,0\\}&\text{if }i>r\geq j.\end{cases}$

We see that if $i>r\geq j$ then

$N+a_{i}-a_{j}-M\leq N+a_{r}-a_{r+1}\leq 0$

so $a_{ij}=0$, and otherwise $a_{ij}\geq N$, so

$(K_{0}\cap hK_{N}h^{-1})K_{m}=\begin{pmatrix}1_{r\times
r}&0_{r\times(n-r)}\\\ *&1_{(n-r)\times(n-r)}\end{pmatrix}\cdot K_{m},$

which is independent of the choice of $M$.

In particular, $g\in S$ if and only if $h\in S$. Thus, if $g\in S$ then

$ZK_{0}\backslash S/K_{N}\supset ZK_{0}\backslash\bigcup_{M\geq
0}ZK_{0}(\varpi^{a_{1}+M},\cdots,\varpi^{a_{r}+M},\varpi^{a_{r+1}},\cdots,\varpi^{a_{n}})K_{0}/K_{N},$

which is clearly infinite, contradicting $ZK_{0}\backslash S/K_{N}$ being
finite.

Finally, putting together (B.1), (B.2), and Lemma B.2, we obtain:

$\displaystyle\dim(\rho^{K_{N}})$ $\displaystyle\sim|{\mathbf{K}}\backslash
S/K_{N}|$ $\displaystyle\sim\sum_{\begin{subarray}{c}a_{1}\geq\cdots\geq
a_{n}\\\ a_{i}-a_{i+1}<N\end{subarray}}q^{\sum_{i<j}\min\\{a_{i}-a_{j},N\\}}$
$\displaystyle\sim q^{\frac{n(n-1)}{2}N},$

concluding the proof of Proposition B.3 that for $\rho$ a supercuspidal
representation of $\GL_{n}(F)$,

$\dim_{GK}(\rho)=\frac{n(n-1)}{2}.\qed$

######  Remark.

The technique in the proof of Proposition B.3 is quite general, once one knows
a representation is induced from a compact open modulo center subgroup. For
instance, [7, Thm 2.1] shows any supercuspidal representation $\rho$ of
${\mathrm{SL}}_{n}(F)$ is induced from a compact open subgroup, so again,

$\dim(\rho^{K_{N}\cap{\mathrm{SL}}_{n}(F)})\sim q^{\frac{n(n-1)}{2}N}.$

A more precise bound is given in Appendix A.

##  Appendix C Level zero representations of $\GL_{2}$ and $\GL_{3}$

An irreducible representation $\pi$ of $G$ is _of level zero_ if
$\pi^{K_{1}}\neq 0$. Such representations which are cuspidal are classified by
the following:

######  Proposition C.1 ([5, Section 1.2]).

Let $\rho$ be an irreducible cuspidal representation of level zero. Then,
there exists a representation $\Lambda$ of $K_{0}Z$ such that
$\Lambda|_{K_{0}}$ is inflated from a cuspidal representation of $\GL_{n}(k)$,
such that

$\rho\cong\operatorname{c-Ind}_{K_{0}Z}^{G}\Lambda.$

The growth polynomial of level zero representations may be calculated using
the techniques in the proof of Proposition B.3. A general formula for
$\dim(\rho^{K_{N}})$, for $\rho$ supercuspidal of level zero is an immediate
consequence of Proposition 5.9:

######  Corollary C.2.

If $\rho$ is a supercuspidal level zero representation, then

$G_{\rho}(X)=\sum_{{\mathcal{O}}}(-1)^{n+r}n(r-1)!w_{\mathcal{O}}^{-1}\begin{bmatrix}n\\\
n_{1},\dots,n_{r}\end{bmatrix}_{q}X^{\frac{1}{2}\dim{\mathcal{O}}},$

where the nilpotent orbit ${\mathcal{O}}$ correspond to the partition
$n=n_{1}+\cdots+n_{r}$, and $w_{\mathcal{O}}$ is the number of permutation of
$r$ letters which fix the $r$-tuple $(n_{1},\dots,n_{r})$.

We offer an alternate, _more direct_ calculation for level zero
representations for small $n$.

Let $\rho=\operatorname{c-Ind}_{K_{0}Z}^{G}\Lambda$ be an irreducible
supercuspidal representation of level zero. By Lemma 3.2,

$\displaystyle\rho^{K_{N}}$ $\displaystyle=\bigoplus_{g\in K_{0}Z\backslash
G/K_{N}}\Lambda^{K_{0}Z\cap gK_{N}g^{-1}}$ $\displaystyle=\bigoplus_{g\in
K_{0}Z\backslash G/K_{N}}\sigma^{(K_{0}\cap gK_{N}g^{-1})/K_{1}},$

where $\Lambda|_{K}$ is inflated from the representation $\sigma$ of
$\GL_{n}(k)$. Since $K_{0}Z$ normalizes $K_{N}$, only the images of $g$ in
$K_{0}Z\backslash G/K_{0}$ matter. That is, we may suppose $g$ is of the form
$(\varpi^{a_{1}},\dots,\varpi^{a_{n}})$ for integers $a_{1}\geq\cdots\geq
a_{n}=0$. Thus,

(C.1) $\dim(\rho^{K_{N}})=\sum_{a_{1}\geq\cdots\geq a_{n}=0}|K_{0}\backslash
K_{0}(\varpi^{a_{1}},\dots,\varpi^{a_{n}})K_{0}/K_{N}|\cdot\dim(\sigma^{(K_{0}\cap
gK_{N}g^{-1})/K_{1}}).$

This gives a polynomial expression for the dimension $\dim(\rho^{K_{N}})$. To
illustrate the techniques, we will give an explicit formula when $n=2,3$ (in
the case $n=2$, this is a direct confirmation of Example 6.4):

######  Proposition C.3.

When $\rho$ is a supercuspidal level zero representation of $\GL_{2}$,

$G_{\rho}(X)=[2!]_{q}X-2$

and when $\rho$ is a supercuspidal level zero representation of $\GL_{3}$,

$G_{\rho}(X)=[3!]_{q}X^{3}-3[3]_{q}X^{2}+3,$

where $X=q^{N-1}$, and $[n]_{q}\colonequals\frac{q^{n}-1}{q-1}$,
$[n!]_{q}\colonequals[1]_{q}\cdots[n]_{q}$.

###### Proof.

When $n=2$ and $g=(\varpi^{a},1)\in K_{0}Z\backslash G/K_{0}$ with $a\geq 0$,
then

$(K_{0}\cap gK_{N}g^{-1})/K_{1}=\begin{pmatrix}1&0\\\
{\mathfrak{p}}^{\max\\{0,N-a\\}}&1\end{pmatrix}/K_{1}=\begin{cases}1&\text{if
}N>a\\\ \begin{pmatrix}1&0\\\ {*}&1\end{pmatrix}&\text{if }N\leq
a,\end{cases}$

so since $\sigma$ is cuspidal,

$\sigma^{(K_{0}\cap gK_{N}g^{-1})/K_{1}}=\begin{cases}\sigma&\text{if }N>a\\\
0&\text{if }N\leq a.\end{cases}$

Thus, since $\dim(\sigma)=q-1$ (see [6, Thm 6.4]), by (C.1) and Lemma B.2 we
have:

$\displaystyle\dim(\rho^{K_{N}})$ $\displaystyle=\sum_{0\leq
a<N}(q-1)|K_{0}\backslash K_{0}(\varpi^{a},1)K_{0}/K_{N}|$
$\displaystyle=(q-1)+\sum_{0<a<N}(q-1)\cdot q^{a-1}(q+1)$
$\displaystyle=(q+1)q^{N-1}-2.$

When $n=3$ and $g=(\varpi^{a},\varpi^{b},1)\in K_{0}Z\backslash G/K_{0}$ with
$a\geq b\geq 0$ then

$(K_{0}\cap gK_{N}g^{-1})/K_{1}=\begin{cases}1&\text{if }N>a\\\
\begin{pmatrix}1\\\ &1\\\ {*}&&1\end{pmatrix}&\text{if }a\geq N,a-b,b<N\\\
\text{Levi subgroup}&\text{otherwise}.\end{cases}$

In the last case, $\sigma^{(K_{0}\cap gK_{N}g^{-1})/K_{1}}=0$ since $\sigma$
is cuspidal. In the first case, $\sigma^{(K_{0}\cap
gK_{N}g^{-1})/K_{1}}=\sigma$. For the second case, decompose $\sigma$ as
follows:

$\sigma\cong\bigoplus_{\chi:N\to{\mathbb{C}}^{\times}}\chi^{m_{\chi}},$

where $N\colonequals\begin{pmatrix}1&\\\ &1\\\ {*}&{*}&1\end{pmatrix}$ is a
nilpotent subgroup. Since $\sigma$ is cuspidal, $m_{1}=0$, and the conjugation
action by $\GL_{2}(k)\times k^{\times}$ on the subgroup $N$ permutes the other
characters transitively so $m_{\chi}$ is constant for nontrivial $\chi$. Thus,
in fact $\sigma\cong\bigoplus_{\chi\neq 1}\chi^{\oplus d/(q^{2}-1)}$, where
$d=(q-1)(q^{2}-1)$ is the dimension of $\sigma$.

Thus, $\sigma^{\begin{pmatrix}1\\\ &1\\\ {*}&&1\end{pmatrix}}$ is of dimension
$d/(q+1)$. Combining,

$\displaystyle\dim(\rho^{K_{N}})=d\sum_{N>a\geq b\geq 0}$
$\displaystyle|K_{0}\backslash K_{0}(\varpi^{a},\varpi^{b},1)K_{0}/K_{N}|$
$\displaystyle+\frac{d}{q+1}\sum_{\begin{subarray}{c}a\geq b\geq 0\\\
a-b,b<N,a\geq N\end{subarray}}|K_{0}\backslash
K_{0}(\varpi^{a},\varpi^{b},1)K_{0}/K_{N}|.$

For each of these sums, we will run casework on what partition $\\{a,b,0\\}$
corresponds to and apply Lemma B.2:

  * •

$3=3$:

$d\sum_{a=b=0}|K_{0}\backslash K_{0}(\varpi^{a},\varpi^{b},1)K_{0}/K_{N}|=d.$

  * •

$3=2+1$:

$\displaystyle d\sum_{N>a=b>0}|K_{0}\backslash
K_{0}(\varpi^{a},\varpi^{b},1)K_{0}/K_{N}|$
$\displaystyle=d\sum_{N>a>0}q^{-2+2a}[3]_{q}$
$\displaystyle=\frac{d[3]_{q}}{q^{2}-1}(q^{2N-2}-1).$

  * •

$3=1+2$: the formula is the same as in $3=1+2$.

  * •

$3=1+1+1$:

$\displaystyle d\sum_{N>a>b>0}|K_{0}\backslash
K_{0}(\varpi^{a},\varpi^{b},1)K_{0}/K_{N}|$
$\displaystyle=d\sum_{N>a>0}(a-1)q^{-3+2a}[3!]_{q}$
$\displaystyle=\frac{d[3!]_{q}}{(q^{2}-1)^{2}}((N-2)q^{2N-1}-(N-1)q^{2N-3}+q).$

For the second term, only the partition $3=1+1+1$ appears, and evaluates as
(this is the main term):

$\displaystyle\frac{d}{q+1}\sum_{\begin{subarray}{c}a>b>0\\\ a-b,b<N,a\geq
N\end{subarray}}q^{-3+a+N}[3!]_{q}$
$\displaystyle=\frac{d}{q+1}\sum_{a=N}^{2N-2}(2N-a-1)q^{-3+a+N}[3!]_{q}$
$\displaystyle=\frac{d[3!]_{q}}{(q+1)(q-1)^{2}}(q^{3N-3}-Nq^{2N-2}+(N-1)q^{2N-3}).$

Combining everything, and using $d=(q-1)(q^{2}-1)$, we obtain:

$\dim(\rho^{K_{N}})=[3!]_{q}\cdot q^{3N-3}-3[3]_{q}q^{2N-2}+3.\qed$

######  Remark.

Combining the sums corresponding to the partition $3=1+1+1$ from both terms
gives a polynomial in $q^{N-1}$, suggesting a clever reorganization of sums
may help simplify and generalize the calculation.

## References

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