# The Newform $K$-Type and $p$-adic Spherical Harmonics

Peter Humphries  Department of Mathematics, University of Virginia,
Charlottesville, VA 22904, USA<EMAIL_ADDRESS>https://sites.google.com/view/peterhumphries/

###### Abstract.

Let $K\coloneqq\operatorname{GL}_{n}(\mathcal{O})$ denote the maximal compact
subgroup of $\operatorname{GL}_{n}(F)$ with $F$ a nonarchimedean local field.
We study the decomposition of the space of square-integrable functions on the
unit sphere in $F^{n}$ into irreducible $K$-modules; for $F=\mathbb{Q}_{p}$,
these are the $p$-adic analogues of spherical harmonics. As an application, we
characterise the newform and conductor exponent of a generic irreducible
admissible smooth representation of $\operatorname{GL}_{n}(F)$ in terms of
distinguished $K$-types. Finally, we compare our results to analogous results
in the archimedean setting.

###### 2010 Mathematics Subject Classification:

20G25 (primary); 11F70, 20G05, 22E50, 33C55 (secondary)

##  1\. Introduction

###  1.1. The Space $L^{2}(S^{n-1})$ as a $K$-Module

Let $F$ be a nonarchimedean local field with ring of integers $\mathcal{O}$,
maximal prime ideal $\mathfrak{p}$, and uniformiser $\varpi$, so that
$\varpi\mathcal{O}=\mathfrak{p}$ and
$\mathcal{O}/\mathfrak{p}\cong\mathbb{F}_{q}$ for some prime power $q$. Thus
either $F$ is a finite extension of the $p$-adic numbers $\mathbb{Q}_{p}$ for
some prime $p$ or $F$ is the field of formal Laurent series
$\mathbb{F}_{q}((t))$, where $\mathbb{F}_{q}$ is the finite field of prime
power order $q$. We let $K_{n}\coloneqq\operatorname{GL}_{n}(\mathcal{O})$
denote the maximal compact subgroup of $\operatorname{GL}_{n}(F)$, which is
unique up to conjugacy; when it is clear from context, we write $K$ in place
of $K_{n}$. Throughout we assume that $n\geq 2$.

The group $K_{n}$ acts transitively on unit sphere $S^{n-1}$ in $F^{n}$ via
the group action $k\cdot x\coloneqq xk$ for $k\in K_{n}$ and $x\in S^{n-1}$,
where

$S^{n-1}\coloneqq\left\\{x=(x_{1},\ldots,x_{n})\in
F^{n}:\max\\{|x_{1}|,\ldots,|x_{n}|\\}=1\right\\};$

here $|\cdot|$ denotes the absolute value on $F$ normalised such that
$|\varpi|=q^{-1}$. The stabiliser of the point
$e_{n}\coloneqq(0,\ldots,0,1)\in S^{n-1}$ is the subgroup

(1.1) $K_{n-1,1}\coloneqq\left\\{\begin{pmatrix}a&b\\\ 0&1\end{pmatrix}\in
K_{n}:a\in K_{n-1},\ b\in\operatorname{Mat}_{(n-1)\times
1}(\mathcal{O})\right\\}$

of $K_{n}$, which has the structure of the outer semidirect product
$K_{n-1}\ltimes\mathcal{O}^{n-1}$. It follows that $S^{n-1}\cong
K_{n-1,1}\backslash K_{n}$. Note that $K_{n-1,1}$ is the maximal compact
subgroup of the mirabolic subgroup

(1.2) $\left\\{\begin{pmatrix}a&b\\\
0&1\end{pmatrix}\in\operatorname{GL}_{n}(F):a\in\operatorname{GL}_{n-1}(F),\
b\in\operatorname{Mat}_{(n-1)\times 1}(F)\right\\}$

of $\operatorname{GL}_{n}(F)$.

Let $R$ denote the right regular representation of $K_{n}\ni k$ on
$L^{2}(S^{n-1})\ni f$; thus $(R(k)\cdot f)(x)\coloneqq f(xk)$ for $x\in
S^{n-1}$. A natural question to ponder is the following.

######  Question.

What is the decomposition of the right regular representation $R$ of $K_{n}$
on $L^{2}(S^{n-1})$ into irreducible smooth representations of $K_{n}$?
Equivalently, which irreducible smooth representations of $K_{n}$ have a
nontrivial $K_{n-1,1}$-fixed vector?

We study this problem in Section 2. While the general classification of
irreducible smooth representations of $K_{n}$ is unknown for $n\geq 3$, we
show that the representations having a nontrivial $K_{n-1,1}$-fixed vector can
be explicitly described; in particular, a precise resolution of this question
is given in Theorem 2.14. The irreducible smooth representations of interest
are indexed by pairs $(\chi,m)$ of characters $\chi$ of $\mathcal{O}^{\times}$
and nonnegative integers $m\geq c(\chi)$, where $c(\chi)$ denotes the
conductor exponent of $\chi$. The character $\chi$ is the central character of
this representation, while $m$ is its level, namely the minimal nonnegative
integer for which this representation factors through
$\operatorname{GL}_{n}(\mathcal{O}/\mathfrak{p}^{m})$. We denote by
$(\tau_{\chi,m},\mathcal{H}_{\chi,m}(S^{n-1}))$ the irreducible smooth
representation associated to such a pair, where
$\mathcal{H}_{\chi,m}(S^{n-1})$ is a finite-dimensional vector subspace of
$L^{2}(S^{n-1})$; this may be thought of as the nonarchimedean analogue of the
vector space of spherical harmonics of degree $m$ on the unit sphere in
$\mathbb{R}^{n}$.

###  1.2. Zonal Spherical Functions

We next study the subspace of locally constant functions in $L^{2}(S^{n-1})$
that are $K_{n-1,1}$-invariant in Section 3. Each irreducible subspace
$\mathcal{H}_{\chi,m}(S^{n-1})$ of $L^{2}(S^{n-1})$ has a one-dimensional
subspace of $K_{n-1,1}$-invariant functions, so that there exists a unique
function $P_{\chi,m}^{\circ}\in\mathcal{H}_{\chi,m}(S^{n-1})$ satisfying
$P_{\chi,m}^{\circ}(xk^{\prime})=P_{\chi,m}^{\circ}(x)$ for all $x\in S^{n-1}$
and $k^{\prime}\in K_{n-1,1}$ and $P_{\chi,m}^{\circ}(e_{n})=1$. We call
$P_{\chi,m}^{\circ}$ the zonal spherical function on $S^{n-1}$ of character
$\chi$ and level $m$. These are the nonarchimedean analogues of zonal
spherical harmonics (or ultraspherical polynomials).

We give an explicit formula for $P_{\chi,m}^{\circ}$ in Proposition 3.4. Using
this, we prove a nonarchimedean analogue of the addition formula for spherical
harmonics in Lemma 3.14. A consequence of this is Corollary 3.20, which states
that $(\dim\mathcal{H}_{\chi,m}(S^{n-1}))P_{\chi,m}^{\circ}$ is the
reproducing kernel for $\mathcal{H}_{\chi,m}(S^{n-1})$, which implies that
each element of $\mathcal{H}_{\chi,m}(S^{n-1})$ is _equal_ to a matrix
coefficient of $\tau_{\chi,m}$.

###  1.3. The Newform and the Conductor Exponent

We apply this theory of $p$-adic spherical harmonics in Section 4 to previous
work of Jacquet, Piatetski-Shapiro, and Shalika [JP-SS81]. They prove the
existence of a distinguished vector, the newform, associated to a given
generic irreducible admissible representation $(\pi,V_{\pi})$ of
$\operatorname{GL}_{n}(F)$ (or more generally to an induced representation of
Langlands type). This vector is invariant under a certain congruence subgroup
$K_{1}(\mathfrak{p}^{m})$ of $K_{n}$ and is the minimal such nontrivial vector
in the sense that there are no nontrivial vectors invariant under
$K_{1}(\mathfrak{p}^{\ell})$ with $\ell<m$. This minimal value of $m$ is
called the conductor exponent of $\pi$ and is denoted by $c(\pi)$.

We give alternative characterisations of the newform and conductor exponent in
Theorem 4.16. We show that the conductor exponent $c(\pi)$ of $\pi$ is the
minimal nonnegative integer $m$ for which there exists a nontrivial
$K_{n-1,1}$-invariant $\tau_{\chi,m}$-isotypic vector in $V_{\pi}$ for some
character $\chi$ of $\mathcal{O}^{\times}$; necessarily, $\chi$ must then be
equal to $\chi_{\pi}$, the restriction from $F^{\times}$ to
$\mathcal{O}^{\times}$ of the central character $\omega_{\pi}$ of $\pi$. We
also show that the newform is precisely the nonzero vector, unique up to
scalar multiplication, that is $K_{n-1,1}$-invariant and
$\tau_{\chi_{\pi},c(\pi)}$-isotypic; for this reason, we name
$\tau_{\chi_{\pi},c(\pi)}$ the newform $K$-type. Our methods also allow us to
describe precisely in Proposition 4.25 the multiplicity with which a
representation $\tau_{\chi,m}$ occurs in the $K$-type decomposition of $\pi$.

These different characterisations open up new avenues of approach to studying
properties of the newform. We give a simple example of one such property in
Corollary 4.34, where we show that the matrix coefficient associated to the
newform may be explicitly described in terms of the zonal spherical function
$P_{\chi_{\pi},c(\pi)}^{\circ}\in\mathcal{H}_{\chi_{\pi},c(\pi)}(S^{n-1})$.

###  1.4. The Archimedean Theory

Finally, in Section 5, we compare our results to analogous results in the
archimedean setting. The archimedean analogue of
$K_{n}=\operatorname{GL}_{n}(\mathcal{O})$ is the orthogonal group
$\operatorname{O}(n)$ if $F=\mathbb{R}$ or the unitary group
$\operatorname{U}(n)$ if $F=\mathbb{C}$. The decomposition of $L^{2}(S^{n-1})$
into irreducible $K_{n}$-modules in these settings is well-known: it is the
classical theory of spherical harmonics. These are the restriction to the unit
sphere of homogeneous harmonic polynomials of a given degree if $F=\mathbb{R}$
or of a given bidegree if $F=\mathbb{C}$. In both cases, there exist analogues
of the the zonal spherical functions $P_{\chi,m}^{\circ}$.

The archimedean analogue of the theory of newforms and conductor exponents of
a generic irreducible admissible smooth representation of
$\operatorname{GL}_{n}(F)$ was recently developed by the author [Hum20]. Over
archimedean local fields, the approach of Jacquet, Piatetski-Shapiro, and
Shalika [JP-SS81] via congruence subgroups is no longer applicable, and
instead the development of the theory of newforms is via distinguished
$K_{n}$-types. Thus our alternative characterisation in the nonarchimedean
setting proven in Theorem 4.16 serves to unify these two different settings.

##  2\. $p$-adic Spherical Harmonics

Our goal is to decompose the right regular representation of $K$ on
$L^{2}(S^{n-1})$ into irreducible smooth representations. In place of
$L^{2}(S^{n-1})$, it suffices to study the subspace
$C^{\infty}(S^{n-1})\subset L^{2}(S^{n-1})$ of locally constant functions
$f:S^{n-1}\to\mathbb{C}$, which is dense in $L^{2}(S^{n-1})$. Our first
observation is that the subspace $C^{\infty}(S^{n-1})$ may be identified with
the induced representation of the trivial representation of $K_{n-1,1}$.

######  Lemma 2.1.

As $K$-modules, the space $C^{\infty}(S^{n-1})$ is isomorphic to
$\operatorname{Ind}_{K_{n-1,1}}^{K}1$, the vector space of locally constant
functions $\phi:K\to\mathbb{C}$ that satisfy $\phi(k^{\prime}k)=\phi(k)$ for
all $k\in K$ and $k^{\prime}\in K_{n-1,1}$, upon which $K$ acts via right
translations.

###### Proof.

Given $f\in C^{\infty}(S^{n-1})$, define $\phi(k)\coloneqq f(e_{n}k)$; then
clearly $\phi\in\operatorname{Ind}_{K_{n-1,1}}^{K}1$. Conversely, given
$\phi\in\operatorname{Ind}_{K_{n-1,1}}^{K}1$, $\phi(k)$ is dependent only on
$e_{n}k$. Since for each $x\in S^{n-1}$, there exists some $k\in K$ for which
$x=e_{n}k$, the function $f(x)\coloneqq\phi(k)$ is well-defined, which gives
us an element of $C^{\infty}(S^{n-1})$. ∎

###  2.1. Reduction to $K_{1}(\mathfrak{p}^{m})$

We next study subspaces of $C^{\infty}(S^{n-1})$ consisting of locally
constant functions invariant under certain congruence subgroups. For each
nonnegative integer $m$, let $K(\mathfrak{p}^{m})$ denote the principal
congruence subgroup of level $m$ of $K$, namely

$K(\mathfrak{p}^{m})\coloneqq\left\\{k\in
K:k-1_{n}\in\operatorname{Mat}_{n\times n}(\mathfrak{p}^{m})\right\\},$

where $1_{n}$ denotes the $n\times n$ identity matrix. This is a normal
subgroup of $K$. These subgroups allow us to construct a filtration of
$C^{\infty}(S^{n-1})$ via the subspaces

$C^{\infty}(S^{n-1})^{K(\mathfrak{p}^{m})}\coloneqq\left\\{f\in
C^{\infty}(S^{n-1}):f(xk)=f(x)\text{ for all $x\in S^{n-1}$ and $k\in
K(\mathfrak{p}^{m})$}\right\\}$

of $K(\mathfrak{p}^{m})$-invariant locally constant functions on $S^{n-1}$. We
observe that $C^{\infty}(S^{n-1})^{K(\mathfrak{p}^{m})}$ is contained in
$C^{\infty}(S^{n-1})^{K(\mathfrak{p}^{\ell})}$ for all nonnegative integers
$m\leq\ell$. Furthermore, the union
$\bigcup_{m=0}^{\infty}C^{\infty}(S^{n-1})^{K(\mathfrak{p}^{m})}$ is dense in
$C^{\infty}(S^{n-1})$.

We also define the congruence subgroup $K_{1}(\mathfrak{p}^{m})$ of $K$ by

$K_{1}(\mathfrak{p}^{m})\coloneqq\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in
K:c\in\operatorname{Mat}_{1\times(n-1)}(\mathfrak{p}^{m}),\
d-1\in\mathfrak{p}^{m}\right\\}$

for each nonnegative integer $m$, so that $K_{1}(\mathfrak{p}^{0})=K$, while
if $m\geq 1$, $\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in
K_{1}(\mathfrak{p}^{m})$ implies that $a\in K_{n-1}$ and
$d\in\mathcal{O}^{\times}$; we make note of the fact that
$K_{1}(\mathfrak{p}^{m})$ contains $K_{1}(\mathfrak{p}^{\ell})$ and
$K(\mathfrak{p}^{\ell})$ as subgroups whenever $\ell\geq m$.

######  Lemma 2.2.

As $K$-modules, $C^{\infty}(S^{n-1})^{K(\mathfrak{p}^{m})}$ is isomorphic to
$\operatorname{Ind}_{K_{1}(\mathfrak{p}^{m})}^{K}1$, the vector space of
locally constant functions $\phi:K\to\mathbb{C}$ that satisfy
$\phi(k_{1}k)=\phi(k)$ for all $k\in K$ and $k_{1}\in
K_{1}(\mathfrak{p}^{m})$, upon which $K$ acts via right translations.

###### Proof.

From Lemma 2.1, $C^{\infty}(S^{n-1})^{K(\mathfrak{p}^{m})}$ is isomorphic as a
$K$-module to the space of locally constant functions $\phi:K\to\mathbb{C}$
that satisfy $\phi(k^{\prime}kk^{\prime\prime})=\phi(k)$ for all
$k^{\prime}\in K_{n-1,1}$, $k\in K$, and $k^{\prime\prime}\in
K(\mathfrak{p}^{m})$. Since $K(\mathfrak{p}^{m})$ is a normal subgroup of $K$,
this is equal to the space of locally constant functions $\phi:K\to\mathbb{C}$
that satisfy $\phi(k^{\prime}k^{\prime\prime}k)=\phi(k)$ for all
$k^{\prime}\in K_{n-1,1}$, $k^{\prime\prime}\in K(\mathfrak{p}^{m})$, and
$k\in K$. The result then follows from the fact that
$K_{n-1,1}K(\mathfrak{p}^{m})=K_{1}(\mathfrak{p}^{m})$, which is clearly true
if $m=0$; for $m\geq 1$, it is immediate that
$K_{n-1,1}K(\mathfrak{p}^{m})\subset K_{1}(\mathfrak{p}^{m})$, while the fact
that $K_{n-1,1}K(\mathfrak{p}^{m})\supset K_{1}(\mathfrak{p}^{m})$ can be seen
directly, since for $\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in
K_{1}(\mathfrak{p}^{m})$, we have that

$\begin{pmatrix}a&b\\\ c&d\end{pmatrix}=\begin{pmatrix}a&abd^{-1}\\\
0&1\end{pmatrix}\begin{pmatrix}1_{n-1}-bd^{-1}c&0\\\ c&d\end{pmatrix}.\qed$

######  Corollary 2.3.

We have that

(2.4) $\dim C^{\infty}(S^{n-1})^{K(\mathfrak{p}^{m})}=\begin{dcases*}1&if
$m=0$,\\\ q^{(m-1)n}(q^{n}-1)&if $m\geq 1$.\end{dcases*}$

###### Proof.

We have that $\dim
C^{\infty}(S^{n-1})^{K(\mathfrak{p}^{m})}=\dim\operatorname{Ind}_{K_{1}(\mathfrak{p}^{m})}^{K}1$
from Lemma 2.2. Since the trivial representation is one-dimensional, the
dimension of the monomial representation
$\operatorname{Ind}_{K_{1}(\mathfrak{p}^{m})}^{K}1$ is simply the index
$[K:K_{1}(\mathfrak{p}^{m})]$ of the subgroup $K_{1}(\mathfrak{p}^{m})$ in
$K$, which is precisely the right-hand side of (2.4). ∎

###  2.2. Reduction to $K_{0}(\mathfrak{p}^{m})$

Next, we decompose $C^{\infty}(S^{n-1})^{K(\mathfrak{p}^{m})}$ further into
subspaces of locally constant functions with a prescribed central character.
Given a character $\chi$ of $\mathcal{O}^{\times}$, we define the subspace of
$K(\mathfrak{p}^{m})$-invariant locally constant functions on $S^{n-1}$ with
central character $\chi$ by

$C^{\infty}(S^{n-1})_{\chi}^{K(\mathfrak{p}^{m})}\coloneqq\left\\{f\in
C^{\infty}(S^{n-1})^{K(\mathfrak{p}^{m})}:f(ax)=\chi(a)f(x)\text{ for all
$a\in\mathcal{O}^{\times}$ and $x\in S^{n-1}$}\right\\}.$

Clearly $C^{\infty}(S^{n-1})_{\chi}^{K(\mathfrak{p}^{m})}$ is trivial if
$m<c(\chi)$, where $c(\chi)$ denotes the conductor exponent of $\chi$, namely
the least nonnegative integer $m$ for which $\chi$ is trivial on
$1+\mathfrak{p}^{m}$. Moreover,
$C^{\infty}(S^{n-1})_{\chi}^{K(\mathfrak{p}^{m})}$ is contained in
$C^{\infty}(S^{n-1})_{\chi}^{K(\mathfrak{p}^{\ell})}$ for all $\ell\geq m\geq
c(\chi)$ and any two subspaces
$C^{\infty}(S^{n-1})_{\chi_{1}}^{K(\mathfrak{p}^{m_{1}})}$ and
$C^{\infty}(S^{n-1})_{\chi_{2}}^{K(\mathfrak{p}^{m_{2}})}$ are mutually
orthogonal whenever $\chi_{1}\neq\chi_{2}$.

For each nonnegative integer $m$, the congruence subgroup
$K_{0}(\mathfrak{p}^{m})$ of $K$ is defined by

$K_{0}(\mathfrak{p}^{m})\coloneqq\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in
K:c\in\operatorname{Mat}_{1\times(n-1)}(\mathfrak{p}^{m})\right\\}$

for each nonnegative integer $m$, so that $K_{0}(\mathfrak{p}^{0})=K$, while
if $m\geq 1$, $\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in
K_{0}(\mathfrak{p}^{m})$ implies that $a\in K_{n-1}$ and
$d\in\mathcal{O}^{\times}$. We observe that $K_{0}(\mathfrak{p}^{m})$ contains
$K_{0}(\mathfrak{p}^{\ell})$, $K_{1}(\mathfrak{p}^{\ell})$, and
$K(\mathfrak{p}^{\ell})$ as subgroups whenever $\ell\geq m$.

Let $\widehat{\mathcal{O}^{\times}}$ denote the set of (continuous) characters
$\chi:\mathcal{O}^{\times}\to\mathbb{C}^{\times}$; necessarily the image of
such a character is in the unit circle $\\{z\in\mathbb{C}^{\times}:|z|=1\\}$.
For $\chi\in\widehat{\mathcal{O}^{\times}}$, let $\psi_{\chi}$ be the
character of $K_{0}(\mathfrak{p}^{c(\chi)})\ni k_{0}=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}$ given by $\psi_{\chi}(k_{0})\coloneqq\chi(d)$, which is a
one-dimensional representation of $K_{0}(\mathfrak{p}^{c(\chi)})$; by
restriction, this is also a one-dimensional representation of
$K_{0}(\mathfrak{p}^{m})$ whenever $m\geq c(\chi)$.

######  Lemma 2.5.

For $\chi\in\widehat{\mathcal{O}^{\times}}$ and $m\geq c(\chi)$,
$C^{\infty}(S^{n-1})_{\chi}^{K(\mathfrak{p}^{m})}$ is isomorphic as a
$K$-module to $\operatorname{Ind}_{K_{0}(\mathfrak{p}^{m})}^{K}\psi_{\chi}$,
the vector space of locally constant functions $f:K\to\mathbb{C}$ that satisfy
$f(k_{0}k)=\psi_{\chi}(k_{0})f(k)$ for all $k\in K$ and $k_{0}\in
K_{0}(\mathfrak{p}^{m})$, upon which $K$ acts via right translations.

###### Proof.

From Lemma 2.2, $C^{\infty}(S^{n-1})_{\chi}^{K(\mathfrak{p}^{m})}$ is
isomorphic as a $K$-module to the space of locally constant functions
$\phi:K\to\mathbb{C}$ that satisfy $\phi(z(a)k_{1}k)=\chi(a)\phi(k)$ for all
$z(a)\coloneqq\operatorname{diag}(a,\ldots,a)\in\mathrm{Z}(\mathcal{O})$, the
centre of $K$, $k_{1}\in K_{1}(\mathfrak{p}^{m})$, and $k\in K$. It remains to
note that
$\mathrm{Z}(\mathcal{O})K_{1}(\mathfrak{p}^{m})=K_{0}(\mathfrak{p}^{m})$. ∎

######  Corollary 2.6.

We have that

(2.7) $\dim
C^{\infty}(S^{n-1})_{\chi}^{K(\mathfrak{p}^{m})}=\begin{dcases*}1&if
$m=c(\chi)=0$,\\\ q^{(m-1)(n-1)}\frac{q^{n}-1}{q-1}&if
$m\geq\max\\{c(\chi),1\\}$,\\\ 0&otherwise.\end{dcases*}$

###### Proof.

We have that $\dim
C^{\infty}(S^{n-1})_{\chi}^{K(\mathfrak{p}^{m})}=\dim\operatorname{Ind}_{K_{0}(\mathfrak{p}^{m})}^{K}\psi_{\chi}$
from Lemma 2.5. As $\psi_{\chi}$ is one-dimensional, the dimension of the
monomial representation
$\operatorname{Ind}_{K_{0}(\mathfrak{p}^{m})}^{K}\psi_{\chi}$ is simply the
index of $K_{0}(\mathfrak{p}^{m})$ in $K$, so that

$\dim\operatorname{Ind}_{K_{0}(\mathfrak{p}^{m})}^{K}\psi_{\chi}=\begin{dcases*}\left[K:K_{0}(\mathfrak{p}^{m})\right]&if
$m\geq c(\chi)$,\\\ 0&otherwise,\end{dcases*}$

which is precisely the right-hand side of (2.7). ∎

######  Corollary 2.8.

We have the orthogonal decomposition

(2.9)
$C^{\infty}(S^{n-1})^{K(\mathfrak{p}^{m})}=\bigoplus_{\begin{subarray}{c}\chi\in\widehat{\mathcal{O}^{\times}}\\\
0\leq c(\chi)\leq
m\end{subarray}}C^{\infty}(S^{n-1})_{\chi}^{K(\mathfrak{p}^{m})}.$

###### Proof.

Since $K_{1}(\mathfrak{p}^{m})$ is a normal subgroup of
$K_{0}(\mathfrak{p}^{m})$ with quotient isomorphic to the finite abelian group
$\mathcal{O}^{\times}/(1+\mathfrak{p}^{m})$, we have that

(2.10)
$\operatorname{Ind}_{K_{1}(\mathfrak{p}^{m})}^{K_{0}(\mathfrak{p}^{m})}1=\bigoplus_{\begin{subarray}{c}\chi\in\widehat{\mathcal{O}^{\times}}\\\
0\leq c(\chi)\leq m\end{subarray}}\psi_{\chi}$

and so by inducing in stages,

$\operatorname{Ind}_{K_{1}(\mathfrak{p}^{m})}^{K}1=\bigoplus_{\begin{subarray}{c}\chi\in\widehat{\mathcal{O}^{\times}}\\\
0\leq c(\chi)\leq
m\end{subarray}}\operatorname{Ind}_{K_{0}(\mathfrak{p}^{m})}^{K}\psi_{\chi}.$

Together with Lemma 2.5, this gives the orthogonal decomposition (2.9). ∎

###  2.3. Irreducible Representations

The spaces $C^{\infty}(S^{n-1})_{\chi}^{K(\mathfrak{p}^{m})}$ are not
irreducible if $m>c(\chi)$ since
$C^{\infty}(S^{n-1})_{\chi}^{K(\mathfrak{p}^{m})}$ contains
$C^{\infty}(S^{n-1})_{\chi}^{K(\mathfrak{p}^{\ell})}$ for all
$\ell\in\\{c(\chi),\ldots,m-1\\}$. Thus we are led to study the orthogonal
complement of $C^{\infty}(S^{n-1})_{\chi}^{K(\mathfrak{p}^{m-1})}$ in
$C^{\infty}(S^{n-1})_{\chi}^{K(\mathfrak{p}^{m})}$. For $m\geq c(\chi)$,
define

(2.11)
$\mathcal{H}_{\chi,m}(S^{n-1})\coloneqq\begin{dcases*}C^{\infty}(S^{n-1})_{\chi}^{K(\mathfrak{p}^{c(\chi)})}&if
$m=c(\chi)$,\\\ C^{\infty}(S^{n-1})_{\chi}^{K(\mathfrak{p}^{m})}\ominus
C^{\infty}(S^{n-1})_{\chi}^{K(\mathfrak{p}^{m-1})}&if
$m>c(\chi)$.\end{dcases*}$

As $K$-modules,

(2.12)
$\mathcal{H}_{\chi,m}(S^{n-1})\cong\begin{dcases*}\operatorname{Ind}_{K_{0}(\mathfrak{p}^{c(\chi)})}^{K}\psi_{\chi}&if
$m=c(\chi)$,\\\
\operatorname{Ind}_{K_{0}(\mathfrak{p}^{m})}^{K}\psi_{\chi}\ominus\operatorname{Ind}_{K_{0}(\mathfrak{p}^{m-1})}^{K}\psi_{\chi}&if
$m>c(\chi)$.\end{dcases*}$

######  Lemma 2.13.

We have that

$\dim\mathcal{H}_{\chi,m}(S^{n-1})=\begin{dcases*}1&if $c(\chi)=m=0$,\\\
q\frac{q^{n-1}-1}{q-1}&if $c(\chi)=0$ and $m=1$,\\\
q^{(c(\chi)-1)(n-1)}\frac{q^{n}-1}{q-1}&if $c(\chi)=m\geq 1$,\\\
q^{(m-2)(n-1)}\frac{(q^{n}-1)(q^{n-1}-1)}{q-1}&if
$m>\max\\{c(\chi),1\\}$.\end{dcases*}$

###### Proof.

This follows immediately from (2.7) and (2.11). ∎

The spaces $\mathcal{H}_{\chi,m}(S^{n-1})$ are $K$-invariant; furthermore, any
two subspaces $\mathcal{H}_{\chi_{1},m_{1}}(S^{n-1})$ and
$\mathcal{H}_{\chi_{1},m_{1}}(S^{n-1})$ are mutually orthogonal whenever
either $\chi_{1}\neq\chi_{2}$ or $m_{1}\neq m_{2}$. We claim that these
subspaces are irreducible, which thereby completes the decomposition of
$C^{\infty}(S^{n-1})$ into irreducible $K$-modules.

######  Theorem 2.14.

For each character $\chi\in\widehat{\mathcal{O}^{\times}}$ and for each
integer $m\geq c(\chi)$, the $K$-module $\mathcal{H}_{\chi,m}(S^{n-1})$ is
irreducible, and we have the orthogonal decompositions

(2.15) $\displaystyle C^{\infty}(S^{n-1})_{\chi}^{K(\mathfrak{p}^{m})}$
$\displaystyle=\bigoplus_{\ell=c(\chi)}^{m}\mathcal{H}_{\chi,\ell}(S^{n-1}),$
(2.16) $\displaystyle C^{\infty}(S^{n-1})$
$\displaystyle=\bigoplus_{m=0}^{\infty}\bigoplus_{\begin{subarray}{c}\chi\in\widehat{\mathcal{O}^{\times}}\\\
0\leq c(\chi)\leq m\end{subarray}}\mathcal{H}_{\chi,m}(S^{n-1}).$

######  Remark 2.17.

Theorem 2.14 is not, in essence, fundamentally new. With
$K_{n}=\operatorname{GL}_{n}(\mathcal{O})$ replaced by
$\operatorname{SL}_{n}(\mathcal{O})$, this decomposition was previously
achieved (with scant proofs) by Petrov [Pet82] in a seemingly neglected paper;
our method follows that sketched by Petrov and achieves the same decomposition
when $n\geq 3$. A closely related result, via slightly different methods, is
due to Hill [Hil94, Proposition 3.3], who studies instead the decomposition of
$L^{2}(\mathrm{Z}(\mathcal{O})K_{n-1,1}\backslash K_{n})$. For $n=2$,
Casselman [Cas73b, Proposition 1] studies the decomposition into irreducible
representations of
$\operatorname{Ind}_{K_{0}(\mathfrak{p}^{m})}^{K}\psi_{\chi}$; see also [BP17,
Section 3.3] for the case $n=2$ and $\chi=1$.

Let $\widehat{K}$ denote the set of equivalence classes of irreducible smooth
representations of $K$, and write $\tau_{\chi,m}$ for the representation in
$\widehat{K}$ given by right translations on the finite-dimensional vector
space $\mathcal{H}_{\chi,m}(S^{n-1})$. We have now classified precisely which
representations in $\widehat{K}$ have a $K_{n-1,1}$-fixed vector.

######  Corollary 2.18.

For every irreducible smooth representation $\tau\in\widehat{K}$,

$\dim\operatorname{Hom}_{K_{n-1,1}}\left(1,\tau|_{K_{n-1,1}}\right)=\begin{dcases*}1&if
$\tau=\tau_{\chi,m}$ for some $\chi\in\widehat{\mathcal{O}^{\times}}$ and
$m\geq c(\chi)$,\\\ 0&otherwise.\end{dcases*}$

In particular, the subspace

$\mathcal{H}_{\chi,m}(S^{n-1})^{K_{n-1,1}}\coloneqq\left\\{f\in\mathcal{H}_{\chi,m}(S^{n-1}):f(xk^{\prime})=f(x)\text{
for all $x\in S^{n-1}$ and $k^{\prime}\in K_{n-1,1}$}\right\\}$

of $K_{n-1,1}$-invariant functions in $\mathcal{H}_{\chi,m}(S^{n-1})$ is one-
dimensional.

###### Proof.

We observe that $\tau_{\chi_{1},m_{1}}$ is isomorphic to
$\tau_{\chi_{2},m_{2}}$ if and only if $\chi_{1}=\chi_{2}$ and $m_{1}=m_{2}$
by examining the dimensions and central characters of these representations.
It follows that $C^{\infty}(S^{n-1})$ is multiplicity-free, so that
$(K_{n},K_{n-1,1})$ is a Gelfand pair. The result then follows via (2.16) and
Frobenius reciprocity. ∎

The proof of Theorem 2.14 requires the following lemma.

######  Lemma 2.19.

For each nonnegative integer $m$, we have the double coset decomposition

$K=\bigsqcup_{\ell=0}^{m}K_{0}(\mathfrak{p}^{m})\begin{pmatrix}1_{n-1}&0\\\
\varpi^{\ell}e_{n-1}&1\end{pmatrix}K_{0}(\mathfrak{p}^{m}).$

For $n=2$, this follows from [Cas73b, Lemma 1] and [Sch02, Lemma 2.1.1], while
the same result with $K_{n}=\operatorname{GL}_{n}(\mathcal{O})$ replaced by
$\operatorname{SL}_{n}(\mathcal{O})$ is implicit in the work of Chang [Cha98].

###### Proof.

Let $\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in K$ with
$a\in\operatorname{Mat}_{(n-1)\times(n-1)}(\mathcal{O})$,
$b\in\operatorname{Mat}_{(n-1)\times 1}(\mathcal{O})$,
$c\in\operatorname{Mat}_{1\times(n-1)}(\mathcal{O})$, and $d\in\mathcal{O}$.
There are three cases to consider.

  1. (1)

If $\max\\{|c_{1}|,\ldots,|c_{n-1}|\\}\leq q^{-m}$, then
$\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in
K_{0}(\mathfrak{p}^{m})=K_{0}(\mathfrak{p}^{m})\begin{pmatrix}1_{n-1}&0\\\
\varpi^{m}e_{n-1}&1\end{pmatrix}K_{0}(\mathfrak{p}^{m})$.

  2. (2)

If $\max\\{|c_{1}|,\ldots,|c_{n-1}|\\}=q^{-\ell}$ for some
$\ell\in\\{1,\ldots,m-1\\}$, then $a\in K_{n-1}$; as $\varpi^{-\ell}ca^{-1}\in
S^{n-2}$, there exists some $\alpha\in K_{n-1}$ such that
$e_{n-1}\alpha^{-1}=\varpi^{-\ell}ca^{-1}$, and we have that

$\begin{pmatrix}a&b\\\ c&d\end{pmatrix}=\begin{pmatrix}\alpha&0\\\
0&1\end{pmatrix}\begin{pmatrix}1_{n-1}&0\\\
\varpi^{\ell}e_{n-1}&1\end{pmatrix}\begin{pmatrix}\alpha^{-1}a&\alpha^{-1}b\\\
0&-ca^{-1}b+d\end{pmatrix}.$

  3. (3)

Finally, if $\max\\{|c_{1}|,\ldots,|c_{n-1}|\\}=1$, then $c\in S^{n-2}$. By
[Cha98, Lemma 3], there exists $\beta\in\operatorname{Mat}_{(n-1)\times
1}(\mathcal{O})$ such that $\det(a-\beta c)\in\mathcal{O}^{\times}$, so that
$a-\beta c\in K_{n-1}$. As $c(a-\beta c)^{-1}\in S^{n-2}$, there exists some
$\alpha\in K_{n-1}$ such that $e_{n-1}\alpha^{-1}=c(a-\beta c)^{-1}$, and we
have that

$\begin{pmatrix}a&b\\\ c&d\end{pmatrix}=\begin{pmatrix}\alpha&\beta\\\
0&1\end{pmatrix}\begin{pmatrix}1_{n-1}&0\\\
e_{n-1}&1\end{pmatrix}\begin{pmatrix}\alpha^{-1}(a-\beta
c)&\alpha^{-1}(b-\beta d)\\\ 0&-c(a-\beta c)^{-1}(b-\beta
d)+d\end{pmatrix}.\qed$

###### Proof of Theorem 2.14.

For $m\geq c(\chi)$, we identify
$\operatorname{End}_{K}(\operatorname{Ind}_{K_{0}(\mathfrak{p}^{m})}^{K}\psi_{\chi})$
with the space of locally constant functions $f:K\to\mathbb{C}$ that satisfy
$f(k_{0}kk_{0}^{\prime})=\psi_{\chi}(k_{0})\psi_{\chi}(k_{0}^{\prime})f(k)$
for all $k\in K$ and $k_{0},k_{0}^{\prime}\in K_{0}(\mathfrak{p}^{m})$. From
Lemma 2.19, we deduce that for each integer $m\geq c(\chi)$,

$\dim\operatorname{End}_{K}\left(\operatorname{Ind}_{K_{0}(\mathfrak{p}^{m})}^{K}\psi_{\chi}\right)=m-c(\chi)+1.$

Since
$\operatorname{Ind}_{K_{0}(\mathfrak{p}^{m})}^{K}\psi_{\chi}\cong\bigoplus_{\ell=c(\chi)}^{m}\mathcal{H}_{\chi,\ell}(S^{n-1})$
from (2.12), we conclude that
$\operatorname{End}_{K}(\mathcal{H}_{\chi,m}(S^{n-1}))$ is one-dimensional by
induction, and hence $\mathcal{H}_{\chi,m}(S^{n-1})$ is irreducible. Finally,
the orthogonal decompositions (2.15) and (2.16) are clear via (2.9) and the
fact that the union of the spaces $C^{\infty}(S^{n-1})^{K(\mathfrak{p}^{m})}$
is dense in $C^{\infty}(S^{n-1})$. ∎

##  3\. Zonal Spherical Functions

Let

$C^{\infty}(S^{n-1})^{K_{n-1,1}}\coloneqq\left\\{f\in
C^{\infty}(S^{n-1}):f(xk^{\prime})=f(x)\text{ for all $x\in S^{n-1}$ and
$k^{\prime}\in K_{n-1,1}$}\right\\}$

denote the subspace of $K_{n-1,1}$-invariant locally constant functions on the
unit sphere. We identify precisely which elements of this lie in
$\mathcal{H}_{\chi,m}(S^{n-1})$ for each character $\chi$ of
$\mathcal{O}^{\times}$ and nonnegative integer $m\geq c(\chi)$. We first
define a $K$-invariant inner product on $C^{\infty}(S^{n-1})\ni f_{1},f_{2}$
via

$\langle
f_{1},f_{2}\rangle\coloneqq\int_{K}f_{1}(e_{n}k)\overline{f_{2}}(e_{n}k)\,dk,$

where $dk$ denotes the Haar probability measure on the compact group $K$.

######  Lemma 3.1.

The subspace $C^{\infty}(S^{n-1})^{K_{n-1,1}}\cap
C^{\infty}(S^{n-1})_{\chi}^{K(\mathfrak{p}^{m})}$ of
$C^{\infty}(S^{n-1})_{\chi}^{K(\mathfrak{p}^{m})}$ has dimension $m-c(\chi)+1$
and is spanned by the functions

(3.2)
$\phi_{\chi,\ell}(x_{1},\ldots,x_{n})\coloneqq\begin{dcases*}\chi(x_{n})&if
$\max\\{|x_{1}|,\ldots,|x_{n-1}|\\}\leq q^{-\ell}$,\\\ 0&if
$q^{-\ell}<\max\\{|x_{1}|,\ldots,|x_{n-1}|\\}\leq 1$,\end{dcases*}$

for $\ell\in\\{c(\chi),\ldots,m\\}$. Furthermore, for $\ell_{1},\ell_{2}\geq
c(\chi)$,

(3.3)
$\left\langle\phi_{\chi,\ell_{1}},\phi_{\chi,\ell_{2}}\right\rangle=\begin{dcases*}1&if
$\ell_{1}=\ell_{2}=0$,\\\
\frac{q-1}{q^{(\max\\{\ell_{1},\ell_{2}\\}-1)(n-1)}(q^{n}-1)}&if
$\max\\{\ell_{1},\ell_{2}\\}\geq 1$.\end{dcases*}$

###### Proof.

That the dimension of this subspace is $m-c(\chi)+1$ is a direct consequence
of (2.15) and Corollary 2.18. It is then straightforward to see that
$\phi_{\chi,\ell}$ is an element of this subspace for each
$\ell\in\\{c(\chi),\ldots,m\\}$ and that these are linearly independent.
Finally, the identity (3.3) is immediate from the definition (3.2) of
$\phi_{\chi,\ell}$, for this implies that

$\left\langle\phi_{\chi,\ell_{1}},\phi_{\chi,\ell_{2}}\right\rangle=\operatorname*{vol}(K_{0}(\mathfrak{p}^{\max\\{\ell_{1},\ell_{2}\\}}))=\frac{1}{\left[K:K_{0}(\mathfrak{p}^{\max\\{\ell_{1},\ell_{2}\\}})\right]},$

which is precisely the right-hand side of (3.3). ∎

It is clear that
$\phi_{\chi,c(\chi)}\in\mathcal{H}_{\chi,c(\chi)}(S^{n-1})^{K_{n-1,1}}$ and
that $\phi_{\chi,c(\chi)}(e_{n})=1$. We show that there exist similar elements
$P_{\chi,m}^{\circ}$ in $\mathcal{H}_{\chi,m}(S^{n-1})^{K_{n-1,1}}$ for each
$m\geq c(\chi)$, which must be a linear combination of the functions
$\phi_{\chi,c(\chi)},\ldots,\phi_{\chi,m}$. We deduce the precise linear
combination by making use of the fact that $P_{\chi,m}^{\circ}$ is orthogonal
to $P_{\chi,j}^{\circ}$ whenever $j\in\\{c(\chi),\ldots,m-1\\}$.

######  Proposition 3.4.

For each $\chi\in\widehat{\mathcal{O}^{\times}}$ and $m\geq c(\chi)$, there
exists a unique locally constant function $P_{\chi,m}^{\circ}$ in
$\mathcal{H}_{\chi,m}(S^{n-1})$ satisfying
$P_{\chi,m}^{\circ}(xk^{\prime})=P_{\chi,m}^{\circ}(x)$ for all $x\in S^{n-1}$
and $k^{\prime}\in K_{n-1,1}$ and $P_{\chi,m}^{\circ}(e_{n})=1$. We have that

(3.5) $P_{\chi,m}^{\circ}(x_{1},\ldots,x_{n})=\begin{dcases*}\chi(x_{n})&if
$\max\\{|x_{1}|,\ldots,|x_{n-1}|\\}\leq q^{-m}$,\\\
\alpha_{\chi,m;m-1}\chi(x_{n})&if $m>c(\chi)$ and
$\max\\{|x_{1}|,\ldots,|x_{n-1}|\\}=q^{-m+1}$,\\\ 0&otherwise,\end{dcases*}$

where

(3.6) $\alpha_{\chi,m;m-1}=\begin{dcases*}-\frac{q-1}{q(q^{n-1}-1)}&if
$m=1$,\\\ -\frac{1}{q^{n-1}-1}&if $m\geq 2$.\end{dcases*}$

In particular, for all $k\in K$, we have that

(3.7) $P_{\chi,m}^{\circ}(e_{n}k)=\overline{P_{\chi,m}^{\circ}}(e_{n}k^{-1}).$

######  Definition 3.8.

We call $P_{\chi,m}^{\circ}$ the zonal spherical function on $S^{n-1}$ of
character $\chi$ and level $m$.

###### Proof of Proposition 3.4.

Since $\mathcal{H}_{\chi,m}(S^{n-1})^{K_{n-1,1}}$ is one-dimensional from
Corollary 2.18, there is a unique function
$P_{\chi,m}^{\circ}\in\mathcal{H}_{\chi,m}(S^{n-1})^{K_{n-1,1}}$ satisfying
$P_{\chi,m}^{\circ}(e_{n})=1$. From (3.2), there exist constants
$\alpha_{\chi,m;\ell}\in\mathbb{C}$ for $\ell\in\\{c(\chi),\ldots,m\\}$ such
that

$\displaystyle P_{\chi,m}^{\circ}(x_{1},\ldots,x_{n})$
$\displaystyle=\sum_{\ell=c(\chi)}^{m}\alpha_{\chi,m;\ell}\varphi_{\chi,m;\ell}(x)$
$\displaystyle=\begin{dcases*}\alpha_{\chi,m;m}\chi(x_{n})&if
$\max\\{|x_{1}|,\ldots,|x_{n-1}|\\}\leq q^{-m}$,\\\
\alpha_{\chi,m;\ell}\chi(x_{n})&if
$q^{-m}<\max\\{|x_{1}|,\ldots,|x_{n-1}|\\}=q^{-\ell}\leq q^{-c(\chi)}$,\\\
0&if $q^{-c(\chi)}<\max\\{|x_{1}|,\ldots,|x_{n-1}|\\}\leq 1$,\end{dcases*}$

where

$\displaystyle\varphi_{\chi,m;\ell}(x_{1},\ldots,x_{n})$
$\displaystyle\coloneqq\begin{dcases*}\phi_{\chi,\ell}(x_{1},\ldots,x_{n})-\phi_{\chi,\ell+1}(x_{1},\ldots,x_{n})&for
$\ell\in\\{c(\chi),\ldots,m-1\\}$,\\\ \phi_{\chi,m}(x_{1},\ldots,x_{n})&for
$\ell=m$,\end{dcases*}$ $\displaystyle=\begin{dcases*}\chi(x_{n})&if
$\ell\in\\{c(\chi),\ldots,m-1\\}$ and
$\max\\{|x_{1}|,\ldots,|x_{n-1}|\\}=q^{-\ell}$,\\\ \chi(x_{n})&if $\ell=m$ and
$\max\\{|x_{1}|,\ldots,|x_{n-1}|\\}\leq q^{-m}$,\\\ 0&otherwise.\end{dcases*}$

Since $P_{\chi,m}^{\circ}(e_{n})=1$, we have that $\alpha_{\chi,m;m}=1$. To
deduce properties of the remaining coefficients, we make use the fact that

(3.9)
$\beta_{\chi,m;\ell}\coloneqq\left\langle\varphi_{\chi,m;\ell},\varphi_{\chi,m;\ell}\right\rangle=\begin{dcases*}\left\langle\phi_{\chi,\ell},\phi_{\chi,\ell}\right\rangle-\left\langle\phi_{\chi,\ell+1},\phi_{\chi,\ell+1}\right\rangle&if
$\ell\in\\{c(\chi),\ldots,m-1\\}$,\\\
\left\langle\phi_{\chi,m},\phi_{\chi,m}\right\rangle&if $\ell=m$\\\
\end{dcases*}$

from (3.3), while

(3.10)
$\left\langle\varphi_{\chi,m;\ell_{1}},\varphi_{\chi,m;\ell_{2}}\right\rangle=0\
\text{ if $\ell_{1}\neq\ell_{2}$}$

since these have disjoint support. The zonal spherical function
$P_{\chi,m}^{\circ}$ is orthogonal to $P_{\chi,j}^{\circ}$ for all
$j\in\\{c(\chi),\ldots,m-1\\}$ as $\mathcal{H}_{\chi,m}(S^{n-1})$ and
$\mathcal{H}_{\chi,j}(S^{n-1})$ are mutually orthogonal. Writing

$P_{\chi,j}^{\circ}(x_{1},\ldots,x_{n})=\sum_{\ell=c(\chi)}^{j-1}\alpha_{\chi,j;\ell}\varphi_{\chi,m;\ell}(x_{1},\ldots,x_{n})+\sum_{\ell=j}^{m}\varphi_{\chi,m;\ell}(x_{1},\ldots,x_{n}),$

we deduce from (3.9) and (3.10) that for each $j\in\\{c(\chi),\ldots,m-1\\}$,

(3.11) $\left\langle
P_{\chi,m}^{\circ},P_{\chi,j}^{\circ}\right\rangle=\sum_{\ell=c(\chi)}^{j-1}\alpha_{\chi,m;\ell}\overline{\alpha_{\chi,j;\ell}}\beta_{\chi,m;\ell}+\sum_{\ell=j}^{m}\alpha_{\chi,m;\ell}\beta_{\chi,m;\ell}=0.$

Using (3.11), we shall prove by induction that

(3.12) $\alpha_{\chi,m;\ell}=\begin{dcases*}1&if $\ell=m$,\\\
-\frac{\beta_{\chi,m;m}}{\beta_{\chi,m;m-1}}&if $m>c(\chi)$ and $\ell=m-1$,\\\
0&otherwise,\end{dcases*}$

which yields (3.5); the identity (3.6) then follows from (3.3) and (3.9). The
base case of (3.12) is $m=c(\chi)$, so that $\ell=c(\chi)$, in which case we
have that $\alpha_{\chi,c(\chi);c(\chi)}=1$ since
$P_{\chi,c(\chi)}^{\circ}(e_{n})=1$. Now we suppose that (3.12) holds with $m$
replaced by $j$ for each $j\in\\{c(\chi),\ldots,m-1\\}$. From (3.11), we have
by the induction hypothesis that for $m\geq c(\chi)+2$,

$\left\langle
P_{\chi,m}^{\circ},P_{\chi,c(\chi)}^{\circ}\right\rangle-\left\langle
P_{\chi,m}^{\circ},P_{\chi,c(\chi)+1}^{\circ}\right\rangle=\alpha_{\chi,m;c(\chi)}\beta_{\chi,m;c(\chi)}\left(1-\frac{\beta_{\chi,c(\chi)+1;c(\chi)+1}}{\beta_{\chi,c(\chi)+1;c(\chi)}}\right)=0.$

Thus $\alpha_{\chi,m;c(\chi)}=0$. Proceeding inductively, we conclude that for
all $j\in\\{c(\chi),\ldots,m-2\\}$,

$\left\langle P_{\chi,m}^{\circ},P_{\chi,j}^{\circ}\right\rangle-\left\langle
P_{\chi,m}^{\circ},P_{\chi,j+1}^{\circ}\right\rangle=\alpha_{\chi,m;j}\beta_{\chi,m;j}\left(1-\frac{\beta_{\chi,j+1;j+1}}{\beta_{\chi,j+1;j}}\right)=0,$

so that $\alpha_{\chi,m;j}=0$. Finally, for $m\geq c(\chi)+1$, we have that

$\left\langle
P_{\chi,m}^{\circ},P_{\chi,m-1}^{\circ}\right\rangle=\alpha_{\chi,m;m-1}\beta_{\chi,m;m-1}+\beta_{\chi,m;m}=0,$

which yields the remaining case $\ell=m-1$ of (3.12).

It remains to prove (3.7). It suffices to show that for all $k\in K$,

(3.13) $\phi_{\chi,\ell}(e_{n}k)=\overline{\phi_{\chi,\ell}}(e_{n}k^{-1})$

for each nonnegative integer $\ell\geq c(\chi)$, since $P_{\chi,m}^{\circ}$
may be written as a linear combination of such functions. The identity (3.13)
is clear if $k\notin K_{0}(\mathfrak{p}^{\ell})$, for then both sides are
zero. If $k\in K_{0}(\mathfrak{p}^{\ell})$, so that $k^{-1}\in
K_{0}(\mathfrak{p}^{\ell})$ as well, then
$e_{n}k\prescript{t}{}{e}_{n}e_{n}k^{-1}\prescript{t}{}{e}_{n}-1\in\mathfrak{p}^{\ell}$
as $kk^{-1}=1_{n}$, and consequently
$\chi(e_{n}k\prescript{t}{}{e}_{n})=\overline{\chi}(e_{n}k^{-1}\prescript{t}{}{e}_{n})$
since $\ell\geq c(\chi)$, which yields the result. ∎

The zonal spherical function $P_{\chi,m}^{\circ}$ is useful for understanding
various properties of the space $\mathcal{H}_{\chi,m}(S^{n-1})$. Our first
application is the following lemma, which may be thought of as the addition
theorem for $\mathcal{H}_{\chi,m}(S^{n-1})$.

######  Lemma 3.14.

Let $\\{Q_{j}\\}$ be an orthonormal basis of $\mathcal{H}_{\chi,m}(S^{n-1})$.
Then for any $x\in S^{n-1}$ and $k\in K$, we have that

(3.15)
$\sum_{j=1}^{\dim\mathcal{H}_{\chi,m}(S^{n-1})}Q_{j}(x)\overline{Q_{j}}(e_{n}k)=\dim\mathcal{H}_{\chi,m}(S^{n-1})P_{\chi,m}^{\circ}(xk^{-1}).$

###### Proof.

Since

(3.16) $Q(xk)=(\tau_{\chi,m}(k)\cdot
Q)(x)=\sum_{j=1}^{\dim\mathcal{H}_{\chi,m}(S^{n-1})}\left\langle\tau_{\chi,m}(k)\cdot
Q,Q_{j}\right\rangle Q_{j}(x)$

for any $Q\in\mathcal{H}_{\chi,m}(S^{n-1})$ and $x\in S^{n-1}$, we have that
for any $k\in K$,

(3.17) $\begin{split}\delta_{\ell_{1},\ell_{2}}&=\left\langle
Q_{\ell_{1}},Q_{\ell_{2}}\right\rangle\\\
&=\left\langle\tau_{\chi,m}(k^{-1})\cdot
Q_{\ell_{1}},\tau_{\chi,m}(k^{-1})\cdot Q_{\ell_{2}}\right\rangle\\\
&=\sum_{j_{1},j_{2}=1}^{\dim\mathcal{H}_{\chi,m}(S^{n-1})}\left\langle\tau_{\chi,m}(k^{-1})\cdot
Q_{\ell_{1}},Q_{j_{1}}\right\rangle\left\langle
Q_{j_{2}},\tau_{\chi,m}(k^{-1})\cdot Q_{\ell_{2}}\right\rangle\left\langle
Q_{j_{1}},Q_{j_{2}}\right\rangle\\\
&=\sum_{j=1}^{\dim\mathcal{H}_{\chi,m}(S^{n-1})}\left\langle\tau_{\chi,m}(k^{-1})\cdot
Q_{\ell_{1}},Q_{j}\right\rangle\left\langle Q_{j},\tau_{\chi,m}(k^{-1})\cdot
Q_{\ell_{2}}\right\rangle\\\
&=\sum_{j=1}^{\dim\mathcal{H}_{\chi,m}(S^{n-1})}\left\langle
Q_{\ell_{1}},\tau_{\chi,m}(k)\cdot
Q_{j}\right\rangle\left\langle\tau_{\chi,m}(k)\cdot
Q_{j},Q_{\ell_{2}}\right\rangle.\end{split}$

For fixed $k\in K$, we now define
$\mathcal{Q}_{e_{n}k}\in\mathcal{H}_{\chi,m}(S^{n-1})$ by

$\mathcal{Q}_{e_{n}k}(x)\coloneqq\sum_{j=1}^{\dim\mathcal{H}_{\chi,m}(S^{n-1})}Q_{j}(x)\overline{Q_{j}}(e_{n}k).$

Then from (3.16) and (3.17), we have that for all $k_{1}\in K$,

(3.18)
$\begin{split}\mathcal{Q}_{e_{n}kk_{1}}(xk_{1})&=\sum_{j=1}^{\dim\mathcal{H}_{\chi,m}(S^{n-1})}Q_{j}(xk_{1})\overline{Q_{j}}(e_{n}kk_{1})\\\
&=\sum_{j,\ell_{1},\ell_{2}=1}^{\dim\mathcal{H}_{\chi,m}(S^{n-1})}\left\langle\tau_{\chi,m}(k_{1})\cdot
Q_{j},Q_{\ell_{1}}\right\rangle\left\langle
Q_{\ell_{2}},\tau_{\chi,m}(k_{1})\cdot Q_{j}\right\rangle
Q_{\ell_{1}}(x)\overline{Q_{\ell_{2}}}(e_{n}k)\\\
&=\sum_{\ell=1}^{\dim\mathcal{H}_{\chi,m}(S^{n-1})}Q_{\ell}(x)\overline{Q_{\ell}}(e_{n}k)\\\
&=\mathcal{Q}_{e_{n}k}(x).\end{split}$

We use (3.18) with $x$ replaced by $xk$ and $k_{1}$ replaced by
$k^{-1}k^{\prime}k$ for $k^{\prime}\in K_{n-1,1}$. Since
$e_{n}k^{\prime}=e_{n}$, we deduce that

$\left(\tau_{\chi,m}(k^{\prime})\cdot\left(\tau_{\chi,m}(k)\cdot\mathcal{Q}_{e_{n}k}\right)\right)(x)=\mathcal{Q}_{e_{n}k}(xk^{\prime}k)=\mathcal{Q}_{e_{n}k}(xk)=\left(\tau_{\chi,m}(k)\cdot\mathcal{Q}_{e_{n}k}\right)(x)$

for all $k^{\prime}\in K_{n-1,1}$. As
$\mathcal{H}_{\chi,m}(S^{n-1})^{K_{n-1,1}}$ is one-dimensional, it follows
that $\tau_{\chi,m}(k)\cdot\mathcal{Q}_{e_{n}k}$ must be a constant multiple
of $P_{\chi,m}^{\circ}$, and this constant is readily seen to be
$\mathcal{Q}_{e_{n}k}(e_{n}k)$ upon taking $x=e_{n}$. So for all $x\in
S^{n-1}$ and $k\in K$,

$\displaystyle\sum_{j=1}^{\dim\mathcal{H}_{\chi,m}(S^{n-1})}Q_{j}(x)\overline{Q_{j}}(e_{n}k)$
$\displaystyle=\mathcal{Q}_{e_{n}k}(x)$
$\displaystyle=\left(\tau_{\chi,m}(k)\cdot\mathcal{Q}_{e_{n}k}\right)(xk^{-1})$
$\displaystyle=\mathcal{Q}_{e_{n}k}(e_{n}k)P_{\chi,m}^{\circ}(xk^{-1})$
$\displaystyle=\sum_{j=1}^{\dim\mathcal{H}_{\chi,m}(S^{n-1})}\left|Q_{j}(e_{n}k)\right|^{2}P_{\chi,m}^{\circ}(xk^{-1}).$

It remains to show that for all $k\in K$,

(3.19)
$\sum_{j=1}^{\dim\mathcal{H}_{\chi,m}(S^{n-1})}\left|Q_{j}(e_{n}k)\right|^{2}=\dim\mathcal{H}_{\chi,m}(S^{n-1}).$

To prove (3.19), we take $x=e_{n}k$ and $k_{1}=k^{-1}$ in (3.18) in order to
see that

$\sum_{j=1}^{\dim\mathcal{H}_{\chi,m}(S^{n-1})}\left|Q_{j}(e_{n}k)\right|^{2}=\sum_{j=1}^{\dim\mathcal{H}_{\chi,m}(S^{n-1})}\left|Q_{j}(e_{n})\right|^{2}$

for all $k\in K$ and hence the left-hand side is a constant. Integrating over
$K\ni k$, we find that

$\displaystyle\sum_{j=1}^{\dim\mathcal{H}_{\chi,m}(S^{n-1})}|Q_{j}(e_{n}k)|^{2}$
$\displaystyle=\int_{K}\sum_{j=1}^{\dim\mathcal{H}_{\chi,m}(S^{n-1})}|Q_{j}(e_{n}k)|^{2}\,dk$
$\displaystyle=\sum_{j=1}^{\dim\mathcal{H}_{\chi,m}(S^{n-1})}\left\langle
Q_{j},Q_{j}\right\rangle$
$\displaystyle=\dim\mathcal{H}_{\chi,m}(S^{n-1}).\qed$

A simple consequence of Lemma 3.14 is the following result showing that
certain matrix coefficients of $\tau_{\chi,m}$ are themselves elements of
$\mathcal{H}_{\chi,m}(S^{n-1})$.

######  Corollary 3.20.

The reproducing kernel for $\mathcal{H}_{\chi,m}(S^{n-1})$ is
$(\dim\mathcal{H}_{\chi,m}(S^{n-1}))P_{\chi,m}^{\circ}$, so that for all
$P\in\mathcal{H}_{\chi,m}(S^{n-1})$ and $k\in K$,

(3.21) $P(e_{n}k)=\dim\mathcal{H}_{\chi,m}(S^{n-1})\left\langle\tau(k)\cdot
P,P_{\chi,m}^{\circ}\right\rangle.$

In particular,

(3.22) $\left\langle
P_{\chi,m}^{\circ},P_{\chi,m}^{\circ}\right\rangle=\frac{1}{\dim\mathcal{H}_{\chi,m}(S^{n-1})}.$

One can also show (3.22) in a more direct fashion by combining (3.3), (3.5),
and (3.6).

###### Proof.

From (3.15) and (3.16), we have that

$\displaystyle P(e_{n}k)$
$\displaystyle=\sum_{j=1}^{\dim\mathcal{H}_{\chi,m}(S^{n-1})}\left\langle
P,Q_{j}\right\rangle Q_{j}(e_{n}k)$ $\displaystyle=\left\langle
P,\sum_{j=1}^{\dim\mathcal{H}_{\chi,m}(S^{n-1})}Q_{j}\overline{Q_{j}}(e_{n}k)\right\rangle$
$\displaystyle=\left\langle
P,\dim\mathcal{H}_{\chi,m}(S^{n-1})\tau(k^{-1})\cdot
P_{\chi,m}^{\circ}\right\rangle$
$\displaystyle=\dim\mathcal{H}_{\chi,m}(S^{n-1})\left\langle\tau(k)\cdot
P,P_{\chi,m}^{\circ}\right\rangle.$

The identity (3.22) then follows upon taking $P=P_{\chi,m}^{\circ}$ and
$k=1_{n}$ since $P_{\chi,m}^{\circ}(e_{n})=1$. ∎

##  4\. The Newform $K$-Type

###  4.1. The Newform and the Conductor Exponent

Let $(\pi,V_{\pi})$ be an induced representation of Langlands type of
$\operatorname{GL}_{n}(F)$. Thus there exist positive integers
$n_{1},\ldots,n_{r}$ for which $n_{1}+\cdots+n_{r}=n$ and essentially square-
integrable representations $(\pi_{j},V_{\pi_{j}})$ of
$\operatorname{GL}_{n_{j}}(F)$ of the form
$\sigma_{j}\otimes\left|\det\right|^{t_{j}}$, where $\sigma_{j}$ is square-
integrable and $t_{j}\in\mathbb{C}$ satisfies
$\Re(t_{1})\geq\cdots\geq\Re(t_{r})$, such that

$\pi=\operatorname{Ind}_{\mathrm{P}(F)}^{\operatorname{GL}_{n}(F)}\mathop{\vphantom{\bigotimes}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxtimes$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxtimes$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxtimes$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxtimes$}}}}}\displaylimits_{j=1}^{r}\pi_{j},$

the representation obtained by normalised parabolic induction from the
standard upper parabolic subgroup
$\mathrm{P}(F)=\mathrm{P}_{(n_{1},\ldots,n_{r})}(F)$ of
$\operatorname{GL}_{n}(F)$. One can take as a model for $\pi$ the space of
smooth functions $f:\operatorname{GL}_{n}(F)\to
V_{\pi_{1}}\otimes\cdots\otimes V_{\pi_{r}}$, upon which $\pi$ acts via right
translations, that satisfy

$f(umg)=\prod_{j=1}^{r}\left|\det
m_{j}\right|^{\frac{1}{2}(n-2(n_{1}+\cdots+n_{j-1})-n_{j})}\left(\bigotimes_{j=1}^{r}\pi_{j}(m_{j})\cdot
f\right)(g)$

for all $u\in\mathrm{N}_{\mathrm{P}}(F)$, the unipotent radical of
$\mathrm{P}(F)$,
$m=\operatorname{blockdiag}(m_{1},\ldots,m_{r})\in\mathrm{M}_{\mathrm{P}}(F)$,
the Levi subgroup of $\mathrm{P}(F)$, and $g\in\operatorname{GL}_{n}(F)$. We
note that every generic irreducible admissible smooth representation of
$\operatorname{GL}_{n}(F)$ is isomorphic to some induced representation of
Langlands type; see, for example, [JS83].

For each nonnegative integer $m$ and each character
$\chi\in\widehat{\mathcal{O}^{\times}}$ for which $0\leq c(\chi)\leq m$, we
define the subspaces

$\displaystyle V_{\pi}^{K_{1}(\mathfrak{p}^{m})}$
$\displaystyle\coloneqq\left\\{v\in V_{\pi}:\pi(k)\cdot v=v\text{ for all
}k\in K_{1}(\mathfrak{p}^{m})\right\\},$ $\displaystyle
V_{\pi}^{K_{0}(\mathfrak{p}^{m}),\chi}$ $\displaystyle\coloneqq\left\\{v\in
V_{\pi}:\pi(k)\cdot v=\psi_{\chi}(k)v\text{ for all }k\in
K_{0}(\mathfrak{p}^{m})\right\\}.$

The former is the subspace of $V_{\pi}$ of $K_{1}(\mathfrak{p}^{m})$-invariant
vectors; the latter is the subspace of
$(K_{0}(\mathfrak{p}^{m}),\psi_{\chi})$-equivariant vectors.

######  Lemma 4.1.

Let $(\pi,V_{\pi})$ be an induced representation of Langlands type of
$\operatorname{GL}_{n}(F)$. For each nonnegative integer $m$, the subspace
$V_{\pi}^{K_{1}(\mathfrak{p}^{m})}$ is the image of the projection map
$\Pi^{K_{1}(\mathfrak{p}^{m})}:V_{\pi}\to V_{\pi}$ given by

(4.2)
$\Pi^{K_{1}(\mathfrak{p}^{m})}(v)\coloneqq\frac{1}{\operatorname*{vol}(K_{1}(\mathfrak{p}^{m}))}\int_{K_{1}(\mathfrak{p}^{m})}\pi(k)\cdot
v\,dk,$

while $V_{\pi}^{K_{0}(\mathfrak{p}^{m}),\chi_{\pi}}$ is the image of the
projection map $\Pi^{K_{0}(\mathfrak{p}^{m}),\chi_{\pi}}:V_{\pi}\to V_{\pi}$
given by

(4.3)
$\Pi^{K_{0}(\mathfrak{p}^{m}),\chi_{\pi}}(v)\coloneqq\begin{dcases*}\int_{K}\pi(k)\cdot
v\,dk&if $m=c(\chi_{\pi})=0$,\\\
\frac{1}{\operatorname*{vol}(K_{0}(\mathfrak{p}^{m}))}\int_{K_{0}(\mathfrak{p}^{m})}\overline{\chi_{\pi}}(e_{n}k\prescript{t}{}{e}_{n})\pi(k)\cdot
v\,dk&if $m>0$.\end{dcases*}$

Finally, we have that

(4.4)
$V_{\pi}^{K_{0}(\mathfrak{p}^{m}),\chi}=\begin{dcases*}V_{\pi}^{K_{1}(\mathfrak{p}^{m})}&if
$\chi=\chi_{\pi}$,\\\ \\{0\\}&otherwise,\end{dcases*}$

where $\chi_{\pi}\coloneqq\omega_{\pi}|_{\mathcal{O}^{\times}}$ with
$\omega_{\pi}:F^{\times}\to\mathbb{C}^{\times}$ the central character of
$\pi$.

###### Proof.

The two identities
$\Pi^{K_{1}(\mathfrak{p}^{m})}(V_{\pi})=V_{\pi}^{K_{1}(\mathfrak{p}^{m})}$ and
$\Pi^{K_{0}(\mathfrak{p}^{m}),\chi_{\pi}}(V_{\pi})=V_{\pi}^{K_{0}(\mathfrak{p}^{m}),\chi}$
are clear. Since
$\mathrm{Z}(\mathcal{O})K_{1}(\mathfrak{p}^{m})=K_{0}(\mathfrak{p}^{m})$ and
$\pi|_{\mathrm{Z}(\mathcal{O})}=\omega_{\pi}|_{\mathcal{O}^{\times}}=\chi_{\pi}$,
the identities
$V_{\pi}^{K_{0}(\mathfrak{p}^{m}),\chi_{\pi}}=V_{\pi}^{K_{1}(\mathfrak{p}^{m})}$
and $V_{\pi}^{K_{0}(\mathfrak{p}^{m}),\chi}=\\{0\\}$ for $\chi\neq\chi_{\pi}$
then follow. ∎

A fundamental result concerning induced representations of Langlands type is
the existence of a newform. Jacquet, Piatetski-Shapiro, and Shalika [JP-SS81]
have shown that each induced representation of Langlands type $(\pi,V_{\pi})$
of $\operatorname{GL}_{n}(F)$ contains a distinguished vector $v^{\circ}\in
V_{\pi}$, the newform, whose complexity is measured in a natural way by a
nonnegative integer $c(\pi)$, the conductor exponent. This generalises a
result of Casselman [Cas73a], who proved this for $n=2$, and observed that
when $F=\mathbb{Q}_{p}$ and $\pi$ is the local component of an automorphic
representation of $\operatorname{GL}_{2}(\mathbb{A}_{\mathbb{Q}})$, this is
the adèlic reformulation of the Atkin–Lehner theory of newforms for classical
modular forms [AL70].

######  Theorem 4.5 ([JP-SS81, Théorème (5)]).

Let $(\pi,V_{\pi})$ be an induced representation of Langlands type of
$\operatorname{GL}_{n}(F)$. There exists a minimal nonnegative integer
$m=c(\pi)$ for which the space
$V_{\pi}^{K_{1}(\mathfrak{p}^{m})}=V_{\pi}^{K_{0}(\mathfrak{p}^{m}),\chi_{\pi}}$
is nontrivial, in which case it is one-dimensional.

######  Definition 4.6.

The nonzero vector $v^{\circ}\in
V_{\pi}^{K_{1}(\mathfrak{p}^{c(\pi)})}=V_{\pi}^{K_{0}(\mathfrak{p}^{c(\pi)}),\chi_{\pi}}$,
unique up to scalar multiplication, is called the newform of $\pi$. The
nonnegative integer $c(\pi)$ is called the conductor exponent of $\pi$.
Elements of
$V_{\pi}^{K_{1}(\mathfrak{p}^{m})}=V_{\pi}^{K_{0}(\mathfrak{p}^{m}),\chi_{\pi}}$
for $m>c(\pi)$ are called oldforms.

######  Remark 4.7.

The proof of [JP-SS81, Théorème (5)] contains a gap; correct proofs were later
independently given by Jacquet [Jac12, Theorem 1] and Matringe [Mat13,
Corollary 3.3].

######  Remark 4.8.

As proven in [JP-SS81], there are other ways to characterise the newform and
the conductor exponent of $\pi$ instead of in terms of the subspace
$V_{\pi}^{K_{1}(\mathfrak{p}^{c(\pi)})}$ of
$K_{1}(\mathfrak{p}^{c(\pi)})$-invariant vectors in $V_{\pi}$. The conductor
exponent $c(\pi)$ is precisely the nonnegative integer for which the epsilon
factor $\varepsilon(s,\pi,\psi)$ associated to $\pi$ is of the form

$\varepsilon(s,\pi,\psi)=\varepsilon\left(\frac{1}{2},\pi,\psi\right)q^{-c(\pi)\left(s-\frac{1}{2}\right)},$

where $\psi:F\to\mathbb{C}^{\times}$ is an unramified additive character of
$F$. The newform is such that when viewed in the Whittaker model
$\mathcal{W}(\pi,\psi)$ of $\pi$, it is the unique Whittaker function
$W^{\circ}$ satisfying $W^{\circ}\left(g\begin{pmatrix}k^{\prime}&0\\\
0&1\end{pmatrix}\right)=W^{\circ}(g)$ for all $g\in\operatorname{GL}_{n}(F)$
and $k^{\prime}\in K_{n-1}$ that is a test vector for the local
$\operatorname{GL}_{n}\times\operatorname{GL}_{n-1}$ Rankin–Selberg integral
whenever the second representation is unramified, so that

$\int\limits_{\mathrm{N}_{n-1}(F)\backslash\operatorname{GL}_{n-1}(F)}W^{\circ}\begin{pmatrix}g&0\\\
0&1\end{pmatrix}W^{\prime\circ}(g)\left|\det
g\right|^{s-\frac{1}{2}}\,dg=L(s,\pi\times\pi^{\prime})$

for all spherical induced representations of Langlands type $\pi^{\prime}$ of
$\operatorname{GL}_{n-1}(F)$ with spherical Whittaker function
$W^{\prime\circ}\in\mathcal{W}(\pi^{\prime},\overline{\psi})$ normalised such
that $W^{\prime\circ}(1_{n-1})=1$.

###  4.2. The Newform $K$-Type

We shall show that the newform lies in a distinguished $K$-type of $\pi$. To
begin, for each irreducible smooth representation $\tau\in\widehat{K}$, we
define the projection map $\Pi^{\tau}:V_{\pi}\to V_{\pi}$ by

$\Pi^{\tau}(v)\coloneqq\int_{K}\xi^{\tau}(k)\pi(k)\cdot v\,dk,$

where

$\xi^{\tau}(k)\coloneqq(\dim\tau)\operatorname{Tr}\tau(k^{-1})$

is the elementary idempotent associated to $\tau$. The image of $V_{\pi}$
under $\Pi^{\tau}$ is the $\tau$-isotypic subspace $V_{\pi}^{\tau}$ of
$V_{\pi}$, which is finite-dimensional since $\pi$ is admissible. We say that
$\tau$ is a $K$-type of $\pi$ if $\operatorname{Hom}_{K}(\tau,\pi|_{K})$ is
nontrivial, in which case $\dim
V_{\pi}^{\tau}=\dim\tau\dim\operatorname{Hom}_{K}(\tau,\pi|_{K})>0$, and we
call $\dim\operatorname{Hom}_{K}(\tau,\pi|_{K})$ the multiplicity of $\tau$ in
$\pi$.

In general, the $K$-type decomposition of an induced representation of
Langlands type $\pi$ is not known except in special cases. For $n=2$, this
follows from work of Casselman [Cas73b], Silberger [Sil70], and Hansen
[Han87]; for $n=3$ and $\pi$ a principal series representation, this problem
has been studied by Campbell and Nevins [CN09, CN10] and Onn and Singla
[OS14].

We do not attempt to determine the full $K$-type decomposition of $\pi$, which
is undoubtedly challenging, since no explicit description of $\widehat{K}$
currently exists for $n>2$. Rather, we study the multiplicity with which
particular representations $\tau\in\widehat{K}$ appear in the $K$-type
decomposition of $\pi$, namely those representations containing a
$K_{n-1,1}$-fixed vector.

To do so, we define the projection map $\Pi^{K_{n-1,1}}:V_{\pi}\to V_{\pi}$
given by

$\Pi^{K_{n-1,1}}(v)\coloneqq\int_{K_{n-1,1}}\pi(k^{\prime})\cdot
v\,dk^{\prime},$

where $dk^{\prime}$ denotes the Haar probability measure on the compact group
$K_{n-1,1}$, so that the image of $\Pi^{K_{n-1,1}}$ is the subspace of
$K_{n-1,1}$-invariant vectors. The composition of the two projections
$\Pi^{\tau}$ and $\Pi^{K_{n-1,1}}$ is the projection

(4.9)
$\left(\Pi^{\tau|_{K_{n-1,1}}}\right)(v)\coloneqq\left(\Pi^{K_{n-1,1}}\circ\Pi^{\tau}\right)(v)=\left(\Pi^{\tau}\circ\Pi^{K_{n-1,1}}\right)(v)=\int_{K}\xi^{\tau|_{K_{n-1,1}}}(k)\pi(k)\cdot
v\,dk$

onto the subspace of $K_{n-1,1}$-invariant $\tau$-isotypic vectors

$V_{\pi}^{\tau|_{K_{n-1,1}}}\coloneqq\left(\Pi^{\tau|_{K_{n-1,1}}}\right)(V_{\pi})=\left\\{v\in
V_{\pi}^{\tau}:\pi(k^{\prime})\cdot v=v\text{ for all $k^{\prime}\in
K_{n-1,1}$}\right\\}.$

Here

$\xi^{\tau|_{K_{n-1,1}}}(k)\coloneqq\int_{K_{n-1,1}}\xi^{\tau}(k^{\prime}k)\,dk^{\prime}=\int_{K_{n-1,1}}\xi^{\tau}(kk^{\prime})\,dk^{\prime}.$

This function has a particularly simple description.

######  Lemma 4.10.

For $\tau\in\widehat{K}$ and $k\in K$, we have that

$\xi^{\tau|_{K_{n-1,1}}}(k)=\begin{dcases*}(\dim\tau_{\chi,m})P_{\chi,m}^{\circ}(e_{n}k^{-1})&if
$\tau=\tau_{\chi,m}$ for some $\chi\in\widehat{\mathcal{O}^{\times}}$ and
$m\geq c(\chi)$,\\\ 0&otherwise.\end{dcases*}$

###### Proof.

This follows from Corollaries 2.18 and 3.20. ∎

From Lemma 4.10, we have the following simple consequences.

######  Corollary 4.11.

We have that

(4.12)
$\displaystyle\sum_{\ell=0}^{m}\sum_{\begin{subarray}{c}\chi\in\widehat{\mathcal{O}^{\times}}\\\
0\leq c(\chi)\leq\ell\end{subarray}}\xi^{\tau_{\chi,\ell}|_{K_{n-1,1}}}(k)$
$\displaystyle=\begin{dcases*}\frac{1}{\operatorname*{vol}(K_{1}(\mathfrak{p}^{m}))}&if
$k\in K_{1}(\mathfrak{p}^{m})$,\\\ 0&otherwise,\end{dcases*}$ (4.13)
$\displaystyle\sum_{\ell=c(\chi)}^{m}\xi^{\tau_{\chi,\ell}|_{K_{n-1,1}}}(k)$
$\displaystyle=\begin{dcases*}1&if $m=c(\chi)=0$ and $k\in K$,\\\
\frac{\overline{\chi}(e_{n}k\prescript{t}{}{e}_{n})}{\operatorname*{vol}(K_{0}(\mathfrak{p}^{m}))}&if
$m>0$ and $k\in K_{0}(\mathfrak{p}^{m})$,\\\ 0&otherwise.\end{dcases*}$

Consequently, for an induced representation of Langlands type $(\pi,V_{\pi})$
of $\operatorname{GL}_{n}(F)$, we have that

(4.14) $\displaystyle\Pi^{K_{1}(\mathfrak{p}^{m})}$
$\displaystyle=\sum_{\ell=0}^{m}\sum_{\begin{subarray}{c}\chi\in\widehat{\mathcal{O}^{\times}}\\\
0\leq c(\chi)\leq\ell\end{subarray}}\Pi^{\tau_{\chi,\ell}|_{K_{n-1,1}}},$
(4.15) $\displaystyle\Pi^{K_{0}(\mathfrak{p}^{m}),\chi}$
$\displaystyle=\sum_{\ell=c(\chi)}^{m}\Pi^{\tau_{\chi,\ell}|_{K_{n-1,1}}}$

for any nonnegative integer $m$ and any character
$\chi\in\widehat{\mathcal{O}^{\times}}$ for which $0\leq c(\chi)\leq m$.

###### Proof.

The identity (4.13) can be seen by combining Proposition 3.4 and Lemmata 2.13
and 4.10, noting that
$1/\operatorname*{vol}(K_{0}(\mathfrak{p}^{m}))=[K:K_{0}(\mathfrak{p}^{m})]$.
The identity (4.12) follows from (4.13) together with character orthogonality,
namely the fact that for $x\in\mathcal{O}^{\times}$,

$\sum_{\begin{subarray}{c}\chi\in\widehat{\mathcal{O}^{\times}}\\\ 0\leq
c(\chi)\leq
m\end{subarray}}\chi(x)=\begin{dcases*}\\#\mathcal{O}^{\times}/(1+\mathfrak{p}^{m})&if
$x-1\in\mathfrak{p}^{m}$,\\\ 0&otherwise,\end{dcases*}$

and noting that
$\\#\mathcal{O}^{\times}/(1+\mathfrak{p}^{m})=[K_{0}(\mathfrak{p}^{m}):K_{1}(\mathfrak{p}^{m})]$.
The identities (4.14) and (4.15) for the projections
$\Pi^{K_{1}(\mathfrak{p}^{m})}$ and $\Pi^{K_{0}(\mathfrak{p}^{m}),\chi}$ in
(4.2) and (4.3) in terms of the projections
$\Pi^{\tau_{\chi,\ell}|_{K_{n-1,1}}}$ in (4.9) are immediate consequences of
(4.12) and (4.13). ∎

Finally, for any nonnegative integer $m$, we set

$V_{\pi}^{K_{n-1,1}(m)}\coloneqq\bigoplus_{\begin{subarray}{c}\tau\in\widehat{K}\\\
c(\tau)=m\end{subarray}}V_{\pi}^{\tau|_{K_{n-1,1}}},$

the subspace of $K_{n-1,1}$-invariant vectors that are $\tau$-isotypic for
some $\tau\in\widehat{K}$ of level $c(\tau)=m$; here the level $c(\tau)$ of
$\tau$ is defined to be the minimal nonnegative integer $m$ for which
$\ker\tau$ contains $K(\mathfrak{p}^{m})$, which is necessarily finite since
$\tau$ is smooth. We now show that the newform and the conductor exponent may
be characterised in terms of $V_{\pi}^{K_{n-1,1}(m)}$.

######  Theorem 4.16.

Let $(\pi,V_{\pi})$ be an induced representation of Langlands type of
$\operatorname{GL}_{n}(F)$. For any nonnegative integer $m$, we have that

(4.17) $\dim
V_{\pi}^{K_{n-1,1}(m)}=\begin{dcases*}\binom{m-c(\pi)+n-2}{n-2}&if $m\geq
c(\pi)$,\\\ 0&otherwise.\end{dcases*}$

In particular, the minimal nonnegative integer $m$ for which
$V_{\pi}^{K_{n-1,1}(m)}$ is nontrivial is $m=c(\pi)$. Furthermore, we have
that

(4.18)
$V_{\pi}^{K_{n-1,1}(m)}=\begin{dcases*}V_{\pi}^{\tau_{\chi_{\pi},m}|_{K_{n-1,1}}}&if
$m\geq c(\pi)$,\\\ \\{0\\}&otherwise,\end{dcases*}$

and that

(4.19)
$V_{\pi}^{K_{1}(\mathfrak{p}^{m})}=V_{\pi}^{K_{0}(\mathfrak{p}^{m}),\chi_{\pi}}=\bigoplus_{\ell=0}^{m}V_{\pi}^{K_{n-1,1}(\ell)}=\begin{dcases*}\bigoplus_{\ell=c(\pi)}^{m}V_{\pi}^{\tau_{\chi_{\pi},\ell}|_{K_{n-1,1}}}&if
$m\geq c(\pi)$,\\\ \\{0\\}&otherwise.\end{dcases*}$

In particular,

(4.20)
$V_{\pi}^{K_{1}(\mathfrak{p}^{c(\pi)})}=V_{\pi}^{K_{0}(\mathfrak{p}^{c(\pi)}),\chi_{\pi}}=V_{\pi}^{K_{n-1,1}(c(\pi))}=V_{\pi}^{\tau_{\chi_{\pi},c(\pi)}|_{K_{n-1,1}}}.$

######  Definition 4.21.

We call $\tau_{\chi_{\pi},c(\pi)}$ the newform $K$-type of $\pi$.

######  Remark 4.22.

This gives alternative characterisations of the newform and conductor exponent
of $\pi$: we may _define_ the conductor exponent $c(\pi)$ of $\pi$ to be the
minimal nonnegative integer $m$ for which the space
$V_{\pi}^{K_{n-1,1}(m)}=V_{\pi}^{\tau_{\chi_{\pi},m}|_{K_{n-1,1}}}$ is
nontrivial, while we may _define_ the newform to be the nonzero vector, unique
up to scalar multiplication, lying in
$V_{\pi}^{K_{n-1,1}(c(\pi))}=V_{\pi}^{\tau_{\chi_{\pi},c(\pi)}|_{K_{n-1,1}}}$.
Of course, this definition is somewhat circular in practice, since, as we
shall shortly see, we use Theorem 4.5 (or rather a consequence thereof) in
order to prove Theorem 4.16.

Theorem 4.16 gives additional information about the newform not immediately
apparent from Theorem 4.5: not only is the newform the unique vector, up to
scalar multiplication, that is $K_{1}(\mathfrak{p}^{c(\pi)})$-invariant (or,
equivalently, $(K_{0}(\mathfrak{p}^{c(\pi)}),\psi_{\chi_{\pi}})$-equivariant),
it is also the unique vector, up to scalar multiplication, that is
$K_{n-1,1}$-invariant and $\tau_{\chi_{\pi},c(\pi)}$-isotypic; moreover, there
are no nontrivial $K_{n-1,1}$-invariant $\tau_{\chi,m}$-isotypic vectors with
$m<c(\pi)$ or $\chi\neq\chi_{\pi}$. We have also shown that the space
$V_{\pi}^{K_{1}(\mathfrak{p}^{m})}$ of oldforms of level $m\geq c(\pi)$
decomposes into the direct sum of the subspaces of $K_{n-1,1}$-invariant
$\tau_{\chi_{\pi},\ell}$-isotypic vectors in $V_{\pi}$ with
$\ell\in\\{c(\pi),\ldots,m\\}$, each of which has dimension
$\binom{\ell-c(\pi)+n-2}{n-2}$ respectively.

###### Proof of Theorem 4.16.

We first note that $V_{\pi}^{\tau_{\chi_{1},m_{1}}|_{K_{n-1,1}}}$ and
$V_{\pi}^{\tau_{\chi_{2},m_{2}}|_{K_{n-1,1}}}$ are mutually orthogonal
whenever either $\chi_{1}\neq\chi_{2}$ or $m_{1}\neq m_{2}$. In conjunction
with (4.4) and (4.15), this implies that

$V_{\pi}^{K_{1}(\mathfrak{p}^{m})}=V_{\pi}^{K_{0}(\mathfrak{p}^{m}),\chi_{\pi}}=\begin{dcases*}\bigoplus_{\ell=c(\chi_{\pi})}^{m}V_{\pi}^{\tau_{\chi_{\pi},\ell}|_{K_{n-1,1}}}&if
$m\geq c(\chi_{\pi})$,\\\ \\{0\\}&otherwise.\end{dcases*}$

Since

$V_{\pi}^{K_{1}(\mathfrak{p}^{m})}=V_{\pi}^{K_{0}(\mathfrak{p}^{m}),\chi_{\pi}}=\\{0\\}\
\text{ whenever $m<c(\pi)$}$

from Theorem 4.5, we deduce that

(4.23) $V_{\pi}^{\tau_{\chi_{\pi},m}|_{K_{n-1,1}}}=\\{0\\}\ \text{ whenever
$c(\chi_{\pi})\leq m<c(\pi)$},$

noting that $c(\chi_{\pi})=c(\omega_{\pi})\leq c(\pi)$, from which (4.19) and
(4.20) both follow.

Next, we have from Lemma 4.10 and the fact that
$\pi|_{\mathrm{Z}(\mathcal{O})}=\chi_{\pi}$ that

$V_{\pi}^{K_{n-1,1}(m)}=\begin{dcases*}V_{\pi}^{\tau_{\chi_{\pi},m}|_{K_{n-1,1}}}&if
$m\geq c(\chi_{\pi})$,\\\ \\{0\\}&otherwise.\end{dcases*}$

Together with (4.23), we deduce (4.18).

Finally, from [Ree91, Theorem 1] (which in turn relies on Theorem 4.5), we
have that

$\dim V_{\pi}^{K_{1}(\mathfrak{p}^{m})}=\dim
V_{\pi}^{K_{0}(\mathfrak{p}^{m}),\chi_{\pi}}=\begin{dcases*}\binom{m-c(\pi)+n-1}{n-1}&if
$m\geq c(\pi)$,\\\ 0&otherwise.\end{dcases*}$

The identity (4.17) then follows from (4.18), (4.19), and induction together
with the fact that

(4.24)
$\sum_{\ell=c(\pi)}^{m}\binom{\ell-c(\pi)+n-2}{n-2}=\binom{m-c(\pi)+n-1}{n-1}.\qed$

We may use Theorem 4.16 to tell us the multiplicity with which an irreducible
smooth representations $\tau_{\chi,m}\in\widehat{K}$ occurs in the restriction
of $\pi$ to $K$; when $\chi_{\pi}$ is trivial, this was observed by Reeder
[Ree94] to follow upon combining results from [Hil94] and [Ree91].

######  Proposition 4.25.

Let $(\pi,V_{\pi})$ be an induced representation of Langlands type of
$\operatorname{GL}_{n}(F)$. For any nonnegative integer $m$ and any character
$\chi\in\widehat{\mathcal{O}^{\times}}$ for which $0\leq c(\chi)\leq m$, we
have that

$\dim\operatorname{Hom}_{K}\left(\tau_{\chi,m},\pi|_{K}\right)=\begin{dcases*}\binom{m-c(\pi)+n-2}{n-2}&if
$m\geq c(\pi)$ and $\chi=\chi_{\pi}$,\\\ 0&otherwise.\end{dcases*}$

While this can be proved directly using Theorem 4.16, we give an alternate
proof via Frobenius reciprocity.

###### Proof.

It is clear that $\operatorname{Hom}_{K}(\tau_{\chi,m},\pi|_{K})$ is trivial
if $\chi\neq\chi_{\pi}$. For $\chi=\chi_{\pi}$, by (2.12) and Frobenius
reciprocity, we have that

$\bigoplus_{\ell=c(\chi_{\pi})}^{m}\operatorname{Hom}_{K}\left(\tau_{\chi_{\pi},\ell},\pi|_{K}\right)\cong\operatorname{Hom}_{K}\left(\operatorname{Ind}_{K_{0}(\mathfrak{p}^{m})}^{K}\psi_{\chi_{\pi}},\pi|_{K}\right)\cong\operatorname{Hom}_{K_{0}(\mathfrak{p}^{m})}\left(\psi_{\chi_{\pi}},\pi|_{K_{0}(\mathfrak{p}^{m})}\right).$

Since
$\operatorname{Hom}_{K_{0}(\mathfrak{p}^{m})}(\psi_{\chi},\pi|_{K_{0}(\mathfrak{p}^{m})})$
is trivial whenever $\chi\neq\chi_{\pi}$, this in turn is isomorphic to

$\displaystyle\bigoplus_{\begin{subarray}{c}\chi\in\widehat{\mathcal{O}^{\times}}\\\
0\leq c(\chi)\leq
m\end{subarray}}\operatorname{Hom}_{K_{0}(\mathfrak{p}^{m})}\left(\psi_{\chi},\pi|_{K_{0}(\mathfrak{p}^{m})}\right)$
$\displaystyle\cong\operatorname{Hom}_{K_{0}(\mathfrak{p}^{m})}\left(\operatorname{Ind}_{K_{1}(\mathfrak{p}^{m})}^{K_{0}(\mathfrak{p}^{m})}1,\pi|_{K_{0}(\mathfrak{p}^{m})}\right)$
$\displaystyle\cong\operatorname{Hom}_{K_{1}(\mathfrak{p}^{m})}\left(1,\pi|_{K_{1}(\mathfrak{p}^{m})}\right)$

by (2.10) and Frobenius reciprocity. By [Ree91, Theorem 1], we have that

$\dim\operatorname{Hom}_{K_{1}(\mathfrak{p}^{m})}\left(1,\pi|_{K_{1}(\mathfrak{p}^{m})}\right)=\begin{dcases*}\binom{m-c(\pi)+n-1}{n-1}&if
$m\geq c(\pi)$,\\\ 0&otherwise.\end{dcases*}$

It follows that

$\sum_{\ell=c(\chi_{\pi})}^{m}\dim\operatorname{Hom}_{K}\left(\tau_{\chi_{\pi},\ell},\pi|_{K}\right)=\begin{dcases*}\binom{m-c(\pi)+n-1}{n-1}&if
$m\geq c(\pi)$,\\\ 0&otherwise,\end{dcases*}$

from which the result via induction in conjunction with the identity (4.24). ∎

###  4.3. Matrix Coefficients

We now study the properties of certain matrix coefficients of $\pi$ via matrix
coefficients of $\mathcal{H}_{\chi_{\pi},c(\pi)}(S^{n-1})$. In order to do so,
we explicitly identity $V_{\pi}^{\tau_{\chi_{\pi},c(\pi)}}$ with
$\mathcal{H}_{\chi_{\pi},c(\pi)}(S^{n-1})$. We first observe an identity for
the newform.

######  Lemma 4.26.

The newform $v^{\circ}\in V_{\pi}^{\tau_{\chi_{\pi},c(\pi)}}$ of an induced
representation of Langlands type $(\pi,V_{\pi})$ satisfies

(4.27)
$v^{\circ}=\dim\tau_{\chi_{\pi},c(\pi)}\int_{K}P_{\chi_{\pi},c(\pi)}^{\circ}(e_{n}k^{-1})\pi(k)\cdot
v^{\circ}\,dk.$

###### Proof.

This is an immediate consequence of Lemma 4.10 and Theorem 4.16. ∎

The finite-dimensional vector space $V_{\pi}^{\tau_{\chi_{\pi},c(\pi)}}$
admits a $K$-invariant inner product $\langle\cdot,\cdot\rangle$. We use such
an inner product to construct an isomorphism between
$\mathcal{H}_{\chi_{\pi},c(\pi)}(S^{n-1})$ and
$V_{\pi}^{\tau_{\chi_{\pi},c(\pi)}}$.

######  Proposition 4.28.

Let $(\pi,V_{\pi})$ be an induced representation of Langlands type with
newform $v^{\circ}\in V_{\pi}^{\tau_{\chi_{\pi},c(\pi)}}$. An explicit
isomorphism between $\mathcal{H}_{\chi_{\pi},c(\pi)}(S^{n-1})\ni P=P_{v}$ and
$V_{\pi}^{\tau_{\chi_{\pi},c(\pi)}}\ni v=v_{P}$ is given by

(4.29) $v=\dim\tau_{\chi_{\pi},c(\pi)}\int_{K}P(e_{n}k^{-1})\pi(k)\cdot
v^{\circ}\,dk.$

For nonzero $v\in V_{\pi}^{\tau_{\chi_{\pi},c(\pi)}}$, the inverse is given by

(4.30) $P(x)=\left(\dim\tau_{\chi_{\pi},c(\pi)}\right)^{2}\langle
P,P\rangle\int_{K}\frac{\left\langle\pi(k^{-1})\cdot
v,v^{\circ}\right\rangle}{\langle
v,v\rangle}\left(\tau_{\chi_{\pi},c(\pi)}(k)\cdot
P_{\chi_{\pi},c(\pi)}^{\circ}\right)(x)\,dk.$

Moreover, for all $P\in\mathcal{H}_{\chi_{\pi},c(\pi)}(S^{n-1})$ and $v\in
V_{\pi}^{\tau_{\chi_{\pi},c(\pi)}}$ that are associated via (4.29) and (4.30),
we have that

(4.31) $\langle P,P\rangle\left\langle\pi(k)\cdot
v^{\circ},v\right\rangle=\frac{\langle
v,v\rangle\overline{P}(e_{n}k^{-1})}{\dim\tau_{\chi_{\pi},c(\pi)}}.$

###### Proof.

We first confirm that for $v$ as in (4.29), we have that
$\Pi^{\tau_{\chi_{\pi},c(\pi)}}(v)=v$, so that $v\in
V_{\pi}^{\tau_{\chi_{\pi},c(\pi)}}$. Indeed, from (4.29),

$\displaystyle\Pi^{\tau_{\chi_{\pi},c(\pi)}}(v)$
$\displaystyle=\int_{K}\xi^{\tau_{\chi_{\pi},c(\pi)}}(k)\pi(k)\cdot v\,dk_{1}$
$\displaystyle=\dim\tau_{\chi_{\pi},c(\pi)}\int_{K}\xi^{\tau_{\chi_{\pi},c(\pi)}}(k_{1})\int_{K}P(e_{n}k_{2}^{-1})\pi(k_{1}k_{2})\cdot
v^{\circ}\,dk_{2}\,dk_{1}$
$\displaystyle=\dim\tau_{\chi_{\pi},c(\pi)}\int_{K}\pi(k_{2})\cdot
v^{\circ}\int_{K}\xi^{\tau_{\chi_{\pi},c(\pi)}}(k_{1})\left(\tau_{\chi_{\pi},c(\pi)}(k_{1})\cdot
P\right)(e_{n}k_{2}^{-1})\,dk_{1}\,dk_{2}$
$\displaystyle=\dim\tau_{\chi_{\pi},c(\pi)}\int_{K}P(e_{n}k^{-1})\pi(k)\cdot
v^{\circ}\,dk$ $\displaystyle=v.$

Next, we show that (4.30) follows from (4.31). By (3.16), (3.21), (3.22), and
Schur orthogonality, we have that for any orthonormal basis $\\{Q_{j}\\}$ of
$\mathcal{H}_{\chi_{\pi},c(\pi)}(S^{n-1})$,

$\displaystyle P(x)$
$\displaystyle=\sum_{j=1}^{\dim\mathcal{H}_{\chi_{\pi},c(\pi)}(S^{n-1})}\left\langle
P,Q_{j}\right\rangle Q_{j}(x)$
$\displaystyle=\sum_{j=1}^{\dim\mathcal{H}_{\chi_{\pi},c(\pi)}(S^{n-1})}Q_{j}(x)\left(\dim\tau_{\chi_{\pi},c(\pi)}\right)^{2}$
$\displaystyle\hskip
56.9055pt\times\int_{K}\left\langle\tau_{\chi_{\pi},c(\pi)}(k)\cdot
P_{\chi_{\pi},c(\pi)}^{\circ},Q_{j}\right\rangle\left\langle\tau_{\chi_{\pi},c(\pi)}(k^{-1})\cdot
P,P_{\chi_{\pi},c(\pi)}^{\circ}\right\rangle\,dk$
$\displaystyle=\dim\tau_{\chi_{\pi},c(\pi)}\int_{K}P_{\chi_{\pi},c(\pi)}^{\circ}(xk)P(e_{n}k^{-1})\,dk.$

Assuming (4.31), this is precisely (4.30).

It remains to prove (4.31). We first note that

(4.32)
$v=\left(\dim\tau_{\chi_{\pi},c(\pi)}\right)^{2}\int_{K}\left\langle\tau_{\chi_{\pi},c(\pi)}(k^{-1})\cdot
P,P_{\chi_{\pi},c(\pi)}^{\circ}\right\rangle\pi(k)\cdot v^{\circ}\,dk$

via (3.21) and (4.29). By (3.21), (4.29), (4.32), and Schur orthogonality, we
therefore have that

$\displaystyle\langle P,P\rangle v$
$\displaystyle=\left(\dim\tau_{\chi_{\pi},c(\pi)}\right)^{2}\langle
P,P\rangle\int_{K}\left\langle\tau_{\chi_{\pi},c(\pi)}(k^{-1})\cdot
P,P_{\chi_{\pi},c(\pi)}^{\circ}\right\rangle\pi(k)\cdot v^{\circ}\,dk$
$\displaystyle=\left(\dim\tau_{\chi_{\pi},c(\pi)}\right)^{3}\int_{K}\pi(k_{1})\cdot
v^{\circ}$
$\displaystyle\qquad\times\int_{K}\left\langle\tau_{\chi_{\pi},c(\pi)}(k_{2})\cdot\left(\tau_{\chi_{\pi},c(\pi)}(k_{1}^{-1})\cdot
P\right),P\right\rangle\left\langle\tau_{\chi_{\pi},c(\pi)}(k_{2}^{-1})\cdot
P,P_{\chi_{\pi},c(\pi)}^{\circ}\right\rangle\,dk_{2}\,dk_{1}$
$\displaystyle=\left(\dim\tau_{\chi_{\pi},c(\pi)}\right)^{3}\int_{K}\left\langle\tau_{\chi_{\pi},c(\pi)}(k_{1}^{-1})\cdot
P,P\right\rangle$
$\displaystyle\qquad\times\int_{K}\left\langle\tau_{\chi_{\pi},c(\pi)}(k_{2}^{-1})\cdot
P,P_{\chi_{\pi},c(\pi)}^{\circ}\right\rangle\pi(k_{1}k_{2})\cdot
v^{\circ}\,dk_{2}\,dk_{1}$
$\displaystyle=\dim\tau_{\chi_{\pi},c(\pi)}\int_{K}\left\langle\tau_{\chi_{\pi},c(\pi)}(k^{-1})\cdot
P,P\right\rangle\pi(k)\cdot v\,dk.$

We also have from (4.29) that

(4.33) $\langle
v,v\rangle=\dim\tau_{\chi_{\pi},c(\pi)}\int_{K}\left\langle\pi(k)\cdot
v^{\circ},v\right\rangle P(e_{n}k^{-1})\,dk.$

We therefore have by (3.21), (4.27), (4.32), (4.33), and Schur orthogonality
that

$\displaystyle\langle P,P\rangle\left\langle\pi(k)\cdot
v^{\circ},v\right\rangle$
$\displaystyle=\left(\dim\tau_{\chi_{\pi},c(\pi)}\right)^{3}\int_{K}\int_{K}\left\langle\tau_{\chi_{\pi},c(\pi)}(k_{1}^{-1})\cdot
P_{\chi_{\pi},c(\pi)}^{\circ},P_{\chi_{\pi},c(\pi)}^{\circ}\right\rangle\left\langle\tau_{\chi_{\pi},c(\pi)}(k_{2})\cdot
P,P\right\rangle$ $\displaystyle\hskip
113.81102pt\times\left\langle\pi(kk_{1})\cdot v^{\circ},\pi(k_{2})\cdot
v\right\rangle\,dk_{1}\,dk_{2}$
$\displaystyle=\left(\dim\tau_{\chi_{\pi},c(\pi)}\right)^{3}\int_{K}\left\langle\pi(k_{1})\cdot
v^{\circ},v\right\rangle\int_{K}\left\langle\tau_{\chi_{\pi},c(\pi)}(k_{2})\cdot
P,\tau_{\chi_{\pi},c(\pi)}(k^{-1})\cdot P\right\rangle$ $\displaystyle\hskip
28.45274pt\times\left\langle\tau_{\chi_{\pi},c(\pi)}(k_{2}^{-1})\cdot
P_{\chi_{\pi},c(\pi)}^{\circ},\tau_{\chi_{\pi},c(\pi)}(k_{1})\cdot
P_{\chi_{\pi},c(\pi)}^{\circ}\right\rangle\,dk_{2}\,dk_{1}$
$\displaystyle=\left(\dim\tau_{\chi_{\pi},c(\pi)}\right)^{2}\left\langle
P_{\chi_{\pi},c(\pi)}^{\circ},\tau_{\chi_{\pi},c(\pi)}(k^{-1})\cdot
P\right\rangle$ $\displaystyle\hskip
85.35826pt\times\int_{K}\left\langle\pi(k_{1})\cdot
v^{\circ},v\right\rangle\left\langle\tau_{\chi_{\pi},c(\pi)}(k_{1}^{-1})\cdot
P,P_{\chi_{\pi},c(\pi)}^{\circ}\right\rangle\,dk_{1}$
$\displaystyle=\frac{\langle
v,v\rangle\overline{P}(e_{n}k^{-1})}{\dim\tau_{\chi_{\pi},c(\pi)}}.\qed$

An immediate consequence of the matrix coefficient identity (4.31) is the
following.

######  Corollary 4.34.

Let $(\pi,V_{\pi})$ be an induced representation of Langlands type with
newform $v^{\circ}\in V_{\pi}^{\tau_{\chi_{\pi},c(\pi)}}$. We have that

$\displaystyle\frac{\left\langle\pi(k)\cdot
v^{\circ},v^{\circ}\right\rangle}{\langle v^{\circ},v^{\circ}\rangle}$
$\displaystyle=P_{\chi_{\pi},c(\pi)}^{\circ}(e_{n}k)$
$\displaystyle=\begin{dcases*}\chi_{\pi}(e_{n}k\prescript{t}{}{e}_{n})&if
$k\in K_{0}(\mathfrak{p}^{c(\pi)})$,\\\
\alpha_{\chi_{\pi},c(\pi);c(\pi)-1}\chi_{\pi}(e_{n}k\prescript{t}{}{e}_{n})&if
$c(\pi)>c(\chi_{\pi})$ and $k\in K_{0}(\mathfrak{p}^{c(\pi)-1})\setminus
K_{0}(\mathfrak{p}^{c(\pi)})$,\\\ 0&otherwise,\end{dcases*}$

where $\alpha_{\chi_{\pi},c(\pi);c(\pi)-1}$ is as in (3.6).

###### Proof.

We take $v=v^{\circ}$ in (4.31) and apply (3.5), (3.7), and (3.22). ∎

######  Remark 4.35.

From [Tem14, Lemma 3.3], we have that $c(\pi)=c(\chi_{\pi})$ if and only if
$\pi$ is a twist-minimal principal series representation, so that
$\pi=\operatorname{Ind}_{\mathrm{P}(F)}^{\operatorname{GL}_{n}(F)}\mathop{\vphantom{\bigotimes}\mathchoice{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxtimes$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxtimes$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxtimes$}}}}{\vbox{\hbox{\leavevmode\resizebox{0.0pt}{}{$\boxtimes$}}}}}\displaylimits_{j=1}^{n}\pi_{j}$
with each $\pi_{j}$ a character of $F^{\times}$ and $c(\pi_{j})=0$ for all but
at most one $j\in\\{1,\ldots,n\\}$.

######  Remark 4.36.

Suppose that $\pi$ is tempered. One may take as a model for $\pi$ the
Whittaker model $\mathcal{W}(\pi,\psi)$; the $K$-invariant inner product on
the $\tau_{\chi_{\pi},c(\pi)}$-isotypic subspace of $\mathcal{W}(\pi,\psi)$ is
then given by

$\langle
W_{1},W_{2}\rangle\coloneqq\int\limits_{\mathrm{N}_{n-1}(F)\backslash\operatorname{GL}_{n-1}(F)}W_{1}\begin{pmatrix}g&0\\\
0&1\end{pmatrix}\overline{W_{2}}\begin{pmatrix}g&0\\\ 0&1\end{pmatrix}\,dg.$

(In fact, this is a $\operatorname{GL}_{n}(F)$-invariant inner product.) The
canonical normalisation of the newform $W^{\circ}$ in the Whittaker model is
such that $W^{\circ}(1_{n})=1$. Venkatesh [Ven06, Section 7] has explicitly
calculated $\langle W^{\circ},W^{\circ}\rangle$ with respect to this inner
product; when $\pi$ is spherical, so that $c(\pi)=0$, this is simply
$L(1,\operatorname{ad}\pi)$.

##  5\. Archimedean Analogues

###  5.1. Archimedean Spherical Harmonics

The archimedean analogues of the results in Sections 2 and 3 are well-known;
we briefly survey them for the sake of comparison. In place of a
nonarchimedean field $F$, we instead work with an archimedean field, which is
either $\mathbb{R}$ or $\mathbb{C}$.

####  5.1.1. $F=\mathbb{R}$

The maximal compact subgroup $K_{n}$ of $\operatorname{GL}_{n}(\mathbb{R})$ is
the orthogonal group $\operatorname{O}(n)$. This acts transitively on unit
sphere

$S^{n-1}\coloneqq\left\\{x=(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}:x_{1}^{2}+\cdots+x_{n}^{2}=1\right\\}$

in $\mathbb{R}^{n}$ via the group action $k\cdot x\coloneqq xk$ for $k\in
K_{n}$ and $x\in S^{n-1}$. The stabiliser of the point
$e_{n}\coloneqq(0,\ldots,0,1)\in S^{n-1}$ is the subgroup

(5.1) $\left\\{\begin{pmatrix}k^{\prime}&0\\\ 0&1\end{pmatrix}\in
K_{n}:k^{\prime}\in K_{n-1}\right\\},$

which we freely identify with $K_{n-1}$. It follows that $S^{n-1}\cong
K_{n-1}\backslash K_{n}$. Note that the subgroup (5.1) does not, at first
glance, appear to be the direct archimedean analogue of the subgroup
$K_{n-1,1}$ as in (1.1), since $K_{n-1,1}\cong
K_{n-1}\ltimes\mathcal{O}^{n-1}$. In the archimedean setting, on the other
hand, there is no analogue of $\mathcal{O}^{n-1}$; nonetheless, the subgroup
(5.1), like the nonarchimedean subgroup (1.1), is the maximal compact subgroup
of the mirabolic subgroup (1.2).

The decomposition of the right regular representation of $K_{n}$ on
$L^{2}(S^{n-1})$ is precisely the theory of spherical harmonics. This is best
understood in terms of homogeneous harmonic polynomials, for which we follow
[AH12, Chapter 2].

Given a nonnegative integer $m$, let $\mathcal{H}_{m}(\mathbb{R}^{n})$ denote
the vector space consisting of homogeneous harmonic polynomials of degree $m$,
namely the set of polynomials $P(x_{1},\ldots,x_{n})$ in
$(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}$ that are annihilated by the Laplacian

$\Delta=\sum_{j=1}^{n}\frac{\partial^{2}}{\partial x_{j}^{2}}$

and satisfy $P(\lambda x_{1},\ldots,\lambda
x_{n})=\lambda^{m}P(x_{1},\ldots,x_{n})$ for all $\lambda\in\mathbb{R}$. This
space has dimension $1$ for $n=1$ and $m\in\\{0,1\\}$ and has dimension

$\binom{m+n-2}{n-2}+\binom{m+n-3}{n-3}=\frac{(2m+n-2)(m+n-3)!}{m!(n-2)!}=\frac{2m+n-2}{m+n-2}\binom{m+n-2}{n-2}$

for $n\geq 2$.

Let $\mathcal{H}_{m}(S^{n-1})$ denote the vector space of the restriction of
elements of $\mathcal{H}_{m}(\mathbb{R}^{n})$ to the unit sphere; via the
homogeneity of elements of $\mathcal{H}_{m}(\mathbb{R}^{n})$, these spaces are
isomorphic. Elements of $\mathcal{H}_{m}(S^{n-1})$ are called spherical
harmonics of degree $m$. The group $K_{n}$ acts on
$\mathcal{H}_{m}(\mathbb{R}^{n})$ via right translation, which descends to an
action on $\mathcal{H}_{m}(S^{n-1})$. As a $K_{n}$-module,
$\mathcal{H}_{m}(S^{n-1})$ is irreducible, and for $n\geq 2$, we have the
orthogonal decomposition

$L^{2}(S^{n-1})=\bigoplus_{m=0}^{\infty}\mathcal{H}_{m}(S^{n-1}),$

analogous to the decomposition (2.16) of $C^{\infty}(S^{n-1})$.

We may view $\mathcal{H}_{m}(S^{n-1})$ as the real analogue of
$\mathcal{H}_{\chi,m}(S^{n-1})$. In turn, the real analogue of
$C^{\infty}(S^{n-1})^{K(\mathfrak{p}^{m})}$ is
$\bigoplus_{\ell=0}^{m}\mathcal{H}_{\ell}(S^{n-1})$, the space of spherical
harmonics of degree at most $m$. As
$K_{1}=\operatorname{O}(1)\cong\mathbb{Z}/2\mathbb{Z}$, which is the real
analogue of $\mathcal{O}^{\times}$, characters of $K_{1}$ are of the form
$\operatorname{sgn}^{\kappa}$ for $\kappa\in\\{0,1\\}$, where
$\operatorname{sgn}(x)\coloneqq x/|x|$. In particular, the central character
$\chi$ of the $K_{n}$-module $\mathcal{H}_{m}(S^{n-1})$ is simply
$\operatorname{sgn}^{m\pmod{2}}$, which is determined by the parity of the
nonnegative integer $m$. So the real analogue of
$C^{\infty}(S^{n-1})_{\chi}^{K(\mathfrak{p}^{m})}$, in terms of its orthogonal
decomposition (2.15), is precisely

$\bigoplus_{\begin{subarray}{c}\ell=0\\\
\ell\equiv\kappa\pmod{2}\end{subarray}}^{m}\mathcal{H}_{\ell}(S^{n-1})$

for $\chi=\operatorname{sgn}^{\kappa}$ with $\kappa\in\\{0,1\\}$ and
$m\geq\kappa$, while the analogue the orthogonal decomposition (2.9) of
$C^{\infty}(S^{n-1})^{K(\mathfrak{p}^{m})}$ is simply

$\bigoplus_{\ell=0}^{m}\mathcal{H}_{\ell}(S^{n-1})=\bigoplus_{\kappa\in\\{0,1\\}}\bigoplus_{\begin{subarray}{c}\ell=0\\\
\ell\equiv\kappa\pmod{2}\end{subarray}}^{m}\mathcal{H}_{\ell}(S^{n-1}).$

There exists a unique spherical harmonic
$P_{m}^{\circ}\in\mathcal{H}_{m}(S^{n-1})$ satisfying $P_{m}^{\circ}(e_{n})=1$
and $P_{m}^{\circ}(xk)=P_{m}^{\circ}(x)$ for all $x=(x_{1},\ldots,x_{n})\in
S^{n-1}$ and $k=\begin{pmatrix}k^{\prime}&0\\\ 0&1\end{pmatrix}\in K_{n}$ with
$k^{\prime}\in K_{n-1}$, namely

$P_{m}^{\circ}(x_{1},\ldots,x_{n})\coloneqq\sum_{\begin{subarray}{c}\nu=0\\\
\nu\equiv
0\pmod{2}\end{subarray}}^{m}\frac{i^{\nu}m!\Gamma\left(\frac{n-1}{2}\right)}{2^{\nu}\left(\frac{\nu}{2}\right)!(m-\nu)!\Gamma\left(\frac{\nu+n-1}{2}\right)}\left(x_{1}^{2}+\cdots+x_{n-1}^{2}\right)^{\frac{\nu}{2}}x_{n}^{m-\nu}.$

This is the zonal spherical harmonic on $S^{n-1}$ of degree $m$, which is the
real analogue of the zonal spherical function
$P_{\chi,m}^{\circ}\in\mathcal{H}_{\chi,m}(S^{n-1})$. From this, the addition
theorem for $\mathcal{H}_{m}(S^{n-1})$ takes precisely the same form as the
nonarchimedean result given in Lemma 3.14; similarly,
$(\dim\mathcal{H}_{m}(S^{n-1}))P_{m}^{\circ}$ is the reproducing kernel for
$\mathcal{H}_{m}(S^{n-1})$, akin to Corollary 3.20.

####  5.1.2. $F=\mathbb{C}$

The maximal compact subgroup $K_{n}$ of $\operatorname{GL}_{n}(\mathbb{C})$ is
the unitary group $\operatorname{U}(n)$. This acts transitively on unit sphere

$S^{n-1}\coloneqq\left\\{z=(z_{1},\ldots,z_{n})\in\mathbb{C}^{n}:z_{1}\overline{z_{1}}+\cdots+z_{n}\overline{z_{n}}=1\right\\}$

in $\mathbb{C}^{n}$ via the group action $k\cdot x\coloneqq xk$ for $k\in
K_{n}$ and $x\in S^{n-1}$. (It behoves us to point out that as a real
topological manifold, this should be viewed as the $(2n-1)$-dimensional unit
sphere, but we use the notation $S^{n-1}$ for the sake of consistency.) Just
as for the real case, the stabiliser of the point
$e_{n}\coloneqq(0,\ldots,0,1)\in S^{n-1}$ is the subgroup (5.1), which we
freely identify with $K_{n-1}$, so that $S^{n-1}\cong K_{n-1}\backslash
K_{n}$.

The decomposition of the right regular representation of $K_{n}$ on
$L^{2}(S^{n-1})$ is again the theory of spherical harmonics, with the
additional complexification that one must consider the _bidegree_ of such a
spherical harmonic. This is best understood in terms of homogeneous harmonic
polynomials, for which we follow [Rud08, Chapter 12].

Given a pair of nonnegative integers $m_{1},m_{2}$, let
$\mathcal{H}_{m_{1},m_{2}}(\mathbb{C}^{n})$ denote the vector space consisting
of homogeneous harmonic polynomials of bidegree $(m_{1},m_{2})$, namely the
set of polynomials
$P(z_{1},\ldots,z_{n},\overline{z_{1}},\ldots,\overline{z_{n}})$ in
$(z_{1},\ldots,z_{n})\in\mathbb{C}^{n}$ that are annihilated by the Laplacian

$\Delta=4\sum_{j=1}^{n}\frac{\partial^{2}}{\partial
z_{j}\partial\overline{z_{j}}}$

and satisfy $P(\lambda z_{1},\ldots,\lambda z_{n},\overline{\lambda
z_{1}},\ldots,\overline{\lambda
z_{n}})=\lambda^{m_{1}}\overline{\lambda}^{m_{2}}P(z_{1},\ldots,z_{n},\overline{z_{1}},\ldots,\overline{z_{n}})$
for all $\lambda\in\mathbb{C}$. This space has dimension $1$ for $n=1$ and

$\frac{(m_{1}+m_{2}+n-1)(m_{1}+n-2)!(m_{2}+n-2)!}{m_{1}!m_{2}!(n-2)!(n-1)!}=\frac{m_{1}+m_{2}+n-1}{n-1}\binom{m_{1}+n-2}{n-2}\binom{m_{2}+n-2}{n-2}$

for $n\geq 2$.

Let $\mathcal{H}_{m_{1},m_{2}}(S^{n-1})$ denote the vector space of the
restriction of elements of $\mathcal{H}_{m_{1},m_{2}}(\mathbb{C}^{n})$ to the
unit sphere; via the homogeneity of elements of
$\mathcal{H}_{m_{1},m_{2}}(\mathbb{C}^{n})$, these spaces are isomorphic.
Elements of $\mathcal{H}_{m_{1},m_{2}}(S^{n-1})$ are called spherical
harmonics of bidegree $(m_{1},m_{2})$. The group $K_{n}$ acts on
$\mathcal{H}_{m_{1},m_{2}}(\mathbb{C}^{n})$ via right translation, which
descends to an action on $\mathcal{H}_{m_{1},m_{2}}(S^{n-1})$. As a
$K_{n}$-module, $\mathcal{H}_{m_{1},m_{2}}(S^{n-1})$ is irreducible, and for
$n\geq 2$, we have the orthogonal decomposition

$L^{2}(S^{n-1})=\bigoplus_{m_{1},m_{2}=0}^{\infty}\mathcal{H}_{m_{1},m_{2}}(S^{n-1}),$

analogous to the decomposition (2.16) of $C^{\infty}(S^{n-1})$.

We may view $\mathcal{H}_{m_{1},m_{2}}(S^{n-1})$ as the real analogue of
$\mathcal{H}_{\chi,m}(S^{n-1})$. In turn, the real analogue of
$C^{\infty}(S^{n-1})^{K(\mathfrak{p}^{m})}$ is $\bigoplus_{m_{1}+m_{2}\leq
m}\mathcal{H}_{m_{1},m_{2}}(S^{n-1})$, the space of spherical harmonics of
total degree at most $m$. As
$K_{1}=\operatorname{U}(1)\cong\mathbb{R}/\mathbb{Z}$, which is the complex
analogue of $\mathcal{O}^{\times}$, characters of $K_{1}$ are of the form
$e^{i\kappa\arg}$ for $\kappa\in\mathbb{Z}$, where $e^{i\arg(z)}\coloneqq
z^{1/2}\overline{z}^{-1/2}$. In particular, the central character $\chi$ of
the $K_{n}$-module $\mathcal{H}_{m_{1},m_{2}}(S^{n-1})$ is
$e^{i(m_{1}-m_{2})\arg}$, which is determined by the difference of $m_{1}$ and
$m_{2}$. So the complex analogue of
$C^{\infty}(S^{n-1})_{\chi}^{K(\mathfrak{p}^{m})}$, in terms of its orthogonal
decomposition (2.15), is precisely

$\bigoplus_{\begin{subarray}{c}m_{1}+m_{2}\leq m\\\
m_{1}-m_{2}=\ell\end{subarray}}\mathcal{H}_{m_{1},m_{2}}(S^{n-1})$

for $\chi=e^{i\ell\arg}$ with $m\geq|\ell|$, while the orthogonal
decomposition (2.9) of $C^{\infty}(S^{n-1})^{K(\mathfrak{p}^{m})}$ has the
complex analogue

$\bigoplus_{m_{1}+m_{2}\leq
m}\mathcal{H}_{m_{1},m_{2}}(S^{n-1})=\bigoplus_{|\ell|\leq
m}\bigoplus_{\begin{subarray}{c}m_{1}+m_{2}\leq m\\\
m_{1}-m_{2}=\ell\end{subarray}}\mathcal{H}_{m_{1},m_{2}}(S^{n-1}).$

There exists a unique spherical harmonic
$P_{m}^{\circ}\in\mathcal{H}_{m_{1},m_{2}}(S^{n-1})$ satisfying
$P_{m_{1},m_{2}}^{\circ}(e_{n},e_{n})=1$ and
$P_{m_{1},m_{2}}^{\circ}(zk,\overline{zk})=P_{m_{1},m_{2}}^{\circ}(z,\overline{z})$
for all $z=(z_{1},\ldots,z_{n})\in S^{n-1}$ and
$k=\begin{pmatrix}k^{\prime}&0\\\ 0&1\end{pmatrix}\in K_{n}$ with
$k^{\prime}\in K_{n-1}$, namely

$P_{m_{1},m_{2}}^{\circ}(z,\overline{z})\coloneqq\sum_{\nu=0}^{\min\\{m_{1},m_{2}\\}}\frac{(-1)^{\nu}\binom{m_{1}}{\nu}\binom{m_{2}}{\nu}}{\binom{\nu+n-2}{n-2}}\left(z_{1}\overline{z_{1}}+\cdots+z_{n-1}\overline{z_{n-1}}\right)^{\nu}z_{n}^{m_{1}-\nu}\overline{z_{n}}^{m_{2}-\nu}.$

This is the zonal spherical harmonic on $S^{n-1}$ of bidegree $(m_{1},m_{2})$,
which is the complex analogue of the zonal spherical function
$P_{\chi,m}^{\circ}\in\mathcal{H}_{\chi,m}(S^{n-1})$. Once more, the addition
theorem for $\mathcal{H}_{m_{1},m_{2}}(S^{n-1})$ takes the same form as Lemma
3.14 and $(\dim\mathcal{H}_{m}(S^{n-1}))P_{m}^{\circ}$ is the reproducing
kernel for $\mathcal{H}_{m}(S^{n-1})$.

###  5.2. Archimedean Newform Theory for $\operatorname{GL}_{n}$

Finally, we consider the archimedean analogues of the results in Section 4;
these are due to Popa [Pop08] for $\operatorname{GL}_{2}$ and to the author
[Hum20] for $\operatorname{GL}_{n}$ with $n$ arbitrary.

Let $(\pi,V_{\pi})$ be an induced representation of Langlands type of
$\operatorname{GL}_{n}(F)$, where $F$ is an archimedean local field, so that
$F$ is either $\mathbb{R}$ or $\mathbb{C}$. There is no obvious analogue in
the archimedean setting of the nonarchimedean congruence subgroups
$K_{1}(\mathfrak{p}^{m})$ and $K_{0}(\mathfrak{p}^{m})$ of $K_{n}$ (though cf.
[JN19]). This prevents one from defining the newform and conductor exponent of
$\pi$ via the subspace $V_{\pi}^{K_{1}(\mathfrak{p}^{c(\pi)})}$ of
$K_{1}(\mathfrak{p}^{c(\pi)})$-invariant vectors as in Definition 4.6.

As highlighted in Remark 4.8, one can instead characterise the nonarchimedean
newform, when viewed in the Whittaker model, as the unique Whittaker function
$W^{\circ}\in\mathcal{W}(\pi,\psi)$ that is right $K_{n-1}$-invariant, with
$K_{n-1}$ embedded in $K_{n}$ as the subgroup (5.1), and that is a test vector
for the local $\operatorname{GL}_{n}\times\operatorname{GL}_{n-1}$
Rankin–Selberg integral whenever the second representation is unramified. It
is shown in [Hum20, Theorem 4.17] that this characterisation does indeed also
hold in the archimedean setting, in that there is a unique such Whittaker
function satisfying these two conditions. One would also like to show that the
conductor exponent is characterised via the epsilon factor
$\varepsilon(s,\pi,\psi)$; while this is the case when $F$ is nonarchimedean,
this is unfortunately insufficient when $F$ is archimedean, for then
$\varepsilon(s,\pi,\psi)$ is simply an integral power of $i$, and this integer
is only determined modulo $4$.

Nonetheless, the archimedean analogue of Theorem 4.16 holds; more precisely,
the archimedean analogues of (4.17) and (4.18) are true via [Hum20, Theorems
4.7 and 4.12]. Here the nonarchimedean subgroup $K_{n-1,1}$ of $K_{n}$ is
replaced by the archimedean subgroup (5.1), which both share the property that
they are the stabiliser subgroup of $e_{n}\in F^{n}$; moreover, the notion of
ordering $K$-types by their _level_ in the nonarchimedean setting is replaced
by ordering $K$-types by their _Howe degree_ in the archimedean setting (cf.
[Hum20, Section 4.2]).

More precisely, in [Hum20, Definition 4.8], the author has _defined_ the
conductor exponent $c(\pi)$ of an induced representation of Langlands type
$(\pi,V_{\pi})$ of $\operatorname{GL}_{n}$ over an archimedean field $F$ to be
the minimal nonnegative integer $m$ for which $\pi$ contains a $K_{n}$-type
$\tau$ of Howe degree $m$ having a nontrivial $K_{n-1}$-fixed vector, with
$K_{n-1}$ embedded in $K_{n}$ as the subgroup (5.1). This distinguished
$K_{n}$-type $\tau=\tau^{\circ}$ is the newform $K$-type and appears with
multiplicity one in $\pi$. The author has also _defined_ the newform in the
archimedean setting to be the nonzero vector in $V_{\pi}$, unique up to scalar
multiplication, that is invariant under the subgroup (5.1) and is
$\tau^{\circ}$-isotypic.

We observed in Remark 4.22 that Theorem 4.16 gives alternative
characterisations of the newform and conductor exponent in the nonarchimedean
setting in terms of $K$-types. In conjunction with [Hum20, Theorem 4.7], this
characterisation thereby unifies the nonarchimedean and archimedean treatments
of the newform and the conductor exponent.

### Acknowledgements

Thanks are owed to Subhajit Jana for helpful discussions regarding newforms
and matrix coefficients.

## References

  * [AL70] A. O. L. Atkin and J. Lehner, “Hecke Operators on $\Gamma_{0}(m)$”, Mathematische Annalen 185:1 (1970), 134–160.
  * [AH12] Kendall Atkinson and Weimin Han, Spherical Harmonics and Approximations on the Unit Sphere: An Introduction, Lecture Notes in Mathematics 2044, Springer, Heidelberg, 2012.
  * [BP17] Ehud Moshe Baruch and Soma Purkait, “Hecke Algebras, New Vectors and New Forms on $\Gamma_{0}(m)$”, Mathematische Zeitschrift 287:1–2 (2017), 705–733.
  * [CN09] Peter S. Campbell and Monica Nevins, “Branching Rules for Unramified Principal Series Representations of $\operatorname{GL}(3)$ over a $p$-adic Field”, Journal of Algebra 321:9 (2009), 2422–2444.
  * [CN10] Peter S. Campbell and Monica Nevins, “Branching Rules for Ramified Principal Series Representations of $\operatorname{GL}(3)$ over a $p$-adic Field”, Canadian Journal of Mathematics 62:1 (2010), 34–51.
  * [Cas73a] William Casselman, “On Some Results of Atkin and Lehner”, Mathematische Annalen 201:4 (1973), 301–314.
  * [Cas73b] William Casselman, “The Restriction of a Representation of $\operatorname{GL}_{2}(k)$ to $\operatorname{GL}_{2}(\mathfrak{o})$”, Mathematische Annalen 206:4 (1973), 311–318.
  * [Cha98] Shu-Chu Chang, “On Classification of Special Linear Group over Commutative Rings”, Communications in Algebra 26:12 (1998), 3993–4005.
  * [Han87] Kristina Hansen, “Restriction to $\operatorname{GL}_{2}(\mathcal{O})$ of Supercuspidal Representations of $\operatorname{GL}_{2}(F)$”, Pacific Journal of Mathematics 130:2 (1987), 327–349.
  * [Hil94] Gregory Hill, “On the Nilpotent Representations of $\operatorname{GL}_{n}(\mathcal{O})$”, manuscripta mathematica 82:3–4 (1994), 293–311.
  * [Hum20] Peter Humphries, “Archimedean Newform Theory for $\operatorname{GL}_{n}$”, preprint (2020), 56 pages.
  * [Jac12] Hervé Jacquet, “A Correction to Conducteur des représentations du groupe linéaire”, Pacific Journal of Mathematics 260:2 (2012), 515–525.
  * [JP-SS81] H. Jacquet, I. I. Piatetski-Shapiro, and J. Shalika, “Conducteur des représentations du groupe linéaire”, Mathematische Annalen 256:2 (1981), 199–214.
  * [JS83] Hervé Jacquet and Joseph Shalika, “The Whittaker Model of Induced Representations”, Pacific Journal of Mathematics 109:1 (1983), 107–120.
  * [JN19] Subhajit Jana and Paul D. Nelson, “Analytic Newvectors for $\operatorname{GL}_{n}(\mathbb{R})$”, preprint (2019), 47 pages.
  * [Mat13] Nadir Matringe, “Essential Whittaker Functions for $\operatorname{GL}(n)$”, Documenta Mathematica 18 (2013), 1191–1214.
  * [OS14] Uri Onn and Pooja Singla, “On the Unramified Principal Series of $\operatorname{GL}(3)$ over Non-archimedean Local Fields”, Journal of Algebra 397 (2014), 1–17.
  * [Pet82] E. E. Petrov, “Harmonic Analysis on a $\mathfrak{P}$-adic Sphere”, Soviet Mathematics (Iz. VUZ) 26:11 (1982), 103–106.
  * [Pop08] Alexandru A. Popa, “Whittaker Newforms for Archimedean Representations of $\operatorname{GL}(2)$”, Journal of Number Theory 128:6 (2008), 1637–1645.
  * [Ree91] Mark Reeder, “Old Forms on $\operatorname{GL}_{n}$”, American Journal of Mathematics 113:5 (1991), 911–930.
  * [Ree94] Mark Reeder, review of [Hil94], Mathematical Reviews MR1265002.
  * [Rud08] Walter Rudin, Function Theory in the Unit Ball of $\mathbb{C}^{n}$, Classics in Mathematics, Springer, Heidelberg 2008.
  * [Sch02] Ralf Schmidt, “Some Remarks on Local Newforms for $\operatorname{GL}(2)$”, Journal of the Ramanujan Mathematical Society 17 (2002), 115–147.
  * [Sil70] Allan J. Silberger, $\operatorname{PGL}_{2}$ over the $p$-adics: Its Representations, Spherical Functions, and Fourier Analysis, Springer Lecture Notes in Mathematics 166, Springer, Berlin, 1970.
  * [Tem14] Nicolas Templier, “Large Values of Modular Forms”, Cambridge Journal of Mathematics 2:1 (2014), 91–116.
  * [Ven06] Akshay Venkatesh, “Large Sieve Inequalities for $\operatorname{GL}(n)$-forms in the Conductor Aspect”, Advances in Mathematics 200:2 (2006), 336–356.