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# Toric periods for a $p$-adic quaternion algebra

U. K. Anandavardhanan and Basudev Pattanayak  Department of Mathematics,
Indian Institute of Technology Bombay, Mumbai - 400076, India.
<EMAIL_ADDRESS>Department of Mathematics, Indian Institute of
Technology Bombay, Mumbai - 400076, India<EMAIL_ADDRESS>
###### Abstract.

Let $G$ be a compact group with two given subgroups $H$ and $K$. Let $\pi$ be
an irreducible representation of $G$ such that its space of $H$-invariant
vectors as well as the space of $K$-invariant vectors are both one
dimensional. Let $v_{H}$ (resp. $v_{K}$) denote an $H$-invariant (resp.
$K$-invariant) vector of unit norm in a given $G$-invariant inner product
$\langle~{},~{}\rangle_{\pi}$ on $\pi$. We are interested in calculating the
correlation coefficient

$c(\pi;H,K)=|\langle v_{H},v_{K}\rangle_{\pi}|^{2}.$

In this paper, we compute the correlation coefficient of an irreducible
representation of the multiplicative group of the $p$-adic quaternion algebra
with respect to any two tori. In particular, if $\pi$ is such an irreducible
representation of odd minimal conductor with non-trivial invariant vectors for
two tori $H$ and $K$, then its root number $\varepsilon(\pi)$ is $\pm 1$ and
$c(\pi;H,K)$ is non-vanishing precisely when $\varepsilon(\pi)=1$.

###### 1991 Mathematics Subject Classification:

Primary 20C15; Secondary 20C20, 20C33

##  1\. Introduction

Let $G$ be a compact group with two given subgroups $H$ and $K$ such that any
irreducible representation $\pi$ of $G$ admits at most a one dimensional space
of invariant vectors for $H$ as well as $K$. Let $v_{H}$ (resp. $v_{K}$)
denote an $H$-invariant (resp. $K$-invariant) vector of unit norm in a fixed
$G$-invariant inner product $\langle~{},~{}\rangle_{\pi}$ on $\pi$. Then the
correlation coefficient of $H$ and $K$ in $\pi$ is defined as

$c(\pi;H,K)=|\langle v_{H},v_{K}\rangle_{\pi}|^{2}.$

Note that $c(\pi;H,K)$ is well-defined, since $\langle~{},~{}\rangle_{\pi}$ is
unique up to scalar multiplication, and since $v_{H}$ and $v_{K}$ are unique
up to a complex number of absolute value $1$. The subgroups $H$ and $K$ are
said to be correlated with respect to $\pi$ if $c(\pi;H,K)\neq 0$. The
correlation coefficient appears in a variety of contexts and we refer to
[Gro91, §8, §9] and [Rob99] for more background material.

In this paper, we compute the correlation coefficient for an irreducible
representation $\pi$ of $G=D^{\times}/F^{\times}$, where $D$ is the quaternion
algebra over a $p$-adic field $F$ with $p$ odd, with respect to any two tori
as the given subgroups $H$ and $K$. The restriction of an irreducible smooth
representation of $D^{\times}$ (or ${\rm GL}(2,F)$) to its tori is pioneered
by the work of Tunnell [Tun83, HSY93, Sai93]. Tunnell’s results, which are for
$(SO(3),SO(2))$, and the work of Prasad for $(SO(4),SO(3))$ [Pra90, Pra92] are
the base cases of more general branching laws and the related Gan-Gross-Prasad
conjectures [GGP12]. The results of this paper on the correlation coefficient
are in the basic context of Tunnell’s results and the striking simplicity of
these results (see Theorem 1.1 and Theorem 1.2) makes one hope for analogous
results in the realm of higher rank branching laws.

For an irreducible representation $\pi$ of $D^{\times}$ to have a fixed vector
for any torus, it is necessary that its central character $\omega_{\pi}$ is
trivial. It follows that $\pi$ is self-dual and that its epsilon factor
$\varepsilon(1/2,\pi,\Psi)$ is independent of the choice of the additive
character $\Psi$ of $F$. This is the local root number which we now denote by
$\varepsilon(\pi)$. By our assumptions, we always have
$\varepsilon(\pi)\in\\{\pm 1\\}$. These local constants of arithmetic
significance and their sign are of fundamental importance in the Langlands
program and we highlight at this point that one of the key findings of this
paper is that the vanishing of the correlation coefficient in some of the
situations considered in this paper is determined precisely by the sign of the
local root number. We state these as our first results.

######  Theorem 1.1.

Let $\pi$ be an irreducible representation of $D^{\times}$ of odd minimal
conductor. Let $S$ and $T$ denote the two non-isomorphic tori for which $\pi$
admits non-trivial fixed vectors. Then the local root number
$\varepsilon(\pi)$ is $\pm 1$, and

$c(\pi;S,T)\neq 0\iff\varepsilon(\pi)=1.$

Moreover, when $\varepsilon(\pi)=1$,

$c(\pi;S,T)=\frac{2}{\dim\pi}.$

We make a remark on the assumption of oddness in Theorem 1.1. Under this
assumption, the two non-isomorphic tori $S$ and $T$ admitting fixed vectors
for $\pi$ have the property that the normalizer $N(S)$ of $S$ in $D^{\times}$
intersects $T$ in a non-central element (and vice-versa), and $(S,T)$ is a
compatible pair in the sense of Remark 4 below. If $\pi$ is of even minimal
conductor then the tori admitting fixed vectors need not be compatible and
this has to be part of the assumption in order to have an analogous result.

######  Theorem 1.2.

Let $\pi$ be an irreducible representation of $D^{\times}$ of even minimal
conductor $f(\pi)$. Let $S$ and $T$ denote two distinct tori for which $\pi$
admits non-trivial fixed vectors. Assume that $N(S)$ intersects $T$ in a non-
central element. Then the local root number $\varepsilon(\pi)$ is $\pm 1$, and

$c(\pi;S,T)\neq 0\iff(-1)^{f(\pi)/2}\varepsilon(\pi)=1.$

Moreover, when $\varepsilon(\pi)=(-1)^{f(\pi)/2}$,

$c(\pi;S,T)=\frac{2}{\dim\pi}.$

######  Remark 1.

In particular, Theorem 1.1 (resp. Theorem 1.2) asserts that the invariant
vectors for the two tori are orthogonal precisely when $\varepsilon(\pi)=-1$
(resp. $\varepsilon(\pi)\neq(-1)^{f(\pi)/2}$). One implication follows from
[Pra96, Theorem 4] as well. Indeed, by loc. cit., this condition on
$\varepsilon(\pi)$ would imply that $\pi$ does not have a fixed vector for
$N(S)$. Thus, there exists an element $t\in N(S)\cap T$ that does not fix the
$S$-fixed vector $v_{S}$. Note that $t\cdot v_{S}=\alpha v_{S}$ for some
$\alpha\neq 1$. Now,

$\langle v_{S},v_{T}\rangle=\langle v_{S},t^{-1}\cdot v_{T}\rangle=\langle
t\cdot v_{S},v_{T}\rangle=\alpha\langle v_{S},v_{T}\rangle,$

and it follows that $c(\pi;S,T)=0$.

The tori in $D^{\times}$ correspond to quadratic extensions of $F$. By our
assumption on the residue characteristic being odd, we have only three
distinct such extensions with one of these being unramified. We denote the
unramified quadratic extension by $M$ and the ramified quadratic extensions by
$K$ and $L$. Let us denote the corresponding tori by the same notations as
well.

The irreducible representations of $D^{\times}$ are understood by [GG68, CH77,
How77a, How77b], and we refer to [GK80, KZ80, BF83] for more details and for
the explicit construction of these representations that we make use of in this
paper. The essential points are summarized in Section 2. Suffices here to say
that they are associated to multiplicative characters of quadratic extensions
of $F$ and in this association the representations of even minimal conductor
arise out of $M$ and the representations of odd minimal conductor arise out of
$K$ or $L$. These constructions are available more generally for any tame
division algebra; i.e., when the index of the division algebra and the residue
characteristic are relatively prime, and this is the reason why we assume that
$p$ is odd.

Now we state the other main results of this paper. The first one is when $\pi$
has odd minimal conductor and the second one is when $\pi$ has even minimal
conductor.

######  Theorem 1.3.

Let $\pi$ be an irreducible representation of $D^{\times}$ associated to a
character $\psi$ of $K^{\times}$, where $K$ is a ramified quadratic extension
of $F$. Assume $\psi$ has trivial restriction to $F^{\times}$ so that
$\psi(\varpi)=\pm 1$. Then,

  1. (1)

$c(\pi;M,K)=\begin{cases}\frac{2}{\dim\pi}&\text{if $q\equiv 3\mod 4$ and
$\psi(\varpi)=1$,}\\\ 0&\text{otherwise.}\end{cases}$

  2. (2)

$c(\pi;M,L)=\begin{cases}\frac{2}{\dim\pi}&\text{if $q\equiv 1\mod 4$ and
$\psi(\varpi)=1$,}\\\ 0&\text{otherwise.}\end{cases}$

######  Remark 2.

In the situation of Theorem 1.3, the representation $\pi$ always has an
$M$-fixed vector and it has a $K$-fixed vector precisely when $q\equiv 3\mod
4$ and an $L$-fixed vector precisely when $q\equiv 1\mod 4$. In particular,
$c(\pi;K,L)=0$, $c(\pi;M,K)=0$ when $q\equiv 1\mod 4$, and $c(\pi;M,L)=0$ when
$q\equiv 3\mod 4$ for trivial reasons.

######  Remark 3.

By Remark 2, it follows from Theorem 1.3 that two non-trivial invariant
vectors - one for $M$ and the other for either $K$ or $L$ \- are orthogonal
precisely when a certain sign, namely the value of $\psi(\varpi)$, is $-1$.
Thus, a sign captures and characterizes the orthogonality of invariant
vectors! And, this sign is nothing but the local root number of $\pi$ (cf.
Section 5.4).

######  Remark 4.

Note that $(M,K)$ (and $(M,L)$) is a compatible pair in the sense that any
other embedding of the unramified extension $M$ and the ramified extension $K$
will give rise to a pair, say $(M^{\prime},K^{\prime})$, which is
simultaneously $G$-conjugate to $(M,K)$; i.e, there exists $g\in D^{\times}$
such that $M^{\prime}=gMg^{-1}$ and $K^{\prime}=gKg^{-1}$. Thus, the
correlation coefficient $c(\pi;M,K)$ (and $c(\pi;M,L)$) is independent of the
choice of the embeddings of the field extensions in $D^{\times}$.

######  Theorem 1.4.

Let $\pi$ be an irreducible representation of $D^{\times}$ associated to a
character $\psi$ of $M^{\times}$, where $M$ is the unramified quadratic
extension of $F$. Assume $\psi$ has trivial restriction to $F^{\times}$. Let
$K=F(\varpi)$ and $L=F(\zeta\varpi)$, where $\varpi$ is a fixed uniformizer of
$D$ and $\zeta$ is a fixed primitive $(q^{2}-1)$-st root of unity in
$D^{\times}$ with $q$ being the residue cardinality of $F$. Then,

$c(\pi;K,L)=\frac{|1+\psi(\zeta)|^{2}}{2\dim\pi}.$

######  Remark 5.

In the situation of Theorem 1.4, the representation $\pi$ always has fixed
vectors for both $K$ and $L$ but not for $M$. Thus, $c(\pi;M,K)=c(\pi;M,L)=0$,
for trivial reasons.

######  Remark 6.

Observe that $(K,L)$ is not a compatible pair and thus Theorem 1.4 does depend
on the choice of the embeddings of the field extensions. In other words,
$c(\pi;K,L)$ depends on the field extensions themselves and is not invariant
under $F$-isomorphisms of these extensions.

######  Remark 7.

Note that Theorem 1.4 can also be stated for two ramified extensions which are
$F$-isomorphic. For instance,

$c(\pi;F(\varpi),F(\zeta^{2}\varpi))=\frac{|1+\psi(\zeta^{2})|^{2}}{2\dim\pi},$

and

$c(\pi;F(\varpi),F(\zeta^{\frac{q+1}{2}}\varpi))=\begin{cases}\frac{2}{\dim\pi}&\text{if
$\psi(\zeta^{\frac{q+1}{2}})=1$,}\\\ 0&\text{if
$\psi(\zeta^{\frac{q+1}{2}})=-1$,}\end{cases}$

where the two fields are $F$-isomorphic precisely when $q\equiv 3\mod 4$. The
sign $\psi(\zeta^{\frac{q+1}{2}})$ is related to the local root number of
$\pi$ and in fact
$\psi(\zeta^{\frac{q+1}{2}})=(-1)^{f(\pi)/2}\varepsilon(\pi)$ (cf. Section
5.4).

######  Remark 8.

It is easy to compute the dimension of an irreducible representation $\pi$ of
$D^{\times}$ from its explicit construction and this is given by

$\dim\pi=\begin{cases}2q^{\frac{c-2}{2}}&\text{if $c$ is even,}\\\
(q+1)q^{\frac{c-3}{2}}&\text{if $c$ is odd,}\end{cases}$

where $c$ is the conductor of $\pi$.

We also know how many irreducible representations are there of a given
conductor of $D^{\times}/F^{\times}$ and this is given by [Tun78, Proposition
3.5 and Remark on p. 189]:

$\mbox{Number of~{}}\pi\mbox{~{}of
conductor~{}}c=\begin{cases}\frac{q^{2}-1}{2}q^{\frac{c-4}{2}}&\text{if $c$ is
even,}\\\ 2(q-1)q^{\frac{c-3}{2}}&\text{if $c$ is odd.}\end{cases}$

It is an elementary observation to note that [AJ22, Lemma 2.1]

(1) $\sum_{\pi\in\widehat{G}}\dim\pi\cdot c(\pi;H,K)=\frac{|G||H\cap
K|}{|H||K|},$

whenever $G$ is a finite group. It is easy to verify that Theorem 1.3 and
Theorem 1.4 are compatible with (1) by considering the finite group
$G_{c}=\frac{D^{\times}}{F^{\times}(1+\mathcal{P}_{D}^{c-1})}$, and noting
that

$|G_{c}|=\begin{cases}2(q+1)q^{3j-1}&\text{if $c=2j+1$,}\\\
2(q+1)q^{3j}&\text{if $c=2j+2$},\end{cases}$

and

$|M_{c}|=(q+1)q^{\left[\frac{c-2}{2}\right]},|K_{c}|=|L_{c}|=2q^{(c-2)-\left[\frac{c-2}{2}\right]},$

and that all the three subgroups mutually intersect trivially. We refer to
Section 5.3 for a proof of this verification.

We end the introduction with a few lines about the other results contained in
this paper and the proofs of the theorems. Theorem 1.1 (resp. Theorem 1.2)
follows from Theorem 1.3 (resp. Theorem 1.4) by appealing to results on
epsilon factors due to Gérardin-Kutzko [GK80] and Fröhlich-Queyrut [FQ73], and
we do this in Section 5.4. The proofs of Theorem 1.3 and Theorem 1.4 are
rather straightforward in that we write down the explicit invariant vectors in
the model for an irreducible representation of $D^{\times}$ as an induced
representation (see Section 2). These invariant vectors are given in Corollary
3.2, Corollary 3.4, Corollary 3.6, Corollary 4.2, and Corollary 4.4. Making
use of these explicit invariant vectors, the corresponding correlation
coefficients are calculated in Section 5. In fact, our methods for writing
down the invariant vectors prove much more and we can write down equivariant
vectors for a large number of characters of the tori. These results are stated
as Proposition 3.1, Proposition 3.3, Proposition 3.5, Proposition 4.1, and
Proposition 4.3. The number of characters for which we can do so is quantified
in Remark 9, Remark 10, Remark 12, Remark 14, and Remark 15.

##  2\. Representations of $D^{\times}$

Let $D$ be the quaternion division algebra over a $p$-adic field $F$, where we
assume that $p$ is odd. Let $\mathcal{O}_{D}$ (resp. $\mathcal{O}_{F}$) be the
ring of integers of $D$ (resp. $F$) and let $\mathcal{P}_{D}$ (resp.
$\mathcal{P}_{F}$) denote the unique maximal ideal of $\mathcal{O}_{D}$ (resp.
$\mathcal{O}_{F}$). Let $\zeta$ be a primitive $(q^{2}-1)$-st root of unity in
$D$, where $q$ is the residue cardinality of $F$. We denote by $\mathbb{D}$
the set of coset representatives of $\mathcal{O}_{D}/\mathcal{P}_{D}$ given by
$\left\langle\zeta\right\rangle\cup\\{0\\}$. Note that a set of coset
representatives of $\mathcal{O}_{F}/\mathcal{P}_{F}$ is given by
$\mathbb{F}=\mathbb{D}\cap F=\\{\zeta^{i(q+1)}:0\leq i\leq q-2\\}\cup\\{0\\}$.
We fix a uniformizer $\varpi$ of $D$ such that $\varpi^{2}=\varpi_{F}$ is a
uniformizer of $F$ and $\varpi\zeta\varpi^{-1}=\zeta^{q}$. Let
$\mathrm{Tr}_{\mathbb{D}/\mathbb{F}}:\mathbb{D}\rightarrow\mathbb{F}$ be the
trace map and let
$\mathbb{T}=\\{\gamma\in\mathbb{D}\mid\mathrm{Tr}_{\mathbb{D}/\mathbb{F}}(\gamma)=0\\}$.
Note that $\mathbb{D}=\mathbb{T}\bigoplus\mathbb{F}$.

The irreducible representations of $D^{\times}$ are classified in [GG68].
These representations, in fact not just for quaternions but for tame division
algebras of any degree, can be constructed in two different ways (see [BF83,
Chapter 8] for more details); via the theory of admissible pairs due to Corwin
and Howe [CH77] and via a more elementary and explicit approach due to Koch
and Zink [KZ80]. For the purposes of this paper, it is the latter approach
which is most suitable and we summarize the constructions in [KZ80, BF83] and
here our exposition closely follows [HSY93, §1]. In the quaternionic case,
these latter constructions are also there in [GK80, §3.4, §3.5, §3.6, §4.5].

The construction of an irreducible representation of $D^{\times}$ involves a
character $\chi$ of $E^{\times}$ where $E$ is a quadratic extension of $F$.
Recall that the conductor $f(\chi)$ of $\chi$ is the smallest non-negative
integer $\nu$ such that $\rho|_{1+\mathcal{P}_{E}^{\nu}}=1$. Let
$\mathrm{Nm}_{E/F}$ and $\mathrm{Tr}_{E/F}$ be the norm and trace maps from
$E$ to $F$. A character $\chi$ of $E^{\times}$ is called minimal if
$f(\chi)\leq f(\chi\cdot\phi\circ\mathrm{Nm}_{E/F})$ for all characters $\phi$
of $F^{\times}$. We also fix an additive character $\psi_{0}$ of $F$ of
conductor $0$ and thus $\psi_{0}(\mathcal{O}_{F})=1$ but
$\psi_{0}(\mathcal{P}_{F}^{-1})\neq 1$. Define $\psi_{E}=\psi_{0}\circ{\rm
Tr}_{E/F}$ and $\psi_{D}=\psi_{0}\circ{\rm Tr}_{D/F}$, where
$\mathrm{Tr}_{D/F}$ is the reduced trace.

If $\pi$ is an irreducible representation of $D^{\times}$ then its conductor
$f(\pi)$ is the smallest non-negative integer $\nu$ such that
$\rho|_{1+\mathcal{P}_{D}^{\nu-1}}=1$. The representation $\pi$ is said to be
minimal if $f(\pi)\leq f(\pi\otimes\phi\circ\mathrm{Nm}_{D/F})$, where
$\mathrm{Nm}_{D/F}$ is the reduced norm. Suppose $\pi$ arises out of
$(E,\chi)$ and is minimal then it turns out that $f(\pi)$ is odd if and only
if $E/F$ is ramified. We say that $\pi$ is ramified (resp. unramified) when
$E/F$ is ramified (resp. unramified).

###  2.1. Ramified representations

For more details of the construction in this case, we refer to [HSY93, p.
133]. Suppose $K$ is a ramified quadratic extension of $F$ whose ring of
integers is $\mathcal{O}_{K}$ with its unique maximal ideal $\mathcal{P}_{K}$.
Let $\psi$ be a minimal character of $K^{\times}$ with even conductor. We
write $f(\psi)=m+1$ where $m$ is an odd integer. In this case, we have
$f(\psi|_{F^{\times}})\leq f(\psi)/2$.

There exists $c\in\mathcal{P}_{K}^{-m-2}$ such that $\psi(1+x)=\psi_{K}(cx)$
for $1+x\in 1+\mathcal{P}_{K}^{\frac{m+1}{2}}$. Now define the character
$\psi_{c}$ of $1+\mathcal{P}_{D}^{\frac{m+1}{2}}$ by
$\psi_{c}(1+x)=\psi_{D}(cx)$. Note that $\psi_{c}(1+\mathcal{P}_{D}^{m})\neq
1$. and $\psi_{c}(1+\mathcal{P}_{D}^{m+1})=1$. The inertial subgroup of
$\psi_{c}$ is $H=K^{\times}(1+\mathcal{P}_{D}^{\frac{m+1}{2}})$. Extend
$\psi_{c}$ to $H$ by $\psi^{\\#}(y(1+x))=\psi(y)\psi_{c}(1+x)$.

Then the representation

$\pi_{\psi}={\rm Ind}^{D^{\times}}_{H}\psi^{\\#}$

is irreducible with $f(\pi_{\psi})=m+2$ and its central character is
$\psi|_{F^{\times}}$. Note that
$\dim\pi_{\psi}=|D^{\times}/H|=(q+1)q^{\frac{m-1}{2}}$. Any minimal
irreducible representation of $D^{\times}$ of odd conductor arises this way
from one of the two ramified quadratic extensions of $F$.

###  2.2. Unramified representations

For more details of the construction in this case, we refer to [HSY93, p.
126]. Suppose $M$ is the unramified quadratic extension of $F$ whose ring of
integers is $\mathcal{O}_{M}$ with its unique maximal ideal $\mathcal{P}_{M}$.
Let $\psi$ be a minimal character of $M^{\times}$ with conductor $\geq 1$. We
write $f(\psi)=m+1$. In this case, we have $f(\psi|_{F^{\times}})\leq
f(\psi)$.

There exists
$c\in\mathcal{P}_{M}^{-m-1}-\mathcal{P}_{M}^{m+1}\cap(F+\mathcal{P}_{M}^{-m})$
such that $\psi(1+x)=\psi_{M}(cx)$ for each
$x\in\mathcal{P}_{M}^{[\frac{m+2}{2}]}$. We also have $M=F(c)$. Define the
character $\psi_{c}$ of $1+\mathcal{P}_{D}^{m+1}$ by
$\psi_{c}(1+x)=\psi_{D}(cx)$. Note that $\psi_{c}(1+\mathcal{P}_{D}^{2m})\neq
1$. and $\psi_{c}(1+\mathcal{P}_{D}^{2m+1})=1$. The inertial subgroup of
$\psi_{c}$ is $H=M^{\times}(1+\mathcal{P}_{D}^{m})$.

We define a representation $\theta_{\psi}^{\\#}$ of $H$ as follows.

  1. (1)

When $m$ is even, note that
$M^{\times}(1+\mathcal{P}_{D}^{m})=M^{\times}(1+\mathcal{P}_{D}^{m+1})$. Now
$\psi_{c}$ is extended by setting
$\theta_{\psi}^{\\#}(y(1+x))=\psi(y)\psi_{c}(1+x)$.

  2. (2)

When $m$ is odd, let
$H^{\prime\prime}=F^{\times}(1+\mathcal{P}_{M})(1+\mathcal{P}_{D}^{m+1})$ and
$H^{\prime}=F^{\times}(1+\mathcal{P}_{M})(1+\mathcal{P}_{D}^{m})$. We first
extend $\psi$ to a character $\psi^{\prime\prime}$ of $H^{\prime\prime}$ by
setting $\psi^{\prime\prime}(a(1+y)(1+x))=\psi(a)\psi(1+y)\psi_{c}(1+x)$. Now
$H^{\prime\prime}$ is a normal subgroup of $H^{\prime}$ with
$H^{\prime}/H^{\prime\prime}$ being an elementary abelian $p$-group with order
$q^{2}$. Using [BF83, Proposition 8.3.3], we have a maximal abelian subgroup
$H^{\prime}_{1}$ of $H^{\prime}$ containing $H^{\prime\prime}$ such that
extending $\psi^{\prime\prime}$ to a character $\psi^{\prime}$ of
$H^{\prime}_{1}$, there exists a uniquely determined irreducible
$q$-dimensional representation $\theta_{\psi}$ of $H^{\prime}$ defined by
$\theta_{\psi}=\mathrm{Ind}^{H^{\prime}}_{H_{1}^{\prime}}\psi^{\prime}$, and
$\theta_{\psi}|{H^{\prime\prime}}$ contains $\psi^{\prime\prime}$. Then by
[BF83, Proposition 8.4.1], this representation $\theta_{\psi}$ can be extended
to an irreducible degree $q$ representation $\theta_{\psi}^{\\#}$ of $H$
uniquely under the condition
$\mathrm{det}[\theta_{\psi}^{\\#}|_{\langle\zeta\rangle}]=\psi^{q}|_{\langle\zeta\rangle}$.

Then the representation

$\pi_{\psi}={\rm Ind}^{D^{\times}}_{H}\theta_{\psi}^{\\#}$

is irreducible with $f(\pi_{\psi})=2m+2$ and its central character is
$\psi|_{F^{\times}}$. Note that
$\dim\pi_{\psi}=|D^{\times}/H|\dim\theta_{\psi}^{\\#}=2q^{m}$. Any minimal
irreducible representation of $D^{\times}$ of even conductor arises this way
from the unramified quadratic extension of $F$.

##  3\. Construction of invariant vectors: Ramified Case

Let $K/F$ be a quadratic ramified extension and let
$H=K^{\times}(1+\mathcal{P}_{D}^{\frac{m+1}{2}})$, where $m$ is an odd
positive integer. Let

$\mathcal{S}=\left\\{(\alpha_{1},\alpha_{2},\dots,\alpha_{\frac{m-1}{2}})\mid\alpha_{r}\in\mathbb{F},1\leq
r\leq\frac{m-1}{2}\right\\}.$

In what follows, by $S^{\prime}$ (resp. $S^{\prime\prime}$) we denote the
subset of $\mathcal{S}$ consisting of those tuples
$\underline{\alpha}=(\alpha_{1},\alpha_{2},\dots,\alpha_{\frac{m-1}{2}})$
where $\alpha_{r}=0$ when $r$ is odd (resp. even).

###  3.1. Restriction to the unramified torus

Let $M$ be the unramified quadratic extension over $F$. Then $M$ is generated
by $\zeta$ over $F$. For each integer $i$ with $0\leq i\leq q$ and
$\underline{\alpha}=(\alpha_{1},\alpha_{2},...,\alpha_{\frac{m-1}{2}})\in\mathcal{S}$,
let

$d_{\underline{\alpha},i}=(1+\sum\limits_{r=1}^{\frac{m-1}{2}}\alpha_{r}\zeta^{\frac{q+1}{2}}\varpi^{r})\cdot\zeta^{i}.$

Then the set
$\\{d_{\underline{\alpha},i}\mid\underline{\alpha}\in\mathcal{S},0\leq i\leq
q\\}$ forms a set of coset representatives for $D^{\times}/H$; i.e.,

$D^{\times}=\bigsqcup_{\begin{subarray}{c}\underline{\alpha}\in\mathcal{S}\\\
0\leq i\leq q\end{subarray}}H\cdot d_{\underline{\alpha},i}.$

For each pair $(\underline{\alpha},i)$ with $\underline{\alpha}\in\mathcal{S}$
and $0\leq i\leq q$, we define a complex valued function on $D^{\times}$ by

$f_{\underline{\alpha},i}(h\cdot
d_{\underline{\beta},j})=\begin{cases}\psi^{\\#}(h)&\text{if
$\underline{\beta}=\underline{\alpha}$ and $j=i$,}\\\
0&\text{otherwise,}\end{cases}$

for $h\in H$. Note that

$\\{f_{\underline{\alpha},i}\mid\underline{\alpha}\in\mathcal{S},0\leq i\leq
q\\}$

forms a basis for the irreducible representation
$\pi_{\psi}=\mathrm{Ind}^{G}_{H}\psi^{\\#}$.

######  Proposition 3.1.

Let $\mu:M^{\times}\rightarrow\mathbb{C}^{\times}$ be a smooth character of
$M^{\times}$ with $\mu|_{H\cap M^{\times}}=\psi^{\\#}|_{H\cap M^{\times}}$.
Then $\mu$ occurs with multiplicity one in the restriction of $\pi_{\psi}$ to
$M^{\times}$ and the vector

$v_{\mu}=\sum\limits_{\begin{subarray}{c}\underline{\alpha}\in S^{\prime}\\\
0\leq i\leq
q\end{subarray}}\mu(d_{\underline{\alpha},i})f_{\underline{\alpha},i}$

lies in the $\mu$-isotypic component of the representation $\pi_{\psi}$.

###### Proof.

The first part of the assertion is just elementary Mackey theory. We show that
$v_{\mu}$ is a $\mu$-equivariant vector. To this end, fix $x=h\cdot
d_{\underline{\delta},i_{0}}\in D^{\times}$, where $h\in H$,
$\underline{\delta}=(\delta_{1},\dots,\delta_{\frac{m-1}{2}})\in\mathcal{S}$,
and $i_{0}\in\\{0,1,\dots,q\\}$. Clearly,

$v_{\mu}(x)=\sum\limits_{\begin{subarray}{c}\underline{\alpha}\in
S^{\prime}\\\ 0\leq i\leq
q\end{subarray}}\mu(d_{\underline{\alpha},i})f_{\underline{\alpha},i}(h\cdot
d_{\underline{\delta},i_{0}})=\begin{cases}\psi^{\\#}(h)\mu(d_{\underline{\delta},i_{0}})&\text{if
$\underline{\delta}\in S^{\prime}$,}\\\ 0&\text{otherwise.}\end{cases}$

If $z\in M^{\times}$ then we write

$z=a\cdot\zeta^{j_{0}}\cdot(1+\sum\limits_{\begin{subarray}{c}r\geq 2\\\
r\text{ is
even}\end{subarray}}\beta_{r}\cdot\zeta^{\frac{q+1}{2}}\cdot\varpi^{r})$

for some $a\in F^{\times}$, $j_{0}\in\\{0,1,\dots,q\\}$, and
$\beta_{r}\in\mathbb{F}$ for all $r$. Then,

$\displaystyle\pi_{\psi}(z)v_{\mu}(x)$ $\displaystyle=v_{\mu}(xz)$
$\displaystyle=v_{\mu}\left(ah\left(1+\sum\limits_{r=1}^{\frac{m-1}{2}}\delta_{r}\zeta^{\frac{q+1}{2}}\varpi^{r}\right)\cdot\left(1+\sum\limits_{\begin{subarray}{c}r\geq
2\\\ r\text{ is
even}\end{subarray}}\beta_{r}\cdot\zeta^{\frac{q+1}{2}}\cdot\varpi^{r}\right)\zeta^{i_{0}+j_{0}}\right).$

Observe that the element

$\textbf{y}=\left(1+\sum\limits_{r=1}^{\frac{m-1}{2}}\delta_{r}\zeta^{\frac{q+1}{2}}\varpi^{r}\right)\cdot\left(1+\sum\limits_{\begin{subarray}{c}r\geq
2\\\ r\text{ is
even}\end{subarray}}\beta_{r}\cdot\zeta^{\frac{q+1}{2}}\cdot\varpi^{r}\right)\zeta^{i_{0}+j_{0}}$

lies in the support of $v_{\mu}$ if and only if $\delta_{r}=0$ for all odd
integers $r\in\\{1,\dots,\frac{m-1}{2}\\}$. Thus, $v_{\mu}(\textbf{y})=0$ if
$\underline{\delta}\notin S^{\prime}$.

Now assume that $\underline{\delta}\in S^{\prime}$. Let $m^{\prime}$ be the
largest even integer in $\\{1,2,\dots,\frac{m-1}{2}\\}$. Then, the element

$\textbf{y}=\left(1+\sum\limits_{\begin{subarray}{c}r=2\\\ r\text{ is
even}\end{subarray}}^{m^{\prime}}\delta_{r}\zeta^{\frac{q+1}{2}}\varpi^{r}\right)\cdot\left(1+\sum\limits_{\begin{subarray}{c}r\geq
2\\\ r\text{ is
even}\end{subarray}}\beta_{r}\cdot\zeta^{\frac{q+1}{2}}\cdot\varpi^{r}\right)\zeta^{i_{0}+j_{0}}$

can be written as

$\textbf{y}=b_{0}\cdot h_{0}\cdot d_{\underline{\delta}^{\prime},i_{0}+j_{0}}$

where

$b_{0}=1+\sum\limits_{\begin{subarray}{c}r=4\\\ r\text{ is
even}\end{subarray}}^{m^{\prime}}\xi_{r}\cdot\varpi^{r}$

with

$\xi_{r}=\left(\sum\limits_{\begin{subarray}{c}t=2\\\ t\text{ is
even}\end{subarray}}^{r-2}\delta_{t}\beta_{r-t}\right)\zeta^{q+1}$

for $4\leq r\leq m^{\prime}$, and

$\underline{\delta}^{\prime}=(\delta_{1}^{\prime},...,\delta_{\frac{m-1}{2}}^{\prime})\in
S^{\prime}$

with

$\delta_{r}^{\prime}=\delta_{r}+\beta_{r}-\sum\limits_{\begin{subarray}{c}t=2\\\
t\text{ is even}\end{subarray}}^{r-2}\delta_{t}^{\prime}\xi_{r-t}$

for even $r\leq m^{\prime}$, and

$h_{0}=1+\sum\limits_{\begin{subarray}{c}r=m^{\prime}+2\\\ r\text{ is
even}\end{subarray}}^{m-1}(\gamma_{r}\zeta^{\frac{q+1}{2}}+\xi_{r})\varpi^{r}+\cdots$

with

$\gamma_{r}=\beta_{r}-\sum\limits_{\begin{subarray}{c}t=2\\\ t\text{ is
even}\end{subarray}}^{m^{\prime}}\delta_{t}^{\prime}\xi_{r-t}-\sum\limits_{\begin{subarray}{c}t=m^{\prime}+2\\\
t\text{ is even}\end{subarray}}^{r-2}\gamma_{t}\xi_{r-t}\in\mathbb{F},$

for $r\geq m^{\prime}+2$, and

$\xi_{r}=\left(\sum\limits_{\begin{subarray}{c}t=2\\\ t\text{ is
even}\end{subarray}}^{m^{\prime}}\delta_{t}\beta_{r-t}-\sum\limits_{\begin{subarray}{c}t=m^{\prime}+2\\\
t\text{ is
even}\end{subarray}}^{r-2}\delta_{r-t}^{\prime}\gamma_{t}\right)\zeta^{q+1}-\sum\limits_{\begin{subarray}{c}t=m^{\prime}+2\\\
t\text{ is even}\end{subarray}}^{r-2}\xi_{t}\xi_{r-t}$

for $m^{\prime}+2\leq r\leq m-1$.

Observe that $b_{0}\in F^{\times}$ and $h_{0}\in H\cap M^{\times}$. By our
assumption on $\mu$, we have $\psi^{\\#}(h_{0})=\mu(h_{0})$. Recall our
assumption that $\underline{\delta}\in\mathcal{S}^{\prime}$. Now,

$\displaystyle\pi_{\psi}(z)v_{\mu}(x)$ $\displaystyle=v_{\mu}\left(ah\cdot
b_{0}\cdot h_{0}\cdot d_{\underline{\delta^{\prime}},i_{0}+j_{0}}\right)$
$\displaystyle=\psi^{\\#}(h)\psi(a\cdot
b_{0})\psi^{\\#}(h_{0})\mu(d_{\underline{\delta^{\prime}},i_{0}+j_{0}})$
$\displaystyle=\psi^{\\#}(h)\mu(a\cdot b_{0}\cdot
h_{0})\mu(d_{\underline{\delta^{\prime}},i_{0}+j_{0}})$
$\displaystyle=\psi^{\\#}(h)\mu(z\cdot d_{\underline{\delta},i_{0}})$
$\displaystyle=\mu(z)v_{\mu}(x).$

Note that we have used the identity $a\cdot b_{0}\cdot h_{0}\cdot
d_{\underline{\delta^{\prime}},i_{0}+j_{0}}=d_{\underline{\delta},i_{0}}\cdot
z$ in the last but one step. Thus,

$\pi_{\psi}(z)v_{\mu}=\mu(z)v_{\mu}$

for $z\in M^{\times}$. ∎

######  Remark 9.

Observe that

$H\cap M^{\times}=F^{\times}(1+\mathcal{P}_{M}^{[\frac{m-1}{4}]+1}).$

and thus we are able to exhibit $\mu$-equivariant vectors in $\pi_{\psi}$ for
$(q+1)q^{[\frac{m-1}{4}]}$ many characters $\mu$.

If $\pi_{\psi}$ has trivial central character then the trivial character of
$M^{\times}$ always occurs in the restriction $\pi_{\psi}$ to $M^{\times}$.
Putting $\mu=1$ in the above proposition, we get

######  Corollary 3.2.

If $\pi_{\psi}$ has trivial central character, an $M^{\times}$-fixed vector in
$\pi_{\psi}$ is given by

$v_{M}=\sum\limits_{\begin{subarray}{c}\underline{\alpha}\in S^{\prime}\\\
0\leq i\leq q\end{subarray}}f_{\underline{\alpha},i}.$

###  3.2. Restricted to ramified tori

Let $K$ and $L$ be the two ramified extensions of $F$. The representation
$\pi_{\psi}$ is associated to the extension $K$. We need to consider its
restriction to $K$ and to $L$. We take $K=F(\varpi)$ and $L=F(\zeta\varpi)$.

####  3.2.1. Restriction to $K^{\times}$

It is easy to observe that the character $\psi$ occurs with multiplicity one
in the restriction $\pi_{\psi}$ to $K^{\times}$ and a $\psi$-fixed vector is
given by $f_{\underline{0},0}$. Similarly, when $g=\zeta^{\frac{q+1}{2}}\in
D^{\times}$, the character $\psi^{g}$ occurs with multiplicity one in the
restriction of $\pi_{\psi}$ and a $\psi^{g}$-fixed vector is given by
$f_{\underline{0},\frac{q+1}{2}}$. In the proposition below, we exhibit many
more equivariant vectors.

For each integer $i$ with $0\leq i\leq q$ and
$\underline{\alpha}=(\alpha_{1},\alpha_{2},\dots,\alpha_{\frac{m-1}{2}})\in\mathcal{S}$,
we define

$d_{\underline{\alpha},i}^{\prime}=(1+\sum\limits_{r=1}^{\frac{m-1}{2}}\alpha_{r}\zeta^{i(1-q)}\varpi^{r})\cdot\zeta^{i}.$

For each pair $(\underline{\alpha},i)$ with $\underline{\alpha}\in\mathcal{S}$
and $0\leq i\leq q$, we define the corresponding functions on $D^{\times}$ by

$f^{\prime}_{\underline{\alpha},i}(h\cdot
d^{\prime}_{\underline{\beta},j})=\begin{cases}\psi^{\\#}(h)&\text{if
$\underline{\beta}=\underline{\alpha}$ and $j=i$,}\\\
0&\text{otherwise,}\end{cases}$

for $h\in H$.

######  Proposition 3.3.

For $1\leq i\leq\frac{q-1}{2}$, pick the element $g=\zeta^{i}\in D^{\times}$
and consider a smooth character $\kappa_{g}$ of $K^{\times}$ such that on
$\kappa_{g}|_{g^{-1}Hg\cap K^{\times}}=\psi^{{\\#}^{g}}$. Then the character
$\kappa_{g}$ occurs with multiplicity one in the restriction of $\pi_{\psi}$
to $K^{\times}$ and $\kappa_{g}|_{F^{\times}}=\psi|_{F^{\times}}$. Let
$\ell\in\\{0,1\\}$ be such that $\kappa_{g}(\varpi)=(-1)^{\ell}\psi(\varpi)$.
The vector

$v_{\kappa_{g}}=\sum\limits_{\underline{\alpha}\in
S^{\prime\prime}}\kappa_{g}(1+\sum\limits_{r}\alpha_{r}\varpi^{r})\left[f_{\underline{\alpha},i}^{\prime}+(-1)^{\ell}\psi(\zeta^{(1-i)(q+1)})f_{\underline{\alpha},q+1-i}^{\prime}\right]$

lies in the $\kappa_{g}$-isotypic component of the representation
$\pi_{\psi}$.

###### Proof.

The first assertion is straightforward Mackey theory. We now show that
$v_{\kappa_{g}}$ is a $\kappa_{g}$-invariant vector. To this end, observe that
$v_{\kappa_{g}}(x)=0$ unless fix $x=h\cdot d_{\underline{\delta},i}^{\prime}$
or $x=h\cdot d_{\underline{\delta},{q+1-i}}^{\prime}$ for $h\in H$,
$\underline{\delta}=(\delta_{1},\dots,\delta_{\frac{m-1}{2}})\in\mathcal{S}^{\prime\prime}$.
Clearly,

$v_{\kappa_{g}}(x)=\begin{cases}\psi^{\\#}(h)\cdot\kappa_{g}(1+\sum\limits_{r}\delta_{r}\varpi^{r})&\text{if
$x=h\cdot d_{\underline{\delta},i}^{\prime}$,}\\\
(-1)^{\ell}\psi(\zeta^{(1-i)(q+1)})\psi^{\\#}(h)\cdot\kappa_{g}(1+\sum\limits_{r}\delta_{r}\varpi^{r})&\text{if
$x=h\cdot d_{\underline{\delta},q+1-i}^{\prime}$,}\\\
0&\text{otherwise.}\end{cases}$

We write $z\in K^{\times}$ as

$z=a\cdot\varpi^{j_{0}}\cdot(1+\sum\limits_{\begin{subarray}{c}j\geq 1\\\
j\text{ is odd}\end{subarray}}\beta_{j}\cdot\varpi^{j})$

for some $a\in F^{\times}$, $j_{0}\in\\{0,1\\}$, and $\beta_{j}\in\mathbb{F}$
for all $r$. Clearly, when $a\in F^{\times}\subset K^{\times}$, we have

$\pi_{\psi}(a)v_{\kappa_{g}}(x)=v_{\kappa_{g}}(xa)=\psi(a)\cdot
v_{\kappa_{g}}(x)=\kappa_{g}(a)v_{\kappa_{g}}(x).$

Now we check the action of the uniformizer $\varpi\in K^{\times}$ and elements
of the form

$z^{\prime}=1+\sum\limits_{\begin{subarray}{c}j\geq 1\\\ j\text{ is
odd}\end{subarray}}\beta_{j}\cdot\varpi^{j}$

with $\beta_{r}\in\mathbb{F}$ for all $r$. Without loss of generality we may
take $x=h\cdot d_{\underline{\delta},i}^{\prime}$, as the proof in the case
$x=h\cdot d_{\underline{\delta},q+1-i}^{\prime}$ being similar. Then,

$\displaystyle\pi_{\psi}(\varpi)v_{\kappa_{g}}(x)$
$\displaystyle=v_{\kappa_{g}}(h\cdot
d_{\underline{\delta},i}^{\prime}\cdot\varpi)$
$\displaystyle=v_{\kappa_{g}}(h\cdot\varpi\cdot\zeta^{(i-1)(q+1)}\cdot
d_{\underline{\delta},q+1-i}^{\prime})$
$\displaystyle=\psi^{\\#}(h)\cdot\psi(\varpi)\cdot\psi(\zeta^{(i-1)(q+1)})\cdot\kappa_{g}(1+\sum\limits_{r}\delta_{r}\varpi^{r})(-1)^{\ell}\psi(\zeta^{(1-i)(q+1)})$
$\displaystyle=(-1)^{\ell}\psi(\varpi)\cdot\psi^{\\#}(h)\kappa_{g}(1+\sum\limits_{r}\delta_{r}\varpi^{r})$
$\displaystyle=\kappa_{g}(\varpi)\cdot v_{\kappa_{g}}(x).$

Now we check the action of $z^{\prime}$. We have

$\displaystyle\pi_{\psi}(z^{\prime})v_{\kappa_{g}}(x)$
$\displaystyle=v_{\kappa_{g}}(h\cdot
d_{\underline{\delta},i}^{\prime}\cdot(1+\sum\limits_{\begin{subarray}{c}j\geq
1\\\ j\text{ is odd}\end{subarray}}\beta_{j}\cdot\varpi^{j}))$
$\displaystyle=v_{\kappa_{g}}\left(h(1+\sum\limits_{\begin{subarray}{c}r=1\\\
r\text{ is
odd}\end{subarray}}^{\frac{m-1}{2}}\delta_{r}\zeta^{i(1-q)}\varpi^{r})\cdot(1+\sum\limits_{\begin{subarray}{c}j\geq
1\\\ j\text{ is
odd}\end{subarray}}\beta_{j}\zeta^{i(1-q)}\varpi^{j})\zeta^{i}\right).$

Let $m^{*}$ be the largest odd integer in $\\{1,2,...,\frac{m-1}{2}\\}$. Then
the element

$\textbf{y}=(1+\sum\limits_{\begin{subarray}{c}r=1\\\ r\text{ is
odd}\end{subarray}}^{m^{*}}\delta_{r}\zeta^{i(1-q)}\varpi^{r})(1+\sum\limits_{\begin{subarray}{c}j\geq
1\\\ j\text{ is
odd}\end{subarray}}\beta_{j}\zeta^{i(1-q)}\varpi^{j})\zeta^{i}$

can be written as

$\textbf{y }=b_{0}\cdot h_{0}\cdot d^{\prime}_{\underline{\delta}^{\prime},i}$

where,

$b_{0}=1+\sum\limits_{\begin{subarray}{c}r=2\\\ r\text{ is
even}\end{subarray}}^{m^{*}+1}\xi_{r}\varpi^{r}$

with

$\xi_{r}=\sum\limits_{\begin{subarray}{c}t=1\\\ t\text{ is
odd}\end{subarray}}^{r-1}\delta_{r-t}\beta_{t}$

for each even $r\leq m^{*}+1$, and

$\underline{\delta}^{\prime}=(\delta_{1}^{\prime},...,\delta_{\frac{m-1}{2}}^{\prime})\in
S^{\prime\prime}$

with

$\delta_{r}^{\prime}=\delta_{r}+\beta_{r}-\sum\limits_{\begin{subarray}{c}t=1\\\
t\text{ is odd}\end{subarray}}^{r-2}\delta_{t}^{\prime}\xi_{r-t},$

for each odd $r\leq m^{*}$, and

$h_{0}=1+\sum\limits_{\begin{subarray}{c}j=m^{*}+2\\\ j\text{ is
odd}\end{subarray}}^{m}\epsilon_{j}\zeta^{i(1-q)}\varpi^{j}+\sum\limits_{\begin{subarray}{c}k=m^{*}+3\\\
k\text{ is even}\end{subarray}}^{m-1}\xi_{k}\varpi^{k}+\cdots$

such that

$\epsilon_{j}=\beta_{j}-\sum\limits_{\begin{subarray}{c}t=1\\\ t\text{ is
odd}\end{subarray}}^{m^{*}}\delta_{t}^{\prime}\xi_{j-t}-\sum\limits_{\begin{subarray}{c}t=m^{*}+2\\\
t\text{ is odd}\end{subarray}}^{j-2}\epsilon_{t}\xi_{j-t},$

for odd $j\geq\frac{m+1}{2}$, and

$\xi_{k}=\sum\limits_{\begin{subarray}{c}t=1\\\ t\text{ is odd
}\end{subarray}}^{m^{*}}\delta_{t}\beta_{k-t}-\sum\limits_{\begin{subarray}{c}t=m^{*}+3\\\
t\text{ is
even}\end{subarray}}^{k-2}\xi_{t}\xi_{k-t}-\sum\limits_{\begin{subarray}{c}t=m^{*}+2\\\
t\text{ is odd}\end{subarray}}^{k-1}\delta_{k-t}^{\prime}\epsilon_{t},$

for even $k\geq\frac{m+1}{2}$. Denote

$\tilde{h}=g^{-1}h_{0}g=1+\sum\limits_{\begin{subarray}{c}j=m^{*}+2\\\ j\text{
is
odd}\end{subarray}}^{m}\epsilon_{j}\varpi^{j}+\sum\limits_{\begin{subarray}{c}k=m^{*}+3\\\
k\text{ is even}\end{subarray}}^{m-1}\xi_{k}\varpi^{k}+\cdots.$

Note that $\tilde{h}\in g^{-1}Hg\cap K^{\times}$. We have

$\displaystyle\pi_{\psi}(z^{\prime})v_{\kappa_{g}}(x)$
$\displaystyle=v_{\kappa_{g}}(h\cdot b_{0}\cdot h_{0}\cdot
d_{\underline{\delta}^{\prime},i}^{\prime})$
$\displaystyle=\psi^{\\#}(h)\cdot\psi(b_{0})\cdot\psi^{\\#^{g}}(\tilde{h})\cdot\kappa_{g}(1+\sum\limits_{\begin{subarray}{c}r=1\\\
r\text{ is odd}\end{subarray}}^{m^{*}}\delta_{r}^{\prime}\varpi^{r}),$
$\displaystyle=\psi^{\\#}(h)\cdot\kappa_{g}\left(b_{0}\cdot\tilde{h}\cdot(1+\sum\limits_{\begin{subarray}{c}r=1\\\
r\text{ is odd}\end{subarray}}^{m^{*}}\delta_{r}^{\prime}\varpi^{r})\right).$

Observe that

$\displaystyle z^{\prime}\cdot(1+\sum\limits_{\begin{subarray}{c}r=1\\\
r\text{ is odd}\end{subarray}}^{m^{*}}\delta_{r}\varpi^{r})$
$\displaystyle=(1+\sum\limits_{\begin{subarray}{c}j\geq 1\\\ j\text{ is
odd}\end{subarray}}\beta_{j}\varpi^{j})\cdot(1+\sum\limits_{\begin{subarray}{c}r=1\\\
r\text{ is odd}\end{subarray}}^{m^{*}}\delta_{r}\varpi^{r})$
$\displaystyle=b_{0}\cdot(1+\sum\limits_{\begin{subarray}{c}j=m^{*}+2\\\
j\text{ is
odd}\end{subarray}}^{m}\epsilon_{j}\varpi^{j}+\sum\limits_{\begin{subarray}{c}k=m^{*}+3\\\
k\text{ is even}\end{subarray}}^{m-1}\xi_{k}\varpi^{k}+\sum\limits_{t\geq
m+1}\gamma_{t}\varpi^{t})\cdot(1+\sum\limits_{\begin{subarray}{c}r=1\\\
r\text{ is odd}\end{subarray}}^{m^{*}}\delta_{r}^{\prime}\varpi^{r})$

for some $\gamma_{t}\in\mathbb{F}$ for $t\geq m+1$. Since
$\kappa_{g}|_{(1+\mathcal{P}_{K}^{m+1})}=1$, we have

$\displaystyle\kappa_{g}\left(b_{0}\cdot\tilde{h}\cdot(1+\sum\limits_{\begin{subarray}{c}r=1\\\
r\text{ is odd}\end{subarray}}^{m^{*}}\delta_{r}^{\prime}\varpi^{r})\right)$
$\displaystyle=\kappa_{g}\left(b_{0}\cdot(1+\sum\limits_{\begin{subarray}{c}j=m^{*}+2\\\
j\text{ is
odd}\end{subarray}}^{m}\epsilon_{j}\varpi^{j}+\sum\limits_{\begin{subarray}{c}k=m^{*}+3\\\
k\text{ is
even}\end{subarray}}^{m-1}\xi_{k}\varpi^{k})\cdot(1+\sum\limits_{\begin{subarray}{c}r=1\\\
r\text{ is odd}\end{subarray}}^{m^{*}}\delta_{r}^{\prime}\varpi^{r})\right)$
$\displaystyle=\kappa_{g}\left(z^{\prime}\cdot(1+\sum\limits_{\begin{subarray}{c}r=1\\\
r\text{ is
odd}\end{subarray}}^{m^{*}}\delta_{r}\varpi^{r})\right)=\kappa_{g}(z^{\prime})\kappa_{g}(1+\sum\limits_{r}\delta_{r}\varpi^{r}).$

Therefore,

$\displaystyle\pi_{\psi}(z^{\prime})v_{\kappa_{g}}(x)$
$\displaystyle=\kappa_{g}(z^{\prime})\cdot\psi^{\\#}(h)\kappa_{g}(1+\sum\limits_{r}\delta_{r}\varpi^{r})$
$\displaystyle=\kappa_{g}(z^{\prime})\cdot v_{\kappa_{g}}(x).$

Hence, $\pi_{\psi}(z)\cdot v_{\kappa_{g}}=\kappa_{g}(z)\cdot v_{\kappa_{g}}$
for all $z\in K^{\times}$. ∎

######  Remark 10.

For each $g=\zeta^{i}$, where $1\leq i\leq\frac{q-1}{2}$, the number of
characters $\kappa_{g}:K^{\times}\rightarrow\mathbb{C}$ satisfying
$\kappa_{g}|_{g^{-1}Hg\cap K^{\times}}=\psi^{{\\#}^{g}}$ is

$\left|\frac{K^{\times}}{F^{\times}(1+\mathcal{P}_{K}^{\frac{m+1}{2}})}\right|=2q^{\frac{m-1}{2}[\frac{m-1}{4}]}.$

Thus, in Proposition 3.3, we have exhibited equivariant vectors in
$\pi_{\psi}$ for

$\frac{q-1}{2}\times
2q^{\frac{m-1}{2}-\left[\frac{m-1}{4}\right]}=(q-1)q^{\frac{m-1}{2}-\left[\frac{m-1}{4}\right]}$

many characters $\kappa_{g}$.

If $q\equiv 3\mod 4$ and $\psi|_{F^{\times}}=1$ then the trivial character
$\textbf{1}_{K^{\times}}$ of $K^{\times}$ occurs in the restriction
$\pi_{\psi}$ to $K^{\times}$. We have the following result as a corollary to
the above proposition.

######  Corollary 3.4.

Suppose $q\equiv 3\mod 4$ and write $q+1=4t_{0}$. Assume that $\pi_{\psi}$ has
trivial central character. Then

$v_{K}=\sum\limits_{\underline{\alpha}\in
S^{\prime\prime}}\left[f_{\underline{\alpha},t_{0}}+\psi(\varpi)f_{\underline{\alpha},3t_{0}}\right]$

is a $K^{\times}$-fixed vector in $\pi_{\psi}$.

###### Proof.

If $\pi_{\psi}$ has trivial central character then $\psi(\varpi)=\pm 1$. Since
$q+1=4t_{0}$, we have

$\zeta^{t_{0}(1-q)}=\eta\cdot\zeta^{\frac{q+1}{2}},$

where $\eta=-\zeta^{(t_{0}-1)(q+1)}\in\mathbb{F}$. Note that
$\zeta^{-t_{0}(1-q)}=-\eta\cdot\zeta^{\frac{q+1}{2}}$. Therefore, for each
$\underline{\alpha}=(\alpha_{1},\alpha_{2},\dots,\alpha_{\frac{m-1}{2}})\in\mathcal{S}^{\prime\prime}$,
we have

$f_{\underline{\alpha},t_{0}}^{\prime}=f_{\underline{\alpha\eta},t_{0}},$

where
$\underline{\alpha\eta}=(\alpha_{1}\eta,\alpha_{2}\eta,\dots,\alpha_{\frac{m-1}{2}}\eta)\in\mathcal{S}^{\prime\prime}$,
and

$f_{\underline{\alpha},q+1-t_{0}}^{\prime}=f_{-\underline{\alpha\eta},3t_{0}},$

where
$-\underline{\alpha\eta}=(-\alpha_{1}\eta,-\alpha_{2}\eta,\dots,-\alpha_{\frac{m-1}{2}}\eta)\in\mathcal{S}^{\prime\prime}$.
When $g=\zeta^{t_{0}}$ and $\psi|_{F^{\times}}=1$, we have
$\kappa_{g}|_{g^{-1}Hg\cap K^{\times}}=\psi^{\\#^{g}}=1$ because $H\cap
gK^{\times}g^{-1}\subset\mathrm{Ker}(\psi^{\\#})$. Now with $g=\zeta^{t_{0}}$
and $\kappa_{g}=1$ in Proposition 3.3, we get

$\displaystyle v_{K}$ $\displaystyle=\sum\limits_{\underline{\alpha}\in
S^{\prime\prime}}\kappa_{g}(1+\sum\limits_{r}\alpha_{r}\varpi^{r})\left[f_{\underline{\alpha},t_{0}}^{\prime}+(-1)^{\ell}\psi(\zeta^{(1-t_{0})(q+1)})f_{\underline{\alpha},q+1-t_{0}}^{\prime}\right]$
$\displaystyle=\sum\limits_{\underline{\alpha}\in
S^{\prime\prime}}\left[f_{\underline{\alpha},t_{0}}^{\prime}+\psi(\varpi)f_{\underline{\alpha},q+1-t_{0}}^{\prime}\right]$
$\displaystyle=\sum\limits_{\underline{\alpha}\in
S^{\prime\prime}}\left[f_{\underline{\alpha\eta},t_{0}}+\psi(\varpi)f_{-\underline{\alpha\eta},3t_{0}}\right]$
$\displaystyle=\sum\limits_{\underline{\alpha}\in
S^{\prime\prime}}\left[f_{\underline{\alpha},t_{0}}+\psi(\varpi)f_{\underline{\alpha},3t_{0}}\right].$

∎

######  Remark 11.

We can state Corollary 3.4 slightly more generally as follows. Let $s$ be an
integer such that $-\frac{q-3}{4}\leq s\leq\frac{q+1}{4}$ and let
$K_{s}=F(\zeta^{2s}\varpi)$. Then, the vector

$v_{K_{s}}=\sum\limits_{\underline{\alpha}\in
S^{\prime\prime}}[f_{\underline{\alpha},t_{0}-s}+\psi(\varpi)f_{\underline{\alpha},3t_{0}-s}]$

is a $K_{s}^{\times}$-fixed vector in $\pi_{\psi}$.

####  3.2.2. Restriction to $L^{\times}$

Let $L$ be the ramified quadratic field extension over $F$ which is not
conjugate to $K=F(\varpi)$. We take $L=F(\zeta\varpi)$.

For each integer $i$ with $0\leq i\leq q$ and
$\underline{\alpha}=(\alpha_{1},\alpha_{2},\dots\alpha_{\frac{m-1}{2}})\in\mathcal{S}$,
we define

$d_{\underline{\alpha},i}^{\prime\prime}=(1+\sum\limits_{r=1}^{\frac{m-1}{2}}\alpha_{r}\zeta^{1+i(1-q)}\varpi^{r})\cdot\zeta^{i}.$

For each pair $(\underline{\alpha},i)$ with $\underline{\alpha}\in\mathcal{S}$
and $0\leq i\leq q$, we define the corresponding functions on $D^{\times}$ by

$f^{\prime\prime}_{\underline{\alpha},i}(h\cdot
d^{\prime\prime}_{\underline{\beta},j})=\begin{cases}\psi^{\\#}(h)&\text{if
$\underline{\beta}=\underline{\alpha}$ and $j=i$,}\\\
0&\text{otherwise.}\end{cases}$

Note that these functions lie in the induced space of
$\pi_{\psi}=\mathrm{Ind}^{G}_{H}\psi^{\\#}$.

######  Proposition 3.5.

For $1\leq i\leq\frac{q+1}{2}$, pick the element $g=\zeta^{i}\in D^{\times}$
and consider a smooth character $\lambda_{g}$ of $L^{\times}$ such that on
$\lambda_{g}|_{g^{-1}Hg\cap L^{\times}}=\psi^{{\\#}^{g}}$. Then the character
$\lambda_{g}$ occurs with multiplicity one in the restriction of $\pi_{\psi}$
to $L^{\times}$ and $\lambda_{g}|_{F^{\times}}=\psi|_{F^{\times}}$. Let
$\ell\in\\{0,1\\}$ be such that and
$\lambda_{g}(\varpi_{L})=\lambda_{g}(\zeta\varpi)=(-1)^{\ell}\sqrt{\psi(\zeta^{q+1})}\psi(\varpi)$.
The vector

$v_{\lambda_{g}}=\sum\limits_{\underline{\alpha}\in
S^{\prime\prime}}\lambda_{g}(1+\sum\limits_{r}\alpha_{r}\zeta\varpi^{r})\left[f_{\underline{\alpha},i}^{\prime\prime}+(-1)^{\ell}\psi(\zeta^{-i(q+1)})\sqrt{\psi(\zeta^{q+1})}f_{\underline{\alpha},q-i}^{\prime\prime}\right]$

lies in the $\lambda_{g}$-isotypic component of the representation
$\pi_{\psi}$.

###### Proof.

The first assertion is straightforward Mackey theory. To show that
$v_{\lambda_{g}}$ is a $\lambda_{g}$-invariant vector, we write each element
of $L^{\times}$ in the form

$z=a\cdot(\zeta\varpi)^{j_{0}}\cdot(1+\sum\limits_{\begin{subarray}{c}j\geq
1\\\ j\text{ is odd}\end{subarray}}\beta_{j}\cdot\zeta\varpi^{j})$

for some $a\in F^{\times}$, $j_{0}\in\\{0,1\\}$, and $\beta_{j}\in\mathbb{F}$
for all $j$. Then the proof proceeds exactly along the lines of the proof of
Proposition 3.3. ∎

######  Remark 12.

For each $g=\zeta^{i}$, where $1\leq i\leq\frac{q+1}{2}$, the number of
characters $\lambda_{g}:L^{\times}\rightarrow\mathbb{C}$ satisfying
$\lambda_{g}|_{g^{-1}Hg\cap L^{\times}}=\psi^{{\\#}^{g}}$ is

$\left|\frac{L^{\times}}{F^{\times}(1+\mathcal{P}_{L}^{\frac{m+1}{2}})}\right|=2q^{\frac{m-1}{2}-\left[\frac{m-1}{4}\right]}.$

Thus, in Proposition 3.5, we have exhibited equivariant vectors for

$\frac{q+1}{2}\times
2q^{\frac{m-1}{2}-\left[\frac{m-1}{4}\right]}=(q+1)q^{\frac{m-1}{2}-\left[\frac{m-1}{4}\right]}$

many characters $\lambda_{g}$.

If $q\equiv 1\mod 4$ and $\psi|_{F^{\times}}=1$ then the trivial character
$\textbf{1}_{L^{\times}}$ of $L^{\times}$ occurs in the restriction
$\pi_{\psi}$ to $L^{\times}$. We have the following result as a corollary to
the above proposition.

######  Corollary 3.6.

Suppose $q\equiv 1\mod 4$ and write $q-1=4r_{0}$. Assume that $\pi_{\psi}$ has
trivial central character. Then

$v_{L}=\sum\limits_{\underline{\alpha}\in
S^{\prime\prime}}\left[f_{\underline{\alpha},r_{0}}+\psi(\varpi)f_{\underline{\alpha},3r_{0}+1}\right]$

is an $L^{\times}$-fixed vector in $\pi_{\psi}$.

###### Proof.

If $q-1=4r_{0}$ then

$\zeta^{1+r_{0}(1-q)}=\eta^{\prime}\cdot\zeta^{\frac{q+1}{2}},$

where $\eta^{\prime}=\zeta^{-r_{0}(q+1)}$. Note that
$\zeta^{q+r_{0}(q-1)}=-\eta^{\prime}\cdot\zeta^{\frac{q+1}{2}}$. Therefore,
for each
$\underline{\alpha}=(\alpha_{1},\alpha_{2},\dots,\alpha_{\frac{m-1}{2}})\in\mathcal{S}^{\prime\prime}$,
we have

$f_{\underline{\alpha},r_{0}}^{\prime\prime}=f_{\underline{\alpha\eta}^{\prime},r_{0}},$

where
$\underline{\alpha\eta}^{\prime}=(\alpha_{1}\eta^{\prime},\alpha_{2}\eta^{\prime},\dots,\alpha_{\frac{m-1}{2}}\eta^{\prime})\in\mathcal{S}^{\prime\prime}$,
and

$f_{\underline{\alpha},q-r_{0}}^{\prime\prime}=f_{-\underline{\alpha\eta}^{\prime},3r_{0}+1},$

where
$-\underline{\alpha\eta}^{\prime}=(-\alpha_{1}\eta^{\prime},-\alpha_{2}\eta^{\prime},\dots,-\alpha_{\frac{m-1}{2}}\eta^{\prime})\in\mathcal{S}^{\prime\prime}$.
The rest of the proof follows from Proposition 3.5 ∎

######  Remark 13.

We can state Corollary 3.6 slightly more generally as follows. Let $s$ be an
such that $-\frac{q-1}{4}\leq s\leq\frac{q+1}{4}$ and let
$L_{s}=F(\zeta^{2s+1}\varpi)$. Then, the vector

$v_{L_{s}}=\sum\limits_{\underline{\alpha}\in
S^{\prime\prime}}[f_{\underline{\alpha},r_{0}-s}+\psi(\varpi)f_{\underline{\alpha},3r_{0}-s+1}]$

is a $L_{s}^{\times}$-fixed vector $\pi_{\psi}$.

##  4\. Construction of invariant vectors: Unramified Case

Recall that we are working with $M=F(\zeta),K=F(\varpi)$, and
$L=F(\zeta\varpi)$.

Let

$\mathcal{R}=\\{(\alpha_{1},0,\alpha_{3},0,\dots,\alpha_{m-1})\mid\alpha_{r}\in\mathbb{D},1\leq
r\leq m-1\\}.$

Let $\mathcal{R}^{\prime}$ denote the subset of $\mathcal{R}$ consisting of
those tuples
$\underline{\alpha}=(\alpha_{1},0,\alpha_{3},0,\dots,\alpha_{m-1})$ such that
$\alpha_{r}\in\mathbb{F}$ for all $r$.

For each integer $i$ with $i\in\\{0,1\\}$ and
$\underline{\alpha}=(\alpha_{1},0,\alpha_{3},0,\dots,\alpha_{m-1})\in\mathcal{R}$,
we define

$\Delta_{i,\underline{\alpha}}=\varpi^{i}\cdot(1+\sum\limits_{r=1}^{m-1}\alpha_{r}\varpi^{r}).$

Then

$\\{\Delta_{i,\underline{\alpha}}\mid\underline{\alpha}\in\mathcal{R},i=0,1\\}$

forms a set of right coset representatives for $D^{\times}/H$, where
$H=M^{\times}(1+\mathcal{P}_{D}^{m})$.

We have

$\pi_{\psi}={\rm Ind}_{H}^{D^{\times}}\theta_{\psi}^{\\#},$

where $\theta_{\psi}^{\\#}$ is a one dimensional character when $m$ is even
and is a $q$-dimensional representation when $m$ is odd. For convenience, let
$W$ denote the space of the representation $\theta_{\psi}^{\\#}$ (in either
case).

When $m$ is even, to each pair $(i,\underline{\alpha})$ with
$\underline{\alpha}\in\mathcal{R}$ and $i\in\\{0,1\\}$, we attach a function
$\phi_{i,\underline{\alpha}}:D^{\times}\rightarrow\mathbb{C}\text{ supported
on }H\cdot\Delta_{i,\underline{\alpha}}$ such that
$\phi_{i,\underline{\alpha}}$ takes value $\psi^{\\#}(h)$ on the element
$h\cdot\Delta_{i,\underline{\alpha}}$ for each $h\in H$.

We also define, for
$\underline{\alpha}=(\alpha_{1},0,\alpha_{3},0,\dots,\alpha_{m-1})\in\mathcal{R}^{\prime}$,

$\Delta^{\prime}_{i,\underline{\alpha}}=(\zeta\varpi)^{i}\cdot(1+\sum\limits_{r=1}^{m-1}\alpha_{r}\zeta\varpi^{r}),$

and corresponding functions
$\phi^{\prime}_{i,\underline{\alpha}}:D^{\times}\rightarrow\mathbb{C}\text{
supported on }H\cdot\Delta^{\prime}_{i,\underline{\alpha}}$ such that
$\phi^{\prime}_{i,\underline{\alpha}}$ takes value $\psi^{\\#}(h)$ on the
element $h\cdot\Delta^{\prime}_{i,\underline{\alpha}}$ for each $h\in H$.

When $m$ is odd, recall that
$H^{\prime\prime}=F^{\times}(1+\mathcal{P}_{M})(1+\mathcal{P}_{D}^{m+1})$ and
$H^{\prime}=F^{\times}(1+\mathcal{P}_{M})(1+\mathcal{P}_{D}^{m})$. Let

$H_{1}^{\prime}=F^{\times}(1+\mathcal{P}_{M})(1+\mathcal{P}_{L}^{m})(1+\mathcal{P}_{D}^{m+1}).$

Note that the set
$\mathcal{D}=\\{1+\alpha\varpi^{m}\mid\alpha\in\mathbb{F}\\}$ forms a set of
right coset representatives for the quotient $H^{\prime}/H_{1}^{\prime}$.

For each $\alpha\in\mathbb{F}$, we define $f_{\alpha}:H\rightarrow\mathbb{C}$
with support $H_{1}^{\prime}(1+\alpha\varpi^{m})$ such that it takes the value
$\psi^{\prime}(h_{1})$ on $h_{1}(1+\alpha\varpi^{m})$ for each $h_{1}\in
H_{1}^{\prime}$. Note that $f_{\alpha}$’s form a basis for $W$. Observe that

(2) $(1+\alpha\varpi^{m})\cdot
f_{\beta}=\theta_{\psi}(1+\alpha\varpi^{m})f_{\beta}=\psi[1+(\beta-\alpha)\alpha\varpi^{2m}]f_{\beta-\alpha},$

and

(3) $(1+\alpha\zeta\varpi^{m})\cdot
f_{0}=\theta_{\psi}(1+\alpha\zeta\varpi^{m})f_{0}=\psi^{\prime}(1+\alpha\zeta\varpi^{m})f_{0}.$

Now for each tuple $(i,\underline{\alpha},\alpha_{m})$ with $i\in\\{0,1\\}$,
$\underline{\alpha}\in\mathcal{R}^{\prime}$, and $\alpha_{m}\in\mathbb{F}$, we
define two $W$-valued functions as follows. The function
$\phi_{i,\underline{\alpha},\alpha_{m}}:D^{\times}\rightarrow W$ is supported
on $H\cdot\Delta_{i,\underline{\alpha}}$ with

$\phi_{i,\underline{\alpha},\alpha_{m}}(h\cdot\Delta_{i,\underline{\alpha}})=\theta_{\psi}^{\\#}(h)(f_{\alpha_{m}})$

and the function
$\phi_{i,\underline{\alpha},\alpha_{m}}^{\prime}:D^{\times}\rightarrow W$ is
supported on $H\cdot\Delta_{i,\underline{\alpha}}^{\prime}$ with

$\phi_{i,\underline{\alpha},\alpha_{m}}^{\prime}(h\cdot\Delta_{i,\underline{\alpha}}^{\prime})=\theta_{\psi}^{\\#}(h)(f_{\alpha_{m}})$

for each $h\in H$.

In this case, we are going to exhibit a number of equivariant vectors for
characters of $K^{\times}$ and $L^{\times}$.

###  4.1. Restricted to $K^{\times}$

We first deal with $K=F(\varpi)$.

######  Proposition 4.1.

Let $\chi$ be a smooth character of $K^{\times}$ with

$\chi|_{H_{1}^{\prime}\cap
K^{\times}}=\chi|_{F^{\times}(1+\mathcal{P}_{K}^{m+1})}=\psi^{\prime}|_{F^{\times}(1+\mathcal{P}_{K}^{m+1})},$

when $m$ is odd, and

$\chi|_{H\cap
K^{\times}}=\chi|_{F^{\times}(1+\mathcal{P}_{K}^{m})}=\theta_{\psi}^{\\#}|_{F^{\times}(1+\mathcal{P}_{K}^{m})},$

when $m$ is even. Then $\chi$ occurs with multiplicity one in the restriction
of $\pi_{\psi}$ to $K^{\times}$. The vector

$v_{\chi}=\begin{cases}\sum\limits_{\begin{subarray}{c}i=0,1\\\
\underline{\alpha}\in\mathcal{R}^{\prime},\alpha_{m}\in\mathbb{F}\end{subarray}}\chi(\Delta_{i,\underline{\alpha}})~{}\chi(1+\alpha_{m}\varpi^{m})\cdot\phi_{i,\underline{\alpha},\alpha_{m}}&\text{if
$m$ is odd,}\\\ \ \sum\limits_{\begin{subarray}{c}i=0,1\\\
\underline{\alpha}\in\mathcal{R}^{\prime}\end{subarray}}\chi(\Delta_{i,\underline{\alpha}})\cdot\phi_{i,\underline{\alpha}}&\text{if
$m$ is even,}\end{cases}$

lies in the $\chi$-isotypic component of the representation $\pi_{\psi}$.

###### Proof.

Suppose, $m$ is odd. Fix $x=h\cdot\Delta_{i_{0},\underline{\delta}}\in
D^{\times}$, where $h\in H$,
$\underline{\delta}=(\delta_{1},0,\delta_{3},0,\dots,\delta_{m-2},0)\in\mathcal{R}$,
and $i_{0}\in\\{0,1\\}$. Note that

$v_{\chi}(x)=\begin{cases}\sum\limits_{\alpha_{m}\in\mathbb{F}}\chi(\Delta_{i_{0},\underline{\delta}})\chi(1+\alpha_{m}\varpi^{m})\cdot\theta_{\psi}^{\\#}(h)(f_{\alpha_{m}})&\text{if
$\underline{\delta}\in\mathcal{R}^{\prime}$,}\\\
0&\text{otherwise.}\end{cases}$

We now assume that $\underline{\delta}\in\mathcal{R}^{\prime}$. Let $z\in
K^{\times}$ and write

$z=a\cdot\varpi^{j_{0}}\cdot(1+\sum\limits_{\begin{subarray}{c}r\geq 1\\\
r\text{ is odd}\end{subarray}}\beta_{r}\varpi^{r})$

for some $a\in F^{\times}$, $j_{0}\in\\{0,1\\}$, and $\beta_{r}\in\mathbb{F}$
for all $r$. Then,

$\pi_{\psi}(z)v_{\chi}(x)=v_{\chi}(xz)=v_{\chi}\left(ah\cdot\varpi^{i_{0}+j_{0}}(1+\sum\limits_{\begin{subarray}{c}r=1\\\
r\text{ is
odd}\end{subarray}}^{m-1}\delta_{r}\varpi^{r})\cdot(1+\sum\limits_{\begin{subarray}{c}r\geq
1\\\ r\text{ is odd}\end{subarray}}\beta_{r}\varpi^{r})\right).$

We write

$\textbf{y}=\varpi^{i_{0}+j_{0}}(1+\sum\limits_{\begin{subarray}{c}r=1\\\
r\text{ is
odd}\end{subarray}}^{m-1}\delta_{r}\varpi^{r})\cdot(1+\sum\limits_{\begin{subarray}{c}r\geq
1\\\ r\text{ is odd}\end{subarray}}\beta_{r}\varpi^{r})$

as

$\textbf{y}=b_{0}\cdot
h_{0}\cdot\Delta_{i_{0}+j_{0},\underline{\delta}^{\prime}},$

where

$b_{0}=1+\sum\limits_{\begin{subarray}{c}r=2\\\ r\text{ is
even}\end{subarray}}^{m-1}\xi_{r}\varpi^{r}$

with

$\xi_{r}=\sum\limits_{\begin{subarray}{c}t=1\\\ t\text{ is
odd}\end{subarray}}^{r-1}\delta_{r-t}\beta_{t}$

for even $r\leq m-1$, and

$\underline{\delta}^{\prime}=(\delta_{1}^{\prime},,0,\delta_{3}^{\prime},0,\dots,\delta_{m-2}^{\prime},0)\in\mathcal{R}^{\prime}$

with

$\delta_{r}^{\prime}=\delta_{r}+\beta_{r}-\sum\limits_{\begin{subarray}{c}t=1\\\
t\text{ is odd}\end{subarray}}^{r-2}\delta_{t}^{\prime}\xi_{r-t}$

for odd $r\leq m-2$, and

$h_{0}=1+\sum\limits_{\begin{subarray}{c}j=m\\\ j\text{ is
odd}\end{subarray}}^{2m-1}\epsilon_{j}\varpi^{j}+\sum\limits_{\begin{subarray}{c}k=m+1\\\
k\text{ is even}\end{subarray}}^{2m}\xi_{k}\varpi^{k}+\cdots$

such that

$\epsilon_{j}=\beta_{j}-\sum\limits_{\begin{subarray}{c}t=1\\\ t\text{ is
odd}\end{subarray}}^{m-2}\delta_{t}^{\prime}\xi_{j-t}-\sum\limits_{\begin{subarray}{c}t=m\\\
t\text{ is odd}\end{subarray}}^{j-2}\epsilon_{t}\xi_{j-t},$

for odd $j\geq m$, and

$\xi_{k}=\sum\limits_{\begin{subarray}{c}t=1\\\ t\text{ is
odd}\end{subarray}}^{m-2}\delta_{t}\beta_{k-t}-\sum\limits_{\begin{subarray}{c}t=m+1\\\
t\text{ is
even}\end{subarray}}^{k-2}\xi_{t}\xi_{k-t}-\sum\limits_{\begin{subarray}{c}t=m\\\
t\text{ is odd}\end{subarray}}^{k-1}\delta_{k-t}^{\prime}\epsilon_{t}$

for even $m+1\leq k\leq 2m$. We take $\delta^{\prime}_{m}=0$ in the above.

Clearly, $b_{0}\in F^{\times}$, and $h_{0}\in H\cap K^{\times}$. We now write

$h_{0}=h_{1}\cdot(1+\epsilon_{m}\varpi^{m}),$

where

$h_{1}=1+\sum\limits_{\begin{subarray}{c}j=m+2\\\ j\text{ is
odd}\end{subarray}}^{2m-1}\epsilon_{j}\varpi^{j}+\sum\limits_{\begin{subarray}{c}k=m+1\\\
k\text{ is even}\end{subarray}}^{2m}\xi_{k}\varpi^{k}+\cdots$

belongs to $H_{1}^{\prime}\cap K^{\times}$. We get,

$\displaystyle\pi_{\psi}(z)v_{\chi}(x)$ $\displaystyle=v_{\chi}\left(ah\cdot
b_{0}\cdot h_{0}\cdot\Delta_{i_{0}+j_{0},\underline{\delta}^{\prime}}\right)$
$\displaystyle=\theta_{\psi}^{\\#}(ah\cdot b_{0}\cdot
h_{0})v_{\chi}\left(\Delta_{i_{0}+j_{0},\underline{\delta}^{\prime}}\right)$
$\displaystyle=\psi(ab_{0})\chi(\Delta_{i_{0}+j_{0},\underline{\delta}^{\prime}})\theta_{\psi}^{\\#}(h)\theta_{\psi}(h_{1})\theta_{\psi}(1+\epsilon_{m}\varpi^{m})\left[\sum\limits_{\alpha_{m}\in\mathbb{F}}\chi(1+\alpha_{m}\varpi^{m})f_{\alpha_{m}}\right].$

We now analyse the vector

$\displaystyle{\bf w}$
$\displaystyle=\theta_{\psi}(h_{1})\theta_{\psi}(1+\epsilon_{m}\varpi^{m})\left[\sum\limits_{\alpha_{m}\in\mathbb{F}}\chi(1+\alpha_{m}\varpi^{m})f_{\alpha_{m}}\right]$
$\displaystyle=\theta_{\psi}(h_{1})\left[\sum\limits_{\alpha_{m}\in\mathbb{F}}\chi(1+\alpha_{m}\varpi^{m})\cdot\psi\left(1+(\alpha_{m}-\epsilon_{m})\epsilon_{m}\varpi^{2m}\right)f_{\alpha_{m}-\epsilon_{m}}\right]$

where in the last step we have made use of (2). Note that, by our assumption
on $\chi$, we have $\theta_{\psi}(h_{1})=\psi^{\prime}(h_{1})=\chi(h_{1})$
(since $h_{1}\in H_{1}^{\prime}\cap K^{\times}$) and also $\chi$ is trivial on
$1+\mathcal{P}_{K}^{2m+1}$ (since $\psi^{\prime}$ is trivial on
$1+\mathcal{P}_{D}^{2m+1}$). Thus, we get

$\displaystyle{\bf w}$
$\displaystyle=\chi(h_{1})\sum\limits_{\alpha_{m}\in\mathbb{F}}\chi(1+\epsilon_{m}\varpi^{m})\cdot\chi(1+(\alpha_{m}-\epsilon_{m})\varpi^{m})f_{\alpha_{m}-\epsilon_{m}}$
$\displaystyle=\chi(h_{0})\sum\limits_{\alpha_{m}\in\mathbb{F}}\chi(1+\alpha_{m}\varpi^{m})f_{\alpha_{m}}$

Therefore,

$\displaystyle\pi_{\psi}(z)v_{\chi}(x)$ $\displaystyle=\chi\left(a\cdot
b_{0}\cdot
h_{0}\cdot\Delta_{i_{0}+j_{0},\underline{\delta}^{\prime}}\right)\theta_{\psi}^{\\#}(h)\left[\sum\limits_{\alpha_{m}\in\mathbb{F}}\chi(1+\alpha_{m}\varpi^{m})f_{\alpha_{m}}\right]$
$\displaystyle=\chi\left(z\cdot\Delta_{i_{0},\underline{\delta}}\right)\sum\limits_{\alpha_{m}\in\mathbb{F}}\chi(1+\alpha_{m}\varpi^{m})\cdot\theta_{\psi}^{\\#}(h)(f_{\alpha_{m}})$
$\displaystyle=\chi(z)\cdot v_{\chi}(x).$

When $m$ is even, $W=\mathbb{C}$ and $\theta_{\psi}^{\\#}$ is a one
dimensional character. Now observe that the above proof in the odd case goes
through, and in fact is simpler, with $\alpha_{m}=0$, $f_{0}=1$, and by taking
$\phi_{i,\underline{\alpha}}=\phi_{i,\underline{\alpha},0}$. ∎

######  Remark 14.

The number of characters $\chi:K^{\times}\rightarrow\mathbb{C}$ satisfying
$\chi|_{H_{1}^{\prime}\cap K^{\times}}=\psi^{\prime}|_{H_{1}^{\prime}\cap
K^{\times}}$ is

$\left|\frac{K^{\times}}{F^{\times}(1+\mathcal{P}_{K}^{m+1})}\right|=2q^{m-\left[\frac{m}{2}\right]}=2q^{\frac{m+1}{2}},$

when $m$ is odd. Similarly, when $m$ is even, the number of characters
$\chi:K^{\times}\rightarrow\mathbb{C}$ which satisfy $\chi|_{H\cap
K^{\times}}=\theta_{\psi}^{\\#}|_{H\cap K^{\times}}$ is

$\left|\frac{K^{\times}}{F^{\times}(1+\mathcal{P}_{K}^{m})}\right|=2q^{m-1-\left[\frac{m-1}{2}\right]}=2q^{\frac{m}{2}}.$

If $\pi_{\psi}$ has trivial central character then the trivial character
$\textbf{1}_{K^{\times}}$ of $K^{\times}$ occurs in the restriction
$\pi_{\psi}$ to $K^{\times}$ and putting $\chi=1$ we get the following
corollary.

######  Corollary 4.2.

If $\pi_{\psi}$ has trivial central character then the vector

$v_{K}=\begin{cases}\sum\limits_{\begin{subarray}{c}i=0,1\\\
\underline{\alpha}\in\mathcal{R}^{\prime},\alpha_{m}\in\mathbb{F}\end{subarray}}\phi_{i,\underline{\alpha},\alpha_{m}}&\text{if
$m$ is odd,}\\\ \sum\limits_{\begin{subarray}{c}i=0,1\\\
\underline{\alpha}\in\mathcal{R}^{\prime}\end{subarray}}\phi_{i,\underline{\alpha}}&\text{if
$m$ is even,}\end{cases}$

is a $K^{\times}$-fixed vector in the representation $\pi_{\psi}$.

###  4.2. Restricted to $L^{\times}$

We take $L=F(\zeta\varpi)$.

######  Proposition 4.3.

Let $\lambda$ be a smooth character of $L^{\times}$ with

$\lambda|_{H_{1}^{\prime}\cap
L^{\times}}=\lambda|_{F^{\times}(1+\mathcal{P}_{L}^{m})}=\psi^{\prime}|_{F^{\times}(1+\mathcal{P}_{L}^{m})},$

when $m$ is odd, and

$\lambda|_{H\cap
L^{\times}}=\lambda|_{F^{\times}(1+\mathcal{P}_{L}^{m})}=\theta_{\psi}^{\\#}|_{F^{\times}(1+\mathcal{P}_{L}^{m})},$

when $m$ is even. Then $\lambda$ occurs with multiplicity one in the
restriction of $\pi_{\psi}$ to $L^{\times}$. The vector

$v_{\lambda}=\begin{cases}\sum\limits_{\begin{subarray}{c}i=0,1\\\
\underline{\alpha}\in\mathcal{R}^{\prime}\end{subarray}}\lambda(\Delta_{i,\underline{\alpha}}^{\prime})\cdot\phi^{\prime}_{i,\underline{\alpha},0}&\text{if
$m$ is odd,}\\\ \sum\limits_{\begin{subarray}{c}i=0,1\\\
\underline{\alpha}\in\mathcal{R}^{\prime}\end{subarray}}\lambda(\Delta_{i,\underline{\alpha}}^{\prime})\cdot\phi_{i,\underline{\alpha}}^{\prime}&\text{if
$m$ is even,}\end{cases}$

lies in the $\lambda$-isotypic component of the representation $\pi_{\psi}$.

###### Proof.

The proof is similar to the proof of Proposition 4.1. Note that we take
$\alpha_{m}=0$ and instead of (2) we need to employ (3). ∎

######  Remark 15.

The number of characters $\lambda:L^{\times}\rightarrow\mathbb{C}$ satisfying
$\lambda|_{H_{1}^{\prime}\cap L^{\times}}=\psi^{\prime}|_{H_{1}^{\prime}\cap
L^{\times}}$ or $\lambda|_{H\cap L^{\times}}=\theta_{\psi}^{\\#}|_{H\cap
L^{\times}}$ is

$\left|\frac{L^{\times}}{F^{\times}(1+\mathcal{P}_{L}^{\left[\frac{m-1}{4}\right]+1})}\right|=2q^{m-1-\left[\frac{m-1}{2}\right]}.$

If $\pi_{\psi}$ has trivial central character then the trivial character
$\textbf{1}_{L^{\times}}$ of $L^{\times}$ occurs in the restriction
$\pi_{\psi}$ to $L^{\times}$ and putting $\lambda=1$ we get the following
corollary.

######  Corollary 4.4.

If $\pi_{\psi}$ has trivial central character then the vector

$v_{L}=\begin{cases}\sum\limits_{\begin{subarray}{c}i=0,1\\\
\underline{\alpha}\in\mathcal{R}^{\prime}\end{subarray}}\phi_{i,\underline{\alpha},0}^{\prime}&\text{if
$m$ is odd,}\\\ \sum\limits_{\begin{subarray}{c}i=0,1\\\
\underline{\alpha}\in\mathcal{R}^{\prime}\end{subarray}}\phi_{i,\underline{\alpha}}^{\prime}&\text{if
$m$ is even,}\end{cases}$

is an $L^{\times}$-fixed vector in the representation $\pi_{\psi}$.

##  5\. Correlation Coefficients

In this section we prove the main theorem of this paper on correlation
coefficients. As earlier, we have

$M=F(\zeta),K=F(\varpi),L=F(\zeta\varpi).$

We take $G=D^{\times}/F^{\times}$ and consider given embeddings of
$M^{\times},K^{\times}$, and $L^{\times}$ in $G$. By abuse of notation, we
denote the corresponding subgroups by $M,K$, and $L$, as this does not cause
any confusion. We are interested in the correlation coefficients

$c(\pi;M,K),c(\pi;M,L),c(\pi;K,L)$

for an irreducible representation $\pi$ of $G$.

As mentioned in the introduction, our methods are straightforward. We work
with a given model of $\pi=\pi_{\psi}$ and we have explicitly written down the
invariant vectors for $M,K$, and $L$ in his model in Sections 3 and 4. Now, in
order to compute the correlation coefficients of the main theorem, we fix a
$G$-invariant inner product on $\pi_{\psi}$, normalize the three invariant
vectors to those of unit length in this inner product, and compute the inner
product between any two of these vectors.

The $G$-invariant inner product is indeed the obvious one. Let $G$ be a
compact group with a closed subgroup $H$. Suppose $(\sigma,W)$ is an
irreducible unitary representation of $H$ with respect to an inner product
$\langle\cdot,\cdot\rangle_{\sigma}$. Then the induced representation ${\rm
Ind}_{H}^{G}\sigma$ is unitary with respect to the inner product

$\langle\phi_{1},\phi_{2}\rangle=\int\limits_{G}\langle\phi_{1}(g),\phi_{2}(g)\rangle_{\sigma}dg,$

where $dg$ is a Haar measure on $G$.

Recall that in Section 3 and also when $m$ is even in Section 3, the inducing
representation $\sigma$ is one dimensional, whereas when $m$ is odd in Section
3, the inducing representation $\sigma$ has dimension $q$.

###  5.1. Ramified case

We are in the context of Section 3, and thus $\pi_{\psi}$ is an irreducible
representation of $D^{\times}$ with odd conductor $m+2$, where $m$ is an odd
positive integer. We further assume that it has trivial central character.
Recall that then it always has a fixed vector for $M$ (cf. Corollary 3.2), it
has a fixed vector for $K$ precisely when $q\equiv 3\mod 4$ (cf. Corollary
3.4), and it has a fixed vector for $L$ precisely when $q\equiv 1\mod 4$ (cf.
Corollary 3.6). Observe that the corresponding unit vectors are:

$\hat{v}_{M}=\frac{1}{\sqrt{(q+1)q^{[\frac{m-1}{4}]}}}\sum\limits_{\begin{subarray}{c}\underline{\alpha}\in
S^{\prime}\\\ 0\leq i\leq q\end{subarray}}f_{\underline{\alpha},i},$
$\hat{v}_{K}=\frac{1}{\sqrt{2q^{[\frac{m+1}{4}]}}}\sum\limits_{\underline{\alpha}\in
S^{\prime\prime}}\left[f_{\underline{\alpha},t_{0}}+\psi(\varpi)f_{\underline{\alpha},3t_{0}}\right],\hat{v}_{L}=\frac{1}{\sqrt{2q^{[\frac{m+1}{4}]}}}\sum\limits_{\underline{\alpha}\in
S^{\prime\prime}}\left[f_{\underline{\alpha},r_{0}}+\psi(\varpi)f_{\underline{\alpha},3r_{0}+1}\right].$

###### Proof of Theorem 1.3.

First assume $q\equiv 3\mod 4$ and in this case we only need to compute
$c(\pi;M,K)$. This is given by

$\displaystyle c(\pi;M,K)]$
$\displaystyle=|\left\langle\hat{v}_{M},\hat{v}_{K}\right\rangle|^{2}$
$\displaystyle=\frac{1}{2(q+1)q^{\frac{m-1}{2}}}\cdot|\left\langle
f_{\underline{0},t_{0}}+f_{\underline{0},3t_{0}},f_{\underline{0},t_{0}}+\psi(\varpi)f_{\underline{0},3t_{0}}\right\rangle|^{2}$
$\displaystyle=\frac{|1+\psi(\varpi)|^{2}}{2(q+1)q^{\frac{m-1}{2}}}$
$\displaystyle=\frac{|1+\psi(\varpi)|^{2}}{2\dim\pi}$
$\displaystyle=\begin{cases}\frac{2}{\dim\pi}&\text{if $\psi(\varpi)=1$,}\\\
0&\text{if $\psi(\varpi)=-1$.}\end{cases}$

Now let $q\equiv 1\mod 4$ and we need to compute $c(\pi;M,L)$. This is given
by

$\displaystyle c(\pi;M,L)$
$\displaystyle=|\left\langle\hat{v}_{M},\hat{v}_{L}\right\rangle|^{2}$
$\displaystyle=\frac{1}{2(q+1)q^{\frac{m-1}{2}}}\cdot|\left\langle
f_{\underline{0},r_{0}}+f_{\underline{0},3r_{0}+1},f_{\underline{0},r_{0}}+\psi(\varpi)f_{\underline{0},3r_{0}+1}\right\rangle|^{2}$
$\displaystyle=\frac{|1+\psi(\varpi)|^{2}}{2(q+1)q^{\frac{m-1}{2}}}$
$\displaystyle=\frac{|1+\psi(\varpi)|^{2}}{2\dim\pi}$
$\displaystyle=\begin{cases}\frac{2}{\dim\pi}&\text{if $\psi(\varpi)=1$,}\\\
0&\text{if $\psi(\varpi)=-1$.}\end{cases}$

∎

######  Remark 16.

Similarly, by making use of Remark 11 and Remark 13, it is easy to compute the
following correlation coefficients. For $q\equiv 3$ mod $4$, we have

$c(\pi;K_{s},K_{s^{\prime}})=0$

for $s\neq s^{\prime}$, and

$c(\pi;M,K_{s})=\frac{|1+\psi(\varpi)|^{2}}{2\dim\pi}.$

For $q\equiv 1$ mod $4$, we have

$c(\pi;L_{s},L_{s^{\prime}})=0$

for $s\neq s^{\prime}$, and

$c(\pi;M,L_{s})=\frac{|1+\psi(\varpi)|^{2}}{2\dim\pi}.$

###  5.2. Unramified case

We are in the context of Section 4, and thus $\pi_{\psi}$ is an irreducible
representation of $D^{\times}$ with even conductor $2m+2$. There are two cases
to consider, namely (i) $m$ is even and (ii) $m$ is odd. We further assume
that $\pi_{\psi}$ has trivial central character. Thus, $\pi_{\psi}$ has fixed
vectors for both $K$ (cf. Corollary 4.2) and $L$ (cf. Corollary 4.4). Observe
that the corresponding unit vectors are:

$\displaystyle\hat{v}_{K}$
$\displaystyle=\frac{1}{\sqrt{2q^{\frac{m}{2}}}}\sum\limits_{\begin{subarray}{c}i=0,1\\\
\underline{\alpha}\in\mathcal{R}^{\prime}\end{subarray}}\phi_{i,\underline{\alpha}},$
$\displaystyle\hat{v}_{L}$
$\displaystyle=\frac{1}{\sqrt{2q^{\frac{m}{2}}}}\sum\limits_{\begin{subarray}{c}i=0,1\\\
\underline{\alpha}\in\mathcal{R}^{\prime}\end{subarray}}\phi_{i,\underline{\alpha}}^{\prime},$

when $m$ is even, and

$\displaystyle\hat{v}_{K}$
$\displaystyle=\frac{1}{\sqrt{2q^{\frac{m+1}{2}}}}\sum\limits_{\begin{subarray}{c}i=0,1\\\
\underline{\alpha}\in\mathcal{R}^{\prime},\alpha_{m}\in\mathbb{F}\end{subarray}}\phi_{i,\underline{\alpha},\alpha_{m}},$
$\displaystyle\hat{v}_{L}$
$\displaystyle=\frac{1}{\sqrt{2q^{\frac{m-1}{2}}}}\sum\limits_{\begin{subarray}{c}i=0,1\\\
\underline{\alpha}\in\mathcal{R}^{\prime}\end{subarray}}\phi_{i,\underline{\alpha},0}^{\prime},$

when $m$ is odd.

###### Proof of Theorem 1.4.

Noting that

$\phi_{0,\underline{0}}^{\prime}=\phi_{0,\underline{0}}~{}\&~{}\phi_{1,\underline{0}}^{\prime}=\psi(\zeta^{-1})\cdot\phi_{1,\underline{0}},$

and

$\phi_{0,\underline{0},\alpha}^{\prime}=\phi_{0,\underline{0},\alpha}~{}\&~{}\phi_{1,\underline{0},\alpha}^{\prime}=\psi(\zeta^{-1})\cdot\phi_{1,\underline{0},\alpha},$

we get, when $m$ is even,

$\displaystyle c(\pi;K,L)$
$\displaystyle=|\left\langle\hat{v}_{K},\hat{v}_{L}\right\rangle|^{2}$
$\displaystyle=\frac{1}{4q^{m}}\cdot|\left\langle\phi_{0,\underline{0}}+\phi_{1,\underline{0}},\phi_{0,\underline{0}}+\psi(\zeta^{-1})\phi_{1,\underline{0}}\right\rangle|^{2}$
$\displaystyle=\frac{|1+\psi(\zeta)|^{2}}{4q^{m}}$
$\displaystyle=\frac{|1+\psi(\zeta)|^{2}}{2\dim\pi},$

and when $m$ is odd,

$\displaystyle c(\pi;K,L)$
$\displaystyle=|\left\langle\hat{v}_{K},\hat{v}_{L}\right\rangle|^{2}$
$\displaystyle=\frac{1}{4q^{m}}\cdot|\left\langle\sum\limits_{\alpha\in\mathbb{F}}\phi_{0,\underline{0},\alpha}+\sum\limits_{\alpha\in\mathbb{F}}\phi_{1,\underline{0},\alpha},\phi_{0,\underline{0},0}+\psi(\zeta^{-1})\phi_{1,\underline{0},0}\right\rangle|^{2}$
$\displaystyle=\frac{|1+\psi(\zeta)|^{2}}{4q^{m}}$
$\displaystyle=\frac{|1+\psi(\zeta)|^{2}}{2\dim\pi}.$

∎

###  5.3. Compatibility with Schur orthogonality relations

In this section, we verify that Theorem 1.3 and Theorem 1.4 agree with the
identity [AJ22, Equation (1) on p. 502]

$\sum_{\pi\in\widehat{G}}\dim\pi\cdot c(\pi;H,K)=\frac{|G||H\cap K|}{|H||K|},$

which is valid for any finite group $G$ with two multiplicity free subgroups
$H$ and $K$. This above identity follows easily from [AJ22, Lemma 2.1] by
Schur orthogonality relations.

The idea, as mentioned in Section 1, is to work with the finite group
$G_{c}=\frac{D^{\times}}{F^{\times}(1+\mathcal{P}_{D}^{c-1})}$, and the
corresponding embeddings $M_{c},K_{c}$, and $L_{c}$ of tori in $G_{c}$. Note
that any two of these subgroups intersect trivially. The cardinalities of
these groups are given by

$|G_{c}|=\begin{cases}2(q+1)q^{3j-1}&\text{if $c=2j+1$,}\\\
2(q+1)q^{3j}&\text{if $c=2j+2$},\end{cases}$

and

$|M_{c}|=(q+1)q^{\left[\frac{c-2}{2}\right]},|K_{c}|=|L_{c}|=2q^{(c-2)-\left[\frac{c-2}{2}\right]}.$

We need to know the number of irreducible representations of $D^{\times}$ of a
given conductor with trivial central character and this is given by [Tun78,
Proposition 3.5 and Remark on p. 189]:

$\mbox{Number of~{}}\pi\mbox{~{}of
conductor~{}}c=\begin{cases}\frac{q^{2}-1}{2}q^{\frac{c-4}{2}}&\text{if $c$ is
even and $c\geq 4$,}\\\ 2(q-1)q^{\frac{c-3}{2}}&\text{if $c$ is odd and $c\geq
3$.}\end{cases}$

Note that the only representation of conductor $1$ is the trivial
representation and the number of irreducible representations of conductor $2$
is the number of two dimensional representations of the dihedral group
$D_{2(q+1)}$ which is $\frac{q-1}{2}$.

We also note that many correlation coefficients vanish because of lack of an
invariant vector. More precisely, when $\pi$ has odd conductor, $c(\pi;M,K)=0$
when $q\equiv 1\mod 4$, $c(\pi;M,L)=0$ when $q\equiv 3\mod 4$, and
$c(\pi;K,L)=0$ always (cf. Remark 2). When $\pi$ has even conductor,
$c(\pi;M,K)=c(\pi;M,L)=0$ (cf. Remark 5).

We first verify the identity involving $c(\pi;M,K)$ and the verification for
the one involving $c(\pi;M,L)$ being similar. Only representations with odd
conductor, say $c=2j+1$, contribute to the sum on the left hand side. Half the
representations of a given odd conductor arise from $K$ and the other half
from $L$. Those representations arising from $L$ do not contribute if $q\equiv
3\mod 4$ and those arising from $K$ do not contribute if $q\equiv 1\mod 4$.
And amongst the representations $\pi_{\psi}$ that can potentially contribute
to the sum, precisely half of these contribute non-trivially, namely
$\frac{2}{\dim\pi}$, and the other half do not contribute, since half the time
$\psi(\varpi)=1$ and half the time $\psi(\varpi)=-1$. Thus,

$\displaystyle\sum_{\pi\in\widehat{G}}\dim\pi\cdot c(\pi;M,K)$
$\displaystyle=1+\sum_{i=3}^{c}\frac{2(q-1)q^{\frac{i-3}{2}}}{4}\cdot\dim\pi\cdot\frac{2}{\dim\pi}$
$\displaystyle=1+(q-1)\left(1+q+\dots+q^{\frac{c-3}{2}}\right)$
$\displaystyle=q^{\frac{c-1}{2}}=q^{j}=\frac{2(q+1)q^{3j-1}}{(q+1)q^{j-1}\cdot
2q^{j}}$ $\displaystyle=\frac{|G_{c}|}{|M_{c}||K_{c}|}.$

Next we verify the identity involving $c(\pi;K,L)$. This time, only
representations with even conductor, say $c=2j+2$, will matter. Note that

$\sum_{\psi}|1+\psi(\zeta)|^{2}=\sum_{\psi}(2+\psi(\zeta)+\psi(\zeta^{-1}))=\begin{cases}q-1&\text{if
$i=0$,}\\\ (q^{2}-1)q^{i-1}&\text{if $i\geq 1$,}\end{cases}$

where the sum is over minimal characters of $M^{\times}$ of conductor $i+1$.
We have

$\displaystyle\sum_{\pi\in\widehat{G}}\dim\pi\cdot c(\pi;K,L)$
$\displaystyle=1+\frac{q-1}{2}+\sum_{i=1}^{j}\frac{q^{2}-1}{2}q^{i-1}$
$\displaystyle=\frac{q+1}{2}q^{j}=\frac{2(q+1)q^{3j}}{2q^{j}\cdot 2q^{j}}$
$\displaystyle=\frac{|G_{c}|}{|K_{c}||L_{c}|}.$

###  5.4. Proofs of Theorem 1.1 and Theorem 1.2

Let $\pi$ be an irreducible representation of $D^{\times}$ which has fixed
vectors for some tori, it follows that $\omega_{\pi}=1$. In particular,

$\varepsilon(\pi,\Psi_{a})=\omega_{\pi}(a)\varepsilon(\pi,\Psi)=\varepsilon(\pi,\Psi)$

is independent of the the additive character $\Psi$ of $F$, so we denote the
epsilon factor by $\varepsilon(\pi)$. We have $\pi\equiv\pi^{\vee}$, since
$\omega_{\pi}=1$. It now follows from the identity

$\varepsilon(\pi)\varepsilon(\pi^{\vee})=\omega_{\pi}(-1)$

that $\varepsilon(\pi)\in\\{\pm 1\\}$.

Suppose $\pi=\pi_{\psi}$ is constructed from a minimal (primordial in the
terminology of [GK80]) character $\psi$ of $\Omega^{\times}$ as in Section 2,
where $\Omega$ is a quadratic extension of $F$ . Then, by [GK80, Theorem 3.7
and Proposition 4.5], we have

$\varepsilon(\pi_{\psi},\Psi)=(-1)^{f(\psi)}\varepsilon(\psi,\Psi\circ{\rm
Tr}_{\Omega/F}).$

Note that $\psi|_{F^{\times}}=\omega_{\pi}=1$, and therefore by [FQ73, Theorem
3], we get

$\varepsilon(\psi,\Psi\circ{\rm Tr}_{\Omega/F})=\psi(\Delta),$

where $\Delta$ is any element of $\Omega^{\times}$ such that
Tr${}_{\Omega/F}(\Delta)=0$.

Note that we may take $\Omega=K$ and $\Delta=\varpi$ when $\pi$ has odd
minimal conductor. Note also that $f(\psi)$ is even in this case. It follows
that

$\varepsilon(\pi)=\psi(\varpi).$

It is easy to see either from the character formulae for irreducible
representations of $D^{\times}$ or from Tunnell’s result [Tun83, Sai93] that
such a $\pi$ always has a non-trivial $M$-fixed vector and it has a non-
trivial $K$-fixed (resp. $L$-fixed) vector when $q\equiv 3\mod 4$ (resp.
$q\equiv 1\mod 4$). Now Theorem 1.1 follows from Theorem 1.3.

Similarly, Theorem 1.2 follows from Theorem 1.4 and Remark 7 by noting that
$\Delta=\zeta^{\frac{q+1}{2}}$ when $\Omega=M$ and in this case we have
$f(\psi)=f(\pi)/2$.

## Acknowledgements

The authors are grateful to Dipendra Prasad for suggesting several ideas, many
fruitful discussions, and for his encouragement for this work. The second
author thanks the Institute of Eminence Cell at IIT Bombay for supporting his
postdoctoral position with an Institute Post-doctoral Fellowship.

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