where more explicitly $v_{0}=m(e_{1}+e_{2}+...+e_{r})-(mr\cdot e_{0})/2$. The
action of $e_{r}(a)\in\mathfrak{Q}^{\dagger}$ on
$\widehat{\mathfrak{Q}_{T_{0}}}$ is given multiplication by the character

$\chi_{e_{r}(a)}:\mathfrak{Q}_{T_{0}}\longrightarrow\mathbf{C}^{\times},$

which takes the form

$\chi_{e_{r}(a)}(v_{0}(b))=(b,a)_{n}^{B_{Q}(e_{r},v_{0})}=(b,a)_{n}^{mQ(\alpha_{1}^{\vee})}=(b,a)_{2}.$

From this it follows that $\widehat{\mathfrak{Q}_{T_{0}}}$ is a
$\mathfrak{Q}^{\dagger}$-torsor in this case.

We give a summary of the results for ${\rm GSpin}_{2r+1}$ below.

######  Proposition 2.22.

Let $\overline{{\rm GSpin}}_{2r+1}$ be a Brylinski–Deligne cover associated
with a quadratic form $Q$ such that $Q(\alpha_{r}^{\vee})=2$.

  1. (i)

If $\wp(n)\geqslant 3$, then ${\rm
Ind}_{\overline{M}^{\dagger}}^{\overline{M}}\tau\boxtimes\rho$ is irreducible
for every $\tau\boxtimes\rho\in{\rm Irr}_{\rm gen}(\overline{M}^{\dagger})$.
Moreover, if $4\nmid(2r-n)$ with $n$ even, then ${\rm
Ind}_{\overline{T}^{\dagger}}^{\overline{T}}\tau\boxtimes\rho$ is irreducible
for $\tau\boxtimes\rho\in{\rm Irr}_{\rm gen}(\overline{T}^{\dagger})$.

  2. (ii)

If $n$ is odd or $4|(2r-n)$, then the restriction of every $\pi\in{\rm
Irr}_{\rm gen}(\overline{T})$ to $\overline{T}_{0}$ is an isotypic sum.
Moreover, if $\wp(n)=2$ and $2|r$, then the restriction
$\pi|_{\overline{T}^{\dagger}}$ is already an isotypic sum.

###  2.5. Restriction of genuine principal series

In this subsection, we consider exclusively the restriction of genuine
principal series (especially in the tame case) by using a direct analysis from
the perspective of Mackey’s theory.

The notations are the same as before, $\overline{G}$ is a cover of $G$ and
$\overline{G}_{0}=\overline{G}_{\rm der}$ the derived subgroup of
$\overline{G}$. Let $\overline{T}_{0}\subset\overline{G}_{0}$ be the covering
torus.

Every genuine principal series is parabolically induced from an irreducible
representation of $\overline{T}$ of the form

${\rm Ind}_{\overline{A}}^{\overline{T}}(\tilde{\chi})$

where $\overline{A}\subset\overline{T}$ is a maximal abelian subgroup,
necessarily containing $Z(\overline{T})$, and

$\tilde{\chi}:\ \overline{A}\longrightarrow\mathbf{C}^{\times}$

is an extension of the central character

$\chi:Z(\overline{T})\longrightarrow\mathbf{C}^{\times}.$

For every $\gamma\in\overline{T}_{0}\backslash\overline{T}/\overline{A}$,
denote

$\overline{A}_{\gamma}=(\gamma\overline{A}\gamma^{-1})\cap\overline{T}_{0}.$

Define the character

${}^{\gamma}\tilde{\chi}:\overline{A}_{\gamma}\longrightarrow\mathbf{C}^{\times}$

by

${}^{\gamma}\tilde{\chi}(g)=\tilde{\chi}(\gamma^{-1}g\gamma)\text{ for
}g\in\overline{A}_{\gamma}.$

The Mackey theory gives a decomposition

$\left({\rm
Ind}_{\overline{A}}^{\overline{T}}(\tilde{\chi})\right)|_{\overline{T}_{0}}=\bigoplus_{\gamma\in\overline{T}_{0}\backslash\overline{T}/\overline{A}}{\rm
Ind}_{\overline{A}_{\gamma}}^{\overline{T}_{0}}({}^{\gamma}\tilde{\chi}).$

We have the following inclusions between various groups

${\overline{A}}$${\overline{T}}$${\overline{T}_{0}}$${\overline{A}_{\gamma}}$${Z(\overline{T})}$${Z(\overline{T})\cap\overline{T}_{0}}$${Z(\overline{T}_{0}).}$

Here the dashed arrow indicates that there may not be an inclusion
$Z(\overline{T}_{0})\subset\overline{A}_{\gamma}$; in particular,
$\overline{A}_{\gamma}$ may not be a maximal abelian subgroup of
$\overline{T}_{0}$.

######  Definition 2.23.

A pair $(\overline{G},\overline{G}_{0})$ is called an isotypic-pair if any of
the following equivalent conditions (see Corollary 2.12) is satisfied:

  1. (i)

the equality $Y_{0}\cap Y_{Q,n}=Y_{0,Q,n}$ holds;

  2. (ii)

the inclusion $Z(\overline{T}_{0})\subset Z(\overline{T})$ holds.

To continue the analysis, we specialize to the tame case when $p\nmid n$. In
this case, we fix a splitting of $\overline{G}$ over $K:={\bf G}(O)$ denoted
by

$s_{K}:\ K\longrightarrow\overline{G}.$

This gives inherited splittings of ${\bf G}_{0}(O),{\bf T}(O)$ and ${\bf
T}_{0}(O)$ in the respective covering groups. If no confusion arises, we will
omit $s_{K}$ and simply write $K\subset\overline{G}$. The group

$\overline{A}=Z(\overline{T})\cdot{\bf T}(O)$

is a maximal abelian subgroup of $\overline{T}$. One has

$\overline{A}_{\gamma}=(Z(\overline{T})\cap\overline{T}_{0})\cdot({}^{\gamma}{\bf
T}(O)\cap\overline{T}_{0})=(Z(\overline{T})\cap\overline{T}_{0})\cdot{}^{\gamma}{\bf
T}_{0}(O)\subset Z(\overline{T}_{0})\cdot{\bf T}_{0}(O),$

where ${}^{\gamma}{\bf T}_{0}(O)=\gamma{\bf T}_{0}(O)\gamma^{-1}$ for every
$\gamma$. Note that

$\overline{A}_{0}:=Z(\overline{T}_{0})\cdot{\bf T}_{0}(O)$

is a maximal abelian subgroup of $\overline{T}_{0}$. Defining

$\overline{A}^{\natural}=(Z(\overline{T})\cap\overline{T}_{0})\cdot{\bf
T}_{0}(O),$

We write every

$\tilde{\chi}:\overline{A}\longrightarrow\mathbf{C}^{\times}$

as

$\tilde{\chi}=\chi\boxtimes\tilde{\chi}_{O},$

where $\chi$ is the central character mentioned above, and

$\tilde{\chi}_{O}:{\bf T}(O)\longrightarrow\mathbf{C}^{\times}$

is a character such that $\chi$ and $\tilde{\chi}_{O}$ agree on the
intersection $Z(\overline{T})\cap{\bf T}(O)$. Thus the character
${}^{\gamma}\tilde{\chi}$ of $\overline{A}_{\gamma}$ is just

${}^{\gamma}\tilde{\chi}=({}^{\gamma}\chi)\boxtimes({}^{\gamma}\tilde{\chi}_{O})=\chi\boxtimes({}^{\gamma}\tilde{\chi}_{O}),$

where

${}^{\gamma}\tilde{\chi}_{O}:{}^{\gamma}{\bf
T}(O)\cap\overline{T}_{0}={}^{\gamma}{\bf
T}_{0}(O)\longrightarrow\mathbf{C}^{\times}$

is given by

${}^{\gamma}\tilde{\chi}_{O}({}^{\gamma}k_{0})=\tilde{\chi}_{O}(k_{0})$

for every $k_{0}\in{\bf T}_{0}(O)$. In view of the identification

$\overline{A}_{\gamma}=\overline{A}^{\natural}\text{ given by
}z\cdot{}^{\gamma}k_{0}\mapsto(z\cdot[\gamma,k_{0}])\cdot k_{0},$

where $z\in Z(\overline{T})\cap\overline{T}_{0}$ and $k_{0}\in{\bf T}_{0}(O)$,
we can identify

${}^{\gamma}\tilde{\chi}=\chi\boxtimes{}^{\gamma}\tilde{\chi}_{O}:\overline{A}_{\gamma}\longrightarrow\mathbf{C}^{\times}$

with a character

${}^{\gamma}\tilde{\chi}=\chi\boxtimes([\gamma,-]^{-1}\tilde{\chi}_{O}):\overline{A}^{\natural}\longrightarrow\mathbf{C}^{\times}.$

Thus, we have

$\left({\rm
Ind}_{\overline{A}}^{\overline{T}}(\tilde{\chi})\right)|_{\overline{T}_{0}}=\bigoplus_{\gamma\in\overline{T}/(\overline{T}_{0}\overline{A})}{\rm
Ind}_{\overline{A}^{\natural}}^{\overline{T}_{0}}({}^{\gamma}\tilde{\chi}),$

where

(2.17) $\overline{T}/(\overline{T}_{0}\overline{A})\simeq
Y/(Y_{0}+Y_{Q,n})=:\mathscr{X}_{Q,n}^{\Gamma}.$

Now we analyze the decomposition of ${\rm
Ind}_{\overline{A}^{\natural}}^{\overline{T}_{0}}({}^{\gamma}\tilde{\chi})$.
By induction in stages, one has

${\rm
Ind}_{\overline{A}^{\natural}}^{\overline{T}_{0}}({}^{\gamma}\tilde{\chi})={\rm
Ind}_{\overline{A}_{0}}^{\overline{T}_{0}}\ {\rm
Ind}_{\overline{A}^{\natural}}^{\overline{A}_{0}}({}^{\gamma}\tilde{\chi}).$

Denote by

$\mathscr{E}({}^{\gamma}\tilde{\chi};\overline{A}^{\natural},\overline{A}_{0})$

the set of all the possible extensions of ${}^{\gamma}\tilde{\chi}$ to
$\overline{A}_{0}$; then it is exactly the set of irreducible constituents of
${\rm
Ind}_{\overline{A}^{\natural}}^{\overline{A}_{0}}({}^{\gamma}\tilde{\chi})$.
We thus have

${\rm
Ind}_{\overline{A}^{\natural}}^{\overline{T}_{0}}({}^{\gamma}\tilde{\chi})=\bigoplus_{({}^{\gamma}\tilde{\chi})^{\prime}\in\mathscr{E}({}^{\gamma}\tilde{\chi};\overline{A}^{\natural},\overline{A}_{0})}{\rm
Ind}_{\overline{A}_{0}}^{\overline{T}_{0}}(({}^{\gamma}\tilde{\chi})^{\prime}).$

Recall that $\overline{T}_{0}$ is a Heisenberg-type group and thus irreducible
genuine representation is determined by the central character. In particular,
the isomorphism class of ${\rm
Ind}_{\overline{A}_{0}}^{\overline{T}_{0}}(({}^{\gamma}\tilde{\chi})^{\prime})$
is in fact determined by the restriction of
$({}^{\gamma}\tilde{\chi})^{\prime}$ to $Z(\overline{T}_{0})$. We thus
consider

$\mathscr{E}(\chi,{}^{\gamma}\tilde{\chi}_{O};Z(\overline{T}_{0}))=\left\\{\begin{array}[]{cc}\omega_{\gamma,j}:Z(\overline{T}_{0})\longrightarrow\mathbf{C}^{\times}:\\\
\bullet\quad\omega_{\gamma,j}\text{ extends
}\chi|_{Z(\overline{T})\cap\overline{T}_{0}},\\\
\bullet\quad\omega_{\gamma,j}\text{ extends
}([\gamma,\cdot]^{-1}\tilde{\chi}_{O})|_{{\bf T}_{0}(O)\cap
Z(\overline{T}_{0})}.\end{array}\right\\}.$

We note that $\chi$ and $[\gamma,\cdot]^{-1}\tilde{\chi}_{O}$ are compatible
on the intersection of $Z(\overline{T})\cap\overline{T}_{0}$ and ${\bf
T}_{0}(O)\cap Z(\overline{T}_{0})$. It is easy to see that there is a natural
bijection

$\mathscr{E}(\chi,{}^{\gamma}\tilde{\chi}_{O};Z(\overline{T}_{0}))\longrightarrow\mathscr{E}({}^{\gamma}\chi;\overline{A}^{\natural},\overline{A}_{0})$

with size

$\left|\mathscr{E}(\chi,{}^{\gamma}\tilde{\chi}_{O};Z(\overline{T}_{0}))\right|=\left|Y_{0,Q,n}/(Y_{0}\cap
Y_{Q,n})\right|.$

This gives the bounds for the index $j$ in $\omega_{\gamma,j}$ as

$1\leqslant j\leqslant\left|Y_{0,Q,n}/(Y_{0}\cap Y_{Q,n})\right|.$

We also have

(2.18) ${\rm
Ind}_{\overline{A}^{\natural}}^{\overline{T}_{0}}({}^{\gamma}\tilde{\chi})=\bigoplus_{\omega_{\gamma,j}\in\mathscr{E}(\chi,{}^{\gamma}\tilde{\chi}_{O};Z(\overline{T}_{0}))}{\rm
Ind}_{\overline{A}_{0}}^{\overline{T}_{0}}(\tilde{\omega}_{\gamma,j})=\bigoplus_{\omega_{\gamma,j}\in\mathscr{E}(\chi,{}^{\gamma}\tilde{\chi}_{O};Z(\overline{T}_{0}))}i(\omega_{\gamma,j}),$

where in the middle term

$\tilde{\omega}_{\gamma,j}:\overline{A}_{0}\longrightarrow\mathbf{C}^{\times}$

is any extension of $\omega_{\gamma,j}$ to $\overline{A}_{0}$. Here
$i(\omega_{\gamma,j})$ denotes the isomorphism class of ${\rm
Ind}_{\overline{A}_{0}}^{\overline{T}_{0}}(\tilde{\omega}_{\gamma,j})$, which
only depends on the central character $\omega_{\gamma,j}$. Note that the
decomposition in (2.18) is multiplicity-free. We thus get

(2.19) $\left({\rm
Ind}_{\overline{A}}^{\overline{T}}(\tilde{\chi})\right)|_{\overline{T}_{0}}=\bigoplus_{\gamma\in\mathscr{X}_{Q,n}^{\Gamma}}\bigoplus_{\omega_{\gamma,j}\in\mathscr{E}(\chi,{}^{\gamma}\tilde{\chi}_{O};Z(\overline{T}_{0}))}i(\omega_{\gamma,j}),$

where we recall that

$\mathscr{X}_{Q,n}^{\Gamma}=Y/(Y_{0}+Y_{Q,n})$

by definition in (2.17).

To simplify the formula (2.19), we consider the group homomorphism

(2.20) ${\mathfrak{c}:\ \mathscr{X}_{Q,n}^{\Gamma}}$${{\rm Hom}({\bf
T}_{0}(O)\cap Z(\overline{T}_{0}),\mu_{n})}$

given by

$\mathfrak{c}(y)(k_{0}):=[y(\varpi),k_{0}]\in\mu_{n}.$

Define

$Y^{\mathfrak{c}}=\left\\{y\in Y:B_{Q}(y,y^{\prime})\in n\mathbf{Z}\text{ for
all }y^{\prime}\in Y_{0,Q,n}\right\\}\subset Y.$

Note that ${\bf T}_{0}(O)\cap Z(\overline{T}_{0})=Y_{0,Q,n}(O)$, and thus

$\mathfrak{c}(y)(y^{\prime}(u))=(\varpi,u)_{n}^{B_{Q}(y,y^{\prime})}$

for every $y^{\prime}\in Y_{0,Q,n}$. Hence,

$\mathfrak{c}(y)=\mathbbm{1}$

if and only if $y\in Y^{\mathfrak{c}}$. It is easy to see that

$Y_{0}+Y_{Q,n}\subset Y^{\mathfrak{c}}$

and thus we have the well-defined group

(2.21)
$\mathscr{X}_{Q,n}^{\mathfrak{c}}:=Y^{\mathfrak{c}}/(Y_{Q,n}+Y_{0})\subset\mathscr{X}_{Q,n}^{\Gamma}.$

We summarize the above discussion to give the following:

######  Theorem 2.24.

Assume $p\nmid n$. Let $i(\chi)={\rm
Ind}_{\overline{A}}^{\overline{T}}(\tilde{\chi})\in{\rm Irr}_{\rm
gen}(\overline{T})$ be with central character $\chi$. Let $I(\chi)$ be the
associated genuine principal series of $\overline{G}$.

  1. (i)

One has ${\rm Ker}(\mathfrak{c})=\mathscr{X}_{Q,n}^{\mathfrak{c}}$ and thus

$i(\chi)|_{\overline{T}_{0}}=\left|\mathscr{X}_{Q,n}^{\mathfrak{c}}\right|\cdot\bigoplus_{\gamma\in\mathscr{X}_{Q,n}^{\Gamma}/\mathscr{X}_{Q,n}^{\mathfrak{c}}}\bigoplus_{\omega_{\gamma,j}\in\mathscr{E}(\chi,{}^{\gamma}\tilde{\chi}_{O};Z(\overline{T}_{0}))}i(\omega_{\gamma,j}),$

where $i(\omega_{\gamma,j})$ appears with multiplicity one in the double sum
of the right hand side.

  2. (ii)

Consequently,

$I(\chi)|_{\overline{G}_{0}}=\left|\mathscr{X}_{Q,n}^{\mathfrak{c}}\right|\cdot\bigoplus_{\gamma\in\mathscr{X}_{Q,n}^{\Gamma}/\mathscr{X}_{Q,n}^{\mathfrak{c}}}\bigoplus_{\omega_{\gamma,j}\in\mathscr{E}(\chi,{}^{\gamma}\tilde{\chi}_{O};Z(\overline{T}_{0}))}I(\omega_{\gamma,j})$

for the restriction of genuine principal series $I(\chi)$. Here generically,
the $I(\omega_{\gamma,j})$’s appear multiplicity-free in the above
decompostion.

  3. (iii)

Assume that $I(\chi)$ is a $(K,s_{K})$-unramified genuine principal series.
Then $I(\omega_{\gamma,j})$ is $(K_{0},s_{K})$-unramified if and only if
$\omega_{\gamma,j}$ belongs to the set
$\mathscr{E}(\chi,\tilde{\chi}_{O};Z(\overline{T}_{0}))$, i.e., with
$\gamma=0\in\mathscr{X}_{Q,n}^{\Gamma}/\mathscr{X}_{Q,n}^{\mathfrak{c}}$ being
the trivial class.

######  Corollary 2.25.

Assume $p\nmid n$. If $(\overline{G},\overline{G}_{0})$ is an isotypic-pair,
then

$\mathscr{X}_{Q,n}^{\Gamma}=\mathscr{X}_{Q,n}^{\mathfrak{c}};$

in this case

$i(\chi)|_{\overline{T}_{0}}=i(\omega)^{\oplus\left|\mathscr{X}_{Q,n}^{\Gamma}\right|}\text{
and
}I(\chi)|_{\overline{G}_{0}}=I(\omega)^{\oplus\left|\mathscr{X}_{Q,n}^{\Gamma}\right|},$

are both isotypic sums with multiplicity
$\left|\mathscr{X}_{Q,n}^{\Gamma}\right|$, where
$\omega=\chi|_{Z(\overline{T}_{0})}$ is the unique character in
$\mathscr{E}(\chi,\tilde{\chi}_{O};Z(\overline{T}_{0}))$.

###### Proof.

Under the assumption, we have $Y^{\mathfrak{c}}=Y$ and thus
$\mathscr{X}_{Q,n}^{\mathfrak{c}}=\mathscr{X}_{Q,n}^{\Gamma}$. The rest is
clear in view of Theorem 2.24. ∎

######  Remark 2.26.

If $I(\chi)$ is $(K,s_{K})$-unramified, then it is $(\gamma\cdot K,\gamma\cdot
s_{K})$-unramified as well. Here $\gamma\cdot K=\gamma K\gamma^{-1}$ and
$\gamma\cdot s_{K}$ is a splitting of $\overline{G}$ over $\gamma\cdot K$
given by

$(\gamma\cdot s_{K})(\gamma\cdot k):=\gamma\cdot s_{K}(k)\cdot\gamma^{-1}.$

Then every $I(\omega_{\gamma,j})$ is in fact $(\gamma\cdot K_{0},\gamma\cdot
s_{K})$-unramified. Note that $\gamma\cdot K_{0}$ and $K_{0}$ may not be in
the same $G_{0}$-conjugacy classes of maximal compact subgroup of $G_{0}$. See
§5.2 for a general discussion of this, and the reader may also refer to §5.4
and §6 for examples.

##  3\. L-parameters and functoriality

We continue to assume in this section that $Z({\bf G})$ is connected as in
§2.3.

###  3.1. Relations among several L-groups

Recall that from Lemma 2.7 the group $Z(\overline{G})$ is associated with the
sublattice

$Y_{z}:=Y_{Q,n}\cap Y_{c}.$

We have the L-group extension

${Z(\overline{G})^{\vee}}$${{}^{L}Z(\overline{G})}$${\text{WD}_{F}}$

where $Z(\overline{G})^{\vee}:={\rm Hom}(Y_{z},\mathbf{C}^{\times})$ denotes
the dual group of $Z(\overline{G})$. Similarly, we have an L-group for
$\overline{Z(G)}$ as an extension

${\overline{Z(G)}^{\vee}}$${{}^{L}\overline{Z(G)}}$${\text{WD}_{F}.}$

Here $\overline{Z(G)}^{\vee}:={\rm Hom}(Y_{c,Q,n},\mathbf{C}^{\times})$. It is
easy to see that

(3.1) $Y_{z}\subset Y_{c,Q,n}$

and thus there is a natural homormorphism

(3.2) $f_{c,z}:{}^{L}\overline{Z(G)}\longrightarrow{}^{L}Z(\overline{G})$

such that

$f_{c,z}|_{\overline{Z(G)}^{\vee}}:\overline{Z(G)}^{\vee}\longrightarrow
Z(\overline{G})^{\vee}$

is just the restriction map

${\rm Hom}(Y_{c,Q,n},\mathbf{C}^{\times})\longrightarrow{\rm
Hom}(Y_{z},\mathbf{C}^{\times})$

induced from (3.1).

On the other hand, the dual group for $\overline{G}$ has root datum
$(Y_{Q,n},\Phi_{Q,n}^{\vee};X_{Q,n},\Phi_{Q,n})$, where $Y_{Q,n}$ is the
character lattice for $\overline{G}^{\vee}$. We see that there is a natural
homomorphism

$f_{G,z}:\overline{G}^{\vee}\longrightarrow Z(\overline{G})^{\vee}$

which extends to a homomorphism of L-groups

(3.3) $f_{G,z}:{}^{L}\overline{G}\longrightarrow{}^{L}Z(\overline{G}),$

depending on the choice of a distinguished character $\chi_{\psi}$. We give
some details of the map $f_{G,z}$ as follows. Let ${\bf G}_{Q,n}$ be the split
linear algebraic group over $F$ such that the Langlands dual group of
$G_{Q,n}={\bf G}_{Q,n}(F)$ is equal to $\overline{G}^{\vee}$, i.e.,

$G_{Q,n}^{\vee}=\overline{G}^{\vee}.$

One can restrict the Brylisnki–Deligne data $(D,\eta)$ to the cocharacter
lattice $Y_{Q,n}$ of $G_{Q,n}$ and to the coroot lattice $Y_{Q,n}^{sc}$
respectively to obtain an $n$-fold cover $\overline{G}_{Q,n}$. As the
restriction of $B_{Q}$ to $Y_{Q,n}$ is trivial modulo $n$, the cover
$\overline{G}_{Q,n}$ is almost a trivial cover over $G_{Q,n}$. In particular,
the covering torus $\overline{T}_{Q,n}\subset\overline{G}_{Q,n}$ is abelian.
In fact, we have

$\overline{G}_{Q,n}^{\vee}=G_{Q,n}^{\vee}=\overline{G}^{\vee},$

where $\overline{G}_{Q,n}^{\vee}$ is the dual group for the $n$-fold cover
$\overline{G}_{Q,n}$. One also has

$Z(\overline{G}_{Q,n})=\overline{Z(G_{Q,n})}\cap
Z(\overline{T}_{Q,n})=\overline{Z(G_{Q,n})}.$

Recall the isogeny

$i_{Q,n}:T_{Q,n}\longrightarrow T$

induced from the inclusion $Y_{Q,n}\subset Y$. By pull-back, one naturally has
the map (by abuse of notation) denoted by

$i_{Q,n}:\overline{T}_{Q,n}\longrightarrow\overline{T}.$

######  Lemma 3.1.

Keeping notations as above, we have

(3.4) $i_{Q,n}(Z(\overline{G}_{Q,n}))=Z(\overline{G}).$

Hence, $Z(\overline{G})^{\vee}=Z(G_{Q,n})^{\vee}$ and the map
$f_{G,z}:\overline{G}^{\vee}\longrightarrow Z(\overline{G})^{\vee}$ is just

$f_{G,z}:\overline{G}^{\vee}\longrightarrow\overline{G}^{\vee}/[\overline{G}^{\vee},\overline{G}^{\vee}],$

where $[\overline{G}^{\vee},\overline{G}^{\vee}]\subset\overline{G}^{\vee}$ is
the derived subgroup.

###### Proof.

For simplicity, writing $i$ for $i_{Q,n}$. One has a commutative diagram

${X/X^{sc}}$${X}$${X_{Q,n}/X_{Q,n}^{sc}}$${X_{Q,n},}$

which gives the commutative diagram

${Z(G)}$${T}$${Z(G_{Q,n})}$${T_{Q,n}.}$$\scriptstyle{i}$$\scriptstyle{i}$

It follows that $i(Z(\overline{G}_{Q,n}))\subset Z(\overline{G})$. To show the
other inclusion, we consider the exact sequence

${Z(G_{Q,n})}$${T_{Q,n}}$${{\rm Hom}(X_{Q,n}^{sc},F^{\times})}$${{\rm
Ext}(X_{Q,n}/X_{Q,n}^{sc},F^{\times})}$${Y_{z}\otimes
F^{\times},}$$\scriptstyle{r}$$\scriptstyle{i_{z}}$

where $i_{z}$ is the map induced from the inclusion $Y_{z}\subset Y_{Q,n}$. It
suffices to show that $r\circ i_{z}=0$. For every $y\otimes a\in Y_{z}\otimes
F^{\times}$ and $x\in X_{Q,n}^{sc}$, one has

$r\circ i_{z}(y\otimes a)(x)=a^{{\left\langle x,y\right\rangle}},$

where ${\left\langle-,-\right\rangle}:X_{Q,n}\times
Y_{Q,n}\longrightarrow\mathbf{Z}$ denotes the canonical pairing. However,
since $x\in X_{Q,n}^{sc}$ and $y\in Y_{c}\cap Y_{Q,n}$, we have ${\left\langle
x,y\right\rangle}=0$. This shows that $r\circ i_{z}=0$ and completes the proof
of (3.4). It follows that $Z(\overline{G})^{\vee}=Z(G_{Q,n})^{\vee}$; as
$\overline{G}^{\vee}=G_{Q,n}^{\vee}$, the rest is clear. ∎

To extend $f_{G,z}$ to be an L-homomorphism. We choose a distinguished
character

$\chi_{\psi}:Z(\overline{T})\longrightarrow\mathbf{C}^{\times}$

which gives an isomorphism
${}^{L}\overline{G}\simeq\overline{G}^{\vee}\times\text{WD}_{F}$, see §2.1.3.
Note that by pull-back to $\overline{T}_{Q,n}\subset\overline{G}_{Q,n}$ and
further restriction to $Z(\overline{G}_{Q,n})$, one has a distinguished
genuine character of $Z(\overline{G}_{Q,n})$, with respect to which we also
have

${}^{L}Z(\overline{G})={}^{L}Z(\overline{G}_{Q,n})=Z(\overline{G})^{\vee}\times\text{WD}_{F}.$

From this, we have an L-map as in (3.3).

Next, we want to understand the relation between ${}^{L}\overline{G}$ and
${}^{L}\overline{G}_{0}$. Our exposition here follows closely that of [GG,
§12.3]. Let $Y_{0}$ be the cocharacter lattice of $G_{0}$. One has

$Y^{sc}\subset Y_{0}\subset Y.$

The root datum of $G_{0}$ is

$(X_{0},\Phi;Y_{0},\Phi^{\vee}),$

where $X_{0}\supset X^{sc}$ is the character lattice. By pull-back, one has a
cover $\overline{G}_{0}$ of $G_{0}$. Then dual group $\overline{G}_{0}^{\vee}$
of $\overline{G}_{0}$ has root datum

$(Y_{0,Q,n},\Phi_{Q,n}^{\vee};X_{0,Q,n},\Phi_{Q,n}).$

Unlike the linear algebraic case, there might be no group homomorphism from
$\overline{G}^{\vee}$ to $\overline{G}_{0}^{\vee}$ in general. However, these
two groups are related as follows.

First, we have

$Y_{0}\cap Y_{Q,n}\subset Y_{0,Q,n}.$

Let ${\bf H}$ be the split algebraic group over $F$ whose Langlands dual group
$H^{\vee}$ has root datum

$(Y_{0}\cap Y_{Q,n},\ \Phi_{Q,n}^{\vee};\ \text{dual lattice of }Y_{0}\cap
Y_{Q,n},\ \Phi_{Q,n}).$

We obtain two homomorphisms of algebraic groups valued in $H^{\vee}$

(3.5) $\overline{G}^{\vee}\longrightarrow
H^{\vee}\longleftarrow\overline{G}_{0}^{\vee}.$

By restricting the Brylinski–Deligne data $D$ and $\eta$ to $Y_{0}\cap
Y_{Q,n}$ and $Y_{Q,n}^{sc}$ respectively, one obtains an $n$-fold cover
$\overline{H}$ of $H$ such that

$\overline{H}^{\vee}\simeq H^{\vee}\simeq\overline{G}^{\vee}.$

Suppose there exist distinguished genuine characters for $\overline{G}$ and
$\overline{G}_{0}$, then we can extend the homomorphism in (3.5) to obtain two
homomorphisms of L-groups,

$f_{G,H}:{}^{L}\overline{G}\longrightarrow{}^{L}\overline{H}$

and

$f_{G_{0},H}:{}^{L}\overline{G}_{0}\longrightarrow{}^{L}\overline{H}.$

Note that if $(\overline{G},\overline{G}_{0})$ is an isotypic-pair, then

$\overline{G}_{0}^{\vee}=\overline{H}^{\vee}.$

In particular, this is true when $\overline{G}_{0}$ is a saturated cover,
i.e., when $\overline{G}_{0}^{\vee}$ is of adjoint type; in this case,
$f_{G_{0},H}$ is an isomorphism.

###  3.2. L-parameters and some speculations

To every parameter

$\phi:\text{WD}_{F}\longrightarrow{}^{L}\overline{G}$

we associate the group of connected components of the centralizer of ${\rm
Im}(\phi)$:

$\mathcal{S}_{\phi}=\mathcal{S}(\phi):=\pi_{0}\left({\rm
Cent}_{\overline{G}^{\vee}}({\rm Im}(\phi)/Z(\overline{G}^{\vee}))\right).$

Let $\Phi({}^{L}\overline{G})$ be the set of $\overline{G}^{\vee}$-conjugacy
classes of L-parameters of $\text{WD}_{F}$ valued in ${}^{L}\overline{G}$. Let
$\Phi^{\rm en}({}^{L}\overline{G})$ be the set of enhanced parameters of the
form

$(\phi,\theta)\text{ with }\phi\in\Phi({}^{L}\overline{G})\text{ and
}\theta\in{\rm Irr}(\mathcal{S}_{\phi}).$

We assume that there is a local Langlands correspondence

${{\rm Irr}_{\rm gen}(\overline{G})}$${\Phi^{\rm en}({}^{L}\overline{G}),}$

assigning to every $\pi\in{\rm Irr}_{\rm gen}(\overline{G})$ a pair
$(\phi_{\pi},\theta_{\pi})\in\Phi^{\rm en}({}^{L}\overline{G})$. For every
$\phi\in\Phi({}^{L}\overline{G})$, denote by

$\mathcal{L}(\phi):=\left\\{\pi\in{\rm Irr}_{\rm gen}(\overline{G}):\
\phi_{\pi}=\phi\right\\}\subset{\rm Irr}_{\rm gen}(\overline{G})$

the hypothetical L-packet associated with $\phi$. Besides the one for linear
algebraic groups given in [Bor], there are several additional desiderata for
such a correspondence as given in [GG, §12], some of which we will also
explain below. These desiderata encode natural properties expected on both the
representation and parameter side.

We expect on the parameter side relations among various L-groups, and also the
compatibility of certain L-parameters, as depicted in the following diagram.

(3.6)
${{}^{L}\overline{Z(G)}}$${{}^{L}Z(\overline{G})}$${\text{WD}_{F}}$${{}^{L}\overline{G}}$${{}^{L}\overline{G}_{0}}$${{}^{L}H.}$$\scriptstyle{f_{c,z}}$$\scriptstyle{\phi_{\pi}}$$\scriptstyle{\phi_{\tau}}$$\scriptstyle{\phi_{\rho}}$$\scriptstyle{f_{G,z}}$$\scriptstyle{f_{G,H}}$$\scriptstyle{f_{G_{o},H}}$

To explain this, for every $\phi\in\Phi({}^{L}\overline{G})$, we denote

$\phi^{\diamondsuit}=f_{G,H}\circ\phi$

and set

$\Phi({}^{L}\overline{G}_{0};\phi)=\left\\{\phi_{0}\in\Phi({}^{L}\overline{G}_{0}):\
f_{G_{0},H}\circ\phi_{0}=\phi^{\diamondsuit}\right\\}.$

Consider the induced homomorphism

$\mathcal{S}_{\phi}\hookrightarrow\mathcal{S}_{\phi^{\diamondsuit}},$

which is actually injective (see [Sol20, Proposition 5.4]). For every
$\phi_{0}\in\Phi({}^{L}\overline{G}_{0};\phi)$, there is also an embedding

$\mathcal{S}_{\phi_{0}}\hookrightarrow\mathcal{S}_{\phi^{\diamondsuit}}.$

The following are some (refined) speculations from [GG, §12.1–§12.3] on
central characters and restriction of representations to the derived subgroup.

######  Conjecture 3.2.

Let $\pi\in{\rm Irr}_{\rm gen}(\overline{G})$ be with central character
$\omega_{\pi}$ and associated enhanced parameter
$(\phi_{\pi},\theta_{\pi})\in\Phi^{\rm en}({}^{L}\overline{G})$. Let
$\tau\boxtimes\rho$ be an irreducible representation of
$\overline{G}^{\dagger}=\overline{Z(G)}\cdot\overline{G}_{0}$ that occurs in
the restriction of $\pi$.

  1. (i)

The L-parameters $\phi_{\omega_{\pi}},\phi_{\tau}$ and $\phi_{\rho}$ satisfy
the following:

    1. $\bullet$

$f_{G,z}\circ\phi_{\pi}=f_{c,z}\circ\phi_{\tau}$ and is equal to the parameter
$\phi_{\omega_{\pi}}$ associated with the central character $\omega_{\pi}$ of
$\pi$;

    2. $\bullet$

$\phi_{\pi}^{\diamondsuit}=f_{G_{0},H}\circ\phi_{\rho}$.

  2. (ii)

There exists $e(\pi)\in\mathbf{N}$ such that for every $\tau\in{\rm Irr}_{\rm
gen}(\overline{G}_{0})$, one has

$\dim{\rm Hom}_{\overline{G}_{0}}(\pi,\tau)=\begin{cases}0&\text{ if
}\phi_{\tau}\notin\Phi({}^{L}\overline{G}_{0};\phi_{\pi}),\\\
e(\pi)\cdot{\left\langle{\rm
Ind}_{\mathcal{S}_{\phi_{\pi}}}^{\mathcal{S}_{\phi^{\diamondsuit}}}\theta_{\pi},{\rm
Ind}_{\mathcal{S}_{\phi_{\tau}}}^{\mathcal{S}_{\phi^{\diamondsuit}}}\theta_{\tau}\right\rangle}_{\mathcal{S}_{\phi^{\diamondsuit}}}&\text{
if }\phi_{\tau}\in\Phi({}^{L}\overline{G}_{0};\phi_{\pi}),\end{cases}$

where ${\left\langle-,-\right\rangle}$ denotes the pairing of two
representations of $\mathcal{S}_{\phi^{\diamond}}$. In particular, if
$(\overline{G},\overline{G}_{0})$ is an isotypic-pair, then

$\pi|_{\overline{G}_{0}}=e(\pi)\cdot\bigoplus_{\tau\in\mathcal{L}(\phi_{\pi}^{\diamondsuit})}{\left\langle\theta_{\tau}|_{\mathcal{S}_{\phi_{\pi}}},\theta_{\pi}\right\rangle}_{\mathcal{S}_{\phi_{\pi}}}\cdot\tau,$

where $(\phi_{\pi}^{\diamondsuit},\theta_{\tau})\in\Phi^{\rm
en}({}^{L}\overline{G}_{0})$ is the enhanced parameter associated with
$\tau\in\mathcal{L}(\phi_{\pi}^{\diamondsuit})$.

  3. (iii)

Let $\tau\in{\rm Irr}_{\rm gen}(\overline{Z(G)})$ and $\rho\in{\rm Irr}_{\rm
gen}(\overline{G}_{0})$ be compatible representations. Then every irreducible
constituent $\pi\in{\rm Irr}_{\rm gen}(G)$ of ${\rm
Ind}_{\overline{G}^{\dagger}}^{\overline{G}}(\tau\boxtimes\rho)$ has a
parameter $\phi_{\pi}$ which fits into a commutative diagram (3.6).

Part (ii) is clearly motivated from the case of linear algebraic groups,
especially the work of Silberger [Sil79], Keys [Key3], Gelbart–Knapp [GeKn1,
GeKn2], Adler–Prasad [AP1]; see also that of Ban, Choiy and Goldberg [BCG18,
Cho19].

###  3.3. Functoriality for genuine principal series

We first briefly recall the local Langlands correspondence for covering tori.
The description here follows closely [We6, §10]. It gives the construction of
the L-group ${}^{L}\overline{T}$ of $\overline{T}$ and establish the local
Langlands correspondence for $\overline{T}$ along the same line of the
construction.

Let $T=Y\otimes F^{\times}$ be a linear torus and $\overline{T}$ an $n$-fold
cover associated with a quadratic form $Q:Y\to\mathbf{Z}$. Let

$i_{Q,n}:T_{Q,n}\to T$

be the isogeny induced from the inclusion $Y_{Q,n}\subset Y$. Recall that the
preimage of

$T^{\dagger}:={\rm Im}(i_{Q,n})\subset T$

inside $\overline{T}$ is just the center $Z(\overline{T})$. Using the fixed
embedding $\varepsilon:\mu_{n}\hookrightarrow\mathbf{C}^{\times}$ we obtain
the push-out $\varepsilon_{*}(Z(\overline{T}))$ of $Z(\overline{T})$. At the
same time, any genuine character $\chi$ of $Z(\overline{T})$ gives rise to a
splitting

${s_{\chi}:\varepsilon_{*}(Z(\overline{T}))}$${\mathbf{C}^{\times}}$

given by

${s_{\chi}:[(z,\overline{t})]}$${z\cdot\chi^{-1}(\overline{t}),}$

where $[(z,\overline{t})]$ denote the class of

$(z,\overline{t})\in\mathbf{C}^{\times}\times Z(\overline{T})$

in $\varepsilon_{*}(Z(\overline{T}))$. We illustrate this by using the
following diagram

${\mu_{n}}$${Z(\overline{T})}$${T^{\dagger}}$${\mathbf{C}^{\times}}$${\varepsilon_{*}(Z(\overline{T}))}$${T^{\dagger}.}$$\scriptstyle{\varepsilon}$$\scriptstyle{\chi}$$\scriptstyle{s_{\chi}}$

Clearly $s_{\chi}$ entails a splitting $\phi_{\chi}$ as in

${\mathbf{C}^{\times}}$${\varepsilon_{*}(Z(\overline{T}))}$${T^{\dagger}}$$\scriptstyle{\phi_{\chi}}$

given by

${\phi_{\chi}:\ t}$${{[(\chi(\overline{t}),\overline{t})]},}$

where $\overline{t}\in Z(\overline{T})$ is any lifting of
$t\in\overline{T}^{\dagger}$.

By definition

$\overline{T}^{\vee}=X_{Q,n}\otimes\mathbf{C}^{\times},$

where $X_{Q,n}={\rm Hom}(Y_{Q,n},\mathbf{Z})$ is the lattice dual to
$Y_{Q,n}$. By abuse of notation, we still use $i_{Q,n}$ to denote the
naturally induced map

$i_{Q,n}:X_{Q,n}\otimes T_{Q,n}\longrightarrow X_{Q,n}\otimes T^{\dagger}.$

Consider the composite

${\mathfrak{m}:\text{WD}_{F}}$${F^{\times}}$${X_{Q,n}\otimes T_{Q,n}\simeq{\rm
Hom}(Y_{Q,n},T_{Q,n}),}$$\scriptstyle{\rm rec}$$\scriptstyle{f}$

where the first map is the reciprocity map of class field theory sending a
geometric Frobenius to the uniformizer $\varpi\in F^{\times}$ and trivial on
${\rm SL}_{2}(\mathbf{C})\subset\text{WD}_{F}$, and the second map is given by

$f(a)(y)=y\otimes{\rm rec}(a),\ y\in Y_{Q,n}.$

The L-group ${}^{L}\overline{T}$ is defined to be the pull-back of
$X_{Q,n}\otimes\varepsilon_{*}(Z(\overline{T}))$ via
$i_{Q,n}\circ\mathfrak{m}$:

(3.7)
${\overline{T}^{\vee}}$${X_{Q,n}\otimes\varepsilon_{*}(Z(\overline{T}))}$${X_{Q,n}\otimes
T^{\dagger}}$${\overline{T}^{\vee}}$${{}^{L}\overline{T}}$${\text{WD}_{F}.}$$\scriptstyle{\phi_{\chi}}$$\scriptstyle{i_{Q,n}\circ\mathfrak{m}}$$\scriptstyle{\phi_{\chi}}$

Here, the bottom splitting $s_{\chi}$ of ${}^{L}\overline{T}$ over
$\text{WD}_{F}$ is the one inherited from the splitting of the top extension.

The above construction

${{\rm Irr}_{\rm gen}(\overline{T})}$${\Phi({}^{L}\overline{T})}$

given by

$i(\chi)\mapsto\phi_{\chi}$

is the local Langlands correspondence (LLC) for covering torus, which is an
injective map.

We want to obtain functoriality of LLC with respect to the restriction to a
subgroup of $\overline{T}$, which is associated with a $W$-stable lattice
$J\subset Y$. Note that $J_{Q,n}$ may not be a sublattice of $Y_{Q,n}$ in
general, and we have

$J_{Q,n}\supset(J\cap Y_{Q,n})\subset Y_{Q,n}.$

Let $T_{J}$ be the torus associated with $J$ and $T_{J,\natural}$ the torus
associated with $J\cap Y_{Q,n}$. We have a relation

${Z(\overline{T}_{J})}$${Z(\overline{T}_{J,\natural})=\overline{T}_{J,\natural}}$${Z(\overline{T}).}$

######  Lemma 3.3.

Let $i(\chi_{J})\in{\rm Irr}_{\rm gen}(\overline{T}_{J})$ and $i(\chi)\in{\rm
Irr}_{\rm gen}(\overline{T})$. Then $i(\chi_{J})\subset
i(\chi)|_{\overline{T}_{J}}$ if and only if $\chi_{J}$ and $\chi$ agree on
$\overline{T}_{J,\natural}$, or equivalently, the following diagram

${Z(\overline{T}_{J})}$${\overline{T}_{J,\natural}}$${Z(\overline{T})}$${\mathbf{C}^{\times}}$$\scriptstyle{\chi_{J}}$$\scriptstyle{\chi}$

commutes.

###### Proof.

The only if part is clear, and it suffices to prove the if part. Assume
$\chi_{J}$ and $\chi$ agree on $\overline{T}_{J,\natural}$, then we get a
genuine character

$\chi_{J}\otimes\chi:\ Z(\overline{T}_{0})\cdot
Z(\overline{T})\longrightarrow\mathbf{C}^{\times}.$

Let $\overline{A}_{0}\subset\overline{T}_{0}$ be a maximal abelian subgroup of
$\overline{T}_{0}$. We extend $\chi_{J}\otimes\chi$ to a character

$\chi_{0}^{\prime}\otimes\chi:\ \overline{A}_{0}\cdot
Z(\overline{T})\longrightarrow\mathbf{C}^{\times},$

where

$\chi_{0}^{\prime}:\overline{A}_{0}\longrightarrow\mathbf{C}^{\times}$

is an extension of $\chi_{J}$. We can find a maximal abelian subgroup
$\overline{A}\subset\overline{T}$ containing $\overline{A}_{0}\cdot
Z(\overline{T})$, and let

$\chi^{\prime}:\ \overline{A}\longrightarrow\mathbf{C}^{\times}$

be extending $\chi_{0}^{\prime}\otimes\chi$. Note that up to isomorphism
class, we have

$i(\chi_{J})\simeq{\rm
Ind}_{\overline{A}_{0}}^{\overline{T}_{0}}(\chi_{0}^{\prime})\text{ and
}i(\chi)\simeq{\rm Ind}_{\overline{A}}^{\overline{T}}(\chi^{\prime}).$

Now Frobenius reciprocity gives that

${\rm Hom}_{\overline{T}_{0}}(i(\chi_{J}),i(\chi))={\rm
Hom}_{\overline{A}_{0}}(\chi_{0}^{\prime},{\rm
Ind}_{\overline{A}}^{\overline{T}}(\chi^{\prime}))\neq 0,$

as $\chi^{\prime}$ extends $\chi_{0}^{\prime}\otimes\chi$. This shows that
$i(\chi_{J})\subset i(\chi)|_{\overline{T}_{0}}$. ∎

If $\chi_{J}$ and $\chi$ agree on $\overline{T}_{J,\natural}$, then we denote
$\chi_{J,\natural}=\chi|_{\overline{T}_{J,\natural}}$. By the construction of
L-groups, we have

${{}^{L}\overline{T}_{J}}$${{}^{L}\overline{T}_{J,\natural}}$${{}^{L}\overline{T}.}$

If $\chi$ and $\chi_{J}$ are compatible, then by the LLC for covering torus,
the two parameters $\phi_{\chi_{J}}$ and $\phi_{\chi}$ are compatible, i.e.,
the following diagram commutes:

(3.8)
${{}^{L}\overline{T}_{J}}$${{}^{L}\overline{T}_{J,\natural}}$${{}^{L}\overline{T}}$${\text{WD}_{F},}$$\scriptstyle{\phi_{\chi_{J}}}$$\scriptstyle{\phi_{\chi}}$

where the middle vertical arrow is the parameter associated with
$\chi_{J,\natural}$.

######  Proposition 3.4.

Let $\pi:=I(\chi)$ and $\rho:=I(\chi_{0})$ be irreducible genuine principal
series of $\overline{G}$ and $\overline{G}_{0}$ respectively. Let $\tau\in{\rm
Irr}_{\rm gen}(\overline{Z(G)})$.

  1. (i)

If $\tau\boxtimes\rho\in{\rm Irr}(\overline{G}^{\dagger})$ occurs in
$I(\chi)|_{\overline{G}^{\dagger}}$, then the L-parameters $\phi_{\tau}$ and
$\phi_{\rho}$ are such that the diagram in (3.6) commutes. Moreover, if
${}^{L}\overline{G}_{0}={}^{L}\overline{H}$, then
$I(\chi)|_{\overline{G}_{0}}$ is an isotypic sum of a genuine principal series
of $\overline{G}_{0}$.

  2. (ii)

The parameter $\phi_{\pi}$ of every irreducible constituent $\pi$ of ${\rm
Ind}_{\overline{G}^{\dagger}}^{\overline{G}}(\tau\boxtimes\rho)$ satisfies a
commutative diagram (3.6) involving $\phi_{\tau}$ and $\phi_{\rho}$.

###### Proof.

For (i), we first consider $\phi_{\pi}$ and $\phi_{\tau}$. Note that we have a
natural map ${}^{L}\overline{T}\longrightarrow{}^{L}\overline{G}$, see §2.1.3.
Taking $J=Y_{c}$, we thus have a commutative diagram from (3.8):

(3.9)
${{}^{L}\overline{Z(G)}}$${\text{WD}_{F}}$${{}^{L}Z(\overline{G})}$${{}^{L}\overline{T}}$${{}^{L}\overline{G}.}$$\scriptstyle{\phi_{\tau}}$$\scriptstyle{\phi_{\chi}}$$\scriptstyle{\phi_{\omega_{\pi}}}$$\scriptstyle{f_{G,z}}$

Similarly, taking $J=Y_{0}$ coupled with (3.8) give another diagram:

(3.10)
${{}^{L}\overline{T}}$${{}^{L}\overline{G}}$${\text{WD}_{F}}$${{}^{L}\overline{T}_{Y_{0},\natural}}$${{}^{L}\overline{H}}$${{}^{L}\overline{T}_{0}}$${{}^{L}\overline{G}_{0},}$$\scriptstyle{f_{G,H}}$$\scriptstyle{\phi_{\chi}}$$\scriptstyle{\phi_{\chi_{J}}}$$\scriptstyle{\phi_{\chi_{0}}}$$\scriptstyle{f_{G_{0},H}}$

where we have $\phi_{\chi_{0}}:=\phi_{\chi_{J}}$. The above two diagrams show
the commutativity of (3.6). If ${}^{L}\overline{G}_{0}={}^{L}\overline{H}$,
then $\phi_{\chi_{0}}$ is uniquely determined by $\phi_{\chi}$, and thus

$i(\chi)|_{\overline{T}_{0}}=e\cdot i(\chi_{0})$

is an isotypic sum of $i(\chi_{0})\in\overline{T}_{0}$. This shows that
$I(\chi)|_{\overline{G}_{0}}=e\cdot I(\chi_{0})$ is an isotypic sum. The proof
of (i) is completed.

For (ii), assume $\pi=I(\chi)\subset{\rm
Ind}_{\overline{G}^{\dagger}}^{\overline{G}}(\tau\boxtimes I(\chi_{0}))$, it
is easy to see that (3.9) commutes. On the other hand, using Frobenius
reciprocity, one has $I(\chi_{0})\subset I(\chi)|_{\overline{G}_{0}}$ and thus
$i(\chi_{0})\subset i(\chi)$. The commutativity of (3.10) then follows from
Lemma 3.3. This concludes the proof. ∎

###  3.4. Metaplectic tensor product for $\overline{{\rm GSp}}_{2r}$

In this subsection, we investigate for $\overline{{\rm GSp}}_{2r}$ an analogue
of the metaplectic tensor product construction for Kazhdan–Patterson covers
studied in [Mez04, Tak3, Tak4, Cai1]. The parameter side interpretation for
such a construction is given in [Gan17] using the formalism of L-groups as in
[We3, We6]. More precisely, consider a Levi subgroup

$M={\rm GL}_{r_{1}}\times{\rm GL}_{r_{2}}\times...\times{\rm
GL}_{r_{k}}\subset{\rm GL}_{r}.$

It is well-known (by checking on the covering tori of ${\rm GL}_{r_{i}}$’s for
example) that the blocks $\overline{{\rm GL}}_{r_{i}}\subset\overline{M}$ do
not commute. Thus, the representation of $\overline{M}$ can not be simply
reduced to that of each $\overline{{\rm GL}}_{r_{i}}$. However, given with
$\pi_{i}\in{\rm Irr}_{\rm gen}(\overline{{\rm GL}}_{r_{i}})$ satisfying
certain condition constrained by the central character of $Z(\overline{G})$,
Mezo [Mez04] gave a natural construction of a representation of
$\overline{M}$. Coarsely, the construction goes through several steps as
follows:

  1. (1)

For every $i$ and $n\in\mathbf{N}$, we denote

${\rm GL}_{r_{i}}^{\left\langle n\right\rangle}=\left\\{g\in{\rm
GL}_{r_{i}}:\det(g)\in F^{\times n}\right\\}.$

One can check that for $i\neq j$, the two covering groups $\overline{{\rm
GL}}_{r_{i}}^{\left\langle n\right\rangle}$ and $\overline{{\rm
GL}}_{r_{j}}^{\left\langle n\right\rangle}$ commute with each other. Hence one
can define

$\overline{M}^{\left\langle n\right\rangle}:=\overline{{\rm
GL}}_{r_{1}}^{\left\langle
n\right\rangle}\times_{\mu_{n}}\times...\times_{\mu_{n}}\overline{{\rm
GL}}_{r_{k}}^{\left\langle n\right\rangle}.$

For each $\pi_{i}\in\overline{{\rm GL}}_{r_{i}}$, let $\sigma_{i}\in{\rm
Irr}_{\rm gen}(\overline{{\rm GL}}_{r_{i}}^{\left\langle n\right\rangle})$ be
an irreducible summand in the restriction of $\pi_{i}$. We have a
representation

$\sigma:=\sigma_{1}\boxtimes...\boxtimes\sigma_{k}$

of $\overline{M}^{\left\langle n\right\rangle}\subset\overline{M}$.

  2. (2)

Pick an irreducible genuine character $\omega:Z(\overline{{\rm
GL}}_{r})\longrightarrow\mathbf{C}^{\times}$ such that

(3.11) $\omega=\sigma\text{ on }Z(\overline{{\rm
GL}}_{r})\cap\overline{M}^{\left\langle n\right\rangle}.$

This gives an irreducible representation $\omega\boxtimes\sigma$ of
$Z(\overline{{\rm GL}}_{r})\cdot\overline{M}^{\left\langle n\right\rangle}$.

  3. (3)

One extends $\omega\boxtimes\sigma$ as much as possible to a representation
$(\omega\boxtimes\sigma)^{\prime}$ of a subgroup $\overline{M}^{\prime}$ such
that the Mackey’s irreducibility criteria are satisfied. We thus obtain

$\tilde{\otimes}_{i}\pi_{i}:={\rm
Ind}_{\overline{M}^{\prime}}^{\overline{M}}((\omega\boxtimes\sigma)^{\prime}),$

which is an irreducible representation of $\overline{M}$.

It is shown in [Mez04] that the representation $\tilde{\otimes}_{i}\pi_{i}$
depends only on $\pi_{i}$’s and $\omega$, and is independent of the
intermediate choices $\sigma_{i}$, $\overline{M}^{\prime}$ and
$(\omega\boxtimes\sigma)^{\prime}$. We highlight that such independence relies
on the crucial property that for every $\pi\in{\rm Irr}_{\rm
gen}(\overline{M})$, one has

${\rm supp}(\Theta_{\pi})\subset Z(\overline{{\rm
GL}}_{r})\cdot\overline{M}^{\left\langle n\right\rangle};$

thus, it follows from (the proof of) Proposition 2.4 that ${\rm
Ind}_{\overline{M}^{\left\langle
n\right\rangle}}^{\overline{M}}(\omega\boxtimes\sigma)$ is an isotypic sum of
an irreducible representation of $\overline{M}$, which is exactly
$\tilde{\otimes}_{i}\pi_{i}$ in (3) above. This observation was explicated in
[Tak3, Proposition 4.6]. In view of this, we could replace (3) above by the
following:

  1. (3)’

One has

${\rm Ind}_{\overline{M}^{\left\langle
n\right\rangle}}^{\overline{M}}(\omega\boxtimes\sigma)=m\cdot\pi\text{ for a
certain }\pi\in{\rm Irr}_{\rm gen}(\overline{M}).$

Now the representation $\tilde{\otimes}_{i}\pi_{i}:=\pi\in{\rm Irr}_{\rm
gen}(\overline{M})$ is the one we seek.

The above construction gives a well-defined surjective map (for the
surjectivity, see [Tak3, Lemma 4.4])

${\tilde{\otimes}:({\rm Irr}_{\rm gen}(\overline{{\rm
GL}}_{r_{1}})\times...\times{\rm Irr}_{\rm gen}(\overline{{\rm
GL}}_{r_{k}})\times{\rm Irr}_{\rm gen}(Z(\overline{{\rm
GL}}_{r})))^{\heartsuit}}$${{\rm Irr}_{\rm gen}(\overline{M}).}$

The superscript $(-)^{\heartsuit}$ indicates the subset of

$(\pi_{1},...,\pi_{r},\omega)$

satisfying the relation (3.11). For every character $\chi_{i}$ of $F^{\times}$
which is trivial on $F^{\times n}$, replacing $\pi$ by
$\pi\otimes(\chi_{i}\circ\det)$ gives the same representation on ${\rm
Irr}_{\rm gen}(\overline{M})$, and this accounts for the non-injectivity of
the map $\tilde{\otimes}$, see [Mez04, Lemma 5.1]. This metaplectic tensor
product construction was further analyzed and refined in [Tak3, Tak4]
especially in the global context, see also [Cai1].

To proceed, in this subsection, we assume $\overline{{\rm GSp}}_{2r}$ is the
type I (similitudes-splitting) $n$-fold cover associated with

$Q(\alpha_{r}^{\vee})=-1\text{ and }Q(e_{0})=0,$

see §2.4.2. One has $B(e_{0},e_{i})=1$ for every $i$, and
$B_{Q}(e_{c},e_{0})=Q(e_{c})=r$. Also,

(3.12) $Y_{Q,n}=\left\\{\begin{array}[]{cc}\sum_{i=0}^{r}y_{i}e_{i}\in Y:\\\
\bullet\quad n|(-2y_{i}+y_{0})\text{ for every }i,\\\ \bullet\quad
n|(y_{1}+y_{2}+...+y_{r}).\end{array}\right\\}.$

On the other hand, $Y_{Q,n}^{sc}$ is spanned by

$\left\\{\alpha_{i,Q,n}^{\vee}=n_{2}\cdot\alpha_{i}^{\vee}:1\leqslant
i\leqslant
r-1\right\\}\cup\left\\{\alpha_{r,Q,n}^{\vee}=n\alpha_{r}^{\vee}\right\\}.$

We consider a partition

${\bf r}=(r_{1},r_{2},...,r_{k};r_{0})$

of $r$ with associated Levi subgroup

$M_{\bf r}={\rm GL}_{r_{1}}\times{\rm GL}_{r_{2}}\times...\times{\rm
GL}_{r_{k}}\times{\rm GSp}_{2r_{0}}.$

For every $j\in\mathbf{N}$, let

${\rm GSp}_{2r}^{\left\langle j\right\rangle}=\left\\{g\in{\rm
GSp}_{2r}:e_{0}^{*}(g)\in F^{\times j}\right\\}$

be the subgroup of ${\rm GSp}_{2r}$ with similitudes lying in $F^{\times j}$,
where

$e_{0}^{*}:{\rm GSp}_{2r}\longrightarrow F^{\times}$

is the similitude map. Also define

$M_{\bf r}^{\left\langle j\right\rangle}={\rm GL}_{r_{1}}\times{\rm
GL}_{r_{2}}\times...\times{\rm GL}_{r_{k}}\times{\rm
GSp}_{2r_{0}}^{\left\langle j\right\rangle}.$

Accordingly, we have the covering subgroups $\overline{{\rm
GSp}}_{2r}^{\left\langle j\right\rangle}\subset\overline{{\rm GSp}}_{2r}$ and
$\overline{M}_{\bf r}^{\left\langle j\right\rangle}\subset\overline{M}_{\bf
r}$. The covering blocks $\overline{{\rm GL}}_{r_{i}}$ in $\overline{M}_{\bf
r}$ commute with each other; however, they may not commute with
$\overline{{\rm GSp}}_{2r_{0}}^{\left\langle j\right\rangle}$ for general $j$.

######  Lemma 3.5.

  1. (i)

The covering group $\overline{{\rm GSp}}_{2r_{0}}^{\left\langle
j\right\rangle}$ commutes with every $\overline{{\rm GL}}_{r_{i}},1\leqslant
i\leqslant k$ if and only if $n|j$.

  2. (ii)

For every genuine representation $\pi$ of $\overline{M}_{\bf r}$, one has

${\rm supp}(\Theta_{\pi})\subset\overline{M}_{\bf r}^{\left\langle
n_{(r)}\right\rangle}$

for the character $\Theta_{\pi}$ of $\pi$.

###### Proof.

For (i), we note that every element in $\overline{{\rm GSp}}_{2r_{0}}$ can be
written as $g_{0}\cdot e_{0}(a)$ with $g_{0}\in\overline{{\rm Sp}}_{2r_{0}}$
and $a\in F^{\times}$. We have block commutativity among the blocks
$\overline{{\rm GL}}_{r_{i}}$ and $\overline{{\rm Sp}}_{2r_{0}}$, thus $g_{0}$
commutes with every element in $\overline{{\rm GL}}_{r_{i}}$. On the other
hand, $e_{0}(a)$ commutes with every $e_{\alpha_{i}}(a),1\leqslant i\leqslant
r-1$, thus $e_{0}(a)$ commutes with $\overline{{\rm GL}}_{r_{i}}$ if and only
if

$[e_{0}(a),e_{i}(x)]=(a,x)_{n}^{B_{Q}(e_{0},e_{i})}$

for every $e_{i}\in Y_{{\rm GL}_{r_{i}}}$ and $x\in F^{\times}$. Since
$B_{Q}(e_{0},e_{i})=1$, the above equality amounts to $a\in F^{\times n}$. For
(ii), the proof is the same as Proposition 2.8 by using the above fact that
$g_{0}$ commutes with $\overline{{\rm GL}}_{r_{i}}$ for every $i$. ∎

Let $M_{\bf r}\subset{\rm GSp}_{2r}$ be a Levi subgroup as above. We assume in
the rest of this subsection that

$\gcd(n,r_{0})=1$

which implies ${\rm supp}(\Theta_{\pi_{0}})\subset\overline{{\rm
GSp}}_{2r_{0}}^{\left\langle n\right\rangle}$ for every $\pi_{0}\in{\rm
Irr}_{\rm gen}(\overline{{\rm GSp}}_{2r_{0}})$, by Lemma 3.5. It is also easy
to check that

(3.13) $\mathbbm{L}(Z(\overline{{\rm GSp}}_{2r})\cap\overline{M}_{\bf
r}^{\left\langle n\right\rangle})=\begin{cases}\mathbf{Z}(ne_{c})&\text{ if
}2\nmid\gcd(n,r),\\\ \mathbf{Z}(me_{c})&\text{ if }2|\gcd(n,r).\end{cases}$

Also,

(3.14) $Z(\overline{{\rm GSp}}_{2r})\cdot\overline{M}_{\bf r}^{\left\langle
n\right\rangle}=\begin{cases}\overline{M}_{\bf r}^{\left\langle
n_{(r)}\right\rangle}&\text{ if }2\nmid\gcd(n,r),\\\ \overline{M}_{\bf
r}^{\left\langle 2n_{(r)}\right\rangle}&\text{ if }2|\gcd(n,r).\end{cases}$

We will discuss the two cases separately: (1) $n$ is odd, (2) $n$ is even.

####  3.4.1. The case of odd $n$

For $n$ odd, it is easy to obtain from (3.12) that

$\left\\{ne_{i}:1\leqslant i\leqslant r\right\\}\cup\left\\{n_{(r)}\cdot
e_{c}\right\\}$

constitutes a $\mathbf{Z}$-basis for $Y_{Q,n}$. This shows that

$\overline{{\rm GSp}}_{2r}^{\vee}=\left\\{(g,a)\in{\rm GSpin}_{2r+1}\times{\rm
GL}_{1}:\lambda(g)=a^{\gcd(n,r)}\right\\},$

where

$\lambda:{\rm GSpin}_{2r+1}\longrightarrow{\rm GL}_{1}$

is the similitude map of ${\rm GSpin}_{2r+1}$ associated with $ne_{c}\in
Y_{Q,n}$. Since we have assumed $\gcd(n,r_{0})=1$, it gives

$\overline{{\rm GSp}}_{2r_{0}}^{\vee}={\rm GSpin}_{2r_{0}+1}.$

Now we describe a construction analogous to the metaplectic tensor product for
$\overline{{\rm GL}}_{r}$. In parallel, there are three steps as follows.

  1. (S1)

Let $\pi_{i}\in{\rm Irr}_{\rm gen}(\overline{{\rm GL}}_{r_{i}})$ and
$\pi_{0}\in{\rm Irr}_{\rm gen}(\overline{{\rm GSp}}_{2r_{0}})$. Let
$\sigma_{0}\subset\pi_{0}$ be an irreducible summand in the restriction of
$\pi_{0}$ to $\overline{{\rm GSp}}_{2r_{0}}^{\left\langle n\right\rangle}$.
The representation

$\sigma:=\pi_{1}\boxtimes...\boxtimes\pi_{k}\boxtimes\sigma_{0}$

is then a representation of $\overline{M}_{\bf r}^{\left\langle
n\right\rangle}$.

  2. (S2)

Let $\omega:Z(\overline{{\rm GSp}}_{2r})\longrightarrow\mathbf{C}^{\times}$ be
a central character such that

(3.15) $\omega=\sigma\text{ on }Z(\overline{{\rm
GSp}}_{2r})\cap\overline{M}_{\bf r}^{\left\langle n\right\rangle}.$

We have $\omega\boxtimes\sigma\in{\rm Irr}(Z(\overline{{\rm
GSp}}_{2r})\cdot\overline{M}_{\bf r}^{\left\langle n\right\rangle})$.

  3. (S3)

One extends $\omega\boxtimes\sigma$ to a representation
$(\omega\boxtimes\sigma)^{\prime}$ of a subgroup $\overline{M}_{\bf
r}^{\prime}\subset\overline{M}_{\bf r}$ such that the representation

$\tilde{\otimes}_{i}\pi_{i}:={\rm
Ind}_{\overline{M}^{\prime}}^{\overline{M}}(\omega\boxtimes\sigma)^{\prime}$

is irreducible. This representation $\tilde{\otimes}_{i}\pi_{i}$ is the sought
metaplectic-tensor product of the $\pi_{i}$’s.

Similar to the case of Kazhdan–Patterson covers $\overline{{\rm GL}}_{r}$
discussed earlier, (S3) can be replaced by an equivalent statement as follows.

  1. (S3’)

It follows from (ii) of Lemma 3.5, the equality (3.14) and Proposition 2.8
that

${\rm Ind}_{{\rm Irr}(Z(\overline{{\rm GSp}}_{2r})\cdot\overline{M}_{\bf
r}^{\left\langle n\right\rangle})}^{\overline{M}_{\bf
r}}(\omega\boxtimes\sigma)=m\cdot\pi$

is an isotypic sum of $\pi\in{\rm Irr}_{\rm gen}(\overline{M}_{\bf r})$. The
representation $\pi$ is just $\tilde{\otimes}_{i}\pi_{i}$ described in (S3)
above. In particular, fixing $\omega\boxtimes\sigma$, the representation
$\tilde{\otimes}_{i}\pi_{i}$ is independent of the choice of
$\overline{M}_{\bf r}^{\prime}$ and the extension
$(\omega\boxtimes\sigma)^{\prime}$.

We have to justify the above construction (S1)-(S3) (or (S1), (S2) and (S3’))
as follows.

######  Lemma 3.6.

The representation $\tilde{\otimes}_{i}\pi_{i}$ is independent of the choice
of $\sigma_{0}\subset\pi_{0}$, the choice of $\overline{M}_{\bf r}^{\prime}$
and the extension $(\omega\boxtimes\sigma)^{\prime}$.

###### Proof.

Again, the argument is essentially the same as in [Mez04]. By the equivalence
between (S3) and (S3’), it suffices to show the independence on the choice of
constituent $\sigma_{0}$ in (S1). Every summand in the restriction of
$\pi_{0}$ to $\overline{{\rm GSp}}_{2r_{0}}$ is of the form ${}^{g}\pi_{0}$,
where $g_{0}\in\overline{{\rm GSp}}_{2r_{0}}/\overline{{\rm
GSp}}_{2r_{0}}^{\left\langle n\right\rangle}\simeq
e_{0}(F^{\times})/e_{0}(F^{\times n})$. Taking $g=e_{0}(a),a\in F^{\times}$,
since $\Theta_{\pi_{i}}\subset\overline{{\rm GL}}_{r_{i}}^{\left\langle
n\right\rangle}$ for every $\pi_{i}\in{\rm Irr}_{\rm gen}(\overline{{\rm
GL}}_{r_{i}})$ by Proposition 2.8, we see that

${}^{g}\Theta_{\pi_{i}}=\Theta_{\pi_{i}}$

and thus ${}^{g}\pi_{i}\simeq\pi_{i}$. It follows that

$\pi_{1}\boxtimes...\boxtimes\pi_{k}\boxtimes{}^{g}\sigma_{0}={}^{g}\sigma.$

If $\omega$ and $\sigma$ agrees on $Z(\overline{{\rm
GSp}}_{2r})\cap\overline{M}_{\bf r}^{\left\langle n\right\rangle}$, then
$\omega={}^{g}\sigma$ on this intersection subgroup as well. Now we have

$\displaystyle{\rm Ind}_{Z(\overline{{\rm GSp}}_{2r})\cdot\overline{M}_{\bf
r}^{\left\langle n\right\rangle}}^{\overline{M}_{\bf
r}}(\omega\boxtimes{}^{g}\sigma)$ $\displaystyle={\rm Ind}_{Z(\overline{{\rm
GSp}}_{2r})\cdot\overline{M}_{\bf r}^{\left\langle
n\right\rangle}}^{\overline{M}_{\bf r}}{}^{g}(\omega\boxtimes\sigma)$
$\displaystyle={}^{g}{\rm Ind}_{Z(\overline{{\rm
GSp}}_{2r})\cdot\overline{M}_{\bf r}^{\left\langle
n\right\rangle}}^{\overline{M}_{\bf r}}(\omega\boxtimes\sigma)$
$\displaystyle={\rm Ind}_{Z(\overline{{\rm GSp}}_{2r})\cdot\overline{M}_{\bf
r}^{\left\langle n\right\rangle}}^{\overline{M}_{\bf
r}}(\omega\boxtimes\sigma)$ $\displaystyle=e\cdot\tilde{\otimes}_{i}\pi_{i},$

where the last equality follows from Proposition 2.8, see also (S3) and (S3’)
above. This shows the independence on ${}^{g}\sigma_{0}$. ∎

We also have a reverse construction from ${\rm Irr}_{\rm
gen}(\overline{M}_{\bf r})$ to $\prod_{i=1}^{k}{\rm Irr}_{\rm
gen}(\overline{{\rm GL}}_{r_{i}})\times{\rm Irr}_{\rm gen}(\overline{{\rm
GSp}}_{2r_{0}})$ as follows.

  1. (RS1)

Let $\Pi\in{\rm Irr}_{\rm gen}(\overline{M}_{\bf r})$ be an irreducible
genuine representation. We pick a summand $\Pi_{0}\subset\Pi$ in the
restriction of $\Pi$ to $\overline{M}_{\bf r}^{\left\langle n\right\rangle}$,
it takes the form

$\Pi_{0}=\pi_{1}\boxtimes\pi_{2}\boxtimes...\boxtimes\pi_{k}\boxtimes\sigma_{0}$

with $\pi_{i}\in{\rm Irr}_{\rm gen}(\overline{{\rm GL}}_{r_{i}})$ and
$\sigma_{0}\in{\rm Irr}_{\rm gen}(\overline{{\rm GSp}}_{2r_{0}}^{\left\langle
n\right\rangle})$. Every summand of $\Pi|_{\overline{M}_{\bf r}^{\left\langle
n\right\rangle}}$ is of the form

${}^{g}(\Pi_{0})={}^{g}\pi_{1}\boxtimes...\boxtimes{}^{g}\pi_{k}\boxtimes{}^{g}\sigma_{0}$

for some $g\in\overline{M}_{\bf r}/\overline{M}_{\bf r}^{\left\langle
n\right\rangle}=e_{0}(F^{\times})/e_{0}(F^{\times n})$. Taking
$g=e_{0}(a),a\in F^{\times}$, since we have ${\rm
supp}(\Theta_{\pi_{i}})\subset Z(\overline{{\rm GSp}}_{2r})\cdot\overline{{\rm
GL}}_{r_{i}}^{\left\langle n\right\rangle}$ by Proposition 2.8, it follows
that

${}^{g}\Theta_{\pi_{i}}=\Theta_{\pi_{i}}$

as $g$ commutes with every elements in $\overline{{\rm
GL}}_{r_{i}}^{\left\langle n\right\rangle}$. This shows that
${}^{g}\pi_{i}\simeq\pi_{i}$. Thus, every constituent
$\Pi_{0}^{\prime}\subset\Pi$ is of the form

$\Pi_{0}^{\prime}=\pi_{1}\boxtimes\pi_{2}\boxtimes...\boxtimes\pi_{k}\boxtimes{}^{g}\sigma_{0}$

for some $g\in e_{0}(F^{\times})$. In particular, we have $\pi_{i}\in{\rm
Irr}_{\rm gen}(\overline{{\rm GL}}_{r_{i}})$ uniquely determined by $\Pi$ for
each $1\leqslant i\leqslant k$.

  2. (RS2)

Extend $\sigma_{0}$ as much as possible to a representation
$\sigma_{0}^{\prime}$ of a subgroup $\overline{{\rm
GSp}}_{2r_{0}}^{\prime}\subset\overline{{\rm GSp}}_{2r_{0}}$, such that the
induced representation

$\pi_{0}:={\rm Ind}_{\overline{{\rm GSp}}_{2r_{0}}^{\prime}}^{\overline{{\rm
GSp}}_{2r_{0}}}(\sigma_{0}^{\prime})$

is irreducible. This gives the desired representation $\pi_{0}\in{\rm
Irr}_{\rm gen}(\overline{{\rm GSp}}_{2r_{0}})$. Using the crucial fact that
$\Theta_{\pi}\subset\overline{{\rm GSp}}_{2r_{0}}^{\left\langle
n\right\rangle}$ for every $\pi\in{\rm Irr}_{\rm gen}(\overline{{\rm
GSp}}_{2r_{0}})$, it follows from Proposition 2.8 that

${\rm Ind}_{\overline{{\rm GSp}}_{2r_{0}}^{\left\langle
n\right\rangle}}^{\overline{{\rm GSp}}_{2r_{0}}}(\sigma_{0})=m\cdot\pi_{0},$

as an isotypic sum of $\pi_{0}\in{\rm Irr}_{\rm gen}(\overline{{\rm
GSp}}_{2r_{0}})$. Thus, $\pi_{0}$ is the desired representation.

######  Proposition 3.7.

Keep notations as above.

  1. (i)

One has $\tilde{\otimes}_{i}\pi_{i}\simeq\tilde{\otimes}_{i}\pi_{i}^{\prime}$
if and only if $\pi_{i}\simeq\pi_{i}^{\prime}$ for every $0\leqslant
i\leqslant k$.

  2. (ii)

Every representation $\Pi\in{\rm Irr}_{\rm gen}(\overline{M}_{\bf r})$ is a
metaplectic tensor product of the form $\tilde{\otimes}_{i}\pi_{i}$, where the
$\pi_{i}$’s are uniquely determined by (RS1)-(RS2).

###### Proof.

For (i), it suffices to prove the “only if” part. Consider the restriction of
$\tilde{\otimes}_{i}\pi_{i}\simeq\tilde{\otimes}_{i}\pi_{i}^{\prime}$ to
$\overline{M}_{\bf r}^{\left\langle n\right\rangle}$, we see that there is
$g\in e_{0}(F^{\times})$ and
$\sigma_{0}\subset\pi_{0},\sigma_{0}^{\prime}\subset\pi_{0}^{\prime}$ such
that

${}^{g}(\pi_{1}\boxtimes...\boxtimes\pi_{k}\boxtimes\sigma_{0})=\pi_{1}^{\prime}\boxtimes...\boxtimes\pi_{k}^{\prime}\boxtimes\sigma_{0}^{\prime},$

where the left hand side is in fact
$\pi_{1}\boxtimes...\boxtimes\pi_{k}\boxtimes{}^{g}\sigma_{0}$. It follows
that

$\pi_{i}\simeq\pi_{i}^{\prime}$

for every $1\leqslant i\leqslant k$, and also
$\sigma_{0}^{\prime}={}^{g}\sigma_{0}$. On the other hand, we have

${\rm Ind}_{\overline{{\rm GSp}}_{2r_{0}}^{\left\langle
n\right\rangle}}^{\overline{{\rm
GSp}}_{2r_{0}}}(\sigma_{0}^{\prime})=e^{\prime}\cdot\pi_{0}^{\prime}\text{ and
}{\rm Ind}_{\overline{{\rm GSp}}_{2r_{0}}^{\left\langle
n\right\rangle}}^{\overline{{\rm GSp}}_{2r_{0}}}(\sigma_{0})=e\cdot\pi_{0}$

for some $e,e^{\prime}\in\mathbf{N}$. Since

${\rm Ind}_{\overline{{\rm GSp}}_{2r_{0}}^{\left\langle
n\right\rangle}}^{\overline{{\rm GSp}}_{2r_{0}}}(\sigma_{0}^{\prime})={\rm
Ind}_{\overline{{\rm GSp}}_{2r_{0}}^{\left\langle
n\right\rangle}}^{\overline{{\rm GSp}}_{2r_{0}}}({}^{g}\sigma_{0})={}^{g}{\rm
Ind}_{\overline{{\rm GSp}}_{2r_{0}}^{\left\langle
n\right\rangle}}^{\overline{{\rm GSp}}_{2r_{0}}}(\sigma_{0})={\rm
Ind}_{\overline{{\rm GSp}}_{2r_{0}}^{\left\langle
n\right\rangle}}^{\overline{{\rm GSp}}_{2r_{0}}}(\sigma_{0}),$

it follows that $e=e^{\prime}$ and $\pi_{0}\simeq\pi_{0}^{\prime}$.

It is clear from the above argument that every $\Pi\in{\rm Irr}_{\rm
gen}(\overline{M}_{\bf r})$ is a metaplectic tensor product with $\pi_{i}$
determined as above. This gives (ii) and completes the proof. ∎

The construction in (S1)-(S3) gives a well-defined map

${\tilde{\otimes}_{i}:(\prod_{i=1}^{k}{\rm Irr}_{\rm gen}(\overline{{\rm
GL}}_{r_{i}})\times{\rm Irr}_{\rm gen}(\overline{{\rm
GSp}}_{2r_{0}})\times{\rm Irr}_{\rm gen}(Z(\overline{{\rm
GSp}}_{2r})))^{\heartsuit}}$${{\rm Irr}_{\rm gen}(\overline{M}_{\bf r}).}$

The superscript $(-)^{\heartsuit}$ indicates the subset of

$(\pi_{1},...,\pi_{k},\pi_{0},\omega)$

satisfying the equality (3.15) in (S2). On the other hand, when restricted to
$Z(\overline{{\rm GSp}}_{2r})\subset\overline{M}_{\bf r}$, the representation
$\Pi$ acts by a character

$\omega_{\Pi}:Z(\overline{{\rm GSp}}_{2r})\to\mathbf{C}^{\times}.$

Thus, the construction in (RS1)-(RS2) gives a “metaplectic restriction”

$\tilde{\rm R}:{\rm Irr}_{\rm gen}(\overline{M}_{\bf
r})\longrightarrow\Big{(}\prod_{i=1}^{k}{\rm Irr}_{\rm gen}(\overline{{\rm
GL}}_{r_{i}})\times{\rm Irr}_{\rm gen}(\overline{{\rm
GSp}}_{2r_{0}})\times{\rm Irr}_{\rm gen}(Z(\overline{{\rm
GSp}}_{2r}))\Big{)}^{\heartsuit}$

given by

$\tilde{\rm R}(\Pi)=(\pi_{1},...,\pi_{k},\pi_{0};\omega_{\Pi}).$

The following is immediate from Proposition 3.7.

######  Theorem 3.8.

Assume $n$ is odd and $\gcd(n,r_{0})=1$. Then the map

$\tilde{\otimes}_{i}:\Big{(}\prod_{i=1}^{k}{\rm Irr}_{\rm gen}(\overline{{\rm
GL}}_{r_{i}})\times{\rm Irr}_{\rm gen}(\overline{{\rm
GSp}}_{2r_{0}})\times{\rm Irr}_{\rm gen}(Z(\overline{{\rm
GSp}}_{2r}))\Big{)}^{\heartsuit}\longrightarrow{\rm Irr}_{\rm
gen}(\overline{M}_{\bf r})$

is a bijection with the inverse given by $\tilde{\rm R}$, i.e. $\tilde{\rm
R}\circ\tilde{\otimes}_{i}={\rm id}=\tilde{\otimes}_{i}\circ\tilde{\rm R}$.

It is expected that for the parameters associated with the representations
involved in the constructions $\tilde{\otimes}$ and $\tilde{\rm R}$, one
should have a corresponding identification. To proceed, we note that the dual
group of $\overline{M}_{\bf r}$ is

$\displaystyle\overline{M}_{\bf r}^{\vee}$
$\displaystyle=\left\\{(g_{1},g_{2},...,g_{k},g_{0};z):\
\lambda(g_{0})\cdot\prod_{i=1}^{k}\det(g_{i})=z^{g(n,r)}\right\\}$
$\displaystyle\subset\prod_{i=1}^{k}{\rm GL}_{r_{i}}\times{\rm
GSpin}_{2r_{0}+1}\times{\rm GL}_{1}(\mathbf{C}).$

Assuming that we have a local Langlands correspondence for the covers, then
associated with $\pi_{i}$’s and $\omega$ we have

$\phi_{i}:\text{WD}_{F}\longrightarrow{\rm GL}_{r_{i}}\text{ for }1\leqslant
i\leqslant k,\quad\phi_{0}:\text{WD}_{F}\longrightarrow{\rm GSpin}_{2r_{0}+1}$

parametrizing $\pi_{i}$’s for $0\leqslant i\leqslant k$, and

$\phi_{\omega}:\text{WD}_{F}\longrightarrow{\rm GL}_{1}(\mathbf{C})$

parametrizing $\omega$. This gives a parameter

$(\phi_{1}\times...\times\phi_{k}\times\phi_{0})\times\phi_{\omega}:\text{WD}_{F}\longrightarrow\left(\prod_{i=1}^{k}{\rm
GL}_{r_{i}}\times{\rm GSpin}_{2r_{0}+1}\right)\times\mathbf{C}^{\times}.$

######  Lemma 3.9.

Assume the local Langlands correspondence with desiderata. Then the
compatibility condition in (3.15) is equivalent to

$\prod_{i=1}^{k}\det(\phi_{i})\cdot\lambda(\phi_{0})=\phi_{\omega}^{\gcd(n,r)}.$

Thus, the parameter
$(\phi_{1}\times...\times\phi_{k}\times\phi_{0})\times\phi_{\omega}$ factors
through $\overline{M}_{\bf r}^{\vee}$.

###### Proof.

It follows from (3.13) that

$Z(\overline{{\rm GSp}}_{2r})\cap\overline{M}_{\bf r}=\overline{Z({\rm
GSp}_{2r})^{n}},$

where $Z(\overline{{\rm GSp}}_{2r})=\overline{Z({\rm GSp}_{2r})^{n_{(r)}}}$.
If we denote by $e_{i,c}\in Y_{{\rm GL}_{r_{i}}}$ the natural generator of
$Y_{{\rm GL}_{r_{i}},c}$, then

$Z(\overline{{\rm GL}}_{r_{i}})=\overline{Z({\rm GL}_{r_{i}})^{n}}\text{ and
}Z(\overline{{\rm GSp}}_{2r_{0}})=\overline{Z({\rm GSp}_{2r_{0}})^{n}}.$

From the desiderata (i) of Conjecture (3.2), the center character

$\omega_{i}:Z(\overline{{\rm GL}}_{r_{i}})\longrightarrow\mathbf{C}^{\times}$

of $\pi_{i}$ has a parameter

${\det(\phi_{i}):\text{WD}_{F}}$${\overline{{\rm
GL}}_{r_{i}}^{\vee}}$${\overline{{\rm GL}}_{r_{i}}^{\vee}/[\overline{{\rm
GL}}_{r_{i}}^{\vee},\overline{{\rm
GL}}_{r_{i}}^{\vee}]\simeq\mathbf{C}^{\times},}$$\scriptstyle{\phi_{\pi}}$$\scriptstyle{\det}$

where the second map is the determinant map of $\overline{{\rm
GL}}_{r_{i}}^{\vee}\simeq{\rm GL}_{r_{i}}$. The center character

$\omega_{0}:Z(\overline{{\rm
GSp}}_{2r_{0}})\longrightarrow\mathbf{C}^{\times}$

of $\pi_{0}$ has a parameter

${\lambda(\phi_{0}):\text{WD}_{F}}$${\overline{{\rm
GSp}}_{2r_{0}}^{\vee}}$${\overline{{\rm GSp}}_{2r_{0}}^{\vee}/[\overline{{\rm
GSp}}_{2r_{0}}^{\vee},\overline{{\rm
GSp}}_{2r_{0}}^{\vee}]\simeq\mathbf{C}^{\times},}$$\scriptstyle{\phi_{\pi}}$$\scriptstyle{\lambda}$

where the second map is the similitude map of $\overline{{\rm
GSp}}_{2r_{0}}^{\vee}\simeq{\rm GSpin}_{2r_{0}+1}$. The identity is thus
clear. ∎

Let

${i_{M}:\overline{M}_{\bf r}^{\vee}}$${\prod_{i=1}^{k}\overline{{\rm
GL}}_{r_{i}}^{\vee}\times\overline{{\rm
GSp}}_{2r_{0}}^{\vee}\times\mathbf{C}^{\times}}$

be the natural inclusion. Assuming the local Langlands correspondence, we
expect that the following diagram

(3.16) ${\text{WD}_{F}}$${\prod_{i=1}^{k}\overline{{\rm
GL}}_{r_{i}}^{\vee}\times\overline{{\rm
GSp}}_{2r_{0}}^{\vee}\times\mathbf{C}^{\times}}$${\overline{M}_{\bf
r}^{\vee}.}$$\scriptstyle{\phi_{1}\times...\times\phi_{k}\times\phi_{0}\times\phi_{\omega}}$$\scriptstyle{\phi_{\Pi}}$$\scriptstyle{i_{M}}$

commutes, when $\phi_{\Pi}$ is related to $\phi_{i}$’s and $\phi_{\omega}$ via
the two metaplectic constructions $\tilde{\otimes}_{i}$ and $\tilde{\rm R}$.
The following is in analogue with [Gan17, §3.2, Conjecture], and in fact is
simpler.

######  Conjecture 3.10.

Keep notations as above.

  1. (i)

The metaplectic tensor product $\tilde{\otimes}_{i}\pi_{i}$ defined above has
an associated $L$-parameter exactly
$(\phi_{1}\times...\times\phi_{k}\times\phi_{0})\times\phi_{\omega}$, which is
a parameter factoring through $i_{M}$ and thus is valued in $\overline{M}_{\bf
r}^{\vee}$.

  2. (ii)

The metaplectic restriction $\tilde{\rm R}(\Pi)$ above has $L$-parameter
exactly $i_{M}\circ\phi_{\Pi}$.

######  Proposition 3.11.

Conjecture 3.10 holds when the $\pi_{i}$’s and $\Pi$ are genuine principal
series.

###### Proof.

We denote by

$\overline{T}_{r_{i}}\subset\overline{{\rm GL}}_{r_{i}}\text{ and
}\overline{T}_{r_{0}}\subset\overline{{\rm GSp}}_{2r_{0}}$

the covering tori of the blocks $\overline{{\rm GL}}_{r_{i}}$ and
$\overline{{\rm GSp}}_{2r_{0}}$ respectively. Every genuine principal series
$\pi_{i}=I(i(\chi_{i}))\in{\rm Irr}_{\rm gen}(\overline{{\rm GL}}_{r_{i}})$ is
associated with

$\chi_{i}:Z(\overline{T}_{r_{i}})\longrightarrow\mathbf{C}^{\times},$

a central genuine character; similarly, a genuine principal series
$\pi_{i}=I(i(\chi_{0}))\in{\rm Irr}_{\rm gen}(\overline{{\rm GSp}}_{2r_{0}})$
is associated with

$\chi_{0}:Z(\overline{{\rm GSp}}_{2r_{0}})\longrightarrow\mathbf{C}^{\times}.$

Denote by $Y_{r_{i}}$ the character lattice of ${\rm GL}_{r_{i}}$, also by
$Y_{r_{0}}$ that of ${\rm GSp}_{2r_{0}}$.

We have

$\mathbbm{L}(Z(\overline{T}_{r_{i}}))=nY_{r_{i}},1\leqslant i\leqslant k$

and

$\mathbbm{L}(Z(\overline{T}_{r_{0}}))=nY_{r_{0}}^{sc}\oplus\mathbf{Z}(ne_{c}),$

where $Y_{r_{0}}^{sc}\subset Y_{r_{0}}$ is the coroot lattice of ${\rm
GSp}_{2r_{0}}$. Since

$\mathbbm{L}(Z(\overline{{\rm GSp}}_{2r}))=\mathbf{Z}n_{(r)}e_{c}\text{ and
}\mathbbm{L}(Z(\overline{T}))=nY^{sc}\oplus\mathbf{Z}\cdot n_{(r)}e_{c},$

it follows that

$\mathbbm{L}(Z(\overline{T}))=\mathbf{Z}n_{(r)}e_{c}+\left(\sum_{i=1}^{k}\mathbbm{L}(Z(\overline{T}_{r_{i}}))+\mathbbm{L}(Z(\overline{T}_{r_{0}}))\right).$

Thus giving a genuine character
$\chi:Z(\overline{T})\longrightarrow\mathbf{C}^{\times}$ is equivalent to
giving

$(\chi_{r_{1}},...,\chi_{r_{k}},\chi_{r_{0}};\omega)\in{\rm Irr}_{\rm
gen}(Z(\overline{T}_{r_{1}}))\times...\times{\rm Irr}_{\rm
gen}(Z(\overline{T}_{r_{k}}))\times{\rm Irr}_{\rm
gen}(Z(\overline{T}_{r_{0}}))\times{\rm Irr}_{\rm gen}(Z(\overline{{\rm
GSp}}_{2r}))$

such that $\chi_{r_{1}}\boxtimes...\chi_{r_{k}}\boxtimes\chi_{r_{0}}$ and
$\omega$ agree on

$\left(Z(\overline{T}_{r_{1}})\times_{\mu_{n}}...\times
Z(\overline{T}_{r_{k}})\times_{\mu_{n}}Z(\overline{T}_{r_{0}})\right)\cap
Z(\overline{{\rm GSp}}_{2r}).$

In particular, $\chi_{r_{i}},0\leqslant i\leqslant k$ and $\omega$ are
obtained from restricting $\chi$ to the corresponding subgroups of
$Z(\overline{T})$. The equality between the parameter $\chi_{\Pi}$ of
$\Pi:=\Pi(\chi)$ and
$\phi_{1}\times...\times\phi_{k}\times\phi_{0}\times\phi_{\omega}$ then easily
follows from functoriality of the local Langlands correspondence for covering
tori with respect to restriction, as discussed in §3.3. ∎

####  3.4.2. Remarks on the even $n$ case

The metaplectic tensor product $\tilde{\otimes}$ and metaplectic restriction
$\tilde{\rm R}$ for odd fold cover of ${\rm GSp}_{2r}$ do not generalize
naively in the even fold cover case. Now, assuming $2|n$, we highlight the
obstacles to the constructions in $\tilde{\otimes}$ and $\tilde{\rm R}$.
Recall that we assume $\gcd(n,r_{0})=1$.

Let $n=2m$. It follows from Proposition 2.8 that ${\rm
supp}(\Theta_{\pi_{i}})\subset\overline{{\rm GL}}_{r_{i}}^{\left\langle
m\right\rangle}$. However, there may exist $\pi_{i}$ such that

${\rm supp}(\Theta_{\pi_{i}})\nsubseteq\overline{{\rm
GL}}_{r_{i}}^{\left\langle n\right\rangle}.$

For such $\pi_{i}$, we may have ${}^{g}\pi_{i}\neq\pi_{i}$. This gives an
obstruction to validating the constructions both in (S1)-(S3) and (RS1)-(RS2),
as the seen from proof of Lemma 3.6. In fact, if $2|\gcd(n,r)$, then it
follows from (3.14) that

$Z(\overline{{\rm GSp}}_{2r})\cdot\overline{M}_{\bf r}^{\left\langle
n\right\rangle}=\overline{M}_{\bf r}^{{\left\langle 2n_{(r)}\right\rangle}},$

which however is not equal to $\overline{M}_{\bf r}^{{\left\langle
n_{(r)}\right\rangle}}$. In this case,

(3.17) ${\rm Ind}_{Z(\overline{{\rm GSp}}_{2r})\cdot\overline{M}_{\bf
r}^{\left\langle n\right\rangle}}^{\overline{M}_{\bf r}}=\bigoplus_{j\in
F^{\times}/2}{\rm Ind}_{\overline{M}_{\bf r}^{{\left\langle
n_{(r)}\right\rangle}}}^{\overline{M}_{\bf
r}}(\omega\boxtimes\sigma)_{j}=\bigoplus_{j}e_{j}\cdot\Pi_{j},$

where $(\omega\boxtimes\sigma)_{j}$ is an extension of
$(\omega\boxtimes\sigma)$ and

$F^{\times}/2:=e_{0}(F^{\times n_{(r)}})/e_{0}(F^{\times 2n_{(r)}})\simeq
F^{\times}/F^{\times 2}.$

The last equality in (3.17) follows from Proposition 2.8 with $\Pi_{j}\in{\rm
Irr}_{\rm gen}(\overline{M}_{\bf r})$. This gives another obstruction in the
case $2|\gcd(n,r)$.

We also remark that the dual side seems to be deceivable when $n$ is even, and
this provides a counterexample where the naive version of functoriality might
fail for general $n$. We illustrate this below according to the parity of
$\gcd(n,r)$.

In the first case, we assume $n=2m$ with $2\nmid\gcd(n,r)$. Setting

$v_{0}:=n_{(r)}\cdot e_{c}\text{ and }v_{i}:=me_{i}+v_{0}/2,1\leqslant
i\leqslant r,$

It is easy to check that

$Y_{Q,n}=\bigoplus_{i=0}^{r}\mathbf{Z}v_{i}$

with $Y_{Q,n}^{sc}$ spanned by $\left\\{m\alpha_{i}^{\vee}:1\leqslant
i\leqslant r-1\right\\}\cup\left\\{n\alpha_{r}^{\vee}\right\\}$. Thus, for
$2\nmid\gcd(n,r)$ one has

$\overline{{\rm GSp}}_{2r}^{\vee}={\rm GSp}_{2r}.$

It follows that (given with the partition ${\bf r}=(r_{1},...,r_{k},r_{0})$ of
$r$)

$\overline{M}_{\bf r}^{\vee}\simeq\prod_{i=1}^{k}{\rm GL}_{r_{i}}\times{\rm
GSp}_{2r_{0}}=\prod_{i=1}^{k}\overline{{\rm
GL}}_{r_{i}}^{\vee}\times\overline{{\rm GSp}}_{2r_{0}}^{\vee}.$

Analyzing the discussion in (S1)-(S3) and (RS1)-(RS2), one can define a
_correspondence_ between $\prod_{i=1}^{k}{\rm Irr}_{\rm gen}(\overline{{\rm
GL}}_{r_{i}})\times{\rm Irr}_{\rm gen}(\overline{{\rm GSp}}_{2r_{0}})$ and
${\rm Irr}_{\rm gen}(\overline{M}_{\bf r})$. However, there is no _map_
(instead of correspondence) from one to the other. This indicates that a naive
analogue of the functoriality might fail when considering representations of
$\overline{M}_{\bf r}$ and those of its covering blocks inside.

In the second case, we assume $2|\gcd(n,r)$, then it follows that

$Y_{Q,n}=Y_{Q,n}^{sc}\oplus\mathbf{Z}\cdot v_{0},$

where $v_{0}=n_{(r)}e_{c}$ as above. In this case,

$\overline{{\rm GSp}}_{2r}^{\vee}={\rm PGSp}_{2r}\times{\rm GL}_{1}$

and thus one has a map

$\prod_{i=1}^{k}\overline{{\rm GL}}_{r_{i}}^{\vee}\times\overline{{\rm
GSp}}_{2r_{0}}^{\vee}\longrightarrow\overline{M}_{\bf r}^{\vee}.$

However, this map does not seem to be associated from the representation-side
by a “metaplectic tensor product” construction, the same as we remarked above
for the $n=2m,2\nmid\gcd(n,r)$ case.

####  3.4.3. General metaplectic tensor product

From the example of Kazhdan–Patterson covers of ${\rm GL}_{r}$ and the odd-
fold cover of ${\rm GSp}_{2r}$ discussed in this subsection, we see that the
formulation of the metaplectic tensor product construction could be carried
out in quite general setting, provided the following constraints are
satisfied. To elaborate, assume for simplicity that

$M=M_{1}\times M_{2}\times...\times M_{k}$

is a Levi subgroup of $G$. We like to have normal subgroups

$M_{j}^{\natural}\subset M_{j},1\leqslant j\leqslant k$

such that every $M_{j}/M_{j}^{\natural}$ is a finite abelian group and satisfy
the following two conditions:

  1. (i)

the blocks $\overline{M}_{j}^{\natural},1\leqslant j\leqslant k$ commute with
each other,

  2. (ii)

${\rm supp}(\Theta_{\pi})\subset\prod_{j}\overline{M}_{j}^{\natural}$ for
every $\pi\in{\rm Irr}_{\rm gen}(\overline{M})$.

If (i) and (ii) are satisfied, then the process described in steps (S1)–(S3)
in §3.4.1 can be applied and one has a metaplectic tensor product construction
following the same procedure. Note that the two conditions (i) and (ii)
counteract against each other: for (i) it is desirable that $M_{j}^{\natural}$
is a smaller subgroup of $M_{j}$; on the other hand, for (ii) to be satisfied
the group $M_{j}^{\natural}$ should be closer to $M_{j}$. Thus, a family
$\left\\{\overline{M}_{j}^{\natural}\right\\}_{j}$ satisfying both (i) and
(ii) is a balancing choice, if it exists.

For $G={\rm GL}_{r}$ and ${\rm GSp}_{2r}$, the natural candidate for
$M_{j}^{\natural}$ is essentially the subgroup of elements with determinant or
similitude lying in $F^{\times n}$, respectively. Another reason of taking
such $M_{j}^{\natural}$ is to ensure we have a natural (even conjectural)
interpretation of the tensor product construction on the dual side.

##  4\. Local coefficients matrix under restriction

###  4.1. Local coefficients matrix

Let $\overline{B}=\overline{T}U$ be the Borel subgroup of $\overline{G}$,
where $U$ is identified as a subgroup of $\overline{G}$ via the canonical
splitting given by $e_{\alpha}(x)\mapsto\overline{e}_{\alpha}(x)$. Let
$\psi:F\longrightarrow\mathbf{C}^{\times}$ be a nontrivial character. By abuse
of notation, denote by

$\psi:U\longrightarrow\mathbf{C}^{\times}$

the unique character such that

$\psi(e_{\alpha}(x))=\psi(x)$

for every $\alpha\in\Delta$ and $x\in F$. Let $(\pi,V_{\pi})\in{\rm Irr}_{\rm
gen}(\overline{G})$ be a genuine irreducible representation of $\overline{G}$.
A functional

$\lambda:V_{\pi}\longrightarrow\mathbf{C}$

is called a $\psi$-Whittaker functional if

$\lambda(\pi(u)v)=\psi(u)\cdot\lambda(v)$

for all $u\in U$ and $v\in V_{\pi}$. Denote by $\text{Wh}_{\psi}(\pi)$ the
space of $\psi$-Whittaker functionals for $\pi$. It follows from the work of
Patel [Pate], which generalizes [MW1] by Mœglin-Waldspurger, that

$\dim{\rm Wh}_{\psi}(\pi)<\infty\text{ for every }\pi\in{\rm Irr}_{\rm
gen}(\overline{G}).$

See [KP, Theorem I.5.2 (i)] also for a proof of this finite dimensionality for
covers of ${\rm GL}_{r}$.

We briefly recall the definition of a local coefficients matrix as in [Szp6,
GSS2]. Let

${\bf P}={\bf M}{\bf N}\subset{\bf G}$

be a parabolic subgroup of ${\bf G}$ associated with $\theta\subset\Delta$.
Let $\sigma\in{\rm Irr}_{\rm gen}(\overline{M})$. We have the parabolic
subgroups $\overline{P}=\overline{M}N\subset\overline{G}$ and
$\overline{P}_{0}=\overline{M}_{0}N\subset\overline{G}_{0}$, see §2.3.1.

By dualizing, the intertwining operator $T(w,\sigma)$ in (2.11), whenever
holomorphic at $\sigma$, gives a linear map between finite dimensional vector
spaces:

${T(w,\sigma)^{*}:\ {\rm
Wh}_{\psi}(I_{\overline{P}^{\prime}}^{\overline{G}}({}^{w}\sigma))}$${{\rm
Wh}_{\psi}(I_{\overline{P}}^{\overline{G}}(\sigma)).}$

Let $\psi_{M}$ be the restriction of $\psi$ to the unipotent radical $U_{M}$
of the Borel subgroup $TU_{M}$ of $M$. Here $\psi_{M}$ and $w$ are compatible
(in the sense of [Sha4, Page 51]). We have a natural isomorphism (see [CS,
Rod1], [Sha2, §3] or [Sha4, §3.3])

$J(\sigma):{\rm Wh}_{\psi_{M}}(\sigma)\longrightarrow{\rm
Wh}_{\psi}(I_{\overline{P}}^{\overline{G}}(\sigma))$

given by

$J(\sigma)(\lambda)(f)=\int_{N^{\prime}}\lambda(f(\overline{w}_{0}^{-1}n^{\prime}))\cdot\psi(n^{\prime})dn^{\prime},$

where $N^{\prime}=w_{0}N^{-}w_{0}^{-1}$ with $w_{0}=w_{l}\cdot w_{l,M}$. Here
$w_{l}$ (resp. $w_{l,M}$) is the longest Weyl element in the Weyl group
$W=W(T,G)$ (resp. $W_{M}=W(T,M)$).

Similarly, one has

$J({}^{w}\sigma):{\rm Wh}_{\psi_{M^{\prime}}}({}^{w}\sigma)\longrightarrow{\rm
Wh}_{\psi}(I_{\overline{P}^{\prime}}^{\overline{G}}(\sigma)).$

At the same time, since $V_{{}^{w}\sigma}=V_{\sigma}$, we also have a
canonical identification

$\iota:{\rm Wh}_{\psi_{M}}(\sigma)={\rm
Wh}_{\psi_{M^{\prime}}}({}^{w}\sigma),$

which we briefly explained as follows. First, note that $l\in{\rm
Wh}_{\psi_{M}}(\sigma)$ if and only if

$l(\sigma(u)(v))=\psi_{M}(u)\cdot v$

for all $u\in U_{M},v\in V_{\sigma}$. For every $u^{\prime}\in U_{M^{\prime}}$
one has $w^{-1}u^{\prime}w\in U_{M}$ and

$\psi_{M^{\prime}}(u^{\prime})=\psi(u^{\prime})=\psi(w^{-1}u^{\prime}w)=\psi_{M}(w^{-1}u^{\prime}w).$

It is verified easily (cf. [GSS2, §3.2]) that

$l({}^{w}\sigma(u^{\prime})(v))=l(\sigma(w^{-1}u^{\prime}w)(v))=\psi_{M}(w^{-1}u^{\prime}w)\cdot
v=\psi_{M^{\prime}}(u^{\prime})\cdot v$

for all $u^{\prime}\in U_{M^{\prime}}$ and $v\in V_{{}^{w}\sigma}=V_{\sigma}$.
This shows that $l\in{\rm Wh}_{\psi_{M^{\prime}}}({}^{w}\sigma)$ and gives the
canonical identification $\iota$.

This gives a natural isomorphism of vector spaces

${\mathcal{C}(w,\sigma)=J({}^{w}\sigma)\circ\iota\circ J(\sigma)^{-1}:\ {\rm
Wh}_{\psi}(I_{\overline{P}}^{\overline{G}}(\sigma))}$${{\rm
Wh}_{\psi}(I_{\overline{P}^{\prime}}^{\overline{G}}({}^{w}\sigma)).}$

We therefore obtain an endomorphism

(4.1) ${\mathcal{T}(w,\sigma)^{*}=T(w,\sigma)^{*}\circ\mathcal{C}(w,\sigma):\
{\rm Wh}_{\psi}(I_{\overline{P}}^{\overline{G}}(\sigma))}$${{\rm
Wh}_{\psi}(I_{\overline{P}}^{\overline{G}}(\sigma))}$

of the finite dimensional vector space ${\rm
Wh}_{\psi}(I_{\overline{P}}^{\overline{G}}(\sigma))$. A local coefficients
matrix associated to $(\overline{P},w,\sigma)$ is the matrix

$\mathcal{M}_{\mathfrak{B}}(w,\sigma)$

representing $\mathcal{T}(w,\sigma)^{*}$ with respect to an ordered basis
$\mathfrak{B}\subset{\rm Wh}_{\psi}(I_{\overline{P}}^{\overline{G}}(\sigma))$.

######  Proposition 4.1.

Fix $\sigma\in{\rm Irr}_{\rm gen}(\overline{M})$. There is a local
coefficients matrix $\mathcal{M}_{\mathfrak{B}}(w,\sigma)$ of the form

$\bigoplus_{\sigma_{i}\subset\sigma|_{\overline{M}_{0}}}\mathcal{M}_{\mathfrak{B}_{i}}(w,\sigma_{i}),$

where $\mathcal{M}_{\mathfrak{B}_{i}}(w,\sigma_{i})$ is a local coefficients
matrix associated with the triple $(\overline{P}_{0},w,\sigma_{i})$ and a
basis $\mathfrak{B}_{i}$ of ${\rm
Wh}_{\psi}(I_{\overline{P}_{0}}^{\overline{G}_{0}}(\sigma_{i}))$. In
particular, for the two invariants trace and determinant, one has

(4.2) ${\rm
Tr}(\mathcal{M}_{\mathfrak{B}}(w,\sigma))=\sum_{\sigma_{i}\subset\sigma|_{\overline{M}_{0}}}{\rm
Tr}(\mathcal{M}_{\mathfrak{B}_{i}}(w,\sigma_{i})),\quad{\rm
det}(\mathcal{M}_{\mathfrak{B}}(w,\sigma))=\prod_{\sigma_{i}\subset\sigma|_{\overline{M}_{0}}}{\rm
det}(\mathcal{M}_{\mathfrak{B}_{i}}(w,\sigma_{i})).$

###### Proof.

Clearly, it suffices to show that $\mathcal{T}(w,\sigma)^{*}$ decomposes as a
direct sum of the operators $\mathcal{T}(w,\sigma_{i})^{*}$. First, we have an
isomorphism of vector spaces

(4.3) ${\rm
Wh}_{\psi}(I_{\overline{P}}^{\overline{G}}(\sigma))\simeq\bigoplus_{\sigma_{i}\subset\sigma|_{\overline{M}_{0}}}{\rm
Wh}_{\psi}(I_{\overline{P}_{0}}^{\overline{G}_{0}}(\sigma_{i})),$

since ${\rm Wh}_{\psi}(-)$ is the dual of the twisted Jacquet module, which
depends only on $\psi:U\to\mathbf{C}^{\times}$ and is an exact functor. It
follows from Lemma 2.14 that

$T(w,\sigma)^{*}=\bigoplus_{\sigma_{i}\subset\sigma|_{\overline{M}_{0}}}T(w,\sigma_{i})^{*}.$

Now we consider the decomposition of $\mathcal{C}(w,\sigma)$ over the
$\mathcal{C}(w,\sigma_{i})$’s with respect to the decomposition in (4.3). For
this purpose, we note first for each
$\sigma_{i}\subset\sigma|_{\overline{M}_{0}}$ the isomorphism

$J(\sigma_{i}):{\rm Wh}_{\psi_{M}}(\sigma_{i})\longrightarrow{\rm
Wh}_{\psi}(I_{\overline{P}_{0}}^{\overline{G}_{0}}(\sigma_{i}))$

and the decomposition

${\rm
Wh}_{\psi_{M}}(\sigma)=\bigoplus_{\sigma_{i}\subset\sigma|_{\overline{M}_{0}}}{\rm
Wh}_{\psi_{M}}(\sigma_{i}).$

It is easy to see from the defining formula of $J(\sigma_{i})$ that

$J(\sigma)=\bigoplus_{\sigma_{i}\subset\sigma|_{\overline{M}_{0}}}J(\sigma_{i}).$

Similar decomposition for $J({}^{w}\sigma)$ holds. Lastly, the canonical
identification $\iota$ also decomposes:

$\iota=\bigoplus_{\sigma_{i}\subset\sigma|_{\overline{M}_{0}}}\iota_{\sigma_{i}},$

where $\iota_{\sigma_{i}}:{\rm Wh}_{\psi_{M}}(\sigma_{i})\to{\rm
Wh}_{\psi_{M^{\prime}}}({}^{w}\sigma_{i})$ is the canonical identification.
Thus, the isomorphism $\mathcal{C}(w,\sigma)$ decomposes as a direct product
of the $\mathcal{C}(w,\sigma_{i})$’s. Hence we have

$\mathcal{T}(w,\sigma)^{*}=\bigoplus_{\sigma_{i}\subset\sigma|_{\overline{M}_{0}}}\mathcal{T}(w,\sigma_{i})^{*},$

which gives the desired result. ∎

The above proposition shows that the invariants associated to a local
coefficients matrix $\mathcal{M}_{\mathfrak{B}}(w,\sigma)$ is a priori
determined by all the local coefficients matrices
$\mathcal{M}_{\mathfrak{B}_{i}}(w,\sigma_{i})$ arising from the restriction of
$\sigma$ to $\overline{M}_{0}$. This has a striking consequence even in the
case where $\sigma=i(\chi)$ is a genuine representation of
$\overline{T}\subset\overline{G}$, where $\chi$ is a central character of
$Z(\overline{T})$. Indeed, in this case, the two invariants

${\rm Tr}(\mathcal{M}_{\mathfrak{B}}(w_{\alpha},\sigma))\text{ and }{\rm
det}(\mathcal{M}_{\mathfrak{B}}(w_{\alpha},\sigma))$

associated with a simple reflection $w_{\alpha}$ are computed in [GoSz, Szp6,
GSS1, GSS2]. They are expressed in terms of the Plancherel measure and gamma
or metaplectic-gamma factors associated with $i(\chi)$; in particular, these
two invariants depend only on the central character $\chi$.

It is expected that ${\rm
Tr}(\mathcal{M}_{\mathfrak{B}_{i}}(w_{\alpha},\sigma_{i}))$ and ${\rm
det}(\mathcal{M}_{\mathfrak{B}_{i}}(w_{\alpha},\sigma_{i}))$ depend on
$\sigma_{i}$ only. Note that even if $\sigma\in\overline{G}$ is an unramified
representation, it is possible to have a ramified constituent
$\sigma_{i}\subset\sigma|_{\overline{T}_{0}}$. In this case, the equalities in
(4.2) entail that there is a cancellation of the ramified information encoded
in the ramified constituents of $\sigma|_{\overline{T}_{0}}$. A prototype of
such examples is the pair $(\overline{{\rm GL}}_{2}^{(n)},\overline{{\rm
SL}}_{2}^{(n)})$ with even $n$, see [GSS2, §8–§9].

A special case is the following.

######  Corollary 4.2.

Let $\sigma\in{\rm Irr}_{\rm gen}(\overline{M})$. Assume that
$\sigma|_{\overline{M}_{0}}=\sigma_{0}^{\oplus m}$ is an isotypic sum of
$\sigma_{0}\in{\rm Irr}_{\rm gen}(\overline{M}_{0})$. Then

$\dim{\rm Wh}_{\psi_{M}}(\sigma)=m\cdot\dim{\rm Wh}_{\psi_{M}}(\sigma)$

and moreover $\mathcal{M}_{\mathfrak{B}_{0}}(w,\sigma_{0})^{\oplus m}$ is a
local coefficients matrix for $\mathcal{T}(w,\sigma)^{*}$, where
$\mathcal{M}_{\mathfrak{B}_{0}}(w,\sigma_{0})$ is a local coefficients matrix
for $\mathcal{T}(w,\sigma_{0})^{*}$. In particular,

${\rm Tr}(\mathcal{M}_{\mathfrak{B}}(w,\sigma))=m\cdot{\rm
Tr}(\mathcal{M}_{\mathfrak{B}_{0}}(w,\sigma_{0})),\quad{\rm
det}(\mathcal{M}_{\mathfrak{B}}(w,\sigma))={\rm
det}(\mathcal{M}_{\mathfrak{B}_{0}}(w,\sigma_{0}))^{m}.$

One immediate application of Corollary 4.2 is the computation of the
multiplicity $m$ by considering the Whittaker dimensions. Indeed, such
multiplicity (or ramification index $e$ in the discussion of §2.2.1) is most
often a subtle quantity to explicate.

###  4.2. Genuine principal series

We assume $p\nmid n$, and fix a splitting $s_{K}$ of $\overline{G}$ over
$K={\bf G}(O)$. We consider a $(K,s_{K})$-unramified genuine principal series
$I(\chi)=I(i(\chi))$ of $\overline{G}$, where

$\chi:Z(\overline{T})\longrightarrow\mathbf{C}^{\times}$

is an unramified character, i.e., it is trivial on $Z(\overline{T})\cap K$. In
this case, as discussed in §2.5, the group

$\overline{A}=Z(\overline{T})\cdot{\bf T}(O)\subset\overline{T}$

is a maximal abelian subgroup. There is a unique extension also denoted by

$\tilde{\chi}:\overline{A}\longrightarrow\mathbf{C}^{\times},$

which is trivial on ${\bf T}(O)$. From now on, we will take

$i(\chi)={\rm Ind}_{\overline{A}}^{\overline{T}}(\tilde{\chi})$

for this $\tilde{\chi}$, which gives the unramified principal series
$I(\chi)=I(i(\chi))$.

It follows from Theorem 2.24 that one has

$I(\chi)|_{\overline{G}_{0}}=\left|\mathscr{X}_{Q,n}^{\mathfrak{c}}\right|\cdot\bigoplus_{\gamma\in\mathscr{X}_{Q,n}^{\Gamma}/\mathscr{X}_{Q,n}^{\mathfrak{c}}}\bigoplus_{\omega_{\gamma,j}\in\mathscr{E}(\chi,{}^{\gamma}\tilde{\chi}_{O};Z(\overline{T}_{0}))}I(\omega_{\gamma,j}),$

where generically $I(\omega_{\gamma,j})$ is a multiplicity-free genuine
principal series of $\overline{G}_{0}$ appearning in the double sum of the
right hand side. We note that

$\dim{\rm Wh}_{\psi}(I(\omega_{\gamma,j}))=\left|\mathscr{X}_{0,Q,n}\right|$

for every
$\omega_{\gamma,j}\in\mathscr{E}(\chi,{}^{\gamma}\tilde{\chi}_{O};Z(\overline{T}_{0}))$,
where

$\mathscr{X}_{0,Q,n}:=Y_{0}/Y_{0,Q,n}.$

One the other hand, we have

$\dim{\rm Wh}_{\psi}(I(\chi))=\left|\mathscr{X}_{Q,n}\right|.$

The relations among various quotients of lattices are illustrated as follows.

(4.4) ${Y_{0,Q,n}/(Y_{0}\cap Y_{Q,n})}$${Y/Y_{0}}$${Y_{0}/(Y_{0}\cap
Y_{Q,n})}$${\mathscr{X}_{Q,n}}$${\mathscr{X}_{Q,n}^{\Gamma}}$${\mathscr{X}_{0,Q,n}.}$$\scriptstyle{f}$

If $Z(\overline{T}_{0})\subset Z(\overline{T})$, or equivalently
$Y_{0,Q,n}=Y_{0}\cap Y_{Q,n}$ by Corollary 2.12, then the above diagram
becomes

(4.5)
${\mathscr{X}_{0,Q,n}}$${\mathscr{X}_{Q,n}}$${\mathscr{X}_{Q,n}^{\Gamma}.}$$\scriptstyle{f}$

Since $\mathscr{X}_{Q,n}$ is the “moduli space” of ${\rm Wh}_{\psi}(I(\chi))$,
i.e., there is a natural parametrization of the Whittaker space by
$\mathscr{X}_{Q,n}$ (see [GSS2, §3.4]), we see that there is a decomposition
of $\mathscr{X}_{Q,n}$ into the disjoint union of the fibers of $f$, and thus
a corresponding decomposition of ${\rm Wh}_{\psi}(I(\chi))$. Each fibre of $f$
is a torsor over $\mathscr{X}_{0,Q,n}$ and in particular of the same size as

$\left|\mathscr{X}_{0,Q,n}\right|=\dim{\rm
Wh}_{\psi}(I(\chi|_{Z(\overline{T}_{0})})).$

This decomposition is compatible with the fact that
$I(\chi)|_{\overline{G}_{0}}=I(\chi|_{Z(\overline{T}_{0})})^{\oplus\left|\mathscr{X}_{Q,n}^{\Gamma}\right|}$
is an isotypic sum, see Corollary 2.25 and Corollary 4.2.

In the remaining of this section, we assume

$\mathfrak{f}(\psi)=O_{F}$

and investigate $\dim{\rm Wh}_{\psi}(\pi)$ and $\dim{\rm Wh}_{\psi}(\pi_{0})$,
where $\pi$ is an irreducible constituent of a regular or unitary unramified
principal series and $\pi_{0}\subset\pi|_{\overline{G}}$.

###  4.3. Regular unramified principal series

In this subsection, we consider the restriction to $\overline{G}_{0}$ of an
irreducible constituent $\pi$ of a regular unramified genuine principal series
$I(\chi)$ of $\overline{G}$. In particular, we will study how the Whittaker
dimension $\dim{\rm Wh}_{\psi}(\pi)$ is distributed into the constituents of
$\pi|_{\overline{G}_{0}}$.

Temporarily, we use $\overline{G}^{\sharp}$ to denote either $\overline{G}$ or
$\overline{G}_{0}$, and $\overline{T}^{\sharp}\subset\overline{G}^{\sharp}$ is
the covering torus. Let

$\chi^{\sharp}:Z(\overline{T}^{\sharp})\longrightarrow\mathbf{C}^{\times}$

be a central character satisfying:

  1. –

if $\overline{T}^{\sharp}=\overline{T}$, then $\chi^{\sharp}$ is unramified;

  2. –

if $\overline{T}^{\sharp}=\overline{T}_{0}$, then $i(\chi^{\sharp})\subset
i(\chi)|_{\overline{T}_{0}}$ for an unramified
$\chi:Z(\overline{T})\longrightarrow\mathbf{C}^{\times}$.

We highlight that in the second case above, $\chi^{\sharp}$ may not be
unramified, unless $(\overline{G},\overline{G}_{0})$ is an isotypic pair;
also, by Lemma 3.3 we have $i(\chi^{\sharp})\subset
i(\chi)|_{\overline{T}_{0}}$ if and only if
$\chi^{\sharp}|_{\overline{T}_{0}\cap
Z(\overline{T})}=\chi|_{\overline{T}_{0}\cap Z(\overline{T})}$. Denoting

$Y_{Q,n}^{\sharp}:=Y_{0}\cap Y_{Q,n}$

and let $\overline{T}_{Q,n}^{\sharp}$ be the preimage in $\overline{T}$ of
${\rm Im}(i_{Q,n}^{\sharp})\subset T$, where

$i_{Q,n}^{\sharp}:\ Y_{Q,n}^{\sharp}\otimes F^{\times}\longrightarrow
Y_{Q,n}\otimes F^{\times}\longrightarrow Y\otimes F^{\times}$

is induced from the inclusions $Y_{Q,n}^{\sharp}\subset Y_{Q,n}\subset Y$. One
has

$\overline{T}_{Q,n}^{\sharp}\subset Z(\overline{T}).$

Similarly, we have the subgroup $\overline{T}_{Q,n}^{sc}\subset
Z(\overline{T})$ associated with $Y_{Q,n}^{sc}\subset Y_{Q,n}$. In any case,
since

$\overline{T}_{Q,n}^{\sharp}=\overline{T}_{0}\cap
Z(\overline{T})=Z(\overline{T}_{0})\cap Z(\overline{T}),$

we see that every $\chi^{\sharp}$ is unramified when restricted to
$\overline{T}_{Q,n}^{\sharp}$. In particular, the restriction

$\chi^{\sharp}|_{\overline{T}_{Q,n}^{sc}}:\overline{T}_{Q,n}^{sc}\longrightarrow\mathbf{C}^{\times}$

is always unramified.

In this subsection, we assume that $\chi^{\sharp}$ is regular, i.e., the
stabilizer subgroup of $\chi^{\sharp}$ in the Weyl group is trivial. Thus the
irreducible constituents of $I(\chi^{\sharp})$ are multiplicity-free. We
denote its Jordan–Holder set by ${\rm JH}(I(\chi^{\sharp}))$. Consider

$\Phi(\chi^{\sharp}):=\left\\{\alpha\in\Phi:\chi^{\sharp}(\overline{h}_{\alpha}(\varpi^{n_{\alpha}}))=q^{-1}\right\\}\subset\Phi.$

Let $\mathscr{P}(\Phi(\chi^{\sharp}))$ be the power set of
$\Phi(\chi^{\sharp})$, and denote by

$\mathscr{C}(X\otimes\mathbf{R};\chi^{\sharp})$

the set of connected components of

$X\otimes\mathbf{R}-\bigcup_{\alpha\in\Phi(\chi^{\sharp})}{\rm
Ker}(\alpha^{\vee}).$

######  Proposition 4.3.

There are bijections between the three sets

$\mathscr{P}(\Phi(\chi^{\sharp}))\longleftrightarrow{\rm
JH}(I(\chi^{\sharp}))\longleftrightarrow\mathscr{C}(X\otimes\mathbf{R};\chi^{\sharp})$

denoted by

$S\leftrightarrow\pi_{S}\leftrightarrow\Gamma_{S},$

which is given as follows. First, we have

$\Gamma_{S}=\left\\{x\in
X\otimes\mathbf{R}:{\left\langle\alpha^{\vee},x\right\rangle}<0\text{ if and
only if }\alpha\in S\right\\}.$

Second, the representation $\pi_{S}$ is characterized by its Jacquet module

$(\pi_{S})_{U}=\bigoplus_{w\in W_{S}}\delta_{B}^{1/2}\cdot
i({}^{w^{-1}}\chi^{\sharp}),$

where

$W_{S}=\left\\{w\in W:\Phi(\chi^{\sharp})^{\vee}\cap
w(\Phi_{-}^{\vee})=S^{\vee}\right\\}\subset W.$

###### Proof.

For linear algebraic group and a regular character $\chi$ of $T\subset G$ (not
necessarily unramified), the result was shown in [Rod4]. For covering groups
and unramified $\chi$, it was extended in [Ga6]. We explain how the argument
in [Ga6], which is adopted from [Rod4], actually works for $\chi^{\sharp}$
here.

Indeed, the key property needed in the proof in [Ga6] is the computation of
the Plancherel measure $\mu(w_{\alpha},i(\chi^{\sharp}))$ associated with
$T(w_{\alpha},i(\chi^{\sharp}))$ with $w_{\alpha}$ being a simple reflection.
For unramified $\chi^{\sharp}$, this was explicitly computed in [Ga1] and used
in [Ga6, Proposition 3.5]; it shows that

(4.6)
$\mu(w_{\alpha},i(\chi^{\sharp}))^{-1}=\frac{1-q^{-1}\chi^{\sharp}(\overline{h}_{\alpha}(\varpi^{n_{\alpha}}))}{1-\chi^{\sharp}(\overline{h}_{\alpha}(\varpi^{n_{\alpha}}))}\cdot\frac{1-q^{-1}\chi^{\sharp}(\overline{h}_{\alpha}(\varpi^{n_{\alpha}}))^{-1}}{1-\chi^{\sharp}(\overline{h}_{\alpha}(\varpi^{n_{\alpha}}))^{-1}}.$

One thus concludes that the poles of $\mu(w_{\alpha},i(\chi^{\sharp}))$ are
achieved when $\alpha\in\Phi(\chi^{\sharp})$ or
$-\alpha\in\Phi(\chi^{\sharp})$. This is the key ingredient used in the proof
in [Rod4], the rest of which applies in the covering setting.

Now if $\chi^{\sharp}$ is for $\overline{G}_{0}$ with $i(\chi^{\sharp})\subset
i(\chi)|_{\overline{T}}$, then we can proceed in two ways in order to
determine $\mu(w_{\alpha},i(\chi^{\sharp}))$. First, one can invoke [GoSz,
Theorem 5.1] directly. Alternatively, it follows from Lemma 2.14 that

$\mu(w_{\alpha},i(\chi^{\sharp}))=\mu(w_{\alpha},i(\chi_{0}))$

for every $\omega:Z(\overline{T}_{0})\longrightarrow\mathbf{C}^{\times}$ such
that $i(\omega)\subset i(\chi)|_{\overline{T}_{0}}$. In particular,
considering the unramified $\omega$ and using the formula of
$\mu(w_{\alpha},i(\omega))$ in [Ga1], one also obtains the same formula (4.6)
for $\mu(w_{\alpha},i(\chi^{\sharp}))$. Now the same argument in [Ga6,
Proposition 3.5] applies to give the desired result. ∎

For every $S\subset\Phi(\chi^{\sharp})$ we denote by

$\pi_{\chi^{\sharp},S}=\pi(\chi^{\sharp})_{S}\in{\rm JH}(I(\chi^{\sharp}))$

the associated irreducible constituent of $I(\chi^{\sharp})$.

######  Corollary 4.4.

Let $\chi:Z(\overline{T})\to\mathbf{C}^{\times}$ be a regular unramified
genuine character.

  1. (i)

For every $\omega:Z(\overline{T}_{0})\longrightarrow\mathbf{C}^{\times}$ such
that $i(\omega)\subset i(\chi)|_{\overline{T}_{0}}$, one has

$\Phi(\chi)=\Phi(\omega).$

  2. (ii)

For every $S\subset\Phi(\chi)$, we have

$(\pi_{\chi,S})|_{\overline{G}_{0}}=\left|\mathscr{X}_{Q,n}^{\mathfrak{c}}\right|\cdot\bigoplus_{\gamma\in\mathscr{X}_{Q,n}^{\Gamma}/\mathscr{X}_{Q,n}^{\mathfrak{c}}}\bigoplus_{\omega_{\gamma,j}\in\mathscr{E}(\chi,{}^{\gamma}\tilde{\chi}_{O};Z(\overline{T}_{0}))}\pi_{\omega_{\gamma,j},S}.$

###### Proof.

Part (i) follows from the proof of Proposition 4.3, since
$\mu(w,i(\chi))=\mu(w,i(\omega))$ for such $\omega$ and depends only on
$\chi|_{\overline{T}_{Q,n}^{sc}}=\omega|_{\overline{T}_{Q,n}^{sc}}$. For (ii),
we note that the representation $\pi_{\chi,S}$ is characterized by its Jacquet
module, and since computing Jacquet module commutes with restriction to
$\overline{G}_{0}$, it suffices to determine

${\rm Res}_{\overline{T}_{0}}^{\overline{T}}(\pi_{\chi,S})_{U}=\bigoplus_{w\in
W_{S}}\delta_{B}^{1/2}\cdot i({}^{w^{-1}}\chi)|_{\overline{T}_{0}}.$

Since

$i(\chi)|_{\overline{T}_{0}}=\left|\mathscr{X}_{Q,n}^{\mathfrak{c}}\right|\cdot\bigoplus_{\gamma\in\mathscr{X}_{Q,n}^{\Gamma}/\mathscr{X}_{Q,n}^{\mathfrak{c}}}\bigoplus_{\omega_{\gamma,j}\in\mathscr{E}(\chi,{}^{\gamma}\tilde{\chi}_{O};Z(\overline{T}_{0}))}i(\omega_{\gamma,j}),$

and similar decomposition for ${}^{w^{-1}}\chi$ for each $w\in W_{S}$ holds,
we get

$\displaystyle{\rm Res}_{\overline{T}_{0}}^{\overline{T}}(\pi_{\chi,S})_{U}$
$\displaystyle=$
$\displaystyle\left|\mathscr{X}_{Q,n}^{\mathfrak{c}}\right|\cdot\bigoplus_{\gamma\in\mathscr{X}_{Q,n}^{\Gamma}/\mathscr{X}_{Q,n}^{\mathfrak{c}}}\bigoplus_{\omega_{\gamma,j}\in\mathscr{E}(\chi,{}^{\gamma}\tilde{\chi}_{O};Z(\overline{T}_{0}))}\bigoplus_{w\in
W_{S}}\delta_{B_{0}}^{1/2}\cdot i({}^{w^{-1}}\omega_{\gamma,j})$
$\displaystyle=$
$\displaystyle\left|\mathscr{X}_{Q,n}^{\mathfrak{c}}\right|\cdot\bigoplus_{\gamma\in\mathscr{X}_{Q,n}^{\Gamma}/\mathscr{X}_{Q,n}^{\mathfrak{c}}}\bigoplus_{\omega_{\gamma,j}\in\mathscr{E}(\chi,{}^{\gamma}\chi_{O};Z(\overline{T}_{0}))}(\pi_{\omega_{\gamma,j},S})_{U}.$

As noted, the representation $\pi_{\omega_{\gamma,j},S}$ is uniquely
determined by its Jacquet module $(\pi_{\omega_{\gamma,j},S})_{U}$, this shows
(ii) and concludes the proof. ∎

For a regular $(K,s_{K})$-unramified $I(\chi)$, we want to determine the
Whittaker space ${\rm Wh}_{\psi}(\pi_{\chi,S})$ in terms of the ${\rm
Wh}_{\psi}(\pi_{\omega_{\gamma,j},S})$’s. As in general there is much
difficulty, in the remaining part of this subsection, we will consider two
special cases:

  1. –

when $Z(\overline{T}_{0})\subset Z(\overline{T})$, and thus
$\pi_{\chi,S}|_{\overline{G}_{0}}=\left|\mathscr{X}_{Q,n}^{\Gamma}\right|\cdot\pi_{\omega,S}$
is an isotypic sum, where $\omega=\chi|_{Z(\overline{T}_{0})}$;

  2. –

when the pair is $(\overline{{\rm GL}}_{2}^{(n)},\overline{{\rm
SL}}_{2}^{(n)})$ with arbitrary $n$.

We hope to illustrate the subtleties on determining even the relation between
$\dim{\rm Wh}_{\psi}(\pi_{\chi,S})$ and $\dim{\rm
Wh}_{\psi}(\pi_{\omega_{\gamma,j},S})$.

####  4.3.1. The isotypic case

If $(\overline{G},\overline{G}_{0})$ is an isotypic pair, then we denote by

$\omega:=\chi|_{Z(\overline{T}_{0})}$

the unramified character of $Z(\overline{T}_{0})$. We have
$\pi_{\chi,S}|_{\overline{G}_{0}}=\left|\mathscr{X}_{Q,n}^{\Gamma}\right|\cdot\pi_{\omega,S}$
and

$\dim{\rm
Wh}_{\psi}(\pi_{\chi,S})=\left|\mathscr{X}_{Q,n}^{\Gamma}\right|\cdot\dim{\rm
Wh}_{\psi}(\pi_{\omega,S}).$

Now we want to give an interpretation of this equality from the perspective of
the short exact sequence (4.5).

######  Lemma 4.5.

Assume that $\overline{G}_{0}$ is a saturated cover. Then

  1. (i)

$\overline{G}$ is also a saturated cover, and thus both $\overline{G}_{0}$ and
$\overline{G}$ are persistent covers;

  2. (ii)

one has $Z(\overline{T}_{0})\subset Z(\overline{T})$ in this case.

###### Proof.

Recall that we have the inclusions

$Y_{Q,n}^{sc}\subset(Y^{sc}\cap Y_{Q,n})\subset(Y_{0}\cap Y_{Q,n})\subset
Y_{0,Q,n}.$

The assumption implies that $Y_{Q,n}^{sc}=Y_{0,Q,n}$. This enforces the
equality $(Y_{0}\cap Y_{Q,n})=Y_{0,Q,n}$ and thus (ii) holds. This also
enforces another equality $Y_{Q,n}^{sc}=Y^{sc}\cap Y_{Q,n}$, which exactly
shows that $\overline{G}$ is saturated and gives (i). ∎

Let $s$ be any splitting, if it exists, of the short exact sequence of finite
abelian groups:

(4.7)
${\mathscr{X}_{0,Q,n}}$${\mathscr{X}_{Q,n}}$${\mathscr{X}_{Q,n}^{\Gamma}.}$$\scriptstyle{f}$$\scriptstyle{s}$

Every such splitting gives a set $s(\mathscr{X}_{Q,n}^{\Gamma})$ of
representatives of $\mathscr{X}_{Q,n}^{\Gamma}$ in $\mathscr{X}_{Q,n}$. We
have

$\mathscr{X}_{Q,n}=\bigsqcup_{z^{\prime}\in\mathscr{X}_{Q,n}^{\Gamma}}f^{-1}(z^{\prime})=\bigsqcup_{z\in
s(\mathscr{X}_{Q,n}^{\Gamma})}(z+\mathscr{X}_{0,Q,n}).$

Assume $\overline{G}_{0}$ is a saturated cover. Also assume that

$\Phi(\chi)\subset\Delta.$

It then follows from Lemma 4.5 and [Ga6, Theorem 6.6] that

$\dim{\rm
Wh}_{\psi}(\pi_{\chi,S})={\left\langle\sigma_{\mathscr{X}},\sigma_{S}\right\rangle}_{W},\quad\dim{\rm
Wh}_{\psi}(\pi_{\omega,S})={\left\langle\sigma_{\mathscr{X}_{0}},\sigma_{S}\right\rangle}_{W}.$

Here $\sigma_{S}$ is a certain Kazhdan–Lusztig representation of $W$ of
dimension $\left|W_{S}\right|$. On the other hand,

$\sigma_{\mathscr{X}}:W\longrightarrow{\rm Perm}(\mathscr{X}_{Q,n})$

is the permutation representation of $W$ given by
$\sigma_{\mathscr{X}}(w)(y):=w[y]$, where

$w[y]=w(y-\rho)+\rho.$

Similarly,

$\sigma_{\mathscr{X}_{0}}:W\longrightarrow{\rm Perm}(\mathscr{X}_{0,Q,n})$

is also given by

$\sigma_{\mathscr{X}_{0}}(w)(y)=w[y]$

for every $y\in\mathscr{X}_{0,Q,n}$. In fact, $\sigma_{\mathscr{X}_{0}}$ is
the subrepresentation of $\sigma_{\mathscr{X}}$ restricted on
$\mathscr{X}_{0,Q,n}\subset\mathscr{X}_{Q,n}$.

For every coset $z+\mathscr{X}_{0,Q,n}\subset\mathscr{X}_{Q,n}$, the action
$w[\cdot]$ is given by

$w[z+y]=w(z)+w[y]=z+(w[y]+w(z)-z)$

for every $y\in\mathscr{X}_{0,Q,n}$, where $w(z)$ denotes the usual action
arising from reflection. As the image of $w(z)-z$ in
$\mathscr{X}_{Q,n}^{\Gamma}$ via $f$ is trivial, we see that

$w(z)-z\in\mathscr{X}_{0,Q,n}.$

Thus, $w[z+y]$ lies in $z+\mathscr{X}_{0,Q,n}$; that is, the action $w[\cdot]$
on $z+\mathscr{X}_{0,Q,n}$ is well-defined. This gives a permutation
representation

$\sigma_{z+\mathscr{X}_{0}}:W\longrightarrow{\rm
Perm}(z+\mathscr{X}_{0,Q,n}).$

On the other hand, we consider the permutation representation

$\sigma_{\mathscr{X}_{0},z}:W\longrightarrow{\rm Perm}(\mathscr{X}_{0,Q,n})$

given by

$\sigma_{\mathscr{X}_{0},z}(w)(y)=w\langle y\rangle:=w[y]+w(z)-z.$

It is easy to check that the action $w\langle\cdot\rangle$ is well-defined.
Thus $\sigma_{\mathscr{X}_{0},z}$ is a well-defined permutation representation
of $W$ on $\mathscr{X}_{0,Q,n}$. In fact, we have:

  1. (i)

The representation $\sigma_{z+\mathscr{X}_{0}}$ is the one obtained from
transporting $\sigma_{\mathscr{X}_{0},z}$ via the translation

$\mathscr{X}_{0,Q,n}\to z+\mathscr{X}_{0,Q,n}$

given by $y\mapsto z+y$. Thus, as representations of the Weyl group,
$\sigma_{z+\mathscr{X}_{0}}$ and $\sigma_{\mathscr{X}_{0},z}$ are equivalent.
In particular, we have

$\sigma_{\mathscr{X}}=\bigoplus_{z\in
s(\mathscr{X}_{Q,n}^{\Gamma})}\sigma_{\mathscr{X}_{0},z}.$

  2. (ii)

Denote by

$\rho_{z}:W\longrightarrow{\rm Perm}(\mathscr{X}_{0,Q,n})$

the representation given by the translation $\rho_{z}(w)(y)=y+w(z)-z$. It is
clear that $\sigma_{\mathscr{X}_{0},z}=\rho_{z}\circ\sigma_{\mathscr{X}_{0}}$.

We note that the exact sequence (4.7) is $W$-equivariant with respect to the
usual action $w(\cdot)$. Also, the action of $W$ on
$\mathscr{X}_{Q,n}^{\Gamma}$ is trivial. In general, the splitting $s$ may not
be $W$-equivariant. However, we believe that in our setting this is indeed the
case:

######  Conjecture 4.6.

Assume that $\overline{G}_{0}$ is a saturated cover. Then there exists a
$W$-equivariant splitting $s:\mathscr{X}_{Q,n}^{\Gamma}\to\mathscr{X}_{Q,n}$
with respect to the Weyl action $w(\cdot)$. Consequently, with respect to such
a splitting $s$, one has

$\sigma_{\mathscr{X}_{0},z}\simeq\sigma_{\mathscr{X}_{0}}$

for every $z\in s(\mathscr{X}_{Q,n}^{\Gamma})$ and thus
$\sigma_{\mathscr{X}}=\left|\mathscr{X}_{Q,n}^{\Gamma}\right|\cdot\sigma_{\mathscr{X}_{0}}$.
In particular, $\dim{\rm
Wh}_{\psi}(\pi_{\chi,S})=\left|\mathscr{X}_{Q,n}^{\Gamma}\right|\cdot\dim{\rm
Wh}_{\psi}(\pi_{\chi_{0},S})$.

If $s$ is $W$-equivariant, then

$w(z)-z=0$

and thus $\rho_{z}(w)={\rm id}$ for every $z\in
s(\mathscr{X}_{Q,n}^{\Gamma})$; hence, it is clear that

$\sigma_{\mathscr{X}_{0},z}=\sigma_{\mathscr{X}_{0}}$

in this case. For a fixed splitting $s$, consider the following three
statements:

  1. (S1)

$\sigma_{\mathscr{X}_{0},z}\simeq\sigma_{\mathscr{X}_{0}}$ for every $z\in
s(\mathscr{X}_{Q,n}^{\Gamma})$;

  2. (S2)

${\left\langle\sigma_{\mathscr{X}_{0},z},\sigma_{S}\right\rangle}_{W}={\left\langle\sigma_{\mathscr{X}_{0}},\sigma_{S}\right\rangle}_{W}$
for every $z\in s(\mathscr{X}_{Q,n}^{\Gamma})$;

  3. (S3)

$\dim{\rm
Wh}_{\psi}(\pi_{\chi,S})=\left|\mathscr{X}_{Q,n}^{\Gamma}\right|\cdot\dim{\rm
Wh}_{\psi}(\pi_{\chi_{0},S})$.

Clearly, we have the implications

${\rm(S1)}\Longrightarrow{\rm(S2)}\Longrightarrow{\rm(S3)}.$

Here Conjecture (4.6) asserts the strongest (S1) in our context, and is
certainly compatible with (S3) which holds a priori.

######  Example 4.7.

Consider the cover $\overline{G}=\overline{{\rm GL}}_{2}^{(n)}$ associated
with $\mathbf{p},\mathbf{q}\in\mathbf{Z}$ such that
$2\mathbf{p}-\mathbf{q}=-1$, as discussed from §2.4.1. Then the cover
$\overline{G}_{0}=\overline{{\rm SL}}_{2}^{(n)}$ is saturated if and only if
$n$ is odd. Thus, we assume that $n$ is odd. An easy computation gives that

$Y_{Q,n}=\left\\{y_{1}e_{1}+y_{2}e_{2}\in\mathbf{Z}e_{1}\oplus\mathbf{Z}e_{2}:\
n|(4\mathbf{p}+1)y_{1}\text{ and }n|(y_{1}-y_{2})\right\\}.$

Setting $d=\gcd(4\mathbf{p}+1,n)$, we have

${\mathscr{X}_{0,Q,n}}$${\mathscr{X}_{Q,n}}$${\mathscr{X}_{Q,n}^{\Gamma},}$

with

$\mathscr{X}_{0,Q,n}=\mathbf{Z}\alpha^{\vee}/(\mathbf{Z}n\alpha^{\vee}),\left|\mathscr{X}_{Q,n}\right|=n^{2}/d\text{
and }\left|\mathscr{X}_{Q,n}^{\Gamma}\right|=n/d.$

We claim that there is a $W$-equivariant splitting $s$ given by

$s(\mathscr{X}_{Q,n}^{\Gamma})=\left\\{ke_{c}:0\leqslant
k\leqslant(n/d-1)\right\\}\subset\mathscr{X}_{Q,n}.$

For this purpose, it suffices to show that

$ke_{c}\notin Y^{sc}+Y_{Q,n}$

for every $1\leqslant k\leqslant(n/d-1)$. Suppose not, then there exists
$m\alpha^{\vee}\in Y^{sc}$ such that $ke_{c}+m\alpha^{\vee}\in Y_{Q,n}$. This
implies that

$(n/d)|(k+m)\text{ and }n|(k+a-k+a).$

As $n$ is odd, this implies that $(n/d)|k$, which is a contradiction. Thus,
Conjecture 4.6 holds for this pair $(\overline{{\rm GL}}_{2},\overline{{\rm
SL}}_{2})$. We remark that when $d=n$ (for example, if $n=3,\mathbf{p}=-1$),
then $\mathscr{X}_{0,Q,n}=\mathscr{X}_{Q,n}$; in this case
$\pi_{\chi,S}|_{\overline{G}_{0}}=\pi_{\chi_{0},S}$ with multiplicity one.

######  Example 4.8.

We consider the cover $\overline{{\rm GSp}}_{4}^{(3)}$ from §2.4.2 such that
$Q(\alpha_{2}^{\vee})=-1$. Note that in this case, $\overline{{\rm Sp}}_{4}$
is saturated, as its dual group is ${\rm SO}_{5}$. Using notations in §2.4.2,
we see that

$Y_{Q,n}=\left\\{\begin{array}[]{cc}y_{1}e_{1}+y_{2}e_{2}+y_{0}e_{0}\in\oplus_{i=0}^{2}\mathbf{Z}e_{i}:\\\
\bullet\quad 3|(-2y_{1}+y_{0})\\\ \bullet\quad 3|(-2y_{1}+y_{0})\\\
\bullet\quad 3|(y_{1}+y_{2}+2y_{0}\cdot Q(e_{0}))\end{array}\right\\}.$

There are two cases as follows.

  1. –

If $Q(e_{0})=\pm 1\mod 3$, then one can check easily that
$\mathscr{X}_{0,Q,n}=\mathscr{X}_{Q,n}$ in this case. Thus, Conjecture 4.6
holds trivially, and we have
$\pi_{\chi,S}|_{\overline{G}_{0}}=\pi_{\chi_{0},S}$ with multiplicity one.

  2. –

If $Q(e_{0})=0\mod 3$, then $Y_{Q,n}=3Y$. In this case, one has

$\mathscr{X}_{0,Q,n}=Y_{0}/3Y_{0}.$

On the other hand, $\mathscr{X}_{Q,n}=Y/3Y$. Thus, a $W$-equivariant splitting
is given by

$s(\mathscr{X}_{Q,n}^{\Gamma})=\left\\{ke_{c}:0\leqslant k\leqslant
2\right\\}.$

In either case, we see that Conjecture 4.6 holds.

####  4.3.2. Covers of ${\rm GL}_{2}$ and ${\rm SL}_{2}$

We consider the pair $(\overline{{\rm GL}}_{2},\overline{{\rm SL}}_{2})$ of
$n$-fold covers associated with $\mathbf{p}$ and $\mathbf{q}$ such that
$Q(\alpha^{\vee})=2\mathbf{p}-\mathbf{q}=-1$. For odd $n$, this was discussed
in Example 4.7; thus, we assume in this subsection that

$n=2m$

is even. Again, we set

$d:=\gcd(4\mathbf{p}+1,n).$

It is easy to see the diagram (4.4) becomes

(4.8)
${\mathbf{Z}(m\alpha^{\vee})/\mathbf{Z}(n\alpha^{\vee})}$${Y/Y_{0}}$${\mathbf{Z}\alpha^{\vee}/\mathbf{Z}(n\alpha^{\vee})}$${\mathscr{X}_{Q,n}}$${\mathscr{X}_{Q,n}^{\Gamma}}$${\mathscr{X}_{Q,n}^{\mathfrak{c}}}$${\mathbf{Z}\alpha^{\vee}/\mathbf{Z}(m\alpha^{\vee})}$${\mathscr{X}_{0,Q,n},}$

where $\left|\mathscr{X}_{Q,n}\right|=n^{2}/d$ and
$\left|\mathscr{X}_{Q,n}^{\Gamma}\right|=n/d$. Also, it is easy to obtain

$Y^{\mathfrak{c}}=\left\\{y_{1}e_{1}+y_{2}e_{2}\in Y:\
2|(y_{1}-y_{2})\right\\},$

and thus we see that

$[\mathscr{X}_{Q,n}^{\Gamma}:\mathscr{X}_{Q,n}^{\mathfrak{c}}]=2.$

It follows that

$\mathscr{X}_{Q,n}^{\Gamma}/\mathscr{X}_{Q,n}^{\mathfrak{c}}=\left\\{ie_{1}:0\leqslant
i\leqslant 1\right\\}.$

For every
$\gamma_{i}\in\mathscr{X}_{Q,n}^{\Gamma}/\mathscr{X}_{Q,n}^{\mathfrak{c}}$, we
have

$\mathscr{E}(\chi,{}^{\gamma}\tilde{\chi}_{O};Z(\overline{T}_{0}))=\left\\{\omega_{\gamma,j}:\
0\leqslant j\leqslant 1\right\\}.$

Setting $\gamma_{i}:=ie_{1}$. It follows from Theorem 2.24 (see also [GSS2,
Theorem 8.15]) that

$I(\chi)|_{\overline{G}_{0}}=(m/d)\cdot\bigoplus_{0\leqslant i\leqslant
1}\bigoplus_{0\leqslant j\leqslant 1}I(\omega^{0}_{\gamma_{i},j}).$

Also, the two representations $I(\omega_{\gamma_{0},j}),j=0,1$ are the two
$(K_{0},s_{K})$-unramified constituents of $I(\chi)$. Again, for every
$S\subset\Phi(\chi)$, we have

(4.9) $\pi_{\chi,S}|_{\overline{G}_{0}}=(m/d)\cdot\bigoplus_{0\leqslant
i\leqslant 1}\bigoplus_{0\leqslant j\leqslant
1}\pi(\omega_{\gamma_{i},j})_{S},$

where we have written $\pi(\omega_{\gamma_{i},j})_{S}$ instead of
$\pi_{\omega_{\gamma_{i},j},S}$ for notational convenience.

In the rest of this subsection, we assume that
$\Phi(\chi)=\Delta^{\vee}=\left\\{\alpha^{\vee}\right\\}$. In this case, we
have

${\pi_{\chi,\emptyset}}$${I(\chi)}$${\pi_{\chi,\Delta},}$

where $\pi_{\chi,\emptyset}$ is the covering analogue of the Steinberg
representation, and $\pi_{\chi,\Delta}$ is the theta representation associated
with $\chi$. Since $\overline{{\rm GL}}_{2}$ is always a saturated cover and
thus persistent (see [Ga6, Lemma 2.7]), it follows from [Ga6, Theorem 6.6]
that

  1. –

$\dim{\rm
Wh}_{\psi}(\pi_{\chi,\emptyset})={\left\langle\sigma_{\mathscr{X}},\mathbbm{1}\right\rangle}_{W}=m(n+1)/d$,
which is equal to the number of $W$-orbits in $\mathscr{X}_{Q,n}$;

  2. –

$\dim{\rm
Wh}_{\psi}(\pi_{\chi,\Delta})={\left\langle\sigma_{\mathscr{X}},\varepsilon_{W}\right\rangle}_{W}=m(n-1)/d$,
which is equal to the number of free $W$-orbits in $\mathscr{X}_{Q,n}$.

Here $\mathbbm{1}$ (resp. $\varepsilon_{W}$) denotes the trivial character
(resp. sign character) of $W$.

######  Theorem 4.9.

Let $\chi:Z(\overline{T})\to\mathbf{C}^{\times}$ be an unramified genuine
character such that $\Phi(\chi)=\Delta$. We always have

$\dim{\rm Wh}_{\psi}(\pi(\omega_{\gamma_{i},j})_{\emptyset})+\dim{\rm
Wh}_{\psi}(\pi(\omega_{\gamma_{i},j})_{\Delta})=\dim{\rm
Wh}_{\psi}(I(\omega_{\gamma_{i},j}))=m.$

Moreover, the following fold.

  1. (i)

If $4|n$ (i.e. $2|m$), then

$\dim{\rm
Wh}_{\psi}(\pi(\omega_{\gamma_{i},j})_{\Delta})=\begin{cases}m/2&\text{ for
}i=0,j\in\left\\{0,1\right\\},\\\ m/2&\text{ for }i=1,j=0,\\\ (m-2)/2&\text{
for }i=1,j=1.\end{cases}$

  2. (ii)

If $n=2m$ with $m$ odd, then

$\dim{\rm
Wh}_{\psi}(\pi(\omega_{\gamma_{i},j})_{\Delta})=\begin{cases}(m+1)/2&\text{
for }i=0,j=0,\\\ (m-1)/2&\text{ for }i=0,j=1,\\\ (m-1)/2&\text{ for
}i=1,j\in\left\\{0,1\right\\}.\end{cases}$

In either case, $\pi(\omega_{\gamma_{0},j})_{\Delta},j=0,1$ are the two
$(K_{0},s_{K})$-unramified theta representations associated with
$I(\omega_{\gamma_{0},j})$.

###### Proof.

We first consider the case $m$ is even. In this case, the cover
$\overline{{\rm SL}}_{2}^{(n)}$ is persistent. As
$I(\omega_{\gamma_{0},j}^{0})$ for $j\in\left\\{0,1\right\\}$ is unramified,
it thus follows from [Ga6, Proposition 6.2] that

(4.10) $\dim{\rm
Wh}_{\psi}(\pi(\omega_{\gamma_{0},j})_{\Delta})={\left\langle\sigma_{\mathscr{X}_{0}},\varepsilon_{W}\right\rangle}_{W}=\frac{m}{2},$

which is equal to the number of free $W$-orbits in $\mathscr{X}_{0,Q,n}$. To
deal with $\dim{\rm Wh}_{\psi}(\pi(\omega_{\gamma_{1},j})_{\Delta})$, we
consider a local coefficients matrix $\mathcal{M}(w,i(\omega_{\gamma_{1},j}))$
associated with the map

$\mathcal{T}(w,i(\omega_{\gamma_{1},j}))^{*}:{\rm
Wh}_{\psi}(I(\omega_{\gamma_{1},j}))\longrightarrow{\rm
Wh}_{\psi}(I(\omega_{\gamma_{1},j})),$

which arises from the intertwining operator

$T(w,i(\omega_{\gamma_{1},j})):I(i(\omega_{\gamma_{1},j}))\longrightarrow
I({}^{w}i(\omega_{\gamma_{1},j})).$

One has

$\dim{\rm Wh}_{\psi}(\pi(\omega_{\gamma_{1},j})_{\Delta})=\text{ rank of
}\mathcal{T}(w,i(\omega_{\gamma_{1},j}))^{*}.$

To have a description of $\mathcal{M}(w,i(\omega_{\gamma_{1},j}))$, we first
reconcile some notations used in [GSS2]. Let

$\xi:F^{\times}\longrightarrow\mathbf{C}^{\times}$

be a character such that

$\xi(x)^{n}=\chi(\overline{h}_{\alpha}(x^{n})).$

Now we have from [GSS2, §9.5.3] (especially the proof of Theorem 9.13 there)
the following.

  1. –

The matrix $\mathcal{M}(w,i(\omega_{\gamma_{1},j}))$, by swapping certain rows
and columns, becomes a block-diagonal matrix with $m/2$-blocks and each block
is of size two by two. In fact, each block is associated with a free $W$-orbit
in $\mathscr{X}_{0,Q,n}$. We label these two by two blocks as

$\mathcal{M}_{k},1\leqslant k\leqslant(m/2).$

Every $\mathcal{M}_{k}$ is a non-zero matrix, and thus in particular ${\rm
rank}(\mathcal{M}_{k})\geqslant 1.$

  2. –

There is a special two by two block labelled by $\mathcal{M}_{1}$, which takes
the form

$\mathcal{M}_{1}=\left(\begin{matrix}0&\gamma(1,\xi^{-n})\\\
\boldsymbol{\beta}(\omega_{\gamma_{1},j},s,\psi)&0\end{matrix}\right),$

where

$\boldsymbol{\beta}(\omega_{\gamma_{1},j},s,\psi)=\frac{(1-q^{-1/2}\cdot(\xi\eta_{u})^{-m_{\gamma_{\psi}}})(1-q^{-1/2}\xi^{m_{\gamma_{\psi}}})}{(1-(\xi\eta_{u})^{m_{\gamma_{\psi}}})(1-\xi^{-m_{\gamma_{\psi}}})}.$

Here $u\in O^{\times}$ is a representative of the unique nontrivial coset of
$O^{\times}/O^{\times 2}$, and $\eta_{u}(a)=(u,a)_{n}$ for $a\in F^{\times}$;
also,

$m_{\gamma_{\psi}}:=m\cdot\gamma_{\psi}(\varpi)\in\left\\{\pm m\right\\},$

where $\gamma_{\psi}$ is the Weil-index associated with $\psi$.

  3. –

For every $2\leqslant k\leqslant(m/2)$, one has

$\det(\mathcal{M}_{k})=_{\mathbf{C}^{\times}}\mu(w,i(\chi))^{-1},$

where $=_{\mathbf{C}^{\times}}$ means the equality holds modulo some elements
in $\mathbf{C}^{\times}$ (in fact modulo an element in
$\left\\{\pm\xi^{-m}\right\\}$ as shown in the proof of [GSS2, Theorem 9.13]),
and $\mu(w,i(\chi))$ denotes the Plancherel measure.

Since $\Phi(\chi)=\Delta$, we see that $\mu(w,i(\chi))^{-1}=0$ and thus

${\rm rank}(\mathcal{M}_{k})=1\text{ for }2\leqslant k\leqslant(m/2).$

Also, we have

$\gamma(1,\xi^{-n})=(1-q^{-1}\xi^{-n})/(1-\xi^{n}),$

which equals $0$ in this case. Let ${\rm sgn}(\xi)\in\left\\{\pm 1\right\\}$
be such that

$\xi(\varpi)^{m}={\rm sgn}(\xi)\cdot q^{-1/2}.$

Then it is not hard to see that

$\boldsymbol{\beta}(\omega_{\gamma_{1},j},s,\psi)=0\Longleftrightarrow{\rm
sgn}(\xi)=\eta_{u,(2)}(\varpi),$

where $\eta_{u,(2)}(x)=(u,x)_{2}$. Thus, we could label
$\omega_{\gamma_{1},0}$ to be such that ${\rm sgn}(\xi)=-\eta_{u,(2)}(\varpi)$
and thus $\boldsymbol{\beta}(\omega_{\gamma_{1},0},s,\psi)\neq 0$; in this
case, ${\rm rank}(\mathcal{M}_{1})=1$ and

$\dim{\rm Wh}_{\psi}(\pi(\omega_{\gamma_{1},0})_{\Delta})=1+(m-2)/2=m/2.$

On the other hand, $\omega_{\gamma_{1},1}$ is such that
$\boldsymbol{\beta}(\omega_{\gamma_{1},1}^{0},s,\psi)=0$, and thus

$\dim{\rm Wh}_{\psi}(\pi(\omega_{\gamma_{1},1})_{\Delta})=(m-2)/2.$

This completes the proof for the case $4|n$.

Now we assume that $m=2l+1$ is odd. Again, we first consider the unramified
theta representation
$\pi(\omega_{\gamma_{0},j})_{\Delta},j\in\left\\{0,1\right\\}$. Note that the
$n$-fold cover ${\rm SL}_{2}$ is not persistent in this case, and thus formula
(4.10) does not hold. The $W$-orbits in
$\mathscr{X}_{0,Q,n}=\mathbf{Z}(\alpha^{\vee})/\mathbf{Z}(m\alpha^{\vee})$ are

$\left\\{\mathcal{O}_{i\alpha^{\vee}}:1\leqslant i\leqslant
l\right\\}\cup\left\\{\mathcal{O}_{-l\alpha^{\vee}}\right\\}$

with

$\mathcal{O}_{i\alpha^{\vee}}=\left\\{i\alpha^{\vee},(1-i)\alpha^{\vee}\right\\}\text{
for }1\leqslant i\leqslant l\text{ and
}\mathcal{O}_{-l\alpha^{\vee}}=\left\\{-l\alpha^{\vee}\right\\}.$

Every free orbit $\mathcal{O}_{i\alpha^{\vee}},1\leqslant i\leqslant l$
supports a one-dimensional subspace of ${\rm
Wh}_{\psi}(\pi(\omega_{\gamma_{0},j})_{\Delta})$ for both
$j\in\left\\{0,1\right\\}$. On the other hand, it follows from [Ga2, Theorem
3.14 and §4] that $\mathcal{O}_{-l\alpha^{\vee}}$ supports a one-dimensional
subspace of ${\rm Wh}_{\psi}(\pi(\omega_{\gamma_{0},j})_{\Delta})$ for exactly
one of the unramified characters in
$\left\\{\omega_{\gamma_{0},j}:j=0,1\right\\}$; we assume without loss of
generality that

${\rm
Wh}_{\psi}(\pi(\omega_{\gamma_{0},0})_{\Delta})_{\mathcal{O}_{-l\alpha^{\vee}}}=1,\text{
and thus }{\rm
Wh}_{\psi}(\pi(\omega_{\gamma_{0},1})_{\Delta})_{\mathcal{O}_{-l\alpha^{\vee}}}=1.$

This gives that

(4.11) ${\rm
Wh}_{\psi}(\pi(\omega_{\gamma_{0},0})_{\Delta})=l+1=(m+1)/2,\text{ and }{\rm
Wh}_{\psi}(\pi(\omega_{\gamma_{0},1})_{\Delta})_{\mathcal{O}_{-l\alpha^{\vee}}}=l=(m-1)/2.$

To determine ${\rm Wh}_{\psi}(\pi(\omega_{\gamma_{1},j})_{\Delta})$, we
consider a local coefficients matrix

$\mathcal{M}^{\prime}(w,i(\omega_{\gamma_{1},j}))$

of size $m\times m$ associated with the map
$\mathcal{T}(w,i(\omega_{\gamma_{1},j}))^{*}$ as above. Again, let
$\xi:F^{\times}\to\mathbf{C}^{\times}$ be the character such that
$\xi(x)^{n}=\chi(\overline{h}_{\alpha}(x^{n}))$ for $x\in F^{\times}$. We may
assume that $\xi$ is ramified and $\xi^{2}$ is unramified. Such a matrix
$\mathcal{M}^{\prime}(w,i(\omega_{\gamma_{1},j}))$ is computed in the proof of
[GSS2, Theorem 9.12] with the following properties.

  1. –

The matrix $\mathcal{M}^{\prime}(w,i(\omega_{\gamma_{1},j}))$, by swapping
certain rows and columns, becomes a block-diagonal matrix with every block
associated to a $W$-orbits in $\mathscr{X}_{0,Q,n}$. Thus, we have
$(m+1)/2$-many blocks $\mathcal{M}_{i}^{\prime}$, each associated with
$\mathcal{O}_{i\alpha^{\vee}},1\leqslant i\leqslant l$ or $i=-l$. Also the
size of $\mathcal{M}_{i}^{\prime}$ is exactly
$\left|\mathcal{O}_{i\alpha^{\vee}}\right|$ for every such $i$.

  2. –

For every $1\leqslant i\leqslant l$, the matrix $\mathcal{M}_{i}^{\prime}$ is
a non zero two by two matrix with

$\det(\mathcal{M}_{i}^{\prime})=_{\mathbf{C}^{\times}}\mu(w,i(\chi))^{-1}.$

On the other hand,

$\mathcal{M}_{-l}^{\prime}=\tilde{\gamma}(1,\xi^{-m},\psi),$

which is the metaplectic-gamma factor defined in [Szp3].

Now, since $\Phi(\chi)=\Delta$, we see that $\mu(w,i(\chi))^{-1}=0$ and thus

${\rm rank}(\mathcal{M}_{i}^{\prime})=1\text{ for every }1\leqslant i\leqslant
l.$

On the other hand, one has from [GoSz, Theorem A.1] (see also [GSS2, Theorem
4.4]) that

$\tilde{\gamma}(1,\xi^{-m},\psi)=_{\mathbf{C}^{\times}}\frac{\gamma(1/2,\xi^{m},\psi)}{\gamma(0,\xi^{n},\psi_{2})},$

where $\psi_{2}(x)=\psi(2x)$. Since $\xi$ is ramified with $\xi^{2}$
unramified, we see that $\xi^{m}$ is ramified and therefore

$\gamma(1/2,\xi^{m},\psi)\in\mathbf{C}^{\times}.$

Also, $\xi^{n}$ is unramified with $\xi^{n}(\varpi)=q^{-1}$, and this implies
that

$\tilde{\gamma}(1,\xi^{-m},\psi)=_{\mathbf{C}^{\times}}\frac{1-q^{-1}\xi^{-n}(\varpi)}{1-\xi^{n}(\varpi)}=0.$

This shows that $\mathcal{M}_{-l}^{\prime}=0$ for either $j=0,1$, and thus

$\dim{\rm Wh}_{\psi}(\pi(\omega_{\gamma_{1},j})_{\Delta})=\frac{m-1}{2}$

for both $j=0,1$.

We remark that the above consideration also applies to the unramified
$\pi(\omega_{\gamma_{0},j})_{\Delta},j=0,1$. Indeed, in this case, we may
assume that $\xi$ is unramified. Thus,

$\tilde{\gamma}(1,\xi^{-m},\psi)=\frac{\gamma(1/2,\xi^{m},\psi)}{\gamma(0,\xi^{n},\psi_{2})}=\frac{1-q^{1/2}\xi^{m}(\varpi)}{1-q^{-1/2}\xi^{-m}(\varpi)}\cdot\frac{1-q^{-1}\xi^{-n}(\varpi)}{1-\xi^{n}(\varpi)}.$

Recall that $\xi^{m}(\varpi)={\rm sgn}(\xi)\cdot q^{-1/2}$, and therefore

$\tilde{\gamma}(1,\xi^{-m},\psi)=0\Longleftrightarrow{\rm sgn}(\xi)=-1.$

This also gives the desired equalities for $\dim{\rm
Wh}_{\psi}(\pi(\omega_{\gamma_{0},j})_{\Delta}),j=0,1$. In any case, the proof
of the case $m$ is odd is completed. ∎

######  Remark 4.10.

We always have the equality

$\sum_{0\leqslant i,j\leqslant 1}\dim{\rm
Wh}_{\psi}(\pi(\omega_{\gamma_{i},j})_{\Delta})=n-1.$

For $n=4$ this equality and (4.10) together enforce $\dim{\rm
Wh}_{\psi}(\pi(\omega_{\gamma_{1},j})_{\Delta})=0$ for both $j=0,1$. For
$n=2$, the above equality coupled with (4.11) also imply that

$\dim{\rm Wh}_{\psi}(\pi(\omega_{\gamma_{1},j})_{\Delta})=0\text{ for }j=0,1.$

However, for $n$ large, it is necessary to have an explicit description of a
local coefficients matrix, as used in the proof of Theorem 4.9.

###  4.4. Unitary unramified principal series

In this subsection, we investigate the decomposition of an irreducible
constituent of a unitary $(K,s_{K})$-unramified principal series under the
restriction from $\overline{G}$ to $\overline{G}_{0}$, and also how the
Whittaker dimension behaves with respect to the restriction.

We first recall some notations and results following [Ga7]. Let $I(\chi)$ be a
unitary $(K,s_{K})$-unramified genuine principal series of $\overline{G}$.
Denote by

$R_{\chi}\subset W(\chi)$

the R-group associated with $I(\chi)$, where $W(\chi)\subset W$ is the
stabilizer subgroup of $\chi$. One has an algebra isomorphism

$\mathbf{C}[R_{\chi}]\simeq{\rm End}_{\overline{G}}(I(\chi)),$

given by

$R_{\chi}\ni w\mapsto\mathcal{A}(w,i(\chi)),$

where

$\mathcal{A}(w,i(\chi))=\gamma(w,\chi)\cdot T(w,i(\chi))$

is the normalized intertwining operator. Here $\gamma(w,\chi)$ is the gamma
factor associated with $w\in W$. One knows that $R_{\chi}$ is abelian (see
[Luo3] or [Ga7, Theorem 4.6]) and thus there is a multiplicity-free
decomposition

$I(\chi)=\bigoplus_{\tau\in{\rm Irr}(R_{\chi})}\pi_{\tau},$

where ${\rm Irr}(R_{\chi})={\rm Hom}(R_{\chi},\mathbf{C}^{\times})$ denotes
the characters of $R_{\chi}$. One has

$\mathcal{A}(w,i(\chi))|_{\pi_{\tau}}=\tau(w)\cdot{\rm id}$

for every $w\in R_{\chi}$.

On the dual side, let

$\phi_{\chi}:\text{WD}_{F}\longrightarrow{}^{L}\overline{T}\longrightarrow{}^{L}\overline{G}$

be the parameter associated with $I(\chi)$. Recall the component group

$\mathcal{S}_{\chi}:=\mathcal{S}_{\phi_{\chi}}$

of the centralizer of ${\rm Im}(\phi_{\chi})$ modulo $Z(\overline{G})^{\vee}$.
By using the results of Keys [Key3], it is shown in [Ga7, Theorem 4.9] that

(4.12) $R_{\chi}\simeq\mathcal{S}_{\chi}.$

Note that the same consideration applies to unitary unramified principal
series of $\overline{G}_{0}$, and analogous results hold.

Consider the decomposition from Theorem 2.24 (using slightly different
notation)

(4.13)
$I(\chi)|_{\overline{G}_{0}}=\left|\mathscr{X}_{Q,n}^{\mathfrak{c}}\right|\cdot\bigoplus_{\gamma_{i}}\bigoplus_{j}I(\omega_{\gamma_{i},j}),$

where

$\mathscr{X}_{Q,n}^{\Gamma}/\mathscr{X}_{Q,n}^{\mathfrak{c}}=\left\\{\gamma_{i}\right\\}_{i}.$

For every $\gamma_{i}$ we have

$\mathscr{E}(\chi,{}^{\gamma_{i}}\tilde{\chi}_{O};Z(\overline{T}_{0}))=\left\\{\omega_{\gamma_{i},j}\right\\}_{j},$

which always has size $\left|Y_{0,Q,n}/(Y_{0}\cap Y_{Q,n})\right|$. For
simplicity of notations, we may even write

$\omega_{i,j}:=\omega_{\gamma_{i},j}.$

For every constituent $I(\omega_{i,j}):=I(\omega_{\gamma_{i},j})$, we have the
L-parameter $\phi_{\omega_{i,j}}$ which fits into the following commutative
diagram:

(4.14)
${W_{F}}$${{}^{L}\overline{G}}$${{}^{L}\overline{G}_{0}}$${{}^{L}\overline{H}.}$$\scriptstyle{\phi_{\chi}}$$\scriptstyle{\phi_{\omega_{i,j}}}$$\scriptstyle{f_{G,H}}$$\scriptstyle{f_{G_{0},H}}$

Recall that we defined

$\phi^{\diamondsuit}=f_{G,H}\circ\phi_{\chi}=f_{G_{0},H}\circ\phi_{\omega_{i,j}}$

and thus have

$\mathcal{S}^{\diamondsuit}:=\mathcal{S}_{\phi^{\diamondsuit}}.$

We denote

$\Omega_{\chi}=\left\\{\omega_{i,j}\in{\rm Hom}_{\rm
gen}(Z(\overline{T}_{0}),\mathbf{C}^{\times}):I(\omega_{i,j})\subset
I(\chi)|_{\overline{G}_{0}}\right\\},$

and have from Lemma 3.3 that

$\Omega_{\chi}=\left\\{\omega_{i,j}\in{\rm Hom}_{\rm
gen}(Z(\overline{T}_{0}),\mathbf{C}^{\times}):\omega_{i,j}|_{\overline{T}_{Q,n}^{\sharp}}=\chi|_{\overline{T}_{Q,n}^{\sharp}}\right\\}.$

We also denote

$\displaystyle\Omega^{\rm un}_{\chi}$
$\displaystyle=\mathscr{E}(\chi,\tilde{\chi}_{O};Z(\overline{T}_{0}))$
$\displaystyle=\left\\{\omega_{i,j}\in{\rm Hom}_{\rm
gen}(Z(\overline{T}_{0}),\mathbf{C}^{\times}):I(\omega_{i,j})\subset
I(\chi)|_{\overline{G}_{0}}\text{ and is $(K_{0},s_{K})$-unramified}\right\\}$

For every $\omega^{\flat}\in\Omega^{\rm un}_{\chi}$, set

$\Omega^{\rm un}_{\chi}(\omega^{\flat}):=\left\\{{}^{w}(\omega^{\flat}):w\in
W\right\\}\cap\Omega^{\rm un}_{\chi}=\left\\{{}^{w}(\omega^{\flat}):w\in
W\right\\}\cap\mathscr{E}(\chi,\tilde{\chi}_{O};Z(\overline{T}_{0})).$

For $\omega^{\flat}\in\Omega^{\rm un}_{\chi}$ there are two natural inclusions
(see §3.2 and [Ga7, §4.3])

${\mathcal{S}_{\chi}}$${\mathcal{S}_{\omega^{\flat}}}$${\mathcal{S}^{\diamondsuit}.}$

Note that $\mathcal{S}^{\diamondsuit}$ depends only on $\phi_{\chi}$ and not
on choice of $\omega^{\flat}$. We write

$\chi_{\rm
re}:=\chi|_{\overline{T}_{Q,n}^{\sharp}}=\omega^{\flat}|_{\overline{T}_{Q,n}^{\sharp}},$

where $\mathbbm{L}(\overline{T}_{Q,n}^{\sharp})=Y_{0}\cap Y_{Q,n}$. Since
$Y_{Q,n}^{sc}\subset(Y_{0}\cap Y_{Q,n})$, it follows from (4.12) and [Ga7,
§4.2] that one has an isomorphism

$\mathcal{S}^{\diamondsuit}/\mathcal{S}_{\omega^{\flat}}\simeq W(\chi_{\rm
re})/W(\omega^{\flat}).$

######  Lemma 4.11.

For every $\omega^{\flat}\in\Omega^{\rm un}_{\chi}$, one has

$\Omega^{\rm un}_{\chi}(\omega^{\flat})=\left\\{{}^{w}(\omega^{\flat}):w\in
W(\chi_{\rm re})\right\\},$

with ${\rm Stab}_{W(\chi_{\rm re})}(\omega^{\flat})=W(\omega^{\flat})$. Thus,
$\Omega^{\rm un}_{\chi}(\omega^{\flat})$ is a torsor over
$\mathcal{S}^{\diamondsuit}/\mathcal{S}_{\omega^{\flat}}$.

###### Proof.

It follows from the definition of $\Omega^{\rm un}_{\chi}(\omega^{\flat})$
that

$\displaystyle{}^{w}(\omega^{\flat})\in\Omega^{\rm un}_{\chi}(\omega^{\flat})$
$\displaystyle\Longleftrightarrow$ ${}^{w}(\omega^{\flat})$ is an unramified
extension of $\chi_{\rm re}$ $\displaystyle\Longleftrightarrow$
$\displaystyle\
{}^{w}(\omega^{\flat})|_{Z(\overline{T})\cap\overline{T}_{0}}=\chi_{\rm
re}=\omega^{\flat}|_{Z(\overline{T})\cap\overline{T}_{0}}$
$\displaystyle\Longleftrightarrow$ $\displaystyle\ w\in W(\chi_{\rm re}).$

It is clear that the stabilizer subgroup of $\omega^{\flat}$ is
$W(\omega^{\flat})\subset W(\chi_{\rm re})$. It follows that $\Omega^{\rm
un}_{\chi}(\omega^{\flat})$ is a torsor over
$\mathcal{S}^{\diamondsuit}/\mathcal{S}_{\omega^{\flat}}$. ∎

We have the short exact sequence

${\mathcal{S}_{\chi}/(\mathcal{S}_{\chi}\cap\mathcal{S}_{\omega^{\flat}})}$${\mathcal{S}^{\diamondsuit}/\mathcal{S}_{\omega^{\flat}}}$${\mathcal{S}^{\diamondsuit}/(\mathcal{S}_{\chi}\cdot\mathcal{S}_{\omega^{\flat}}).}$

The action of $\mathcal{S}_{\chi}$ on $\Omega^{\rm un}_{\chi}(\omega^{\flat})$
factors through
$\mathcal{S}_{\chi}/(\mathcal{S}_{\chi}\cap\mathcal{S}_{\omega^{\flat}})$, and
every such orbit is in fact a torsor over
$\mathcal{S}_{\chi}/(\mathcal{S}_{\chi}\cap\mathcal{S}_{\omega^{\flat}})$ by
Lemma 4.11. Thus, we have a decomposition

$\Omega^{\rm
un}_{\chi}(\omega^{\flat})=\bigsqcup_{k=1}^{\left|\mathcal{S}^{\diamondsuit}/(\mathcal{S}_{\chi}\cdot\mathcal{S}_{\omega^{\flat}})\right|}\mathcal{O}(\omega^{\flat})_{k}$

into a disjoint union of $\mathcal{S}_{\chi}$-orbits, where every
$\mathcal{O}(\omega^{\flat})_{k}$ is a torsor over
$\mathcal{S}_{\chi}/(\mathcal{S}_{\chi}\cap\mathcal{S}_{\omega^{\flat}})$.

To proceed, we analyze the space ${\rm
Hom}_{\overline{G}_{0}}(\pi_{\rho},\pi_{\tau})$ with $\rho\in{\rm
Irr}(\mathcal{S}_{\omega^{\flat}}),\tau\in{\rm Irr}(\mathcal{S}_{\chi})$ in
more detail by adapting ideas from [Key3]. To set up some notations:

  1. –

$f:A\longrightarrow B$ is a ring homomorphism;

  2. –

$W_{A}$ is a $A$-module, and $W_{B},V$ are both $B$-modules;

  3. –

setting $E={\rm End}_{B}(V)$, then ${\rm Hom}_{B}(W_{B},V)$ and ${\rm
Hom}_{A}(W_{A},V_{f})$ are both $E$-modules, where $V_{f}=V$ is view as a
$A$-module via $f$.

It is proved in [Key3, Page 53] that if $W_{B}$ is a direct summand of $V$,
then the natural homomorphism

(4.15) ${\rm Hom}_{A}(W_{A},(W_{B})_{f})\longrightarrow{\rm Hom}_{E}({\rm
Hom}_{B}(W_{B},V),{\rm Hom}_{A}(W_{A},V_{f}))$

of vector spaces is an isomorphism. We apply the above to the following data

$A=C_{c}^{\infty}[\overline{G}_{0}],\ B=C_{c}^{\infty}[\overline{G}],\
V=I(\chi),\ W_{A}=\pi_{\rho},\ W_{B}=\pi_{\tau}$

to obtain

(4.16) ${\rm
Hom}_{\overline{G}}(\pi_{\rho},\pi_{\tau}|_{\overline{G}_{0}})={\rm
Hom}_{\mathbf{C}[\mathcal{S}_{\chi}]}\left(\tau,{\rm
Hom}_{\overline{G}_{0}}(\pi_{\rho},I(\chi)|_{\overline{G}_{0}})\right),$

where we note that

$\tau\simeq{\rm Hom}_{\overline{G}}(\pi_{\tau},I(\chi))$

as $\mathbf{C}[\mathcal{S}_{\chi}]$-module. In view of the decomposition of
$I(\chi)|_{\overline{G}_{0}}$ in (4.13), we first observe that (see [BZ2,
Theorem 2.9])

${\rm Hom}_{\overline{G}_{0}}(\pi_{\rho},I(\omega_{i,j}))\neq
0\Longleftrightarrow\omega_{i,j}\in\Omega^{\rm un}(\omega^{\flat}).$

Denoting

$\Pi(\omega^{\flat}):=\bigoplus_{\omega_{i,j}\in\Omega^{\rm
un}(\omega^{\flat})}I(\omega_{i,j})$

and

$\Pi(\mathcal{O}(\omega^{\flat})_{k}):=\bigoplus_{\omega_{i,j}\in\mathcal{O}(\omega^{\flat})_{k}}I(\omega_{i,j}),$

one clearly has

$\Pi(\omega^{\flat})=\bigoplus_{k}\Pi(\mathcal{O}(\omega^{\flat})_{k}).$

Also,

${\rm
Hom}_{\overline{G}_{0}}(\pi_{\rho},I(\chi)|_{\overline{G}_{0}})=\left|\mathscr{X}_{Q,n}^{\mathfrak{c}}\right|\cdot\bigoplus_{k}{\rm
Hom}_{\overline{G}_{0}}(\pi_{\rho},\Pi(\mathcal{O}(\omega^{\flat})_{k})).$

The action of $w\in\mathcal{S}_{\chi}$ on ${\rm
Hom}_{\overline{G}_{0}}(\pi_{\rho},I(\chi)|_{\overline{G}_{0}})$ is given by
transporting the action of $\mathcal{A}(w,\chi)$ on $I(\chi)$. Also, we have a
decomposition

$\mathcal{A}(w,\chi)=\left|\mathscr{X}_{Q,n}^{\mathfrak{c}}\right|\cdot\bigoplus_{i,j}\mathcal{A}(w,\omega_{i,j}),$

where

$\mathcal{A}(w,\omega_{i,j}):I(\omega_{i,j})\longrightarrow
I({}^{w}\omega_{i,j})$

is the normalized intertwining operator of each $I(\omega_{i,j})$. We see that
the restriction $\mathcal{A}(w,\chi)|_{\Pi(\mathcal{O}(\omega^{\flat})_{k})}$
is well-defined for every $k$. Thus, the space

${\rm
Hom}_{\overline{G}_{0}}(\pi_{\rho},\Pi(\mathcal{O}(\omega^{\flat})_{k}))$

affords a representation of $\mathbf{C}[\mathcal{S}_{\chi}]$ of dimension
$\left|\mathcal{O}(\omega^{\flat})_{k})\right|$, and $\mathcal{S}_{\chi}$ also
acts on the $\left|\Omega^{\rm un}_{\chi}(\omega^{\flat})\right|$-dimensional
space

${\rm
Hom}_{\overline{G}_{0}}(\pi_{\rho},\Pi(\omega^{\flat}))=\bigoplus_{k}{\rm
Hom}_{\overline{G}_{0}}(\pi_{\rho},\Pi(\mathcal{O}(\omega^{\flat})_{k})).$

######  Theorem 4.12.

Let $I(\chi)$ be a unitary $(K,s_{K})$-unramified genuine principal series of
$\overline{G}$ and $I(\omega^{\flat})$ an $(K_{0},s_{K})$-unramified
constituent of $I(\chi)|_{\overline{G}_{0}}$. Keep notations as above. Then as
representations of $\mathcal{S}_{\chi}$,

(4.17) ${\rm
Hom}_{\overline{G}_{0}}(\pi_{\rho},\Pi(\mathcal{O}(\omega^{\flat})_{k}))\simeq{\rm
Ind}_{\mathcal{S}_{\chi}\cap\mathcal{S}_{\omega^{\flat}}}^{\mathcal{S}_{\chi}}(\rho)$

for every $k$. Consequently,

(4.18) $\dim{\rm
Hom}_{\overline{G}_{0}}(\pi_{\rho},\pi_{\tau})=\left|\mathscr{X}_{Q,n}^{\mathfrak{c}}\right|\cdot{\left\langle{\rm
Ind}_{\mathcal{S}_{\omega^{\flat}}}^{\mathcal{S}^{\diamondsuit}}\rho,{\rm
Ind}_{\mathcal{S}_{\chi}}^{\mathcal{S}^{\diamondsuit}}\tau\right\rangle}_{\mathcal{S}^{\diamondsuit}}$

for every $\tau\in{\rm Irr}(\mathcal{S}_{\chi})$ and $\rho\in{\rm
Irr}(\mathcal{S}_{\omega^{\flat}})$.

###### Proof.

First,

${\rm
Hom}_{\overline{G}_{0}}(\pi_{\rho},\Pi(\mathcal{O}(\omega^{\flat})_{k}))=\bigoplus_{\omega_{i,j}\in\mathcal{O}(\omega^{\flat})_{k}}{\rm
Hom}_{\overline{G}_{0}}(\pi_{\rho},I(\omega_{i,j})).$

Since $I(\omega_{i,j})\simeq I(\omega^{\flat})$ for every
$\omega_{i,j}\in\mathcal{O}(\omega^{\flat})_{k}$, it is easy to check that as
representations of $\mathcal{S}_{\chi}\cap\mathcal{S}_{\omega^{\flat}}$, one
has

${\rm
Hom}_{\overline{G}_{0}}(\pi_{\rho},I(\omega_{i,j})\simeq\rho|_{\mathcal{S}_{\chi}\cap\mathcal{S}_{\omega^{\flat}}}.$

Moreover, as $\mathcal{O}(\omega^{\flat})_{k}$ is a torsor over
$\mathcal{S}_{\chi}/(\mathcal{S}_{\chi}\cap\mathcal{S}_{\omega^{\flat}})$ for
every $k$, the representation of $\mathcal{S}_{\chi}$ on ${\rm
Hom}_{\overline{G}_{0}}(\pi_{\rho},\Pi(\mathcal{O}(\omega^{\flat})_{k}))$ is
indeed ${\rm
Ind}_{\mathcal{S}_{\chi}\cap\mathcal{S}_{\omega^{\flat}}}^{\mathcal{S}_{\chi}}\rho$.
This gives (4.17).

Now to prove (4.18), we see that as representations of $\mathcal{S}_{\chi}$,

$\displaystyle{\rm
Hom}_{\overline{G}_{0}}(\pi_{\rho},I(\chi)|_{\overline{G}_{0}})$
$\displaystyle=\left|\mathscr{X}_{Q,n}^{\mathfrak{c}}\right|\cdot\bigoplus_{k}{\rm
Hom}_{\overline{G}_{0}}(\pi_{\rho},\Pi(\mathcal{O}(\omega^{\flat})_{k}))$
$\displaystyle=\left|\mathscr{X}_{Q,n}^{\mathfrak{c}}\right|\cdot\left|\mathcal{S}^{\diamondsuit}/(\mathcal{S}_{\chi}\cdot\mathcal{S}_{\omega^{\flat}})\right|\cdot{\rm
Ind}_{\mathcal{S}_{\chi}\cap\mathcal{S}_{\omega^{\flat}}}^{\mathcal{S}_{\chi}}\rho.$

On the other hand, the Frobenius reciprocity gives

${\left\langle{\rm
Ind}_{\mathcal{S}_{\omega^{\flat}}}^{\mathcal{S}^{\diamondsuit}}\rho,{\rm
Ind}_{\mathcal{S}_{\chi}}^{\mathcal{S}^{\diamondsuit}}\tau\right\rangle}_{\mathcal{S}^{\diamondsuit}}={\left\langle{\rm
Res}_{\mathcal{S}_{\chi}}^{\mathcal{S}^{\diamondsuit}}{\rm
Ind}_{\mathcal{S}_{\omega^{\flat}}}^{\mathcal{S}^{\diamondsuit}}\rho,\tau\right\rangle}_{\mathcal{S}_{\chi}},$

and by Mackey’s theory one has

${\rm Res}_{\mathcal{S}_{\chi}}^{\mathcal{S}^{\diamondsuit}}{\rm
Ind}_{\mathcal{S}_{\omega^{\flat}}}^{\mathcal{S}^{\diamondsuit}}\rho=\bigoplus_{s\in\mathcal{S}^{\diamondsuit}/(\mathcal{S}_{\chi}\cdot\mathcal{S}_{\omega^{\flat}})}{\rm
Ind}_{\mathcal{S}_{\chi}\cap\mathcal{S}_{\omega^{\flat}}}^{\mathcal{S}_{\chi}}\rho=\left|\mathcal{S}^{\diamondsuit}/(\mathcal{S}_{\chi}\cdot\mathcal{S}_{\omega^{\flat}})\right|\cdot{\rm