# principal series representations of metaplectic groups

Shiang Tang  Shiang Tang, Department of Mathematics, University of Utah, Salt
Lake City, UT 84112<EMAIL_ADDRESS>
###### Abstract.

We study the principal series representations of central extensions of a split
reductive algebraic group by a cyclic group of order $n$. We compute the
Plancherel measure of the representation using Eisenstein series and a
comparison method. In addition, we construct genuine central characters of the
metaplectic torus in the simply-laced case.

##  1\. Introduction

Let $F$ be a non-archimedean local field with ring of integers
$\mathcal{O}_{F}$ and assume that $\underline{G}$ is a split reductive group
defined over $F$. Let $n$ be a positive integer. A central extension of
$\underline{G}$ by $\mu_{n}$ is a topological group $G$ that fits in the
following exact sequence:

$1\to\mu_{n}\to G\to\underline{G}\to 1$

The aim of this paper is to study the principal series representations of $G$.
Such representations have been studied by several authors. Kazhdan and
Patterson [5] have studied the case of $GL_{n}$, in which they have carried
out the local theory and global theory, in particular, the theory of
Eisenstein series in great detail. Savin [11] has considered the case when $G$
is simply laced and simply connected. The double cover of a general simply
connected reductive group has been studied by Loke and Savin [6] in which they
construct genuine automorphic representations of the metaplectic torus using
theta series. McNamara [7] has generalized the theory to the case of an
arbitrary reductive group, under the hypothesis that $2n$ is coprime to the
residual characteristic of $F$ and $F^{\times}$ contains $2n$ distinct $2n$-th
roots of unity. In another direction, Weissman [14] has studied extensively
the structure of the metaplectic torus, both in the local and global cases,
and their representations. This paper studies the same objects the above
authors have considered, with the following new results and constructions:

  1. (1)

We remove the tameness assumption on $n$ imposed by McNamara and other
authors.

  2. (2)

We use a comparison method to compute the Plancherel measures of ramified
representations of $G(F)$. More precisely, we compare the covering group with
a well-chosen linear group (motivated by [14]), and transfer the Plancherel
measure from the linear case to the non-linear case. This has been done for
$G$ the $n$-fold cover of $\mathrm{SL}_{2}(F)$ by Goldberg and Szpruch in [4]
using a different approach.

  3. (3)

We give explicit constructions of genuine Weyl group invariant central
automorphic character of the covering torus (in the sense of [14], Definition
4.7).

The organization of this paper is as follows. In Section 2, we review the
local theory, most of the results are known for a long time. In Section 3, we
review the global theory, in particular, we obtain a product formula of local
Plancherel measures which will be used in the comparison method mentioned
above. In Section 4, we carry out the comparison method and prove our main
theorem on the relation between Plancherel measures of non-linear groups and
their associated linear groups, under the assumption on the existence of
genuine Weyl group invariant central automorphic characters of the metaplectic
torus. In Section 5, we construct explicitly such characters for simply-laced
groups, and hence prove their existence.

I would like to thank my advisor Gordan Savin, without whom none of this work
would be possible.

##  2\. central extensions of split reductive groups over local fields

###  2.1. Group theory

Let $p$ be a fixed prime and $n$ be a fixed integer. Let $F$ be a finite
extension of $\mathbb{Q}_{p}$ with ring of integers $R$ and maximal ideal
$\mathfrak{m}$ generated by a uniformizer $\varpi$. We assume that $F$
contains the group $\mu_{n}$ of $n$-th roots of unity. Let $\underline{G}$ be
a split reductive group over $F$ with Lie algebra $\mathfrak{g}$. Let
$\underline{T}$ be a fixed maximal split torus of $\underline{G}$ with Lie
algebra $\mathfrak{t}$. Let $\Phi=\Phi(\underline{G},\underline{T})$ be the
root system associated to $\underline{T}$ and let $\Phi^{\vee}$ the set of
coroots. We fix a Borel subgroup $\underline{B}$ containing $\underline{T}$
with the corresponding set $\Phi^{+}$ of positive roots and let
$\Delta\subset\Phi^{+}$ be the subset of simple roots. Let $X$ be the
character lattice and $Y$ be the cocharacter lattice of $\underline{T}$. We
have $\Phi\subset X$ and $\Phi^{\vee}\subset Y$. The quadruple

$(X,\Phi,Y,\Phi^{\vee})$

is called the root datum of $\underline{G}$, which determines $\underline{G}$
up to isomorphism. Let $H_{\beta}$ ($\beta\in\Delta$), $X_{\alpha}$
($\alpha\in\Phi$) be a Chevalley basis of
$\mathfrak{g}^{der}=[\mathfrak{g},\mathfrak{g}]$. Let $\underline{G}^{sc}$ be
the simply connected Chevalley group with Lie algebra $\mathfrak{g}^{der}$.
There exists a map

$\underline{G}^{sc}\to\underline{G}$

which differentiates to the inclusion map $\mathfrak{g}^{der}\to\mathfrak{g}$
on the level of Lie algebras. The elements $\underline{x}_{\alpha}(t)$,
$\underline{w}_{\alpha}(t)$, $\underline{h}_{\alpha}(t)$ in
$\underline{G}^{sc}$ are defined as in [9]. We use the same symbols to denote
their images in $\underline{G}$. However, $\underline{x}_{\alpha}(t)$ for
$t\in F,\alpha\in\Phi$ no longer generate $\underline{G}$. Let $\underline{K}$
be a maximal subgroup of $\underline{G}$ with the property that the adjoint
action of $\underline{K}$ preserves the $R$-lattice in $\mathfrak{g}^{der}$
spanned by the Chevalley basis. Then $\underline{K}$ is a maximal compact
subgroup of $\underline{G}$. In particular, when $\underline{G}$ is simply-
connected, $\underline{K}$ is the subgroup of $\underline{G}$ generated by
$\underline{x}_{\alpha}(t)$ for $t\in R$, $\alpha\in\Phi$.

We may identify $\underline{T}$ with $Y\otimes F^{\times}$. Let $(,):Y\times
Y\to\mathbb{Z}$ be an integer valued symmetric bilinear form on $Y$ that is
Weyl group invariant and satisfies $(\alpha^{\vee},\alpha^{\vee})=2$ when
$\alpha$ is long. Let $G$ be a nontrivial central extension of $\underline{G}$
by the group of $n$-th roots of unity:

$1\rightarrow\mu_{n}\rightarrow G\to\underline{G}\rightarrow 1$

Let $T$ be the preimage of $\underline{T}$ in $G$. We assume the followings:

  1. (1)

$[y_{1}\otimes t_{1},y_{2}\otimes t_{2}]=(t_{1},t_{2})_{n}^{(y_{1},y_{2})}$

where $y_{i}\otimes t_{i}$ represents an element in the fibre of $y_{i}\otimes
t_{i}\in\underline{T}$ by abuse of notation and $(,)_{n}$ is the Hilbert norm
residue symbol in class field theory.

  2. (2)

If $p$ does not divide $n$, then there is a splitting (not necessarily unique)
section $s:\underline{K}\to G$. The image of $s$ is denoted by $K$.

######  Remark 1.

Item 1 implies that given $t,t^{\prime}\in T$, their commutator
$[t,t^{\prime}]\in\mu_{n}$ only depends on
$\pi(t),\pi(t^{\prime})\in\underline{T}$.

######  Remark 2.

Item 2 implies that on $\underline{K}_{0}$, the subgroup of $\underline{K}$
generated by $\underline{x}_{\alpha}(t)$ with $t\in R$ and $\alpha\in\Phi$,
$s$ is the same as the canonical section which sends
$\underline{x}_{\alpha}(t)$ to its unique lift in $G$.

######  Remark 3.

For any $\alpha\in\Phi$, $t\in F$, $\underline{x}_{\alpha}(t)\in\underline{G}$
lifts uniquely to $G$, and we use $x_{\alpha}(t)$ to denote the lift. Define

$\displaystyle w_{\alpha}(t):=x_{\alpha}(t)x_{-\alpha}(-t^{-1})x_{\alpha}(t)$
$\displaystyle h_{\alpha}(t):=w_{\alpha}(t)w_{\alpha}(-1)$

By [9], when $\Phi$ is simply laced, we have the following relations in $T$:

$h_{\alpha}(t)h_{\alpha}(u)=(t,u)_{n}h_{\alpha}(tu)$
$[h_{\alpha}(t),h_{\beta}(u)]=(t,u)_{n}^{(\alpha^{\vee},\beta^{\vee})}$

However, the $h_{\alpha}$’s do not generate $T$ unless $[G,G]=G$.

###  2.2. Unramified principal series

In this section, we keep the assumptions made in Section 2.1. Suppose we have
an irreducible genuine representation $\sigma$ of the metaplectic torus $T$
which is Weyl group invariant, i.e., its central character is fixed by the
conjugation action of the Weyl group. By [1], Proposition 2.2, there is a
bijection between irreducible genuine representations of $T$ and genuine
central characters of its center $Z_{T}$. We say that $\sigma$ is _unramified_
if $\sigma^{T_{0}}\neq 0$ where $T_{0}:=T\cap K$.

######  Proposition 2.1.

$Z_{T}\cdot T_{0}$ is a maximal abelian subgroup of $T$ containing $T_{0}$.

###### Proof.

Clearly $Z_{T}\cdot T_{0}$ is an abelian group. We need to show that any
element in $Z_{T}(T_{0})$ must belong to $Z_{T}\cdot T_{0}$. Observe that
every element in $T/T_{0}$ is represented by an element in $T$ which is of the
form $x\otimes\varpi$ for some $x\in Y$. Therefore it is enough to show that
if $x\otimes\varpi$ centralizes $T_{0}$, then it must lie in $Z_{T}$. To see
this, note that $[x\otimes\varpi,y\otimes u]=1$ for any $y\in Y$ and any $u\in
R^{\times}$, the LHS of which is
$(\varpi,u)^{(x,y)}=(\bar{u}^{-1})^{(x,y)\cdot\frac{q-1}{n}}$ by Item 1. So
$n$ divides $(x,y)$, for all $y\in Y$. It follows that $x\otimes\varpi\in
Z_{T}$. ∎

######  Proposition 2.2.

If $p$ does not divide $n$ and $\sigma$ is unramified, then
$\dim\sigma^{T_{0}}=1$.

###### Proof.

Let $\chi$ be the central character of $\sigma$. Observe that $\chi$ is
trivial on $Z_{T}\cap K$. In fact, choose $0\neq v_{0}\in\sigma^{T_{0}}$, we
have for any $z\in T_{0}$, $v_{0}=\sigma(z)v_{0}=\chi(z)v_{0}$ which forces
$\chi(z)$ to be $1$. Let $\tilde{\chi}$ be the extension of $\chi$ to
$Z_{T}\cdot T_{0}$ by letting $\chi$ act trivially on $T_{0}$. Form the
induction

$Ind_{Z_{T}T_{0}}^{T}(\tilde{\chi})$

which has central character $\chi$. Frobenius reciprocity gives

$Hom_{T}(Ind_{Z_{T}T_{0}}^{T}(\tilde{\chi}),\sigma)=Hom_{Z_{T}T_{0}}(\tilde{\chi},\sigma|_{Z_{T}T_{0}})$

Since $\sigma$ is unramified, there is a line $L\subset\sigma$ that is fixed
by $T_{0}$, so $\tilde{\chi}$ is a summand of $\sigma|_{Z_{T}T_{0}}$ and hence
there is a nonzero $T$-intertwining map from
$Ind_{Z_{T}T_{0}}^{T}(\tilde{\chi})$ to $\sigma$.

Claim: For any $s\in T-Z_{T}\cdot T_{0}$, $\tilde{\chi}^{s}\neq\tilde{\chi}$
on $T_{0}$.

Once we have proved this claim, it will follow by Mackey’s criterion that
$Ind_{Z_{T}T_{0}}^{T}(\tilde{\chi})$ is irreducible and isomorphic to
$\sigma$, and that $\tilde{\chi}$ has multiplicity one in
$Ind_{Z_{T}T_{0}}^{T}(\tilde{\chi})|_{Z_{T}T_{0}}$, from which the proposition
follows.

To prove the claim, we may assume that $s=x\otimes\varpi$ for some $x\in Y$.
If the claim were false, then $\tilde{\chi}([s,t])=1$ for all $t\in T_{0}$,
that is, $[x\otimes\varpi,y\otimes u]=1$ for any $y\in Y$ and $u\in
R^{\times}$. By the proof of the previous proposition, this would imply $s\in
Z_{T}$, a contradiction. ∎

######  Lemma 2.3.

Suppose $\alpha,\beta$ are roots in $\Phi$ such that $<\beta,\alpha>=-1$
(which implies that $|\alpha|\geq|\beta|$ and $w_{\alpha}\beta=\alpha+\beta$),
then

$h_{\alpha+\beta}(t)=(c(\alpha,\beta)t^{<\alpha,\beta>},t^{n_{\beta}})h_{\alpha}(t^{-<\alpha,\beta>})h_{\beta}(t)$

where $n_{\beta}$ is 1 when $\beta$ is long and is the maximal number of links
between adjacent dots in the Dynkin diagram of $\Phi$ when $\beta$ is short.

###### Proof.

Let $u=t^{-<\alpha,\beta>}$.

$\displaystyle\>\>\>\>\>h_{\alpha+\beta}(t)$
$\displaystyle=h_{w_{\alpha}\beta}(t)$ $\displaystyle=\zeta
w_{\alpha}(u)h_{\beta}(t)w_{\alpha}(-u)$ $\displaystyle=\zeta
w_{\alpha}(u)h_{\beta}(t)x_{\alpha}(-u)x_{-\alpha}(u^{-1})x_{\alpha}(-u)$
$\displaystyle=\zeta
w_{\alpha}(u)x_{\alpha}(-t^{<\alpha,\beta>}u)h_{\beta}(t)x_{-\alpha}(u^{-1})x_{\alpha}(-u)$
$\displaystyle=\zeta
w_{\alpha}(u)x_{\alpha}(-t^{<\alpha,\beta>}u)x_{-\alpha}(t^{-<\alpha,\beta>}u^{-1})h_{\beta}(t)x_{\alpha}(-u)$
$\displaystyle=\zeta
w_{\alpha}(u)x_{\alpha}(-t^{<\alpha,\beta>}u)x_{-\alpha}(t^{-<\alpha,\beta>}u^{-1})x_{\alpha}(-t^{<\alpha,\beta>}u)h_{\beta}(t)$
$\displaystyle=\zeta
w_{\alpha}(u)w_{\alpha}(-t^{<\alpha,\beta>}u)h_{\beta}(t)$
$\displaystyle=\zeta h_{\alpha}(t^{-<\alpha,\beta>})h_{\beta}(t)$

where $\zeta\in\mu_{n}$.

We now compute $\zeta$: By [9],

$w_{\alpha}(u)h_{\beta}(t)w_{\alpha}(-u)=h_{w_{\alpha}\beta}(c(\alpha,\beta)ut)h_{w_{\alpha}\beta}(c(\alpha,\beta)u)^{-1}$
$=(c(\alpha,\beta)u^{-1},c(\alpha,\beta)u)^{n_{\beta}}h_{w_{\alpha}\beta}(c(\alpha,\beta)ut)h_{w_{\alpha}\beta}(c(\alpha,\beta)u^{-1})$
$=(cu^{-1},cu)^{n_{\beta}}(cut,cu^{-1})^{n_{\beta}}h_{w_{\alpha}\beta}(t)=(cu^{-1},t^{-1})^{n_{\beta}}h_{w_{\alpha}\beta}(t)$

So $\zeta=(cu^{-1},t)^{n_{\beta}}=(ct^{<\alpha,\beta>},t)^{n_{\beta}}$. ∎

######  Proposition 2.4.

Suppose that $\Phi$ is not of type $A_{1}$ and $\alpha\in\Phi$ is long, then
$\sigma(h_{\alpha}(t))=1$ for any $\alpha\in\Phi$, $t\in(F^{\times})^{n}$.

###### Proof.

We have

$w_{\alpha}(1)h_{\alpha}(t)w_{\alpha}(1)=h_{\alpha}(t^{-1})$
$w_{\gamma}(1)h_{\alpha}(t)w_{\gamma}(-1)=h_{w_{\gamma}\alpha}(t)$

for any $t\in(F^{\times})^{n}$ and any $\alpha,\gamma\in\Phi$. Since the
central character of $\sigma$ is invariant under the action of $W$, the above
identities imply $\sigma(h_{\alpha}(t))=\pm 1$ for any $t\in(F^{\times})^{n}$
and for all roots $\alpha\in\Phi$ of the same length.

Suppose that $\Phi$ is simply-laced and is not of type $A_{1}$. We have by
2.3, $h_{\alpha+\beta}(t)=h_{\alpha}(t)h_{\beta}(t)$ when $<\beta,\alpha>=-1$
and $t\in(F^{\times})^{n}$. It is clear that such pair of roots exist if
$\Phi$ is not of type $A_{1}$. Pick such pair of roots $\alpha,\beta$, we have
$\sigma(h_{\alpha+\beta}(t))=\sigma(h_{\alpha}(t))=\sigma(h_{\beta}(t))=1$.
Therefore $\sigma(h_{\alpha}(t))=1$, $t\in(F^{\times})^{n}$ for all roots
$\alpha$.

The demonstration for $\Phi$ multi-laced is similar. ∎

######  Remark 4.

2.4 is true for short roots for $\Phi$ of type $G_{2}$ and $3$ not dividing
$n$, and for $\Phi$ of type $F_{4}$ and $2$ not dividing $n$.

######  Example 2.1.

If $\underline{G}=SL_{2}$ and $n=1$, any quadratic character of $T$ is Weyl
group invariant. So 2.4 fails in this case.

Let $\underline{B}$ be a Borel subgroup of $\underline{G}$ corresponding to
$\Phi^{+}$, we have $\underline{B}=\underline{T}N$ where $N$ is the unipotent
radical of $\underline{B}$ which lifts uniquely to $G$. Let $B=TN$ be the
preimage of $B=\underline{T}N$. Define an induced representation (normalized
induction)

$I(\nu)=Ind_{B}^{G}(\nu\otimes\sigma)$

where $\nu$ is an unramified character of $\underline{T}$. For any $w\in
N_{K}(T)/T_{0}$, choose a representative $\hat{w}$ in $N_{K}(T)$. There is a
unique isomorphism

$j_{\hat{w}}:\sigma^{\hat{w}}\to\sigma$

such that $j_{\hat{w}}$ fixes $\sigma^{T_{0}}$ pointwise. In other words,
$j\circ\sigma(\hat{w}^{-1}t\hat{w})=\sigma(t)\circ j$, $\forall t\in T$ and
$j(v_{0})=v_{0}$ for any $v_{0}\in\sigma^{T_{0}}$. If $\hat{w}^{\prime}$ is a
different representative in $N_{K}(T)$, say $\hat{w}^{\prime}=\hat{w}t_{0}$
for some $t_{0}\in T_{0}$, then
$j_{\hat{w}^{\prime}}=j_{\hat{w}}\circ\sigma(t_{0})$.

Fix an Haar measure on $N$. For every $w\in W$, there is an intertwining
operator

$A_{w}=A_{w}(\nu):I(\nu)\to I(\nu^{w})$

defined as follows:

$A_{w}(f)(x)=j_{\hat{w}}\int_{N_{w}}f(\hat{w}^{-1}nx)dn$

where $\hat{w}\in N_{K}(T)$ is a representative of $w$.

######  Lemma 2.5.

$A_{w}$ is well-defined and is independent of the choice of a representative
of $w$.

###### Proof.

$\displaystyle\>\>\>\>\>(A_{w}f)(tx)$
$\displaystyle=j_{\hat{w}}\int_{N_{w}}f(\hat{w}^{-1}ntx)dn$
$\displaystyle=j_{\hat{w}}\int_{N_{w}}f((\hat{w}^{-1}t\hat{w})\hat{w}^{-1}(t^{-1}nt)x)dn$
$\displaystyle=\delta^{1/2}(\hat{w}^{-1}t\hat{w})\chi(\hat{w}^{-1}t\hat{w})j_{\hat{w}}\sigma(\hat{w}^{-1}t\hat{w})\int_{N_{w}}f(\hat{w}^{-1}(t^{-1}nt)x)dn$
$\displaystyle=\delta^{1/2}(\hat{w}^{-1}t\hat{w})\chi(\hat{w}^{-1}t\hat{w})\sigma(t)j_{\hat{w}}\int_{N_{w}}f(\hat{w}^{-1}(t^{-1}nt)x)dn$
$\displaystyle=\delta^{1/2}(t)\chi^{w}(t)\sigma(t)j_{\hat{w}}\int_{N_{w}}f(\hat{w}^{-1}nx)dn$
$\displaystyle=\delta^{1/2}(t)\chi^{w}(t)\sigma(t)(A_{w}f)(x)$

So $A_{w}$ is well-defined. Since
$j_{\hat{w}^{\prime}}=j_{\hat{w}}\circ\sigma(t_{0})$, $A_{w}$ is independent
of the choice of $\hat{w}$.

∎

If $\sigma$ is unramified, then $I(\nu)^{K}$ is nonzero. Moreover, it has
dimension one by 2.2. The induced map

$I(\nu)^{K}\to I(\nu^{w})^{K}$

can be computed explicitly. See [6], Section 6 for the case when $G$ is the
double cover of a simply connected algebraic group over non-archimedean
fields, and see [7], Section 7 for the case when $G$ is an $n$-fold cover of
an algebraic group over non-archimedean fields. In [12], an archimedean
analogue is also established. The following proposition is well-known, we
record it here for the reader’s convenience. We normalize the Haar measure $m$
on $F$ such that $m(R)=d_{F}^{-1/2}$ where $d_{F}$ is the discriminant of $F$
over $\mathbb{Q}_{p}$.

######  Proposition 2.6.

Suppose that $\alpha$ is a long root when $\Phi$ is not of type $A_{1}$ and
suppose $\sigma$ satisfies 2.4 when $\Phi$ is of type $A_{1}$. Let
$f_{\nu}^{o}\in I(\nu)^{K}$ be a function such that $f_{\nu}^{o}(1)=v_{0}$
where $v_{0}$ is a fixed nonzero vector in $\sigma^{T_{0}}$. Let
$w=w_{\alpha}$. Then $A_{w}f_{\nu}^{o}=c(\nu)f_{\nu^{w}}^{o}$ where

$c(\nu)=d_{F}^{-1/2}\frac{1-q^{-1}\nu(h_{\alpha}(\varpi^{n}))}{1-\nu(h_{\alpha}(\varpi^{n}))}$

###### Proof.

This is essentially a calculation in the group $SL_{2}$. To alleviate
notation, we use $f$ to denote $f_{\nu}^{o}$.

$\int_{F}f(w_{\alpha}(-1)x_{\alpha}(t))dt=m(R)v_{0}+\sum_{k\geq
1}\int_{\varpi^{-k}R^{\times}}f(w_{\alpha}(-1)x_{\alpha}(t))dt$

For $t\in F-R$,

$\displaystyle\>\>\>\>\>f(w_{\alpha}(-1)x_{\alpha}(t))$
$\displaystyle=f(x_{\alpha}(t^{-1})w_{\alpha}(-1)x_{\alpha}(t))$
$\displaystyle=f(w_{\alpha}(-1)x_{-\alpha}(-t^{-1})x_{\alpha}(t))$
$\displaystyle=f(w_{\alpha}(-1)x_{-\alpha}(-t^{-1})x_{\alpha}(t)x_{-\alpha}(-t^{-1}))$
$\displaystyle=f(w_{\alpha}(-1)w_{-\alpha}(-t^{-1}))=f(w_{\alpha}(-1)w_{\alpha}(t))$
$\displaystyle=f(w_{\alpha}(-1)h_{\alpha}(t)w_{\alpha}(1))$
$\displaystyle=f(h_{\alpha}(t^{-1}))$

Let $t^{-1}=\varpi^{k}u$, $u\in R^{\times}$, then

$h_{\alpha}(t^{-1})=(\varpi^{k},u)_{n}^{-1}h_{\alpha}(\varpi^{k})h_{\alpha}(u)=(\bar{u})^{k\frac{q-1}{n}}h_{\alpha}(\varpi^{k})h_{\alpha}(u)$

where $(\bar{u})^{k\frac{q-1}{n}}\in\mathbb{F}_{q}^{\times}$ is a $n$-th root
of unity in $\mathbb{F}_{q}$. The expression is well-defined because $n|q-1$.
We have

$\displaystyle\>\>\>\>\>f(h_{\alpha}(t^{-1}))$
$\displaystyle=(\bar{u})^{k\frac{q-1}{n}}\delta^{1/2}\nu(h_{\alpha}(\varpi^{k}))\sigma(h_{\alpha}(\varpi^{k}))v_{0}$
$\displaystyle=(\bar{u})^{k\frac{q-1}{n}}q^{-k}\nu(h_{\alpha}(\varpi^{k}))\sigma(h_{\alpha}(\varpi^{k}))v_{0}$

It is easy to see that, for any $m\in\mathbb{Z}$,
$\sum_{x\in\mathbb{F}_{q}}x^{m}=0$ if $q-1$ does not divide $m$.

Therefore, if $n$ does not divide $k$,

$\int_{\varpi^{-k}R^{\times}}f(h_{\alpha}(t^{-1}))dt=0$

and if $n|k$, then by 2.4,

$f(h_{\alpha}(t^{-1}))=q^{-k}\nu(h_{\alpha}(\varpi^{k}))\sigma(h_{\alpha}(\varpi^{k}))v_{0}=q^{-k}\nu(h_{\alpha}(\varpi^{k}))v_{0}$

So

$\int_{\varpi^{-k}R^{\times}}f(h_{\alpha}(t^{-1}))dt=m(\varpi^{-k}R^{\times})q^{-k}\nu(h_{\alpha}(\varpi^{k}))v_{0}$
$=d_{F}^{-1/2}(1-\frac{1}{q})\nu(h_{\alpha}(\varpi^{k}))v_{0}$

Finally,

$\displaystyle\>\>\>\>\>\int_{F}f(w_{\alpha}(-1)x_{\alpha}(t))dt$
$\displaystyle=d_{F}^{-1/2}v_{0}+\sum_{n|k}d_{F}^{-1/2}\frac{q-1}{q}\nu(h_{\alpha}(\varpi^{k}))v_{0}$
$\displaystyle=d_{F}^{-1/2}v_{0}+d_{F}^{-1/2}\frac{q-1}{q}\frac{\nu(h_{\alpha}(\varpi^{n}))}{1-\nu(h_{\alpha}(\varpi^{n}))}v_{0}$
$\displaystyle=d_{F}^{-1/2}\frac{1-q^{-1}\nu(h_{\alpha}(\varpi^{n}))}{1-\nu(h_{\alpha}(\varpi^{n}))}v_{0}$

∎

##  3\. Central extensions of split reductive groups over the adeles

###  3.1. Group theory

Given a number field $k$ containing $\mu_{n}$, let $\Sigma$ be the set of all
places of $k$. Let $\underline{G}$ be a split simply-connected Chevalley group
defined over $k$. We use $\underline{G}_{v}$ to denote $\underline{G}(k_{v})$.
For each $v$, there is a unique (up to isomorphism) nontrivial central
extension $G_{v}$ of $\underline{G}_{v}$ by $\mu_{n}$ ([9]):

$1\to\mu_{n}\to G_{v}\to\underline{G}_{v}\to 1$

Consider the restricted product

$\mathcal{G}=\prod_{v\in\Sigma}G_{v}$

with respect to the family of compact subgroups $K_{v}$ of $G_{v}$ generated
by $x_{\alpha}(t)$ ($\alpha\in\Phi,t\in r_{v}$). For each $v$, let $Z_{v}$
denote the kernel of the covering projection $G_{v}\to\underline{G}_{v}$. Form
the restricted product (i.e., all but finitely many components of any given
element is 1) $\mathcal{Z}=\prod_{v\in\Sigma}Z_{v}$ and let $\mathcal{Z}_{0}$
be the subgroup of $\mathcal{Z}$ consisting of elements whose coordinates
multiply up to one. Define

$G(\mathbb{A}_{k}):=\mathcal{G}/\mathcal{Z}_{0}$

which fits into a short exact sequence:

$1\to\mu_{n}\to G(\mathbb{A}_{k})\to\underline{G}(\mathbb{A}_{k})\to 1$

######  Proposition 3.1.

Suppose that $\underline{G}$ is a simply-connected Chevalley group. Then there
is an injective homomorphism from $\underline{G}(k)$ to $G(\mathbb{A}_{k})$.

###### Proof.

Let $\mathcal{G}(k)$ be the subgroup of $\mathcal{G}$ generated by
$X_{\alpha}(t)=({x}_{\alpha}(t),{x}_{\alpha}(t),...)$ with $\alpha\in\Phi$,
$t\in k$ and let $H(k)$ be its image in $G(\mathbb{A}_{k})$. We still use
$X_{\alpha}(t)$ to denote the image of $X_{\alpha}(t)$ in $H(k)$. They
generate $H(k)$ by definition. Put
$W_{\alpha}(t)=X_{\alpha}(t)X_{-\alpha}(-t^{-1})X_{\alpha}(t)$ and
$H_{\alpha}(t)=W_{\alpha}(t)W_{\alpha}(-1)$.

Claim:

$\underline{G}(k)\cong H(k)$

To prove the claim, notice that $H_{\alpha}(t)H_{\alpha}(u)=H_{\alpha}(tu)$ by
Hilbert reciprocity. Since $\underline{G}$ is simply connected, there is a
surjection $f:\underline{G}(k)\to H(k)$ mapping $x_{\alpha}(t)$ to
$X_{\alpha}(t)$. On the other hand, there is a surjection
$g:\mathcal{G}(k)\to\underline{G}(k)$ such that
$g(X_{\alpha}(t))=x_{\alpha}(t)$ and that $g$ is trivial on $\mathcal{Z}_{0}$.
So $g$ factors through $H(k)\to\underline{G}(k)$ which is still denoted by
$g$. Now $g\circ f$ is the identity map, which forces $f,g$ to be
isomorphisms. ∎

In general, if $\underline{G}$ is a split connected reductive group defined
over $k$, such an explicit construction is not available to us. Instead, we
will simply impose a number of axioms.

  1. (1)

There is a central extension $G(\mathbb{A}_{k})$ of
$\underline{G}(\mathbb{A}_{k})$ by $\mu_{n}$:

$1\to\mu_{n}\to G(\mathbb{A}_{k})\to\underline{G}(\mathbb{A}_{k})\to 1$

  2. (2)

For each place $v$ of $k$, there is a central extension

$1\to\mu_{n}\to G_{v}\to\underline{G}_{v}\to 1$

such that [2.1, Item 1] holds for all $v$ and [2.1, Item 2] holds for all but
finitely many $v$.

  3. (3)

There is a surjection

$\prod_{v\in\Sigma}G_{v}\to G(\mathbb{A}_{k})$

whose kernel is in the center of $\prod_{v\in\Sigma}G_{v}$.

  4. (4)

There is an injective homomorphism from $\underline{G}(k)$ to
$G(\mathbb{A}_{k})$.

The metaplectic torus $T(\mathbb{A}_{k})$ is the image of the restricted
product of $\\{T(k_{v})\\}_{v\in\Sigma}$ with respect to
$\\{T(r_{v})=T(k_{v})\cap K_{v}\\}_{v\in\Sigma}$ in $G(\mathbb{A}_{k})$.

###  3.2. Principal series representations and intertwining maps

We begin with a definition:

######  Definition 3.1.

We say a character $\chi=\prod\chi_{v}$ of $Z_{T}=Z_{T}(\mathbb{A}_{k})$ is a
genuine automorphic central character (genuine ACC in short) of
$T=T(\mathbb{A}_{k})$ if

1\. $\chi(\zeta z)=\zeta\chi(z)$, $\forall z\in Z_{T}$,
$\forall\zeta\in\mu_{n}$.

2\. $\chi$ is trivial on $\underline{T}(k)\cap Z_{T}$.

3\. $\chi$ is unramified almost everywhere, i.e., for $v$ outside a finite set
of places $S$ which contains all archimedean places and places where the
residue characteristic divides $n$, $\chi_{v}$ is trivial on $Z_{T}(r_{v})$.

Suppose that $\chi=\prod\chi_{v}$ is invariant under the conjugation action of
the Weyl group and $\sigma$ is an irreducible representation of
$T(\mathbb{A}_{k})$ with central character $\chi$. $\sigma$ may be decomposed
as a restricted tensor product of genuine irreducible representations
$\sigma_{v}$ of $T_{v}$ with central character $\chi_{v}$:

$\sigma=\bigotimes_{v\in\Sigma}\sigma_{v}$

where the the restricted product is with respect to a fixed set of vectors
$\\{u_{v}\neq 0|v\notin S,u_{v}\in\sigma_{v}^{T_{v}^{0}}\\}$.

Let $\nu=\prod_{v}\nu_{v}$ be an unramified character of
$\underline{T}(k)\backslash\underline{T}(\mathbb{A}_{k})$. For example, the
adelic absolute value function
$|\cdot|_{\mathbb{A}_{k}}=\prod_{v}|\cdot|_{v}:\mathbb{A}_{k}^{\times}\to\mathbb{R}_{>0}$
is an unramified character of $\mathbb{A}_{k}^{\times}$. Let

$I(\nu)=Ind_{B(\mathbb{A})}^{G(\mathbb{A})}(\nu\otimes\sigma)$

and let

$I(\nu_{v})=Ind_{B(k_{v})}^{G(k_{v})}(\nu_{v}\otimes\sigma_{v})$

We have $\dim I_{v}(\nu)^{K_{v}}=1$ for almost all $v$. For such $v$, let
$f_{\nu}^{v}\in I_{v}(\nu)^{K_{v}}$ be the unique (vector valued) function so
that $f_{\nu}^{v}(1)=u_{v}$. Then $I(\nu)$ is the restricted tensor product of
$I(\nu_{v})$ for $v\in\Sigma$ with respect to this set of functions.

Suppose that $\sigma$ is a submodule of $L^{2}(\underline{T}(k)\backslash
T(\mathbb{A}_{k}))$. For a $w\in W$ and a choice $\hat{w}$ of its
representative in $N_{\underline{G}(k)}\underline{T}(k)$, The natural action
of $\hat{w}$ on $L^{2}(\underline{T}(k)\backslash T(\mathbb{A}_{k}))$ maps
$\sigma$ to $\sigma^{\hat{w}}$ (which has the same underlying vector space but
a twisted $T(\mathbb{A}_{k})$-action). Let $j_{\hat{w}}$ be the vector space
isomorphism

$\displaystyle\sigma^{\hat{w}}\to\sigma$
$\displaystyle\varphi\mapsto\varphi^{\hat{w}}(t)=\varphi(\hat{w}^{-1}t\hat{w})$

We have $j_{\hat{w}}=\otimes_{v}j_{\hat{w}}^{v}$ where
$j_{\hat{w}}^{v}:\sigma_{v}^{\hat{w}}\to\sigma_{v}$ is defined in Section 2.2
for $v\notin S$.

Take a representative $\hat{w}$ of $w$, define for any $f\in I(\nu)$

$\big{(}A_{w}(\nu)(f)\big{)}(x)=j_{\hat{w}}\int_{N_{w}(\mathbb{A}_{k})}f(\hat{w}^{-1}nx)dn$

Here $N_{w}(\mathbb{A}_{k})=(N(\mathbb{A}_{k})\cap
wN(\mathbb{A}_{k})w^{-1})\backslash N(\mathbb{A}_{k})$ and we normalize the
Haar measure on $\mathbb{A}_{k}$ so that it is the product measure of the
Lebesgue measure on $\mathbb{R}$ for $v$ real, twice the Lebesgue measure on
$\mathbb{C}$ for $v$ complex, and the unique Haar measure on $k_{v}$ such that
$r_{v}$ has measure $d_{k_{v}}^{-1/2}$ for $v$ non-archimedean, where
$d_{k_{v}}$ is the absolute discriminant of $k_{v}$.

######  Proposition 3.2.

$A_{w}(\nu)$ maps $I(\nu)$ into $I(\nu^{w})$ and is independent of the choice
of a representative of $w$ in $N_{\underline{G}(k)}\underline{T}(k)$.

###### Proof.

By the definition of $j$, $A_{w}(\nu)$ maps $I(\nu)$ into $I(\nu^{w})$. If
$\hat{w}^{\prime}=\hat{w}t$ for some $t\in\underline{T}(k)$, then
$f(\hat{w}^{\prime-1}nx)=\delta_{N}^{1/2}(t)^{-1}\nu(t)^{-1}\sigma(t)^{-1}f(\hat{w}^{-1}nx)=\sigma(t)^{-1}f(\hat{w}^{-1}nx)$.
On the other hand, $j_{\hat{w}^{\prime}}=j_{\hat{w}}\circ\sigma(t)$ since any
function in the vector space $\sigma$ is left $\underline{T}(k)$-invariant.
Thus $A_{w}(\nu)$ is independent of the choice of a representative of $w$ in
$N_{\underline{G}(k)}\underline{T}(k)$. ∎

######  Proposition 3.3.

$A_{w}(\nu)=\bigotimes_{v}A_{w}(\nu_{v})$

where for each $v$,

$A_{w}(\nu_{v}):I(\nu_{v})\to I(\nu_{v}^{w})$

is an intertwining map between local principal series.

###### Proof.

This follows from the definition of intertwining operators and the identity
$j_{\hat{w}}=\otimes j_{\hat{w}}^{v}$. ∎

######  Remark 5.

The tensor decomposition of $A_{w}(\nu)$ also yields for $v\in S$ a canonical
intertwining operator at $v$ (i.e., it does not depend on the choice of a
representative of $w$).

Let

$C^{\infty}(N(\mathbb{A}_{k})\underline{T}(k)\backslash G(\mathbb{A}_{k}))$

be the space of smooth, left $N(\mathbb{A}_{k})$ invariant, left
$\underline{T}(k)$-invariant, right $K(\mathbb{A}_{k})$-finite functions on
$G(\mathbb{A}_{k})$, and let

$C^{\infty}(N(\mathbb{A}_{k})\underline{T}(k)\backslash
G(\mathbb{A}_{k}))_{\nu\otimes\sigma}$

be the subspace consisting of functions $\phi$ such that for any $k\in
K(\mathbb{A}_{k})$, the function $\phi_{k}(t)=\delta^{-1/2}(t)\phi(tk)$
belongs to the isotypic component of $\nu\otimes\sigma$ in
$L^{2}(\underline{T}(k)\backslash T(\mathbb{A}_{k}))$. (This definition is
from [8], Ch 1, 3.3)

######  Lemma 3.4.

For $f\in I(\nu)$, let $f^{*}$ to be the associated $\mathbb{C}$-valued
function given by $f^{*}(x)=(f(x))(1)$. Then $f^{*}\in
C^{\infty}(N(\mathbb{A})\underline{T}(k)\backslash
G(\mathbb{A}))_{\nu\otimes\sigma}$.

###### Proof.

First, $f^{*}\in C^{\infty}(N(\mathbb{A})\underline{T}(k)\backslash
G(\mathbb{A}))$: $N(\mathbb{A})$-invariance of $f^{*}$ is obvious. For
$\underline{T}(k)$-invariance, we note that for $t\in\underline{T}(k)$,
$f^{*}(tx)=f(tx)(1)=(\sigma(t)f(x))(1)=f(x)(t)$ which is $f(x)(1)$ since
$f(x)\in\sigma$. It remains to check that $\nu^{-1}f^{*}_{k}\in\sigma$ for any
$k\in K(\mathbb{A})$. In fact, $\forall t\in T(\mathbb{A})$,
$\nu^{-1}(t)\delta(t)^{-1/2}f^{*}(tk)=\nu^{-1}(t)\delta(t)^{-1/2}f(tk)(1)=(\sigma(t)f(k))(1)$
$=f(k)(t)$ which shows $\nu^{-1}f^{*}_{k}=f(k)\in\sigma$. ∎

For $\phi\in C^{\infty}(N(\mathbb{A})\underline{T}(k)\backslash
G(\mathbb{A}))_{\nu\otimes\sigma}$ and $x\in G(\mathbb{A})$, whenever the
integral below is convergent, we set

$\big{(}M_{w}(\nu)\phi\big{)}(x)=\int_{N_{w}(\mathbb{A}_{k})}\phi(w^{-1}nx)dn$

note that this integral is independent of the choice of a representative of
$w$.

######  Proposition 3.5.

The integral defined above converges absolutely when $\nu$ belongs to an open
subset of $X^{*}(T)\otimes\mathbb{C}$ and $M_{w}(\nu)$ maps
$C^{\infty}(N(\mathbb{A})\underline{T}(k)\backslash
G(\mathbb{A}))_{\nu\otimes\sigma}$ into
$C^{\infty}(N(\mathbb{A})\underline{T}(k)\backslash
G(\mathbb{A}))_{\nu^{w}\otimes\sigma}$.

###### Proof.

The first half is proved in [8], Ch.2, 1.6 which also proved that $M_{w}(\nu)$
maps the space $C^{\infty}(N(\mathbb{A})\underline{T}(k)\backslash
G(\mathbb{A}))_{\nu\otimes\sigma}$ into
$C^{\infty}(N(\mathbb{A})\underline{T}(k)\backslash
G(\mathbb{A}))_{\nu^{w}\otimes\sigma^{w}}$. But the latter is equal to

$C^{\infty}(N(\mathbb{A})\underline{T}(k)\backslash
G(\mathbb{A}))_{\nu^{w}\otimes\sigma}$

since $\sigma\cong\sigma^{w}$. ∎

######  Theorem 3.6.

([8], Ch.4, 1.10) $\nu\mapsto M_{w}(\nu)$ can be continued in a unique way to
a meromorphic function on $X^{*}(T)\otimes\mathbb{C}$. Moreover, we have the
equality of meromorphic functions given by

$M_{w^{\prime}}(\nu^{w})\circ M_{w}(\nu)=M_{w^{\prime}w}(\nu)$

######  Lemma 3.7.

For any $f\in I(\nu)$, we have

$(A_{w}(\nu)f)^{*}=M_{w}(\nu)f^{*}$

###### Proof.

$\displaystyle\>\>\>\>\>(A_{w}(\nu)f)^{*}(x)$
$\displaystyle=\big{(}j_{\hat{w}}\int_{N_{w}(\mathbb{A}_{k})}f(\hat{w}^{-1}nx)dn\big{)}(1)$
$\displaystyle=\int_{N_{w}(\mathbb{A}_{k})}(j_{\hat{w}}f(\hat{w}^{-1}nx))(1)dn$
$\displaystyle=\int_{N_{w}(\mathbb{A}_{k})}f(\hat{w}^{-1}nx)(1)dn$
$\displaystyle=(M_{w}(\nu)f^{*})(x)$

∎

######  Corollary 3.8.

$A_{w^{\prime}}(\nu^{w})\circ A_{w}(\nu)=A_{w^{\prime}w}(\nu)$

###  3.3. Rank one group

In this section, we assume that ${\mathfrak{g}}^{der}\cong\mathfrak{sl}_{2}$
so
$\mathfrak{g}=\mathfrak{t}\oplus\mathfrak{g}_{\alpha}\oplus\mathfrak{g}_{-\alpha}$
where ${\mathfrak{t}}$ is the maximal abelian subalgebra of ${\mathfrak{g}}$
and $\alpha$ is a root of ${\mathfrak{g}}$. Let $\underline{G}$ be a connected
reductive group with Lie algebra ${\mathfrak{g}}$. Let $\underline{T}$ be the
maximal torus of $\underline{G}$ with Lie algebra ${\mathfrak{t}}$. Let $w$ be
the nontrivial element in the Weyl group $W\cong\mathbb{Z}/2\mathbb{Z}$. There
is a homomorphism from $SL_{2}$ to $\underline{G}$ which differentiates to the
inclusion map among the Lie algebras ${\mathfrak{g}}^{der}\to\mathfrak{g}$. We
use $\underline{h}(t)$ to denote the image of $diag(t,t^{-1})\in SL_{2}$. Let
$\nu:\underline{T}(\mathbb{A}_{k})\to\mathbb{R}_{>0}$ be a character such that
$\nu(\underline{h}(t))=|t|_{\mathbb{A}}^{s}$ for some complex number $s$. Let
$v$ be a place of $k$, for generic $\nu$, $I(\nu_{v})$ is irreducible, hence
by Schur’s lemma, $A(\nu_{v}^{w})\circ A(\nu_{v})$ is a constant, which only
depends on $s$ and we denote by $\lambda_{v}(s)$. Define the Plancherel
measure

$\mu_{v}(s)=\frac{1}{\lambda_{v}(s)}$

By 2.6,

$\displaystyle\mu_{v}(s)$
$\displaystyle=d_{k_{v}}\frac{1-\nu(\underline{h}(\varpi_{v}^{n}))}{1-q_{v}^{-1}\nu(\underline{h}(\varpi_{v}^{n}))}\frac{1-\nu(\underline{h}(\varpi_{v}^{-n}))}{1-q_{v}^{-1}\nu(\underline{h}(\varpi_{v}^{-n}))}$
$\displaystyle=d_{k_{v}}\frac{1-q_{v}^{-ns}}{1-q_{v}^{-ns-1}}\frac{1-q_{v}^{ns}}{1-q_{v}^{ns-1}}$

We record here a basic property of $\mu_{v}$:

######  Proposition 3.9.

For any non-archimedean place $v$ of $k$, $\mu_{v}(s)$ is a rational function
of $q_{v}^{s}$, $\mu_{v}(s)=\mu_{v}(-s)$ and $\mu_{v}(it)\geq 0$ for $t$ real.

######  Proposition 3.10.

Assume that for all $v\notin S$, $\sigma(h(t^{n}))=1$, $\forall t\in
k_{v}^{\times}$. Then

$\left(\prod_{v\notin S}d_{k_{v}}\right)\left(\prod_{v\in
S}\mu_{v}(s)\right)\frac{L_{S}(ns+1)}{L_{S}(ns)}\frac{L_{S}(-ns+1)}{L_{S}(-ns)}=1$

where

$L_{S}(s)=\prod_{v\notin S}\frac{1}{1-q_{v}^{-s}}$

is a partial $L$-function which converges when $Re(s)>>0$ and has a
meromorphic continuation to ${\mathbb{C}}$.

###### Proof.

Take $f_{\nu}\in I(\nu)=\bigotimes_{v}I(\nu_{v})=\otimes_{v}f_{\nu}^{v}$ such
that $f_{\nu}^{v}\in I(\nu_{v})$ for $v\in S$ and that $f_{\nu}^{v}\in
I(\nu_{v})^{K_{v}}$ for $v\notin S$.

$\displaystyle A(\nu)f_{\nu}$ $\displaystyle=\prod_{v\in
S}A(\nu_{v})f_{\nu}^{v}\prod_{v\notin S}A(\nu_{v})f_{\nu}^{v}$
$\displaystyle=\prod_{v\in S}A(\nu_{v})f_{\nu}^{v}\cdot\prod_{v\notin
S}d_{k_{v}}^{-1/2}\frac{L_{S}(ns)}{L_{S}(ns+1)}\prod_{v\notin
S}f_{\nu^{w}}^{v}$

Letting $A(\nu^{w})$ act on both sides, we obtain by 3.8 that

$\displaystyle f_{\nu}$ $\displaystyle=\prod_{v\in
S}\lambda_{v}(s)f_{\nu}^{v}\cdot\left(\prod_{v\notin
S}d_{k_{v}}^{-1}\frac{L_{S}(ns)}{L_{S}(ns+1)}\frac{L_{S}(-ns)}{L_{S}(-ns+1)}\right)\prod_{v\notin
S}f_{\nu}^{v}$ $\displaystyle=\left(\prod_{v\in S}\lambda_{v}(s)\prod_{v\notin
S}d_{k_{v}}^{-1}\frac{L_{S}(ns)}{L_{S}(ns+1)}\frac{L_{S}(-ns)}{L_{S}(-ns+1)}\right)f_{\nu}$

which proves the proposition. ∎

##  4\. A comparison method

###  4.1. A related linear reductive group

Given a connected reductive group $\underline{G}$ defined over a local field
$F$ and a central extension $G$ of $\underline{G}$ by $\mu_{n}$, we will
define a linear reductive group that is closely related to $G$. As in Section
2.1, let $X=X^{*}(\underline{T})$, $Y=X_{*}(\underline{T})$ and fix a non-
degenerate symmetric Weyl group invariant bilinear form $(,)$ on $Y$ such that
$(\alpha^{\vee},\alpha^{\vee})=2$ for all $\alpha^{\vee}\in\Phi^{\vee}$. We
define, following [14], Section 1,

$Y^{\sharp}=\\{y\in Y|(y,y^{\prime})\in n\mathbb{Z},\forall y^{\prime}\in
Y\\}$

It is clear that $Y^{\sharp}\otimes F^{\times}\cong\pi(Z_{T})$ with $\pi$ the
covering map. Let $Y^{\prime}=\frac{1}{n}Y^{\sharp}$ and let $X^{\prime}$ be
the dual lattice of $Y^{\prime}$ under the natural perfect pairing
$<,>:X_{\mathbb{Q}}\times Y_{\mathbb{Q}}\to{\mathbb{Q}}$.

######  Lemma 4.1.

$\Phi\subset X^{\prime}$, $\Phi^{\vee}\subset Y\subset Y^{\prime}$.

###### Proof.

Since $nY\subset Y^{\sharp}$, $Y\subset Y^{\prime}$; For any $\alpha\in\Phi$
and $y^{\prime}\in Y^{\prime}$, write $y^{\prime}=\frac{1}{n}y^{\sharp}$ for
some $y^{\sharp}\in Y^{\sharp}$, we have
$<\alpha,y^{\prime}>=\frac{1}{n}<\alpha,y^{\sharp}>=\frac{1}{n}(\alpha^{\vee},y^{\sharp})$
which is in ${\mathbb{Z}}$ because of $\alpha^{\vee}\in Y$ and the definition
of $Y^{\sharp}$. ∎

It follows that

$(X^{\prime},\Phi,Y^{\prime},\Phi^{\vee})$

is a root datum associated to some (linear) connected reductive group
$G^{\prime}$ with a maximal torus $T^{\prime}$ such that $X^{\prime}$ is the
character lattice of $T^{\prime}$, $Y^{\prime}$ is the cocharacter lattice of
$T^{\prime}$, $\Phi=\Phi(G^{\prime},T^{\prime})$ is the root system of
$G^{\prime}$, $\Phi^{\vee}=\Phi^{\vee}(G^{\prime},T^{\prime})$ is the coroots
of $G^{\prime}$. There is a group homomorphism $\underline{T}\to T^{\prime}$
induced by the inclusion map $Y\subset Y^{\prime}$.

As in setting of Section 3.2, we form the principal series representation of
$G(\mathbb{A}_{k})$ induced from $\nu\otimes\sigma$ where $\nu$ is an
unramified automorphic character of $\underline{T}(\mathbb{A}_{k})$ and
$\sigma$ is an irreducible genuine automorphic representation of $T$ with a
Weyl group invariant genuine automorphic central character. We also consider
the principal series representation of $G^{\prime}(\mathbb{A}_{k})$, denoted
$I^{\prime}(\nu)$, induced from $\nu\otimes\sigma^{\prime}$ where
$\sigma^{\prime}$ denotes the trivial representation of
$T^{\prime}(\mathbb{A}_{k})$. We compare the product formulas (3.10) for these
groups. Let us fix a long root $\alpha$, and suppose
$\nu(\underline{h}_{\alpha}(t))=|t|_{\mathbb{A}_{k}}^{s}$ for some complex
number $s$ depending on $\alpha$ and $\nu$. Let $\mu_{\alpha,v}(s)$ be the
Plancherel measure of $I(\nu_{v})$ corresponding to $\alpha$ and
$\mu^{\prime}_{\alpha,v}(s)$ be the Plancherel measure of
$I^{\prime}(\nu_{v})$ corresponding to $\alpha$. We have for the covering
group $G$:

$\left(\prod_{v\in S}\mu_{\alpha,v}(s)\right)\left(\prod_{v\notin
S}d_{k_{v}}\right)\frac{L_{S}(ns+1)}{L_{S}(ns)}\frac{L_{S}(-ns+1)}{L_{S}(-ns)}=1.$

On the other hand, we have for the associated linear group $G^{\prime}$:

$\left(\prod_{v\in S}\mu^{\prime}_{\alpha,v}(s)\right)\left(\prod_{v\notin
S}d_{k_{v}}\right)\frac{L_{S}(s+1)}{L_{S}(s)}\frac{L_{S}(-s+1)}{L_{S}(-s)}=1.$

Therefore, after making a simple substitution $s\mapsto ns$ in the second
equation, we obtain

######  Theorem 4.2.

Let $\nu$ be an unramified character of
$\underline{T}(k)\backslash\underline{T}(\mathbb{A}_{k})$. Let $\sigma$ be a
Weyl group invariant genuine irreducible representation of $T(\mathbb{A}_{k})$
such that for all $v\notin S$, $\sigma_{v}(h_{\alpha}(t^{n}))=1$,
$\forall\alpha\in\Phi,\forall t\in k_{v}^{\times}$. Let $\alpha$ be a long
root of $\Phi$ such that $\nu(h_{\alpha}(t))=|t|_{\mathbb{A}_{k}}^{s}$, then

$\prod_{v\in S}\mu_{\alpha,v}(s)=\prod_{v\in S}\mu^{\prime}_{\alpha,v}(ns)$

for all but finitely many $s\in\mathbb{C}$.

###  4.2. Application to ramified representations of reductive groups over
local fields

4.2 gives us a way to go from those easy-to-compute Plancherel measures of
linear groups to those unknown Plancherel measures of covering groups. We will
make it precise in this section.

Let $p$ be any prime, and let $n$ be any positive integer. Let $F$ be a finite
extension of $\mathbb{Q}_{p}$ containing $\mu_{n}$. Let $q=p^{f}$ be the order
of its residue field $k_{F}$. Let $G$ be a $n$-fold cover of a split connected
reductive group $\underline{G}$ defined over $F$. Suppose that we have an
irreducible, genuine, Weyl group invariant representation $\sigma_{p}$ of the
metaplectic torus $T$ and an unramified character $\nu_{p}$ of
$\underline{T}$. Form the principal series representation of $G(F)$:

$I(\nu_{p})=Ind_{B}^{G}(\nu_{p}\otimes\sigma_{p})$

For any $w\in W$, let

$A_{w}(\nu_{p}):I(\nu_{p})\to I(\nu_{p}^{w})$

be the standard intertwining operator. Then for generic $\nu_{p}$,
$I(\nu_{p}),I(\nu_{p}^{w})$ are irreducible and hence for any simple
reflection $w=w_{\alpha}$, we have

$A_{w_{\alpha}}(\nu_{p}^{w_{\alpha}})\circ
A_{w_{\alpha}}(\nu_{p})=1/\mu_{\alpha,p}(\nu_{p})\cdot\mathrm{id}$

Suppose that $\nu_{p}(\underline{h}_{\alpha}(t))=|t|_{p}^{s}$ for some
$s\in\mathbb{C}$. Let $\mu_{\alpha,p}(s)=\mu_{\alpha,p}(\nu_{p})$.

Our goal in this section is to find an explicit expression of
$\mu_{\alpha,p}(s)$ in the case when the principal series $I(\nu_{p})$ is
ramified, i.e., $I(\nu_{p})^{K_{p}}=0$. Let $k$ be a global field that has
completions isomorphic to $F$ at places above $p$. Such $k$ is easy to be
constructed. For instance, take $k_{0}$ to be a number field that has exactly
one place ${\mathfrak{p}}$ above $p$ such that $k_{{\mathfrak{p}}}\cong F$.
Then we can take $k$ to be the composite of $k_{0}$ and
${\mathbb{Q}}(\mu_{n})$. The number field $k$ may have multiple places above
$p$. Let $\chi_{p}$ be the central character of $\sigma_{p}$.

######  Lemma 4.3.

Let $v$ be a complex place $k$ and $\alpha$ be a simple root of
$\underline{G}$. Suppose that $\nu_{v}(\underline{h}_{\alpha}(t))=|t|_{v}^{s}$
where $t\in{\mathbb{C}}$, $|\cdot|_{v}$ is the square of the complex absolute
value and $s\in{\mathbb{C}}$. Then $\mu_{\alpha,v}(s)=-s^{2}/4\pi^{2}$.

######  Theorem 4.4 (Main Theorem).

Let $\alpha\in\Delta$ be a long root. Suppose that there is a Weyl group
invariant genuine ACC $\chi$ such that $\chi_{v}=\chi_{p}$ for all $v|p$, and
for all but finitely many places $v$, $\chi_{v}(h_{\alpha}(t^{n}))=1$,
$\forall\alpha\in\Phi,\forall t\in k_{v}^{\times}$. Suppose also that there is
an unramified character $\nu$ of
$\underline{T}(k)\backslash\underline{T}(\mathbb{A}_{k})$ such that
$\nu(\underline{h}_{\alpha}(t))=|t|_{\mathbb{A}}^{s}$. Then

$\mu_{\alpha,p}(s)=c\mu_{\alpha,p}^{\prime}(ns)$

for some constant $c$. Therefore, $\mu_{\alpha,p}(s)$ is a constant multiple
of

$\frac{1-q^{-ns}}{1-q^{-ns-1}}\frac{1-q^{ns}}{1-q^{ns-1}}$

###### Proof.

This will follow from 4.2, together with an elementary argument we are going
to present. Since we are fixing $\alpha$, we write $\mu_{v}$ in place of
$\mu_{\alpha,v}$ for any $v\in\Sigma$, the set of places of $k$. We may assume
that $k$ has no real embeddings. Let $m$ be the number of places of $k$ above
$p$. For any complex place $v$, by 4.3, $\mu_{v}(ns)/\mu_{v}(s)$ is a
constant. So 4.2 and the assumption gives

$\mu_{p}(s)^{m}\prod_{v\in
S_{f}-p}\mu_{v}(s)=C\cdot\mu^{\prime}_{p}(ns)^{m}\prod_{v\in
S_{f}-p}\mu^{\prime}_{v}(ns)\>\>\>\>\>\>(1)$

for some constant $C$. Here $S_{f}-p$ denotes the set of finite places in $S$
that is not above $p$.

By 3.9, we may write (up to constant) that

$\mu_{p}(s)=\prod_{i}\frac{(a_{i}-q^{-s})(a_{i}-q^{s})}{(b_{i}-q^{-s})(b_{i}-q^{s})}$

for finitely many complex numbers $a_{i}$, $b_{i}$, counting multiplicities.
Choose for each $i$ a complex number $s_{i}$ so that $a_{i}=q^{s_{i}}$. Then
the zeros of $\mu_{p}(s)$ are

$\pm s_{i}+\frac{2\pi i}{\log q}\mathbb{Z}=\pm s_{i}+\frac{2\pi i}{f\log
p}\mathbb{Z}$

On the other hand, by 2.6, the zeros of the right hand side of (1) are

$\frac{2\pi i}{n\log q_{v}}\mathbb{Z}=\frac{2\pi i}{nf_{v}\log
p_{v}}\mathbb{Z}$

for some $v\in S_{f}$.

We claim that

$s_{i}\in\frac{2\pi i}{nf\log p}\mathbb{Z}$

for all $i$. In fact, for a fixed $i$, by Pigeonhole Principal, there is some
prime $p^{\prime}$ such that there are infinitely many elements in
$s_{i}+\frac{2\pi i}{f\log p}\mathbb{Z}$ that belong to $\frac{2\pi
i}{nf^{\prime}\log p^{\prime}}\mathbb{Z}$. If $p^{\prime}\neq p$, take two
distinct elements in

$\left(s_{i}+\frac{2\pi i}{f\log p}\mathbb{Z}\right)\cap\left(\frac{2\pi
i}{nf^{\prime}\log p^{\prime}}\mathbb{Z}\right)$

Because their difference also belongs to $\frac{2\pi i}{nf^{\prime}\log
p^{\prime}}\mathbb{Z}$, we obtain a nontrivial ${\mathbb{Q}}$-linear relation
among $\log p$ and $\log p^{\prime}$, which is impossible since $\log p,\log
p^{\prime}$ are linearly independent over ${\mathbb{Q}}$. Therefore,
$p^{\prime}=p$ and in particular, $s_{i}\in\frac{2\pi i}{nf\log p}\mathbb{Z}$,
proves the claim.

It follows that $a_{i}^{n}=1$ and all the zeros of $\mu_{p}(s)$ belong to

$\frac{2\pi i}{nf\log p}\mathbb{Z}$

By the same argument, $b_{i}^{n}=q$ and all the poles of $\mu_{p}(s)$ belong
to

$\pm\frac{1}{n}+\frac{2\pi i}{nf\log p}\mathbb{Z}$

Therefore the zeros and poles of $\mu_{p}(s)^{m}$ are contained in that of
$\mu^{\prime}_{p}(ns)^{m}$. It follows that $\mu_{p}(s)^{m}$ is a constant
multiple of $\mu^{\prime}_{p}(ns)^{m}$, i.e.,
$(\mu_{p}(s)/\mu^{\prime}_{p}(ns))^{m}$ is a constant. On the other hand, by
3.9, both of $\mu_{p}(s)$ and $\mu^{\prime}_{p}(ns)$ are positive on the
imaginary axis, so $\mu_{p}(s)/\mu^{\prime}_{p}(ns)$ must equal to a positive
constant when restricted to the imaginary axis. It follows that
$\mu_{p}(s)/\mu^{\prime}_{p}(ns)$ is a constant. ∎

######  Remark 6.

In certain cases, the assumption in 4.4 can be achieved using the construction
of a Weyl group invariant genuine ACC in Section 5.

##  5\. Construction of Weyl group invariant genuine automorphic central
characters

###  5.1. $n$-fold cover of $SL_{2}$

We let $m$ to be $n$ when $n$ is odd and $n/2$ when $n$ is even. Then
$Y^{\sharp}=mY\cong m\mathbb{Z}$ and hence both $\underline{T}(F)$ and
$\underline{T}^{\sharp}(F)$ are isomorphic to $F^{\times}$. The natural
inclusion $Y^{\sharp}\to Y$ induces a map
$\underline{T}^{\sharp}(F)\to\underline{T}(F)$, $t\to t^{m}$. The covering
torus $T(F)$ is generated by symbols $h(t)$, $t\in F^{\times}$ satisfying
$h(t)h(u)=(t,u)_{n}h(tu)$. Let $\pi:T(F)\to\underline{T}(F)$, $h(t)\mapsto t$
be the covering map. We define $T^{\sharp}(F)$ to be the set
$F^{\times}\times\mu_{n}$ with group law

$(t,\zeta)(t^{\prime},\zeta^{\prime})=(tt^{\prime},\zeta\zeta^{\prime})$

when $n\equiv 0,\pm 1(4)$;

$(t,\zeta)(t^{\prime},\zeta^{\prime})=(tt^{\prime},\zeta\zeta^{\prime}(t,t^{\prime})_{2})$

when $n\equiv 2(4)$. Under this group law, $T^{\sharp}(F)$ is an abelian
group. Moreover, $T^{\sharp}(F)\to\underline{T}^{\sharp}(F)$,
$(t,\zeta)\mapsto t$ is an $n$-to-$1$ homomorphism. We then define a map
$T^{\sharp}(F)\to T(F)$, $(t,\zeta)\mapsto\zeta h(t^{m})$, then it is a group
homomorphism with image $Z(F)$, the center of $T(F)$. It follows that
$Z(F)\cong T^{\sharp}(F)/(\mu_{m}\times 1)$.

Let $\gamma:F^{\times}/(F^{\times})^{2}\to\mu_{\infty}$ be a function such
that

$\gamma(a)\gamma(b)=(a,b)_{F}\gamma(ab)$

and

$\gamma(1)=1,$

where $(a,b)_{F}$ is the quadratic Hilbert symbol.

We now define a character of $T^{\sharp}(F)$ as follows, when $n\equiv 0,\pm
1(4)$, $(t,\zeta)\mapsto\zeta$; when $n\equiv 2(4)$,
$(t,\zeta)\mapsto\zeta\gamma(t)$. Then it factors through $Z(F)$ because the
Weil index $\gamma$ is trivial on $\mu_{m}=(\mu_{n})^{2}$ when $n$ is even. It
is easy to see that this character is Weyl group invariant. We denote this
character by $\gamma^{*}_{p}$ where $p$ indicates the residual characteristics
of $F$.

######  Proposition 5.1.

Any Weyl group invariant genuine character of $T^{\sharp}(F)$ is a product of
$\gamma^{*}_{p}$ and a quadratic character.

###### Proof.

Take two genuine Weyl group invariant characters $\chi,\chi^{\prime}$, let
$\varepsilon=\chi/\chi^{\prime}$. Then $\varepsilon$ factors through
$\underline{T}^{\sharp}=F^{\times}$. Since $\varepsilon$ is Weyl group
invariant, $\varepsilon(t)=\varepsilon(t^{-1})$ and so $\varepsilon(t)^{2}=1$.
∎

Let $\nu_{p}:\underline{T}(F)\to{\mathbb{R}}_{>0}$ be the character defined by
$\nu_{p}(t)=|t|_{F}^{s}$ for $t\in F^{\times}$. Let $\sigma_{p}$ be the
genuine irreducible representation of $T(F)$ corresponding to
$\gamma_{p}^{*}$. Put

$I(s)=I(\nu_{p})={\mathrm{Ind}}_{B}^{G}(\nu_{p}\otimes\sigma_{p})$

######  Lemma 5.2.

The assumptions in are satisfied in the case of $n$-fold cover of $SL_{2}$.

###### Proof.

We take a system $\\{\psi_{v}\\}_{v}$ of additive characters of $k_{v}$, and
let $\gamma_{v}=\gamma_{\psi_{v}}$ be the Weil index attached to the quadratic
form $x^{2}-y^{2}$ and $\psi_{v}$. Let $\gamma_{v}^{*}$ be the genuine central
character of $T^{\sharp}(k_{v})$ associated to $\gamma_{v}$ as described
above. The tensor product

$\gamma_{\mathbb{A}}^{*}:=\prod_{v}\gamma_{v}^{*}$

defines a genuine character of the center $\underline{T}(k)\backslash
T^{\sharp}(\mathbb{A}_{k})$. By construction, $\gamma_{v}^{*}=\gamma_{p}^{*}$
for any $v|p$. When $n$ is odd, $\gamma_{v}^{*}(h(t^{n}))=1$ for $t\in
k_{v}^{*}$ by definition of $\gamma_{v}^{*}$; when $n$ is even, this is also
true since $t^{n}$ is a square in $k_{v}^{*}$ and Weil index is trivial on
squares. ∎

######  Corollary 5.3.

$I(0)$ is irreducible, $I(1/n)$ and $I(-1/n)$ are reducible.

###### Proof.

Theorem implies that

$\mu_{p}(s)=c\frac{1-q^{-ns}}{1-q^{-ns-1}}\frac{1-q^{ns}}{1-q^{ns-1}}$

for some constant $c$. In particular, it has a double zero at $s=0$ and has a
simple pole at $s=\pm 1/n$. It follows that $I(1/n)$ and $I(-1/n)$ are
reducible. By in the appendix, $I(0)$ is irreducible since $\mu_{p}^{-1}(s)$
has a pole at $s=0$. ∎

###  5.2. Two-fold cover of simply-connected Chevalley group of type ADE

In this section, we assume that $\underline{G}$ is a simply-connected, simply-
laced Chevalley group defined over a local field $F$. Let $G$ be its
nontrivial two-fold central extension. Let $T$ be the metaplectic torus of $G$
generated by $h_{\alpha}(t)$ for $\alpha\in\Phi$, $t\in F^{\times}$. Let
$Z_{T}$ be the center of $T$ and $T^{2}$ be the subgroup of $Z_{T}$ generated
by $h_{\alpha}(t^{2})$ for $t\in F^{\times}$, $\alpha\in\Phi$.

The following proposition follows directly from Proposition 4.2 of [11].

######  Proposition 5.4.

Let $S$ be a subset of vertices of the Dynkin diagram of $G$ such that no two
vertices in $S$ are adjacent and every vertex not in $S$ is adjacent to an
even number of vertices in $S$. Then there is no nonempty $S$ if $\Phi$ is of
type $A_{2n},E_{6},E_{8}$, there is exactly one nonempty $S$ when $\Phi$ is of
type $A_{2n-1},D_{2n-1},E_{7}$ and there are three nonempty $S$ when $\Phi$ is
of type $D_{2n}$. Moreover,

$Z(T)/T^{2}$

is generated by

$h_{S}(t)=\prod_{\alpha\in S}h_{\alpha}(t)$

$t\in F^{\times}/(F^{\times})^{2}$.

Let us label the Dynkin diagram of $D_{2n}$ as follows: we label the nodes
from the very left end to the conjunction by $1,2,\cdots,2n-2$ respectively,
and we label the node at the top right of the conjunction by $2n-1$, the node
at the bottom right of the conjunction by $2n$. The the three nonempty $S$ are

$S_{1}=\\{1,3,\cdots,2n-3,2n-1\\}$ $S_{2}=\\{1,3,\cdots,2n-3,2n\\}$
$S_{3}=\\{2n-1,2n\\}$

Thus every element in $Z_{T}/T^{2}$ is of the form

$\pm h_{S_{1}}(t_{1})h_{S_{2}}(t_{2})h_{S_{3}}(t_{3})$

######  Corollary 5.5.

When $\Phi$ is of type $A_{2n},E_{6},E_{8}$, $Z_{T}/T^{2}$ is trivial; When
$\Phi$ is of type $A_{2n-1},D_{2n-1},E_{7}$, $Z_{T}/T^{2}$ is generated by
$\mu_{2}$ and $h_{S}(t)$, $t\in F^{\times}/(F^{\times})^{2}$ satisfying
$h_{S}(t)h_{S}(u)=(t,u)^{|S|}h(tu)$; When $\Phi$ is of type $D_{2n}$,
$Z_{T}/T^{2}$ is generated by $h_{S_{1}}(t)$, $h_{S_{2}}(t)$, $h_{S_{3}}(t)$.

######  Proposition 5.6.

Let $\gamma:F^{\times}/(F^{\times})^{2}\to\mu_{4}$ be a function such that

$\gamma(t)\gamma(u)=(t,u)\gamma(tu)$

and

$\gamma(1)=1.$

(For instance, the normalized Weil index attached to the quadratic form
$x^{2}-y^{2}$ and an additive character $\psi:F\to S^{1}$) Define

$\gamma^{*}(\epsilon h_{S}(t))=\epsilon\gamma(t)^{|S|}$

Then $\gamma^{*}$ is a genuine character of $Z_{T}$.

######  Proposition 5.7.

The Weyl group $W$ fixes $Z_{T}/T^{2}$ pointwisely.

###### Proof.

It is enough to show that $W$ fixes $h_{S}(t)$ for any $S$ and $t\in
F^{\times}/(F^{\times})^{2}$, that is,

$w_{\alpha}(1)h_{S}(t)w_{\alpha}(-1)=h_{S}(t)$

$\forall\alpha\in\Delta,t\in F^{\times}/(F^{\times})^{2}$. By the definition
of $S$, if $\alpha$ does not belong to $S$, then it connects to an even number
of nodes in $S$, if the number is zero, we are done; if not, then it must
connect to two nodes in $S$, say $\beta,\gamma$. It suffices to check that

$w_{\alpha}(1)h_{\beta}(t)h_{\gamma}(t)w_{\alpha}(-1)=h_{\beta}(t)h_{\gamma}(t)$

The LHS is

$(t,c(\alpha,\beta))h_{\alpha+\beta}(t)(t,c(\alpha,\gamma))h_{\beta+\gamma}(t)=(t,c(\alpha,\beta)c(\alpha,\gamma))h_{\alpha+\beta}(t)h_{\alpha+\gamma}(t)$
$=(t,c(\alpha,\beta)c(\alpha,\gamma))(t,c(\beta,\alpha)t)h_{\beta}(t)h_{\alpha}(t)(t,c(\alpha,\gamma)t)h_{\alpha}(t)h_{\gamma}(t)$
$=(t,c(\alpha,\beta)c(\beta,\alpha))(t,t)h_{\beta}(t)h_{\gamma}(t)=(t,-1)(t,t)h_{\beta}(t)h_{\gamma}(t)=h_{\beta}(t)h_{\gamma}(t)$

Finally, if $\alpha\in S$, then since

$w_{\alpha}(1)h_{\alpha}(t)w_{\alpha}(-1)=h_{\alpha}(t^{-1})=h_{\alpha}(t)$

and $\alpha$ is not adjacent to any element in $S$, $w_{\alpha}(1)$ fixes
$h_{S}(t)$. ∎

For a number field $k$, we take a system $\\{\psi_{v}\\}_{v}$ of additive
characters of $k_{v}$ for all places $v$ of $k$, and we let
$\gamma_{v}=\gamma_{\psi_{v}}$ be the Weil index attached to the quadratic
form $x^{2}-y^{2}$ and $\psi_{v}$.

######  Lemma 5.8 ([13]).

$\gamma_{v}$ is unramified for almost all $v$ and

$\prod_{v}\gamma_{v}(x)=1$

$\forall x\in k^{\times}$.

Let $\gamma_{v}^{*}$ be the genuine central character associated to
$\gamma_{v}$, we form the product

$\gamma_{\mathbb{A}}^{*}:=\prod_{v}\gamma_{v}^{*}$

which defines a character of the center $Z_{T}(\mathbb{A}_{k})$ of
$T(\mathbb{A}_{k})$.

######  Lemma 5.9.

$\underline{T}(k)\cap Z_{T}(\mathbb{A}_{k})$

is generated by

$\prod_{\alpha\in\Delta}h_{\alpha}(t_{\alpha}^{2})$

and

$h_{S}(t)=\prod_{\alpha\in S}h_{\alpha}(t)$

where $t_{\alpha},t\in k^{\times}$.

###### Proof.

Any element in the intersection is generated by the elements

$\prod_{\alpha\in\Delta}h_{\alpha}(u_{\alpha})$ $\prod_{\alpha\in
S}h_{\alpha}(t_{\alpha})$

such that $u_{\alpha},t_{\alpha}\in k^{\times}$, $u_{\alpha}$ is a square in
$k_{v}$ for all $v$ and $\alpha\in\Delta$, and $t_{\alpha},t_{\beta}$ differ
by a square in $k_{v}$ for all $\alpha,\beta\in S$ and places $v$. But local-
global principal implies that any element of $k$ that is a square in each
$k_{v}$ must be a square in $k$, proves the lemma. ∎

######  Corollary 5.10.

$\gamma_{\mathbb{A}}^{*}:Z_{T}(\mathbb{A}_{k})\to\mu_{4}$ is trivial on
$\underline{T}(k)\cap Z_{T}(\mathbb{A}_{k})$, hence is a genuine Weyl group
invariant ACC.

######  Remark 7.

By the proof of 5.2, the assumptions of 5.2 are also met if we take the
principal series representation to be one induced from the irreducible genuine
representation of $T(\mathbb{A}_{k})$ associated to $\gamma_{\mathbb{A}}^{*}$.

###  5.3. Two-fold cover of $GL_{n}$

Consider the homomorphism $i:GL_{n}(F)\to SL_{n+1}(F)$ given by mapping $A\in
GL_{n}(F)$ to $\begin{pmatrix}A&0\\\ 0&det(A)^{-1}.\end{pmatrix}$ Let
$\widetilde{SL}_{n+1}(F)$ denote the nontrivial two-fold central extension of
$SL_{n+1}(F)$, and let $\widetilde{GL}_{n}(F)$ denote its pullback, which is a
nontrivial two-fold central extension of $GL_{n}(F)$. Because the standard
maximal torus of $GL_{n}(F)$ is isomorphic to the standard maximal torus of
$SL_{n+1}(F)$ under the map $i$, their pullbacks are also isomorphic. So we
may identify the covering torus of $\widetilde{GL}_{n}(F)$ with the covering
torus of $\widetilde{SL}_{n+1}(F)$, which is generated by
$h_{i}(t)=h_{\alpha_{i}}(t)$ for $1\leq i\leq n$ and $t\in F^{\times}$. By
5.5, $Z_{T}(F)$ is generated by $\mu_{2}$ and $h_{i}(t^{2})$ for $1\leq i\leq
n$, $t\in F^{\times}$ when $n$ is odd, and is generated by $h_{i}(t^{2})$ and
$h_{odd}(t)=h_{1}(t)h_{3}(t)\cdots h_{n}(t)$ for $1\leq i\leq n$, $t\in
F^{\times}$ when $n$ is even. Globally, we may define $\widetilde{GL}_{n}$
over the adeles and its covering torus in the same fashion.

######  Proposition 5.11.

There exists a Weyl group invariant genuine character

$\gamma^{*}:Z_{T}(\mathbb{A}_{k})/T^{2}(\mathbb{A}_{k})\to\mu_{4}$

of the center of the covering torus of $\widetilde{GL}_{n}(\mathbb{A}_{k})$.
Explicitly, $\gamma^{*}$ is given by the isomorphism
$Z_{T}(\mathbb{A}_{k})/T^{2}(\mathbb{A}_{k})\cong\mu_{2}$ when $n$ is odd, and
is given by

$\gamma^{*}(\epsilon h_{odd}(t))=\epsilon\gamma(t)^{(n+1)/2}$

when $n$ is even.

###### Proof.

By 5.10, there is a Weyl group invariant genuine ACC of the covering torus of
$\widetilde{SL}_{n+1}(\mathbb{A}_{k})$ where Weyl group refers to the Weyl
group of $SL_{n+1}$. Pulling back through $i$, we get a genuine ACC of the
covering torus of $\widetilde{GL}_{n}(\mathbb{A}_{k})$. Because the Weyl group
of $GL_{n}$ may be identified with the subgroup of the Weyl group of
$SL_{n+1}$ that leaves the last entry of the torus invariant, the character is
invariant under the action of the Weyl group of $GL_{n}$. The second part
follows directly from 5.6. ∎

##  6\. Appendix

The results in this appendix are due to Gordan Savin.

###  6.1. Introduction

Many of Harish-Chandra’s results on intertwining operators and reducibility of
induced representations are contained in his work on the Plancherel measure
for reductive groups. However, since Harish-Chandra works with algebraic
groups, his results are not applicable to non-linear central extensions of
reductive groups. A purpose of this note is to provide simple proofs of some
of those results in a manner applicable to non-linear groups.

###  6.2. Notation

Let $k$ be a $p$-adic field with residual field of order $q$. Fix a
uniformizing element $\varpi$ and an absolute value on $k$ such that
$|\varphi|=1/q$.

Let $G$ be a reductive group defined over $k$. Let $A$ be a maximal split
torus and $\Delta$ a set of simple roots for the corresponding root system.
For every $\Theta\subseteq\Delta$ we have a standard parabolic subgroup
$P_{\Theta}=M_{\Theta}N_{\Theta}$ (or simply $P=MN$ if $\Theta$ is fixed). In
particular, $P_{\emptyset}$ is the minimal parabolic subgroup, while maximal
parabolic subgroups correspond to $\Theta$ such that $\Delta\setminus\Theta$
has one simple root. Let $A_{\Theta}\subseteq A$ be the connected component of
the intersection of kernels of all $\alpha\in\Theta$. Then $A_{\Theta}$ is a
split torus contained in the center of $M_{\Theta}$ and $A_{\Theta}\cong
X_{\Theta}\otimes k^{\times}$ where $X_{\Theta}$ is the co-character lattice.
Let $W$ be the Weyl group and $W_{\Theta}\subseteq W$ the subgroup generated
by simple reflections in $\Theta$. Let $w^{\ell}$ and $w^{\ell}_{\Theta}$ be
the longest elements in $W$ and $W_{\Theta}$ respectively. Let
${w_{\Theta}}=w^{\ell}w^{\ell}_{\Theta}$. Then
$\bar{\Theta}={w_{\Theta}}(\Theta)\subseteq\Delta$ and $P_{\bar{\Theta}}$ (or
simply $\bar{P}$) is the associated parabolic subgroup.

Now assume that $P_{\Theta}$ is maximal and $P_{\Theta}={P_{\bar{\Theta}}}$.
Then $w_{\Theta}$ is of order $2$ and normalizes $M_{\Theta}$. Let
$Y_{\Theta}\cong{\mathbb{Z}}$ be the lattice of characters of $M_{\Theta}$
trivial on the center of $G$. Fix a generator $\chi$ of this group of
characters. Then $\chi^{w_{\Theta}}=\chi^{-1}$. The image of natural pairing
between $X_{\Theta}$ and $Y_{\Theta}$ is $m{\mathbb{Z}}$, for some positive
integer $m$. Let $a_{\Theta}\in A_{\Theta}$ be such that
$\chi(a_{\Theta})=\varpi^{m}$.

###  6.3. Induced representations

Let $P=MN$ be a maximal parabolic subgroup corresponding to
$\Theta\subset\Delta$. Let $\delta_{N}$ be the modular character. Let
$(\pi,E)$ be a supercuspidal representation of $M$. Let $I_{P}^{G}(\pi)$ be
the (normalized) induced representation. It consists of locally constant
functions $f:G\rightarrow E$ such that $f(nmg)=\delta_{N}^{1/2}(m)\pi(m)f(g)$
for all $n\in N$, $m\in M$ and $g\in G$.

If $\bar{P}\neq P$ then $I_{P}^{G}(\pi)_{N}\cong\delta_{N}^{1/2}\pi$ (the
Jacquet functor) and $I_{P}^{G}(\pi)$ is always irreducible. Henceforth we
assume that $\bar{P}=P$. For example, this is true for every maximal parabolic
in $Sp_{2n}(k)$. Using the Bruhat decomposition $G=\cup_{w\in
W_{\Theta}\backslash W/W_{\Theta}}PwP$ define a filtration of $I_{P}^{G}(\pi)$
by $P$-invariant subspaces

$J(\pi)\subseteq J^{\prime}(\pi)\subseteq I_{P}^{G}(\pi)$

where $J(\pi)$ is subspace consisting of functions in $I_{P}^{G}(\pi)$
supported on $Pw_{\Theta}P=Pw_{\Theta}N$, and $J^{\prime}(\pi)$ is the
subspace of functions supported on $G\setminus P$. Then
$J(\pi)_{N}\cong\delta_{N}^{1/2}\pi^{w}$ by integrating over $w_{\Theta}N$,
$(J^{\prime}(\pi)/J(\pi))_{N}=0$, and
$(I_{P}^{G}(\pi)/J^{\prime}(\pi))_{N}\cong\delta_{N}^{1/2}\pi$ by evaluating
functions at the identity of $G$.

The representation $I_{P}^{G}(\pi)$ sits in a family of induced
representations. More precisely, for every $s\in\mathbb{C}$ let $(\pi_{s},E)$
be the twist of $\pi$ by $|\chi|^{s}$. This representation depends only on
$q^{s}\in{\mathbb{C}}^{\times}$. In this way we have an algebraic family
$\mathcal{F}$ of cuspidal representations of $M$. From now on, rather than
fixing one particular member of $\mathcal{F}$, $\pi$ will usually denote any
member of $\mathcal{F}$. However, if $\pi$ is fixed, then we can identify
$\mathcal{F}$ with ${\mathbb{C}}^{\times}$ or with ${\mathbb{C}}$, if it is
more convenient.

Let $K$ be a special maximal compact subgroup of $G$. Since $G=PK$, by
restricting functions in $I_{P}^{G}(\pi)$ to $K$, we can identify all induced
representations in $\mathcal{F}$ with $X={\mathrm{Ind}}^{K}_{K\cap P}(E)$. In
particular, an element $f\in X$ defines a map $\pi\mapsto f(\pi)\in
I_{P}^{G}(\pi)$ for all $\pi\in\mathcal{F}$. We shall call this map a constant
section. A regular section is a linear combination $\sum a_{i}f_{i}$ where
$a_{i}$ are regular functions on $\mathcal{F}$, and $f_{i}$ are constant
sections. If $f$ is a constant section such that $f(\pi)$ is in $J(\pi)$ (or
in $J^{\prime}(\pi)$) for one $\pi\in\mathcal{F}$, then $f(\pi)$ is in
$J(\pi)$ (or in $J^{\prime}(\pi)$, respectively) for all $\pi$. Thus we have a
corresponding filtration of constant sections

$Y\subseteq Y^{\prime}\subseteq X.$

We have a rational family of intertwining operators
$A(\pi):I_{P}^{G}(\pi)\rightarrow I_{P}^{G}(\pi^{w_{\Theta}})$ traditionally
defined (on an open set in $\mathcal{F}$) by the integral

$A(\pi)f(g)=\int_{N}f(w_{\Theta}ng)~{}dn.$

More precisely, there exists a rational function $a$ on $\mathcal{F}$ such
that for every constant section $f\in X$, $\pi\mapsto a(\pi)\cdot A(\pi)(f)$
is a regular section. Via the Frobenius reciprocity, the intertwining operator
$A(\pi)$ corresponds to a $P$-intertwining map

$\ell(\pi):I_{P}^{G}(\pi)\rightarrow\delta_{N}^{1/2}\pi^{w_{\Theta}}$

which, when restricted to $J(\pi)$, is given by the integral over
${w_{\Theta}}N$ and is therefore non-zero. We view $\ell$ as a rational map
$\pi\mapsto\ell(\pi)\in{\mathrm{Hom}}_{{\mathbb{C}}}(X,E)$. The analytic
behaviour of $A$ and $\ell$ is the same, in particular:

  1. (1)

For every constant section $f\in X$ the map $\pi\mapsto
a(\pi)\cdot\ell(\pi)(f)\in E$ is regular.

  2. (2)

For every constant section $f\in Y$ the map $\pi\mapsto\ell(\pi)(f)\in E$ is
regular.

  3. (3)

For every $\pi$ there is $f\in Y$ such that $\ell(\pi)(f)\neq 0$.

######  Lemma 6.1.

We have:

  1. (1)

For every constant section $f\in Y^{\prime}$ the map
$\pi\mapsto\ell(\pi)(f)\in E$ is regular.

  2. (2)

If $\pi$ is not isomorphic to $\pi^{w_{\Theta}}$ (which is always true if
$\pi$ is not unitary) then for every constant section $f\in X$, the map
$\pi\mapsto\ell(\pi)(f)\in E$ is regular at $\pi$.

###### Proof.

1) Assume that $\ell$, when applied to all sections in $Y^{\prime}$, has a
pole of (maximal) order $n$ at $\pi$. Using $\pi$ as a basepoint, we identify
the family $\mathcal{F}$ with ${\mathbb{C}}$. Then $s^{n}\ell$ defines a non-
zero $P$-intertwining map from $J^{\prime}(\pi)/J(\pi)$ to
$\delta_{N}^{1/2}\pi^{w_{\Theta}}$. But there is no such map since
$(J^{\prime}(\pi)/J(\pi))_{N}=0$. This is a contradiction. 2) Similarly if
$\ell$, when applied to all sections in $X$, has a pole of order $n$ at $\pi$,
then $s^{n}\ell$ defines a non-zero $P$-intertwining map from
$I_{P}^{G}(\pi)/J^{\prime}(\pi)$ to $\delta_{N}^{1/2}\pi^{w_{\Theta}}$. Since
$(I_{P}^{G}(\pi)/J^{\prime}(\pi))_{N}\cong\delta_{N}^{1/2}\pi$, we obtain an
isomorphism of $\pi$ and $\pi^{w_{\Theta}}$. This is a contradiction. The
lemma is proved. ∎

Since $I_{P}^{G}(\pi)$ is irreducible for a generic $\pi\in\mathcal{F}$, the
composition $A(\pi^{w_{\Theta}})\circ A(\pi)$ is a necessarily equal to
$\mu^{-1}(\pi)\cdot 1_{X}$ for a rational function $\mu(\pi)$, called
Plancherel measure. The following is contained in Casselman’s notes:

######  Proposition 6.2.

We have:

  1. (1)

If $\pi$ is not unitary, then $I_{P}^{G}(\pi)$ is reducible iff
$\mu^{-1}(\pi)=0$.

  2. (2)

If $\pi$ is unitary and $\pi$ is not isomorphic to $\pi^{w_{\Theta}}$ then
$I_{P}^{G}(\pi)$ is irreducible.

###### Proof.

1) Assume that $I_{P}^{G}(\pi)$ reduces. We claim that $I_{P}^{G}(\pi)$ has a
non-split composition series of length 2. Let $V^{\prime}$ be an irreducible
submodule. Then, by the Frobenius reciprocity, $\delta_{N}^{1/2}\pi$ must be a
quotient of $V^{\prime}$. Since $\pi$ is not isomorphic to $\pi^{w}$,
$\delta_{N}^{1/2}\pi$ appears with multiplicity one in $I_{P}^{G}(\pi)_{N}$.
This shows that $V^{\prime}$ is unique submodule,
$V^{\prime}_{N}\cong\delta_{N}^{1/2}\pi$ and
$V^{\prime\prime}_{N}\cong\delta_{N}^{1/2}\pi^{w}$ where
$V^{\prime\prime}=I_{P}^{G}(\pi)/V^{\prime}$. It follows, from the Frobenius
reciprocity, that $V^{\prime\prime}$ is a submodule of $I_{P}^{G}(\pi^{w})$
and $V^{\prime}$ is a quotient. Thus $A(\pi^{w})\circ A(\pi)=0$. Conversely,
since $A(\pi)$ and $A(\pi^{w})$ are defined and non-zero, $\mu^{-1}(\pi)=0$
must imply that $I_{P}^{G}(\pi)$ and $I_{P}^{G}(\pi^{w})$ both reduce. 2) If
$I_{P}^{G}(\pi)$ is reducible then, arguing as in 1), it has a unique
submodule and a unique quotient. But $I_{P}^{G}(\pi)$ is unitary and thus
semisimple. This is a contradiction. ∎

Assume that $\pi$ is unitary and that $\pi$ is isomorphic to
$\pi^{w_{\Theta}}$. Let $t_{w}:\pi^{w_{\Theta}}\rightarrow\pi$ be a map
intertwining the two actions of $M$. Let $\pi_{s}=\pi\otimes|\chi|^{s}$, where
$s\in{\mathbb{C}}$. Let $I(s)$ be the induced representation of $G$. We have
an intertwining map $A(s):I_{P}^{G}(s)\rightarrow I_{P}^{G}(-s)$ defined by
the integral

$A(s)f(g)=t_{w}\int_{N}f(w_{\Theta}ng)~{}dn.$

Let $\mu^{-1}(s)$ be the function obtained by composing $A(s)$ and $A(-s)$.
The following is not contained in Casselman’s notes:

######  Proposition 6.3.

Assume that $\pi$ is unitary and that $\pi$ is isomorphic to
$\pi^{w_{\Theta}}$. Then the following three are equivalent:

  1. (1)

$I_{P}^{G}(\pi)$ is irreducible.

  2. (2)

The exact sequence

$0\rightarrow\delta_{N}^{1/2}\pi\rightarrow
I_{P}^{G}(\pi)_{N}\rightarrow\delta_{N}^{1/2}\pi\rightarrow 0$

does not split.

  3. (3)

The function $\mu^{-1}(s)$ has a pole, necessarily of order 2, at $s=0$.

###### Proof.

Since $I_{P}^{G}(\pi)$ is unitary, it is irreducible if and only if
${\mathrm{Hom}}_{G}(I_{P}^{G}(\pi),I_{P}^{G}(\pi))\cong{\mathbb{C}}$. By the
Frobenius reciprocity,
${\mathrm{Hom}}_{G}(I_{P}^{G}(\pi),I_{P}^{G}(\pi))\cong{\mathrm{Hom}}_{M}(I_{P}^{G}(\pi),\delta_{N}^{1/2}\pi)$.
Thus $I_{P}^{G}(\pi)$ is irreducible if and only if $I_{P}^{G}(\pi)_{N}$ does
not split. This proves the equivalence of 1) and 2). If $\ell$ is regular at
$\pi$ then it defines a splitting of the exact sequence, and $I_{P}^{G}(\pi)$
reduces. In order to finish, we need to show that if $\mu^{-1}$ has a pole at
$\pi$, then it is of order $2$ and $I_{P}^{G}(\pi)_{N}$ does not split.

Let $a_{\Theta}$ be the element in the center of $M$, defined in the previous
section. Let $\epsilon$ be the central character of $\pi$. Let $s\mapsto
f_{s}\in I_{P}^{G}(\pi_{s})$ be a constant section. Then

$f_{s}^{\prime}=f_{s}-\epsilon^{-1}(a_{\Theta})\delta_{N}^{-1/2}(a_{\Theta})|\chi(a_{\Theta})|^{-s}\cdot\pi_{s}(a_{\Theta})(f_{s})$

is a regular section such that $f^{\prime}_{s}\in J^{\prime}(\pi_{s})$ for all
$s$. In particular, by Lemma 6.1, $s\mapsto\ell(f_{s}^{\prime})$ is regular.
If $s\neq 0$ but $|s|$ is sufficiently small, then $\ell$ is defined on the
whole $I_{P}^{G}(\pi_{s})$, hence

$\ell(f_{s}^{\prime})=\ell(f_{s})-q^{2ms}\ell(f_{s})=(1-q^{2ms})\ell(f_{s}).$

Solving for $\ell(f_{s})$ we see that $\ell$ can have the pole of order at
most one at $s=0$, and that happens if and only if
$\ell(\pi)(f_{0}^{\prime})\neq 0$ for some constant section $f_{s}$. Now we
can finish easily. Assume that $\ell$ has a pole at $s=0$. We want to show
that $I_{P}^{G}(\pi)_{N}$ does not split. If it splits, then $\ell(\pi)$
extends to $I_{P}^{G}(\pi)$. Let $\tilde{\ell}(\pi)$ be an extension. Then,
for every constant section $f_{s}$, the above computation shows
$\ell(\pi)(f_{0}^{\prime})=(1-q^{0})\tilde{\ell}(\pi)(f_{0})=0$, a
contradiction. Thus we have seen that the intertwining operator $A$ can have a
pole of order at most one at $I_{P}^{G}(\pi)$, and if it does, then
$I_{P}^{G}(\pi)$ is irreducible. It follows that the residue of $A$ acts on
$I_{P}^{G}(\pi)$ as multiplication by a non-zero scalar, so $\mu^{-1}$ has a
double pole. ∎

######  Corollary 6.4.

Assume that $\pi$ is unitary, $\pi$ is isomorphic to $\pi^{w_{\Theta}}$, and
$I_{P}^{G}(\pi)$ reduces (necessarily into two non-isomorphic summands). Then
$\mu(0)^{-1}\neq 0$ and $A(0)$ acts on the two summands by different scalars
$\pm\mu(0)^{-1/2}$.

###### Proof.

We know that $A(0)$ is well defined and non-zero since $\ell(\pi)$ is non-zero
on $J(\pi)$. Let $\lambda_{1}$ and $\lambda_{2}$ be the eigenvalues of $A(0)$
on the two summands. Since $\ell(\pi)$ is not given by evaluation of functions
at identity, the operator $A(0)$ is not a multiple of the identity operator.
Thus $\lambda_{1}\neq\lambda_{2}$. Since $A(0)^{2}=\mu(\pi)^{-1}$, the
eigenvalues must be non-zero and of opposite signs. ∎

Remark: If $G^{\prime}$ is a central extension of $G$ by a group of $n$-th
roots of 1, then all of the above arguments go through provided that we
replace $a_{\Theta}$ by an inverse image in $G^{\prime}$ of $a^{n}_{\Theta}$.
(That element is clearly contained in the in the center of $M^{\prime}$.)

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