\documentstyle{amsppt}
\magnification\magstep1
\hfuzz5pt
\parindent=0mm
\define\R{\Bbb R}
\define\C{\Bbb C}
\define\Z{\Bbb Z}
\define\Q{\Bbb Q}
\define\W{\Bbb W}
\define\K{\Bbb K}
\define\tsp{\widetilde{Sp}(2n,\R)}
\define\WR{W_{\Bbb R}}
\redefine\Sp{Sp(2n,\R)}
\define\sgn{\text{sgn}}
\define\tildeGL#1{\widetilde{GL}(#1,\R)}

\centerline{Lifting of Characters on Orthogonal and}

\centerline{Metaplectic Groups}

\centerline{Jeffrey Adams}
\medskip
Notes of my talk at the AMS conference at Northeastern
University, Boston, October 7-8, 1995. 
These are the transparencies of my talk, condensed and slightly
edited. They are available on the web at 
http://www.wam.umd.edu/~jda/preprints.
The main results appear in a preprint {\it Lifting of Characters on
Orthogonal and Metaplectic Groups\/}.

\medskip
\centerline{ABSTRACT}
\medskip

We define lifting of characters between the split orthogonal group 
$SO(n+1,n)$ and the non-linear metaplectic group $\tsp$.
This is analogous to endoscopic lifting from the the quasisplit form 
of a linear group $G$. For tempered representations it agrees with
theta-lifting. It is analogous to Kazhdan and Patterson's theory of
metaplectic forms.

\document
\bigskip

$\Sp$:=symplectic group in $2n$ variables over $\R$,

$\tsp$:=metaplectic (non-linear, two-fold) cover of $\Sp$,

$O(p,q)$:=orthogonal group in $2n+1$ variables, signature $p,q$,

$SO(p,q)$:= special orthogonal group

The following theorem confirms the real case of a conjecture of Kudla
and generalizes a result of Waldspurger
[Forum. Math. 1991].

\proclaim{Theorem} (joint with D. Barbasch):

The dual pair correspondence induces a bijection 
$$
\tsp_{genuine}\sphat
\longleftrightarrow
\bigcup_{p,q}
SO(p,q)\sphat
$$
between the irreducible, admissible, genuine representations of $\tsp$
and the  irreducible admissible representations of the groups
$SO(p,q)$ (here and elsewhere the union is over the real forms of
$SO(2n+1)$, i.e. $p+q=2n+1$ with $(-1)^q$ fixed.)
\endproclaim

Question: Suppose $\pi\leftrightarrow\pi'$. What is the relation
between the global characters of $\pi$ and $\pi'$?

\proclaim{Lemma}

(1) There is a natural bijection (the orbit correspondence) between
the strongly regular semisimple conjugacy classes of $\Sp$ and 
$\{SO(p,q)\}$.

(2) This bijection induces a bijection between {\it stable} strongly
regular semisimple conjugacy classes. Two such classes correspond if
and only if they have the same non-trivial eigenvalues.
\endproclaim

This is the basic setup of lifting.

For (2), the real forms of $SO(2n+1)$ are taken together.

Notation: $g\longleftrightarrow g'$.

Definition: $G_0$ denotes the strongly regular semisimple elements of
$G$.

We identify a representation $\pi$ with its global character
$\Theta_\pi$, and $\Theta_\pi$ with the function on $G_0$ representing
it.

\noindent Example: Discrete Series

Let $\pi_{SO}(\lambda)$ be the discrete series representation of $SO(n+1,n)$ 
with Harish-Chandra parameter
$$
\lambda=(a_1,\dots, a_k;b_1,\dots,b_\ell)
$$
($a_i,b_j\in\Z+\frac12,a_1>\dots>a_k> 0,b_1>\dots>
b_\ell>0$).

Then the theta-lift of $\pi_{SO}(\lambda)$ is the discrete series
representation  $\pi_{Sp}(\lambda')$  of $\tsp$ with Harish-Chandra parameter
$$
\lambda'=(a_1,\dots, a_k,-b_\ell,\dots,-b_1).
$$

The global characters of $\pi_{Sp}(\lambda')$ and $\pi_{SO}(\lambda)$
have the following formulas on a compact Cartan subgroup:

$$
\align
\pi_{SO}(\lambda)&=
\frac
{\sum_{w\in W_K}\sgn(w)e^{w\lambda}}
{D_{SO}}\\
\pi_{Sp}(\lambda')&=\frac
{\sum_{w\in W_{K'}}\sgn(w)e^{w\lambda'}}
{D_{Sp}}
\endalign
$$

where 
$W_K$ is the Weyl group of $K=S(O(n+1)\times O(n))$,
$W_{K'}$ is the Weyl group of $K'=U(n)$,
and 
$$
D=\prod(e^{\alpha/2}-e^{-\alpha/2})
$$
is the Weyl denominator.

\smallskip

Remark: This is very imprecise: the Weyl denominators
depend on a choice of positive roots, and $D_{SO}$ is only
well-defined
on a covering group.

Stabilize:  define

$$
\align
\overline\pi_{SO}(\lambda)&=\sum_{w\in W_K\backslash
W}\pi_{SO}(w\lambda)\\
\overline\pi_{Sp}(\lambda')&=\sum_{w\in W_{K'}\backslash
W}\pi_{Sp}(w\lambda')
\endalign
$$
where $W$
is the Weyl group of type
$B_n/C_n$.

These have character formulas (on the compact Cartans):
$$
\align
\overline\pi_{SO}(\lambda)&=
\frac
{\sum_{w\in W}\sgn(w)e^{w\lambda}}
{D_{SO}}\\
\overline\pi_{Sp}(\lambda')&=\frac
{\sum_{w\in W}\sgn(w)e^{w\lambda'}}
{D_{Sp}}
\endalign
$$

The compact Cartan subgroups of  $SO(n+1,n)$ and $\tsp$ are isomorphic,
and this isomorphism can be chosen to take $\lambda$ to $\lambda'$.

Thus $\overline\pi_{Sp}(\lambda')$ may be obtained from
$\overline\pi_{SO}(\lambda)$ by multiplying by the {\it transfer
factor}
$$
\frac{D_{SO}}{D_{Sp}}
$$

\smallskip
Problem: Make this precise (cf. the remark), 
extend to other Cartan subgroups and
other representations.

Let $\omega=\omega_{even}\oplus\omega_{odd}$ be the oscillator
representation of $\tsp$, with global character 
$\Omega=\Omega_{even}+\Omega_{odd}$.

\proclaim{Definition}
$$
\Phi=\Omega_{even}-\Omega_{odd}
$$
(a function on the regular elements).
\endproclaim
\proclaim{Proposition}
For all regular semisimple $g\in\tsp$:
$$
\left|\frac{D_{SO}(g')}{D_{Sp}(g)}\right|=|\Phi(g)|
\quad(g'\longleftrightarrow g).
$$
\endproclaim

\smallskip
Remarks:

(1) This is a basic requirement of transfer factors 
[Shelstad, Math. Annalen 1982].

(2)  $\Phi(g)^2=\dfrac{\pm1}{det(1+g)}$ (Howe).
Without the absolute value both sides of the proposition are in
$\R^*\cup i\R^*$, so there is equality up to a fourth root of $1$.

(3) By (2) $\Phi(g)$ is non-singular near the identity.

(4) $\Phi$ is stable (stable: an ad hoc definition for $\tsp$).

(Properties (3) and (4) do not hold for $\Omega$).

By the Proposition it makes sense to {\it define} the transfer factor
to be $\Phi$ on any Cartan subgroup.

\proclaim{Main Definition:}

Let $\Theta$ be a stable, invariant function on
$SO(n+1,n)_0$.

For $g'\in\Sp_0$ let
$$
\tau(\Theta)(g')=\Theta(g)\quad(g\longleftrightarrow g')
$$
(independent of the choice of $g$).

For $g'\in\tsp_0$ let
$$
\Gamma(\Theta)(g')=\tau(\Theta)(p(g'))\Phi(g')
$$
This is a genuine, stable invariant function 
on $\tsp_0$.
\endproclaim

In other words, letting $\Theta'=\Gamma(\Theta)$,
$$
\Theta'(g')=\Phi(g')\Theta(g)
\quad(g'\longleftrightarrow g).
$$

For example taking $\Theta=1$ gives $\Theta'=\Omega_e-\Omega_o$.
This is a useful principal: given any lifting theory of this form,
taking $\Theta=1$ shows that the transfer factor is the character of a
(virtual) representation.

\proclaim{Main Theorem}

(1)  The map $\Gamma:\Theta\rightarrow\Theta'=\Gamma(\Theta)$ is a bijection between
stable invariant eigendistributions on $SO(n+1,n)$ and genuine stable invariant
eigendistributions on $\tsp$.

(2) $\Theta'$ is a virtual character if and only if $\Theta$ is a
virtual character.

(3) $\Theta'$ is tempered if and only if $\Theta$ is
tempered.

(4) $\Gamma$ takes the (stable) discrete series of $SO(2n+1)$ to the
(stable) genuine discrete series of $\tsp$.

(5) $\Gamma$ commutes with parabolic induction and coherent continuation.

(6) If $\Theta$ is a tempered virtual character 
then $\Theta'$ is the (normalized) theta-lift of $\Theta$.
\endproclaim

\proclaim{Conjecture} An analogous result holds over a p-adic field.
\endproclaim

This result holds, and is very easy, in the case of complex groups.

\smallskip
Remarks:

(1) The proof uses the {\it acceptable} group
$Spin(p,q)$: the inverse image of $SO(p,q)$ in $Spin(2n+1,\C)$. 

The Cartan subgroups of $Spin(p,q)$ and $\tsp$ are 
{\it isomorphic}.

(2) Key Fact:
$$
\text{$\Theta$ and $\Theta'$ have 
the same (normalized) Weyl numerators}
$$
This follows from:
$$
\Phi(g)=\frac{D'_{SO}(g')}{D'_{Sp}(g)}
\quad (g'\longleftrightarrow g)
$$
where $D'$ is the normalized Weyl denominator.

(3) The fact that $\Gamma$ only agrees with theta-lifting for tempered
representations is analogous to the fact that the sum of the
representations in an arbitrary L-packet is not stable.
In other words, $\Gamma$ has a nice formula on standard modules,
whereas theta-lifting is defined on irreducibles --- only in the tempered
case do these coincide.

(4) Since $\Phi$ is non-singular near the identity, $\Gamma$
preserves the order 
[Barbasch-Vogan, JFA 1980]
of an invariant
eigendistribution.

(5) A stable virtual character on $SO(n+1,n)$ may be identified with a
stable virtual character on $\{SO(p,q)\}$.
The relationship with theta-lifting uses the latter interpretation.

(6) This lifting is similar to character lifting coming from endoscopic
groups 
[Shelstad, Math. Annalen 1982],
$SO(n+1,n)$ playing the role of the quasisplit inner form of
$\tsp$. In endoscopy  there is a map 
$$
f\rightarrow f_H
$$
from $C^\infty_c(G)$ to $C^\infty_c(H)$ (well defined on quotients of
these spaces). This map satisfies a matching of orbital integrals, and
is dual to a lifting of characters - character formulas on each Cartan
subgroup are a consequence. 

\proclaim{Conjecture}
There is a map $f_{Sp}\rightarrow f_{SO}$,
defining a matching of orbital integrals, such that 
$\Gamma(\Theta)(f_{Sp})=\Theta(f_{SO})$.
\endproclaim
A plausible candidate for $f_{Sp}\rightarrow f_{SO}$ has been given by Rallis.

(7) In the case $n=1$ this is related, but not identical, to 
Flicker's lifting [Inventiones 1980]
between $GL(2)$ and its two-fold cover 
$\widetilde{GL}(2)$ (via the isomorphism $PGL(2,\R)\simeq SO(2,1)$).

Kazhdan-Patterson [Advances 1986]
have conjectural generalizations of Flicker's result
to $\widetilde{GL}(n)$. The same methods used in the proof of the Main
Theorem should yield a proof of some or all of these conjectures over $\R$.

\medskip

\hrule

\medskip

Sketch of the proof of the Main Theorem.

(a) $\Theta'$ satisfies Hirai's matching conditions
[Japanese Journal 1976]
(using (2) above). Therefore $\Gamma$ takes invariant
eigendistributions to invariant eigendistributions.

(b) Discrete series: determined (among invariant eigendistributions)
by the restriction to a compact Cartan.

(c) Induction: induced character formula as in endoscopic lifting
[Shelstad, Compositio 1979].

(d) Therefore $\Gamma$ takes standard modules to standard modules these span
the virtual representations.

\enddocument
\end