\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