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Bessel Models for General Admissible Induce Representations

The calculation of the space of Whittaker models is one of the main links between Number Theory and Representation Theory. A well known example of its applications is the proof of the multiplicity one theorem for $ GL(n)$ using the uniqueness of local Whittaker models given by Jacquet and Langlands [J-L, 1970] for $ n=2$ and independently by Shalika [S-1974] and Piateski-Shapiro [PS-1979] for $ n > 2$ .

One of the reasons why the space of Whittaker models is so useful in the study of $ GL(n)$ is that all the representations appearing in its cuspidal spectrum have a Whittaker model, something that's not always true for other groups. For example for $ GSp(4)$ ??? have shown that some automorphic cuspidal representations do not have a Whittaker model. One way around this is to work instead with the space of Bessel Models, that is, linear functionals in the unipotent radical of the Siegel parabolic. Since $ N$ is abelian all its irreducible representations are one-dimensional, and hence all cuspidal representations will have a Bessel Model. A lot a work has been done to study this space of Bessel Models. In the real case the more general results can be found in [W-86] where a multiplicity one result is proved for a class of parabolic subgroups, later characterized to be the very nice parabolic subgroups [W-recent]. Here multiplicity one means the following: Let $ P=MAN$ be a very nice parabolic subgroup, with given Langlands decomposition. Let $ (\sigma, H_{\sigma})$ be a finite dimensional representation of $ M$ , and let $ \nu$ be a complex valued linear functional on $ \mathfrak{a}=Lie(A)$ . Let $ I_{P,\sigma,\nu}^{\infty}$ be the smooth representation induced from $ \sigma_{\nu}$ , and let $ \chi$ be a generic [W86] character of $ N$ . Let

$\displaystyle Wh_{\chi}(I_{P,\sigma,\nu}^{\infty})=\{\lambda:I_{P,\sigma,\nu}^{... ...\longrightarrow \mathbb{C} \, \vert \, \lambda(n \cdot f)=\chi(n)\lambda(f)\} $

be the space of Whittaker functionals of $ I_{P,\sigma,\nu}^{\infty}$ . Then

$\displaystyle \dim{Wh_\chi(I_{P,\sigma,\nu}^{\infty})}=\dim{H_{\sigma}}.$ (1)

During his visit to UCSD during the academic year 2007-2008 Dipendra Prasad asked if a similar result was true in the case where $ (\sigma, H_{\sigma})$ is an admissible, smooth, Frechet, moderate growth representation of $ M$ . In this case the statement about dimensions in equation (1) has to be replaced by an $ M_{\chi}$ -intertwiner isomorphism between $ H_{\sigma}$ and $ Wh_\chi(I_{P,\sigma,\nu}^{\infty})$ , where

$\displaystyle M_{\chi}=\{m\in M \, \vert\,$   $\displaystyle \mbox{$\chi(mnm^{-1})=\chi(n)$, for all $n \in N$}$$\displaystyle \}. $

In my joint work with Wallach [G-W] we use the theory of the transverse symbol of a vector valued distribution developed by Kolk and Varadajan [K-V] to prove this result in the case where $ M_{\chi}$ is compact. More formally we have the following result:

Let $ G$ be a connected simple Lie group with finite center and let $ K$ be maximal compact subgroup. We assume that $ G/K$ is Hermitian symmetric of tube type. Up to covering homomorphisms there is a one-to-one correspondence between the set of simple Jordan algebras and the set of Lie groups satisfying this conditions [G-W]. Let $ P=MAN$ be a parabolic subgroup, with given Langlands decomposition, such that $ N$ is abelian. Let $ I_{M\cap K,\sigma\vert _{M\cap K}}^{\infty}$ be the representation induced by $ \sigma\vert _{M\cap K}$ from $ M\cap K$ to $ K$ . Given $ f\in I_{M\cap K,\sigma\vert _{M\cap K}}^{\infty}$ define

$\displaystyle f_{P,\sigma,\nu}(namk)=a^{\nu+\rho}\sigma(m)f(k). $

The map $ f\mapsto f_{P,\sigma,\nu}$ defines a $ K$ -equivariant linear isomorphism from $ I_{M\cap K,\sigma\vert _{M\cap K}}^{\infty}$ to $ I_{P,\sigma,\nu}^{\infty}$ [W,vol1]. Let $ w_{M}$ be an element in $ N_{K}(A)$ that conjugates $ P$ to $ \bar{P}$ and consider the integrals

$\displaystyle J_{P,\sigma,\nu}^{\chi}(f)=\int_{N}\chi(n)^{-1}f_{P,\sigma,\nu}(w_{M}n)\, dn. $

These integrals are called generalized Jacquet integrals and converge absolutely and uniformly on compacta for $ \operatorname{Re} \nu \gg 0$ [W-86]. Let $ \mu \in H_{\sigma}'$ and define $ \gamma_{\mu}(\nu)=\mu\circ J_{P,\sigma,\nu}^{\chi}$ . Observe that if $ \operatorname{Re} \nu \gg 0$ then $ \gamma_{\mu}$ defines a weakly holomorphic map into $ (I_{M\cap K,\sigma\vert _{M\cap K}}^{\infty})'$ .

Theorem 1   Assume that $ M_{\chi}$ is compact.
i)
$ \gamma_{\mu}$ extends to a weakly holomorphic map from $ \mathfrak{a}'$ to $ (I_{M\cap K,\sigma\vert _{M\cap K}}^{\infty})'$
ii)
Given $ \nu\in\mathfrak{a}'$ define

$\displaystyle \lambda_{\mu}(f_{P,\sigma,\nu})=\gamma_{\mu}(\nu)(f), \qquad f\in I_{M\cap K,\sigma\vert _{M\cap K}}^{\infty}. $

Then $ \lambda_{\mu}\in Wh_{\chi}(I_{P,\sigma,\nu}^{\infty})$ and the map $ \mu \mapsto \lambda_{\mu}$ defines an $ M_{\chi}$ -equivariant isomorphism between $ H_{\sigma}'$ and $ Wh_{\chi}(I_{P,\sigma,\nu}^{\infty})$ .

This theorem is essentially theorem 12 in [G-W].

In [G1] I look at the case when $ M_{\chi}$ is not compact. In this case the above theorem as it is stated is false. The main reason for this is that in this case the orbits of the symmetric space $ X:=M_{\chi}\backslash M$ under the action of a minimal parabolic are much more complicated that in the case where $ M_{\chi}$ is compact [Matsuki]. However something can still be said about $ Wh_{\chi}(I_{P,\sigma,\nu}^{\infty})$ . Assume $ M_{\chi}={}^{\circ}M_{\chi}$ [W,Vol1], and let $ (\tau,V_{\tau})$ be an irreducible, admissible infinite dimensional representation of $ M_{\chi}$ . Let $ \sigma^{w_{M}}$ be the twisting of $ \sigma$ by $ w_{M}$ and let $ \mu \in Hom(H_{\sigma^{w_{M}}},V_{\tau})$ (Observe that $ \sigma^{w_{M}}\vert M\cap K \cong \sigma$ so this modification was unnecessary in the compact stabilizer case). As in the case were $ M_{\chi}$ is compact, define $ \gamma_{\mu}(\nu)=\mu\circ J_{P,\sigma,\nu}^{\chi}$ and observe that if $ \operatorname{Re} \nu \gg 0$ then $ \gamma_{\mu}$ defines a weakly holomorphic map into $ Hom(I_{M\cap K,\sigma\vert _{M\cap K}}^{\infty},V_{\tau})$ . Let

$\displaystyle Wh_{\chi,\tau}(I_{P,\sigma,\nu}^{\infty})=\{\lambda:I_{P,\sigma,\nu}^{\infty}\longrightarrow V_{\tau} \, \vert \,$   $\displaystyle \mbox{$\lambda(mn\cdot f)=\chi(n)\tau(m)\lambda(f)$, for all $m\in M$, $n\in N$}$$\displaystyle \}. $

Theorem 2   With assumptions as above.
i)
$ \gamma_{\mu}$ extends to a weakly holomorphic map from $ \mathfrak{a}'$ to $ Hom(I_{M\cap K,\sigma\vert _{M\cap K}}^{\infty},V_{\tau})$ .
ii)
Given $ \nu\in\mathfrak{a}'$ define

$\displaystyle \lambda_{\mu}(f_{P,\sigma,\nu})=\gamma_{\mu}(\nu)(f), \qquad f\in I_{M\cap K,\sigma\vert _{M\cap K}}^{\infty}. $

Then $ \lambda_{\mu}\in Wh_{\chi,\tau}(I_{P,\sigma,\nu}^{\infty})$ and the map $ \mu \mapsto \lambda_{\mu}$ defines an isomorphism between $ Hom(H_{\sigma^{w_{M}}},V_{\tau})$ and $ Wh_{\chi,\tau}(I_{P,\sigma,\nu}^{\infty})$ .

In [G1] I prove an equivalent formulation of this result.

In [J-S-Z] Jian, Sun and Zhu prove a uniqueness result for the space of Bessel Models of a large class of groups. However the space of Bessel models defined there is different than the one defined here, except for the special case of $ SO(n,2)_{\circ}$ . In [J-S-Z] the parabolic subgroups considered keep a strong similitude with minimal parabolic subgroups in their structure. In the extreme cases this result reduces to the result of Sun and Zhu and independently Gourevitch and Aizenbud in one hand and the results on the uniqueness of Whittaker models for minimal parabolic subgroups in the other extreme.

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Next: The Main Example Up: Research Statement Previous: Introduction
Raul Gomez 2010-09-22


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