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The Main Example

Let $ G$ be a reductive group and le $ H\subset G$ be a subgroup such that $ X:=H\backslash G$ is a spherical variety, i.e., the Borel subgroup of $ G$ has an open orbit in $ X$ . In [S-V] Sakellaridis and Venkatesh give a conjecture describing the support of the Plancherel measure of $ X$ , and a Plancherel formula reducing this conjecture to the discrete spectrum (using a method of Bernstein). This conjecture generalizes the results of Harish-Chandra on $ L^2(G)$ and of Delorme, Schlichtkrull and Van Der Ban for $ L^2(X)$ where $ X$ is a symmetric space ( $ X=H \backslash G$ is a symmetric space if there exists an involution $ \sigma$ of $ G$ such that $ G^{\sigma}_{\circ} \subset H \subset G^{\sigma}$ where $ G^{\sigma}$ is the set of fixed points of $ \sigma$ and $ G^{\sigma}_{\circ}$ is its connected component). The description of the support of the Plancherel measure of $ X$ in [S-V] is given in terms of the ``dual group'' associated to the variety $ X$ by Gaisgory and Nadler [G-N]. This dual group construction formalizes an observation of Jacquet and many others [Ref?] and sets the basis for a ``relative'' Langlands program [Ref some paper of Sakellaridis talking about this].

With this point of view the results of howe [H] describing the Plancherel measure of $ L^2(O(p-1,q)\backslash O(p,q))$ in terms of $ SL(2,\mathbb{R})$ , and of Ørsted and Zhang [OZ 1,2] describing the Plancherel measure of $ L^2(U(p-1,q)\backslash U(p,q))$ in terms of $ U(1,1)$ can be considered as Archimedian examples of Venkatesh and Sakellaridis result. Observe that in Howe's example $ SL(2,\mathbb{R})$ is not an algebraic group, however work has been done to include metaplectic groups into the Langlands Program [Adams, Mc].

The goal at the beginning of my thesis was to generate examples similar to Howe's result, but that lie outside of the theory being develped by [S-V]. This is done in the following way: Let $ P=MAN$ be the Siegel parabolic of $ Sp(m,\mathbb{R})$ , with given Langlands decomposition, and let $ \chi_{r,s}$ , $ r+s=m$ , be the character of $ N$ given by

$\displaystyle \chi_{r,s}\left(\left[\begin{array}{cc} 1 & X \\ & 1 \end{array}\right]\right) = \chi(\operatorname{tr} I_{r,s}X) $

where

$\displaystyle I_{r,s}=\left[\begin{array}{cc} I_r & \\ & -I_s \end{array}\right] $

and $ \chi$ is some fixed unitary character of $ \mathbb{R}$ . Let

$\displaystyle L^2(N\backslash Sp(m,\mathbb{R});\chi_{r,s})=\left\lbrace f:Sp(m,... ...R})} \vert f(g)\vert^2 \, dNg < \infty$} \end{array} \right\rbrace\right. . $

Observe that $ MA\cong GL(m,\mathbb{R})$ in a natural way, and in the notation of section 2 $ M_{\chi_{r,s}}\cong O(r,s)$ . In [G-Par] I give the follwoing decomposition of $ L^2(N\backslash Sp(m,\mathbb{R});\chi_{r,s})$

Theorem 3   As an $ O(r,s)\times Sp(m,\mathbb{R})$ -module with $ O(r,s)$ acting on the left, and $ Sp(m,\mathbb{R})$ acting on the right

$\displaystyle L^2(N\backslash Sp(m,\mathbb{R});\chi_{r,s}) \cong \int_{\hat{Sp(... ...\chi_{r,s},\tau}(\pi)\otimes \tau^{\ast} \otimes\pi \, d\nu(\tau) \, d\mu(\pi).$ (2)

where $ \nu$ is the Plancherel measure of $ O(r,s)$ , $ \mu$ is the Plancherel measure of $ Sp(m,\mathbb{R})$ , and $ W_{\chi_{r,s},\tau}(\pi)$ is some multiplicity space yet to be determined.

Also in [G-Par] I have the following proposition that generalizes and simplifies the work of Wolf and Howe [Wolf,Howe,...]

Proposition 4   For $ \mu$ -almost all tempered unitary representations $ \pi$ of $ Sp(m,\mathbb{R})$

$\displaystyle \pi^{\ast}\vert _{P} \cong \bigoplus_{r+s=m} \int_{\hat{O(r,s)}} ... ...eratorname{Ind}_{O(r,s) N}^{P} \tau^{\ast} {\chi_{r,s}^{\ast}} \, d\nu(\tau). $

Now consider the dual pair $ (Sp(m,\mathbb{R})\times O(p,q)) \subset Sp(mn,\mathbb{R})$ , $ p+q=n$ , and assume that $ p\geq q\geq m$ . The last condition states that we are in the stable range. Howe showed that in the stable range the Oscillator representation $ (\xi,L^2(\mathbb{R}^{mn}))$ of $ Sp(mn,\mathbb{R})$ decomposes in the following way when restricted to $ Sp(m,\mathbb{R})\times O(p,q)$

$\displaystyle L^2\mathbb{R}^{mn})\cong \int_{\hat{Sp(m,\mathbb{R})}} \pi\otimes \Theta(\pi) \, d\mu(\pi). $

where $ \mu$ is the Plancherel measure and $ \Theta(\pi)$ is a representation of $ O(p,q)$ called the $ \Theta$ -lift of $ \pi$ . A lot of work has been done to describe the explicit $ \Theta$ -correspondence and in the stable range case this correspoindence can be described using the work of Jian-Shu Li [JL] among others. Using this results an the explicit formulas for the action of $ (Sp(m,\mathbb{R})\times O(p,q))$ on $ L^2(\mathbb{R}^{mn}$ given in [Rao, Admas,Roberts] I obtain the following description of $ L^2(O(p-r,q-s)\backslash O(p,q))$ in [G-Ex]

Theorem 5   As an $ O(r,s)\times O(p,q)$ -module with $ O(r,s)$ acting on the left, and $ O(p,q)$ acting on the right

$\displaystyle L^2(O(p-r,q-s)\backslash O(p,q))\cong \int_{\hat{Sp(m,\mathbb{R})... ...\pi)\otimes \tau^{\ast} \otimes \Theta(\pi^{\ast}) \, d\nu(\tau) \, d\mu(\pi) $

where $ \nu$ and $ \mu$ are the Plancherel measures of $ O(r,s)$ and $ Sp(m,\mathbb{R})$ respectively.

Observe that when $ m=1$ we regain Howe's result [H]. Also observe that in this case the decomposition given in equation (2) is contained in Wallach's work on the Plancherel-Whittaker measure for minimal parabolic subgroups. In this case Wallach result says that $ W_{\chi_{r,s},\tau}(\pi)\cong Wh_{\chi_{r,s},\tau}(\pi)$ . It's therefore natural to state the following conjecture

Conjecture 6   With notation as above

$\displaystyle W_{\chi_{r,s},\tau}(\pi)=Wh_{\chi_{r,s},\tau}(\pi) $

For all $ r,s$ and all $ \tau,\pi$ tempered and irreducible.

An important step in the direction of proving the conjecture is the calculation of the space $ Wh_{\chi_{r,s}}(\pi)$ where $ \pi$ is an induced representation which is done in [G-W] when $ M_{\chi_{r,s}}\cong O(r,s)$ is compact, i.e., $ r=0$ or $ s=0$ , and in [G-1] in all the other cases. Using the results in [G-W] and following the ideas given in chapter 15 of [W,vol2] I have been able to calculate the Bessel-Plancherel measure of $ L^2(N\backslash G;\chi)$ in the case where $ M_{\chi}$ is compact. The calculations and the explicit Plancherel measure and intertwiners can be found in [G-2] and prove the conjecture in the compact stabilizer case. The full conjecture should be proved in a similar way, but first some complications of functional analysis nature should be solved. I will talk a little bit more about this difficulties in the next section.

next up previous
Next: Further Research Up: Research Statement Previous: Bessel Models for General
Raul Gomez 2010-09-22


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