The spectrum of the symplectic Grassmannian and $\mathrm{Mat}_{n,m}$

Abstract

Let $\mathbf{G}$ be a reductive group and $\mathbf{X}$ a spherical $\mathbf{G}$-variety over a local non-archimedean field $\mathbb{F}$. We denote by $S(\mathbf{X}(\mathbb{F}))$ the Schwartz-functions on $\mathbf{X}(\mathbb{F})$. In this paper we offer a new approach on how to obtain bounds on \[\dim_{\mathbb{C}}\mathrm{Hom}_{\mathbf{G}(\mathbb{F})}(S(\mathbf{X}(\mathbb{F})),\pi)\]for an irreducible smooth representation $\pi$ of $\mathbf{G}(\mathbb{F})$. Our strategy builds on the theory of $\rho$-derivatives and the Local Structure Theorem for spherical varieties. Currently, we focus on the case of the symplectic Grassmannian and the space of matrices. In particular, we obtain a new proof of Howe duality in type II as well as an explicit description of the local Miyawaki-liftings in the Hilbert-Siegel case. Furthermore, we manage to extend previous results of the author regarding the conservation relation in the theta correspondence to metaplectic covers of symplectic groups. Finally, we use our new proof of Howe duality in type II to relate the order of the poles of Godement-Jacquet $L$-functions to the geometry of the space of matrices and the order of poles of certain intertwining operators.