The spectrum of the symplectic Grassmannian and Mat_n,m

2025/06/16 by Johannes Droschl, Droschl, Johannes
Mathematics · #11F70 #22E46 #22E50 #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2506.13530

openalex publication_date 2025/06/16 · openalex created_date 2025/10/13 · openalex updated_date 2026/07/28

Abstract

Let G be a reductive group and X a spherical G-variety over a local non-archimedean field \mathbbF. We denote by S(X(\mathbbF)) the Schwartz-functions on X(\mathbbF). In this paper we offer a new approach on how to obtain bounds on dim_ℂHom_G(\mathbbF)(S(X(\mathbbF)),π)for an irreducible smooth representation π of G(\mathbbF). Our strategy builds on the theory of ρ-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.

Citations

• A note on the Howe Duality conjecture for symplectic-orthogonal and unitary pairs
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• Classification of irreducible representations of metaplectic covers of the general linear group over a non-archimedean local field
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• The explicit Zelevinsky-Aubert duality
• A theory of Miyawaki liftings: The Hilbert-Siegel case
• On the geometry of spherical varieties
• Periods and harmonic analysis on spherical varieties
• Sur l'irréductibilité d'une induite parabolique

Related

• Harmonic Analysis and Gamma Functions on Symplectic Groups
• The ρ-Fourier transform
• A Conjecture of Sakellaridis-Venkatesh on the Unitary Spectrum of Spherical Varieties
• The stack of spherical Langlands parameters
• Spherical varieties, functoriality, and quantization