On the Jacquet functor of Symplectic groups

TL;DR

This paper computes the Jacquet modules of certain representations of symplectic groups over non-Archimedean fields, revealing that for some classes, the multiplicity is at most two, advancing understanding of their structure.

Contribution

It provides explicit calculations of Jacquet modules for specific classes of symplectic group representations, showing bounded multiplicity results.

Findings

Jacquet modules computed for particular symplectic group representations

Multiplicity of Jacquet modules is at most 2 for a subclass of representations

Enhances understanding of the structure of symplectic group representations

Abstract

Let $\F$ be a non-Archimedean local field.~Consider $\G_{n}:= \Sp_{2n}(\F)$ and let $\M:= \GL_l \times \G_{n-l}$ be a maximal Levi subgroup of $\G_{n}$.~This paper undertakes the computation of the Jacquet module of representations of $\G_{n}$ with respect to the maximal Levi subgroup, belonging to a particular class. Finally, we conclude that for a subclass of representations of $\G_{n},$ multiplicity of the Jacquet module does not exceed 2.