Towards a categorical analogue of Gelfand-Kazhdan Theorem

TL;DR

This paper explores a categorical analogue of the Gelfand-Kazhdan theorem, proving it for a specific case involving cuspidal categorical representations of PGL_2 over a local field.

Contribution

It formulates a conjecture extending the Gelfand-Kazhdan theorem to categorical representations and proves it for a particular example involving PGL_2.

Findings

Confirmed the conjecture for a specific cuspidal categorical representation of PGL_2

Established a foundation for categorical analogues of classical representation theorems

Provided insights into the structure of categorical representations over local fields

Abstract

A celebrated theorem by Gelfand-Kazhdan states that the restriction of any cuspidal irreducible representations of $GL_n(\mathcal{K})$ over local field to the mirabolic subgroup $P$ is isomorphic to the standard irreducible representation of $P$. We formulate a conjecture that an analogous statement should hold for categorical representations. In this note we prove this for a particular example of an irreducible cuspidal categorical representation of $PGL_2(\mathcal{K})$.