Tame Supercuspidal Representations of $\GL_n$ Distinguished by Orthogonal Involutions

TL;DR

This paper characterizes how tame supercuspidal representations of GL_n over a p-adic field embed into functions on symmetric matrices, revealing their distinguished properties under orthogonal involutions.

Contribution

It provides a detailed description of the embeddings of tame supercuspidal representations into symmetric matrix spaces, highlighting their distinction by orthogonal involutions.

Findings

Explicit embedding descriptions for tame supercuspidal representations.

Identification of conditions for representations to be distinguished by orthogonal involutions.

Advancement in understanding symmetry properties of supercuspidal representations.

Abstract

For a $p$-adic field $F$, the embeddings of a tame supercuspidal representation of $G= {\rm GL}_n (F)$ in the space of smooth functions on the set of symmetric matrices in $G$ are determined.