Supercuspidal representations of $\mathrm{GL}_{n}(F)$ distinguished by an orthogonal involution

TL;DR

This paper characterizes when supercuspidal representations of GL(n,F) are distinguished by an orthogonal subgroup, revealing conditions on the representation's central character and its embedding in functions on symmetric matrices.

Contribution

It provides a complete characterization of distinguished supercuspidal representations of GL(n,F) with respect to orthogonal involutions, extending Hakim's results to non-tame cases.

Findings

Non-zero distinguished space characterized by central character at -1

Dimension of embedding space is four under specific conditions

Generalizes Hakim's results to broader class of supercuspidal representations

Abstract

Let $F$ be a non-archimedean locally compact field of residue characteristic $p\neq2$, let $G=\mathrm{GL}_{n}(F)$ and let $H$ be an orthogonal subgroup of $G$. For $\pi$ a complex smooth supercuspidal representation of $G$, we give a full characterization for the distinguished space $\mathrm{Hom}_{H}(\pi,1)$ being non-zero and we further study its dimension as a complex vector space, which generalizes a similar result of Hakim for tame supercuspidal representations. As a corollary, the embeddings of $\pi$ in the space of smooth functions on the set of symmetric matrices in $G$, as a complex vector space, is non-zero and of dimension four, if and only if the central character of $\pi$ evaluating at $-1$ is $1$.