Distinguished Tame Supercuspidal Representations and Odd Orthogonal Periods

TL;DR

This paper refines the theory of distinguished tame supercuspidal representations of p-adic groups and applies it to classify when certain representations of GL_n are distinguished by orthogonal groups, establishing uniqueness of the distinguishing form.

Contribution

It simplifies the existing theory for tame supercuspidal representations and provides a precise criterion for distinction with respect to orthogonal groups in the case of GL_n with odd n.

Findings

Characterizes when supercuspidal representations are distinguished by orthogonal groups.

Shows the space of distinguishing linear forms is one-dimensional.

Extends the theory to finite reductive groups and simplifies previous frameworks.

Abstract

We further develop and simplify the general theory of distinguished tame supercuspidal representations of reductive $p$-adic groups due to Hakim and Murnaghan, as well as the analogous theory for finite reductive groups due to Lusztig. We apply our results to study the representations of ${\rm GL}_n(F)$, with $n$ odd and $F$ a nonarchimedean local field, that are distinguished with respect to an orthogonal group in $n$ variables. In particular, we determine precisely when a supercuspidal representation is distinguished with respect to an orthogonal group and, if so, that the space of distinguishing linear forms has dimension one.