On the wavefront sets associated with theta representations

TL;DR

This paper investigates a conjectural formula for wavefront sets of theta representations over p-adic groups, confirming its consistency with known cases and relating it to archimedean analogs.

Contribution

It proposes and verifies a conjectural formula for the maximal elements in wavefront sets of theta representations, connecting p-adic and archimedean cases.

Findings

The formula agrees with existing literature for various groups.

The archimedean case results support the conjectural p-adic formula.

The study advances understanding of wavefront sets in representation theory.

Abstract

We study a conjectural formula for the maximal elements in the wavefront set associated with a theta representation of a covering group over $p$-adic fields. In particular, it is shown that the formula agrees with the existing work in the literature for various families of groups. We also recapitulate the results of an analogous formula in the archimedean case, which motivated the conjectural formula in the $p$-adic setting.