Fourier Coefficients for Theta Representations on Covers of General Linear Groups

TL;DR

This paper investigates theta representations on covers of general linear groups, analyzing their Fourier coefficients to understand their structure and the associated unipotent orbits, extending previous work on double covers.

Contribution

It characterizes the support of certain Fourier coefficients on these theta representations and determines the unipotent orbit attached to them.

Findings

Semi-Whittaker coefficients vanish or not depending on the cover

Explicit determination of unipotent orbits for theta representations

Extension of coefficients analysis to general covers

Abstract

We show that the theta representations on certain covers of general linear groups support certain types of unique functionals. The proof involves two types of Fourier coefficients. The first are semi-Whittaker coefficients, which generalize coefficients introduced by Bump and Ginzburg for the double cover. The covers for which these coefficients vanish identically (resp. do not vanish for some choice of data) are determined in full. The second are the Fourier coefficients associated with general unipotent orbits. In particular, we determine the unipotent orbit attached, in the sense of Ginzburg, to the theta representations.