Quasi-admissible, raisable nilpotent orbits, and theta representations

TL;DR

This paper investigates the properties of nilpotent orbits in covering groups, specifically their quasi-admissibility and raisability, and computes wavefront sets and character expansion coefficients for theta representations.

Contribution

It determines the degrees of covers for which certain nilpotent orbits are quasi-admissible and non-raisable, and explicitly computes wavefront sets and Harish-Chandra coefficients for theta representations.

Findings

Identified degrees of covers with quasi-admissible, non-raisable nilpotent orbits.

Explicitly computed wavefront sets of theta representations.

Determined leading coefficients in Harish-Chandra character expansions.

Abstract

We study the quasi-admissibility and raisablility of some nilpotent orbits of a covering group. In particular, we determine the degree of the cover such that a given split nilpotent orbit is quasi-admissible and non-raisable. The speculated wavefront sets of theta representations are also computed explicitly, and are shown to be quasi-admissible and non-raisable. Lastly, we determine the leading coefficients in the Harish-Chandra character expansion of theta representations of covers of the general linear groups.