Geometric wavefront sets of genuine Iwahori-spherical representations

TL;DR

This paper investigates the geometric wavefront sets of genuine Iwahori-spherical representations of central covers, establishing bounds, equalities in specific cases, and analyzing character expansions to deepen understanding of their structure.

Contribution

It proves an upper bound inequality for geometric wavefront sets and shows the equality is attained in certain types of covers and representations, extending previous work.

Findings

Upper bound inequality for geometric wavefront sets

Equality of wavefront sets in type A covers and some exceptional groups

Determination of leading coefficients in character expansions

Abstract

For Iwahori-spherical genuine representations of central covers with positive real Satake parameters, we prove the upper bound inequality for their geometric wavefront sets, formulated for general genuine representations in an earlier work by Gao--Liu--Lo--Shahidi. Meanwhile, we show the equality is attained for covers of type A groups and for some representations of covers of the exceptional groups. We also verify the equality for certain Iwahori-spherical representations occurring in regular unramified principal series; this uses and generalizes the earlier work of Karasiewicz--Okada--Wang on theta representations. Lastly, we determine the leading coefficients in the Harish-Chandra character expansion of a theta representation when its geometric wavefront set is of a special type.