Wavefront sets for genuine representations of $\rm GL$-covers of Kazhdan--Patterson or Savin types

TL;DR

This paper studies wavefront sets of irreducible genuine representations of certain $p$-adic covering groups, linking them to Bernstein--Zelevinsky derivatives and local Langlands correspondence.

Contribution

It determines the wavefront sets for irreducible genuine representations of Kazhdan--Patterson and Savin covers, connecting them to derivatives and duality theories.

Findings

Wavefront sets expressed via iterated degrees of derivatives.

Classification of genuine spectrum for specific covers.

Reinterpretation of wavefront sets through local Langlands correspondence.

Abstract

First, we consider general Brylinski--Deligne covers of the $p$-adic general linear groups, and discuss the theory of Bernstein--Zelevinsky derivatives. We also recall the Zelevinsky-type classification of the irreducible genuine spectrum for the Kazhdan--Patterson and Savin covers. Following this, for these two special families of covers, we determine the wavefront sets of their irreducible genuine representations, expressed in terms of the iterated degrees of the highest Bernstein--Zelevinsky derivatives. Finally, for Kazhdan--Patterson covers, we reinterpret this result on the wavefront set using a version of the local Langlands correspondence and the covering Barbasch--Vogan duality.