Covering Barbasch-Vogan duality and wavefront sets of genuine representations

TL;DR

This paper introduces a covering Barbasch-Vogan duality for genuine representations of p-adic covering groups, proposes an upper bound conjecture for wavefront sets, and proves it for certain linear groups.

Contribution

It defines a new covering duality, formulates a generalized wavefront set conjecture, and proves it for Kazhdan-Patterson coverings of general linear groups.

Findings

Defined a covering Barbasch-Vogan duality.

Formulated an upper bound conjecture for wavefront sets.

Proved the conjecture for Kazhdan-Patterson coverings.

Abstract

In this paper, we start by defining a covering Barbasch-Vogan duality and prove some of its properties. Then, for genuine representations of $p$-adic covering groups we formulate an upper bound conjecture for their wavefront sets using this covering Barbasch-Vogan duality and reduce it to anti-discrete representations. The formulation generalizes that of Ciubotaru-Kim and Hazeltine-Liu-Lo-Shahidi for linear algebraic groups. We prove this upper bound conjecture for Kazhdan-Patterson coverings of general linear groups.