Weyl groups, the Dirac inequality, and isolated unitary unramified representations

TL;DR

This paper applies the Dirac inequality to identify isolated unitary representations and spectral gaps in unramified principal series, especially linking nilpotent orbits to unitary representations in a p-adic context.

Contribution

It introduces a novel application of the Dirac inequality to classify isolated unitary representations associated with large nilpotent orbits in the unramified setting.

Findings

Identification of spectral gaps in unramified principal series

Connection between nilpotent orbits and unitary representations

p-adic analogue of Salamanca-Riba's classification

Abstract

I present several applications of the Dirac inequality to the determination of isolated unitary representations and associated "spectral gaps" in the case of unramified principal series. The method works particularly well in order to attach irreducible unitary representations to the large nilpotent orbits (e.g., regular, subregular) in the Langlands dual complex Lie algebra. The results could be viewed as a $p$-adic analogue of Salamanca-Riba's classification of irreducible unitary $(\mathfrak g,K)$-modules with strongly regular infinitesimal character.