Twisted Kazhdan-Lusztig conjecture for $p$-adic general linear group

TL;DR

This paper classifies irreducible representations of twisted p-adic general linear groups using enhanced Langlands parameters and proves a twisted Kazhdan-Lusztig conjecture for multiplicities.

Contribution

It introduces a classification framework for twisted p-adic representations and proves the conjecture using graded Hecke algebra methods, extending Lusztig's work.

Findings

Classification of irreducible representations via enhanced Langlands parameters

Proof of the twisted Kazhdan-Lusztig conjecture for multiplicities

Compatibility of parametrization with Whittaker normalization

Abstract

We use enhanced Langlands parameters to obtain a classification for irreducible representations of twisted $p$-adic general linear groups in unramified principal series. We give the definition of standard representations and prove the twisted Kazhdan-Lusztig conjecture for the multiplicities in the Grothendieck group. We mainly follow Lusztig's work in the connected case using graded Hecke algebra. We show that the parametrization is compatible with the Whittaker-normalized one.