A stacky generalized Springer correspondence and rigid enhancements of L-parameters

TL;DR

This paper advances the geometric understanding of the generalized Springer correspondence for disconnected reductive groups using stacks, and explores implications for rigid L-parameter enhancements in the Langlands program.

Contribution

It provides a geometric reformulation of the Springer correspondence and applies this to analyze Kaletha's rigid enhancements of L-parameters.

Findings

Established a geometric version of the Bernstein-Zelevinsky Lemma

Compared group and Lie algebra correspondences via quasi-logarithms

Showed the existence of a cuspidal support map for rigid L-parameters

Abstract

Motivated by applications to the Langlands program, Aubert-Moussaoui-Solleveld extended Lusztig's generalized Springer correspondence to disconnected reductive groups. We use stacks to give a more geometric account of their theory, in particular, formulating a truly geometric version of the (relevant analogue of the) Bernstein-Zelevinsky Geometrical Lemma and explaining how to compare the correspondence on the group and the Lie algebra using quasi-logarithms. As an application, we study Kaletha's rigid enhancements of L-parameters and draw the same conclusions as Aubert-Moussaoui-Solleveld for this enhancement: there exists a cuspidal support map and its fibers are parameterized by irreducible representations of twisted group algebras.