Bi-orbital sheaves and affine Hecke algebras at roots of unity

TL;DR

This paper advances the understanding of perverse sheaves on nilpotent cones, introducing supercuspidal sheaves and demonstrating their role in classifying bi-orbital sheaves, with applications to affine Hecke algebras at roots of unity.

Contribution

It introduces the concept of supercuspidal sheaves on nilpotent cones and shows they generate bi-orbital sheaves via parabolic induction, linking geometric and algebraic representation theories.

Findings

Bi-orbital sheaves are produced by parabolic induction from supercuspidal sheaves.

Provides a proof of Grojnowski's theorem on affine Hecke algebra modules at roots of unity.

Establishes a parametrization of simple modules via nilpotent singular support.

Abstract

Continuing the study of perverse sheaves on the nilpotent cone of a $\mathbb{Z}/m$-graded Lie algebra initiated by Lusztig--Yun, we study in this work the parabolic induction and introduce the notion of supercuspidal sheaves on the nilpotent cone. One of our main results shows that simple perverse sheaves with nilpotent singular support (called bi-orbital sheaves) are produced by parabolic induction from supercuspidal sheaves. As application, we provide a proof for a theorem announced by I. Grojnowski on the parametrisation of simple modules of affine Hecke algebras at roots of unity via Deligne--Langlands--Lusztig parameters with nilpotent singular support.