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

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.