Let G be a complex, connected, reductive, algebraic group, and χ : ℂ× → G be a fixed cocharacter that defines a grading on 𝔤, the Lie algebra of G. In my first talk I have talked about the grading, derived category of equivariant perverse sheaves, bijection between the simple objects and some pairs that we are familiar with. In positive characteristic parity sheaves will play an important role. In this talk I will define parabolic induction and restriction both on nilpotent cone and graded setting. We will dive into the results of Lusztig in characteristic 0 in the graded setting. Under some assumptions on the field k and the group G we will recover some results of Lusztig in characteristic 0. These assumptions together with Mautner’s cleanness conjecture will play a vital role. The main result is that every parity sheaf occurs as a direct summand of the parabolic induction of some cuspidal pair. Lusztig’s work on ℤ-graded Lie algebras is related to representations of affine Hecke algebras, so a long term goal of this work will be to interpret parity sheaves in the context of affine Hecke algebras.
This is the second part of two talks, the first of which may be found here.
This video is part of the University of Georgia‘s Algebra seminar.
