Seminar: (Deligne)-Lusztig and Other Related Varieties

Abstract:  I'll explain work with J. Hong and D. Lee on the geometry of Deligne-Lusztig varieties and Lusztig varieties. These are two related (but definitely different) subvarieties of generalized flag varieties. The first were introduced by Deligne and Lusztig in their paper on the representations of a reductive group over a finite field, and the second were introduced by Lusztig to take the Deligne-Lusztig study further.  If G is a reductive group over a finite field F, then G(F) acts on the etale cohomology of certain sheaves on the Deligne-Lusztig varieties giving rise to representations of the finite group G(F). On the other hand, the Lusztig varieties live in a family of subvarieties of the flag variety fibered over G, and, via the sheaf-function correspondence, this family gives rise to characters of G(F).

One of the results of our work is that Deligne-Lusztig varieties are affine as are Lusztig varieties over regular semisimple elements.  This was known in many cases going back to the original paper of Deligne and Lusztig and continuing to a 2021 preprint of Lusztig, which showed that many Lusztig varieties are affine.  I'll explain why the role our result plays in the Deligne-Lusztig theory.  I'll also explain the main idea behind the proof along with some of the related results in our work (e.g., how it relates to Hessenberg varieties).

Faculty Host: Matt Kerr