Lusztig's strata are locally closed

TL;DR

This paper proves that Lusztig's strata in a connected reductive algebraic group are locally closed and characterizes their irreducible components as certain sheets of the group action, extending results to arbitrary characteristic.

Contribution

It extends the known properties of Lusztig's strata to arbitrary characteristic and characterizes their irreducible components as specific sheets of the group action.

Findings

Lusztig's strata are locally closed in G.

Irreducible components of a stratum are contained in specific sheets.

Results hold in arbitrary characteristic.

Abstract

Let G be a connected reductive algebraic group over an algebraically closed field k. We consider the strata in G defined by Lusztig as fibers of a map given in terms truncated induction of Springer representations. We extend to arbitrary characteristic the following two results: Lusztig's strata are locally closed and the irreducible components of a stratum X are those sheets for the G-action on itself that are contained in X.