Lusztig’s strata are locally closed

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 of truncated induction of Springer representations. We elaborate on the existing proof of the following two results in order to show that it extends to arbitrary characteristic: 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.