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 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.