Lusztig’s strata are locally closed

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.

A parametrization of sheets of conjugacy classes in bad characteristic

“…Sheets in a reductive algebraic group G are the irreducible components of the locally closed subsets of G consisting of conjugacy classes of the same dimension. They occur also as irreducible components of the strata in the partition of G , defined in [10] in terms of Springer representations with trivial local system (see [2, 3]). One of the most fascinating features of strata is that they are parametrized by a family of irreducible representations of the Weyl group which depends on the root system of G and not on the characteristic of the base field.…”

“…A description of sheets in good characteristic, and a parametrization of sheets in terms of G -conjugacy classes of triples where M is the identity component of the centralizer of a semisimple element in G , is a suitable coset in the component group , and is a rigid unipotent conjugacy class in M was given in [4] in good characteristic, and extended to the case of bad characteristic in [14]. A refinement of this parametrization in terms of pairs where M and are as above was given in [3] under the assumption that G is simple of adjoint type and the characteristic of the base field is good for G . The present paper answers a question by G .…”

“…We therefore elaborate upon results in [5, 6, 8] in the spirit of [13] in order to provide a combinatorial description of the root systems of connected centralizers of semisimple elements, which will allow us to retrieve most ingredients that were necessary for the proof of [3, Theorem 4.1]. As an example, we show that the number of sheets in type is the same in good characteristic and in characteristic , but it is smaller in characteristic , showing a difference in the behavior of sheets as opposed to strata.…”

Untitled

“…In [Lus15] we have defined for any r ∈ {0} ∪ P a surjective map κ (r) : G (r) → S(W ) whose fibres are called the strata of G (r) ; each stratum is a union of conjugacy classes of the same dimension, independent of r and, according to [Car20], is locally closed in G (r) . If g ∈ G (r) is unipotent, then κ (r) (g) is the same as the image of the conjugacy class of g under ι (r) .…”

Distinguished strata in a reductive group

“…In [L15] we have defined a partition of G into finitely many strata; each stratum is locally closed [Ca20] and a union of G-conjugacy classes of fixed dimension. The set Str(G) of strata of G can be viewed as an enlargement of U(G) (a unipotent class of G is contained in exactly one stratum).…”