On Sheets of Conjugacy Classes in Good Characteristic

Abstract

We show that the sheets for a connected reductive algebraic group G over an algebraically closed field in good characteristic acting on itself by conjugation are in bijection with G-conjugacy classes of tripleswhere M is the connected centralizer of a semisimple element in G, Z(M ) • t is a suitable coset in Z(M )/Z(M ) • and O is a rigid unipotent conjugacy class in M . Any semisimple element is contained in a unique sheet S and S corresponds to a triple with O = {1}. The sheets in G containing a unipotent conjugacy class are precisely those corresponding to triples for which M is a Levi subgroup of a parabolic subgroup of G and such a class is unique.

Lusztig’s partition and sheets (with an Appendix by M. Bulois)

“…In other words, the extension φ G of Springer's correspondence is constant along sheets. This is a direct consequence of the results in [9] together with compatibility of induction of unipotent conjugacy classes with truncated induction [30]. The image of φ G contains more irreducible representations of W than those obtained by the Springer correspondence for a trivial local system: this shows once more that, as opposed to the Lie algebra case, where every sheet contains a unique nilpotent orbit [3], not every sheet of conjugacy classes contains a unipotent one.…”

“…This suggests that the parametrization in [9] may be improved, and this is in fact the case. We show that the second term in the triple parametrizing sheets may be dropped when G is simple and of adjoint type.…”

“…Sheets for the action of a complex connected reductive algebraic group on its Lie algebra are very well understood [3,2]. Along similar lines, a parametrization and a description of sheets of conjugacy classes in a connected reductive algebraic group G over an algebraically closed field of zero or good characteristic has been given in [9].…”

“…Sections 3 and 4 are devoted to the refinement of some results in [9]. There, sheets were parametrized by G-conjugacy classes of triples (M, Z(M)…”

Quotients for Sheets of Conjugacy Classes

“…More precisely, a Jordan class J = J(su) is dense in a sheet if and only if it is not contained in (J ′ ) reg for any Jordan class J ′ different from J. We recall from [8,Proposition 4.8] that…”