Sheets of conjugacy classes in simple algebraic groups

Abstract

For a connected reductive algebraic group G defined over an algebraically closed field of characteristic p the sheets of conjugacy classes have been parametrized by G. Carnovale and F. Esposito when p is good for G. We show that the method is independent of characteristic and that a similar parametrization is possible for all p. MSC 2010. 14L10, 14L35, 14L40.

A parametrization of sheets of conjugacy classes in bad characteristic

“…It was observed in [14,Remark 3.3] that for G of type B 2 , the number of sheets is independent of the characteristic and it was suggested this to hold in general for G connected and simply connected. This fails in general because there exist sheets that are obtained from one another by multiplication by a central element, and such central element might no longer exist in bad characteristic: for example, in G = SL 2 (k), the sheets are: {id}, {−id}, and G re g for p ≠ 2, and {id} and G re g for p = 2.…”

“…The sheets in G are the locally closed sets of the form where J is maximal with respect to . Hence, the set of sheets in G is in bijection with the set consisting of maximal Jordan classes (see [4, Proposition 5.1], [14, Section 3]).…”

“…If as above, then [3, Lemma 2.1] and [4, Proposition 4.8] give where is Lusztig–Spaltenstein’s induced unipotent conjugacy class (see [11]). The maximal Jordan classes are precisely those for which the class of u is rigid in , i. e., it is not induced from any unipotent class in a proper Levi subgroup of a parabolic subgroup of (see [4, Proposition 5.3], [14, Lemma 2.4]).…”

“…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 .…”