Lusztig's partition and sheets (with an appendix by M. Bulois)

TL;DR

This paper explores the structure of Lusztig's partition of algebraic groups, showing that its parts are unions of conjugacy class sheets, and refines the parametrization of these sheets with combinatorial descriptions and new insights into their closures.

Contribution

It demonstrates that Lusztig's strata are unions of sheets, refines the parametrization for simple adjoint groups, and provides a combinatorial description linking spherical classes and Weyl group involutions.

Findings

Lusztig's strata are unions of sheets of conjugacy classes.

A bijection between spherical conjugacy classes and Weyl group involutions.

Closure of a stratum may not be a union of strata.

Abstract

We show that, for a connected reductive algebraic group G over an algebraically closed field of zero or good characteristic, the parts, called strata, in the partition of G recently introduced by Lusztig are unions of sheets of conjugacy classes. For G simple and adjoint we refine the parametrization of such sheets obtained in previous work with F. Esposito. We give a simple combinatorial description of strata containing spherical conjugacy classes, showing that Lusztig's correspondence induces a bijection between unions of spherical conjugacy classes and unions of classes of involutions in the Weyl group. Using ideas from the Appendix by M. Bulois, we show that the closure of a stratum is not necessarily a union of strata.