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

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 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.

Keywords

conjugacy class, sheet, Lusztig’s partition, Bruhat decomposition, spherical conjugacy classes, involutions in the Weyl group

2010 Mathematics Subject Classification

Primary 20E45, 20G15. Secondary 20F55.

Full Text (PDF format)

Accepted 1 September 2014

Published 20 May 2015