Spherical conjugacy classes and involutions in the Weyl group

Abstract

Let G be a simple algebraic group over an algebraically closed field of characteristic zero or positive odd, good characteristic. Let B be a Borel subgroup of G. We show that the spherical conjugacy classes of G intersect only the double cosets of B in G corresponding to involutions in the Weyl group of G. This result is used in order to prove that for a spherical conjugacy class O with dense B-orbit v(0). BwB there holds l(w) + rk(1-w) = dim O extending to the case of groups over fields of odd, good characteristic a characterization of spherical conjugacy classes obtained by Cantarini, Costantini and the author

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

“…More precisely, such strata are in bijection with conjugacy classes in the Weyl group W containing a maximum w m , and a spherical conjugacy class γ lies in such a stratum if and only if Bw m B ∩ γ is dense in γ. This result is a consequence of the combinatorial description of spherical conjugacy classes [5,6,8,24] and the alternative description of strata in terms of the Bruhat decomposition of G in [29]. Through this alternative description it is proved in Theorem 5.8 that spherical strata correspond to unions of classes of involutions in W having w m as a maximum.…”

“…If γ is spherical and char(k) = 2 then γ ⊂ w 2 =1 BwB by [6,Theorem 2.7]. For char(k) = 2, Φ is of type A and we invoke Lemma 5.7(1).…”

“…1. For char(k) = 2 this is [6,Theorem 2.7]. If char(k) = 2, then Φ is of type A and spherical classes are described in Lemma 5.1.…”

On the coordinate ring of spherical conjugacy classes

“…We have w = w 0 w J , the subset J is invariant under ϑ, where ϑ is the symmetry of Π induced by −w 0 , and w 0 and w J act in the same way on Φ J (see [10] the discussion at the end of section 3, Corollary 4.2, Remark 4.3 and Proposition 4.15).…”

“…Moreover w is always an involution (see [9], Remark 4, [10], Theorem 2.7). From this result we conjectured that, for a spherical O, the decomposition of the ring C[O] of regular functions on O (to which we refer as to the coordinate ring of O) as a G-module should be strictly related to w(O).…”