On Lusztig's map for spherical unipotent conjugacy classes

Abstract

We provide an alternative description of the restriction to spherical unipotent conjugacy classes, of Lusztig's map Ψ from the set of unipotent conjugacy classes in a connected reductive algebraic group to the set of conjugacy classes of its Weyl group. For irreducible root systems, we analyze the image of this restricted map and we prove that a conjugacy class in a finite Weyl group has a unique maximal length element if and only if it has a maximum.

A classification of spherical conjugacy classes

“…If p is zero, or good and odd this is [6,Theorem 2.7]. The same proof holds as long as p = 2 (see also [9,Theorem 2.1]). Remark 3.…”

“…We have established the information presented in Table 5 and 6, where Table 7, 8, the unipotent classes are represented in [14,Table 8,9].…”

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.…”

On Lusztig's map for spherical unipotent conjugacy classes in disconnected groups

“…We analyze the image of the restriction of $$\nu$$ to the set $$\underline{G}_{\theta ,sph}$$ of spherical unipotent conjugacy classes in $$G\theta$$. The case where $$\theta =1$$ is dealt with in [7, Remark 2]: here we only consider the case $$\theta \not=1$$. Proposition The restriction of $$\nu$$ to $$\underline{G}_{\theta ,sph}$$ is injective.…”