On intersections of conjugacy classes and bruhat cells

Abstract

For a connected semisimple algebraic group G over an algebraically closed field k and a fixed pair (B, B − ) of opposite Borel subgroups of G, we determine when the intersection of a conjugacy class C in G and a double coset BwB − is nonempty, where w is in the Weyl group W of G. The question comes from Poisson geometry, and our answer is in terms of the Bruhat order on W and an involution mC ∈ W associated to C. We prove that the element mC is the unique maximal length element in its conjugacy class in W , and we classify all such elements in W . For G = SL(n + 1, k), we describe mC explicitly for every conjugacy class C, and when w ∈ W ∼ = S n+1 is an involution, we give an explicit answer to when C ∩ (BwB) is nonempty.

On Lusztig's map for spherical unipotent conjugacy classes

“…Proof. Statement (i) is Corollary 2.11 in [7], the proof of which is characteristic-free. Statement (ii) follows from the fact that any w is conjugate to w −1 [6,Theorem C].…”

“…We recall some basic facts. Theorem 2.3 [7]. Let γ be a conjugacy class in G and let w γ and C γ be as above.…”

“…In addition, spherical conjugacy classes in good, odd characteristic are also characterized as those classes intersecting only Bruhat double cosets corresponding to involutions [3,4]. Combining all these properties with the analysis of the elements w γ in [7] leads us to the proof of our main result:…”

“…We also give some results on the map ι : G → W defined by ι(γ) = W · w γ . This map can be defined on the set of all conjugacy classes in G. It was observed in [7,Remark 3] that the image of the set of all conjugacy classes and of the set of all spherical conjugacy classes through this map, in characteristic zero or good and odd characteristic, is the set W m of classes in W having a unique maximal length element. We analyze the image of the restriction of ι to the set G sph of spherical unipotent conjugacy classes.…”

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

“…We claim that for every class C with a maximum m C there always exists a spherical conjugacy class γ 0 such that m γ 0 = m C . If char(k) = 2 this is [10,Remark 3]. If char(k) = 2 then Φ is of type A.…”