On spherical twisted conjugacy classes

Abstract

Let G be a simple algebraic group over an algebraically closed eld of good odd characteristic, and let be an automorphism of G arising from an involution of its Dynkin diagram. We show that the spherical -twisted conjugacy classes are precisely those intersecting only Bruhat cells corresponding to twisted involutions in the Weyl group. We show how the analogue of this statement fails in the triality case. As a byproduct, we obtain a dimension formula for spherical twisted conjugacy classes that was originally obtained by J.-H. Lu in characteristic zero

On theT-leaves and the ranks of a Poisson structure on twisted conjugacy classes

“…Remark 4.6. From Proposition 4.5 and the linear algebra fact that if t → A(t) ∈ End(V ) is a continuous map from an open neighborhood of 0 ∈ C to End(V ) then rank(A(t)) ≥ rank(A(0)) for t closed to 0, one concludes that Generalizing a result in [5] of N. Cantarini, G. Carnovale, and M. Costantini for θ = Id G , the following characterization of spherical θ-twisted conjugacy classes is proved in [22] for the case of characteristic zero and in [7] for good odd characteristics. Theorem 5.2.…”

“…Generalizing a result in [5] of N. Cantarini, G. Carnovale, and M. Costantini for θ = Id G , the following characterization of spherical θ-twisted conjugacy classes is proved in [22] for the case of characteristic zero and in [7] for good odd characteristics. Corollary 5.3.…”

“…where m C is the unique element in W such that C ∩ (Bm C B) is dense in C. The elements m C play an important role in the study of spherical conjugacy classes and their applications to the theory of representations of quantum groups (see works of N. Cantarini, G. Carnovale, and M. Costantini in [5,6,7,11]). For G simple and θ = Id G , the list of all m C 's, as C runs over all conjugacy classes in G, is given in [9, §3] using results from [5].…”

“…[5,7,22] For θ ∈ Aut(G) such that θ(B) = B and θ(T ) = T , a θ-twisted conjugacy class C in G is spherical if and only if dim C = l(m C ) + rank(1 − m C θ).…”

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

“…Proof If $${\rm char}(k)\ne 2$$, this is [6, Theorem 6] (note that in the paper the base field is of zero or good odd characteristic, but the arguments used in Section 2 only use $${\rm char}(k)\ne 2$$). If $${\rm char}(k)=2$$, this is Theorem 3.6.$$\Box$$…”