On intersections of conjugacy classes and Bruhat cells

الأصل · EN

For a connected complex semi-simple Lie group G 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 non-empty, 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 ∈ W associated to C. We study properties of the elements. For G = SL(n+1,), we describe explicitly for every conjugacy class C, and for the case when w ∈ W is an involution, we also give an explicit answer to when C ∩ (BwB) is non-empty.