On intersections of conjugacy classes and bruhat cells
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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 ∈ 2 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 ≌ Sn+1 is an involution, we give an explicit answer to when C ∩ (BwB) is nonempty.
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