Homology of generalized Steinberg varieties and Weyl group invariants

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Let G be a complex, connected, reductive algebraic group. In this paper we show analogues of the computations by Borho and MacPherson of the invariants and anti-invariants of the cohomology of the Springer fibres of the cone of nilpotent elements, N, of Lie(G) for the Steinberg variety Z of triples. Using a general specialization argument we show that for a parabolic subgroup Wp x W Q of W x W the space of Wp x W Q -invariants and the space of Wp x W Q -anti-invariants of H 4n (Z) are isomorphic to the top Borel-Moore homology groups of certain generalized Steinberg varieties introduced by Douglass and Rohrle (2004). The rational group algebra of the Weyl group W of G is isomorphic to the opposite of the top Borel-Moore homology H 4n (Z) of Z, where 2n = dim N. Suppose Wp x W Q is a parabolic subgroup of W x W. We show that the space of Wp x W Q -invariants of H 4n (Z) is e Q QWe p , where ep is the idempotent in the group algebra of Wp affording the trivial representation of Wp and e Q is defined similarly. We also show that the space of Wp x W Q -anti-invariants of H 4n (Z) is ∈ Q QW∈ p , where ep is the idempotent in the group algebra of Wp affording the sign representation of Wp and ∈ Q is defined similarly.