Homology of Generalized Steinberg Varieties and Weyl Group Invariants

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 antiinvariants 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 ×WQ of W × W the space of WP × WQ-invariants and the space of WP × WQ-anti-invariants of H4n(Z) are isomorphic to the top Borel-Moore homology groups of certain generalized Steinberg varieties introduced in [5]. The rational group algebra of the Weyl groupW ofG is isomorphic to the top Borel-Moore homology H4n(Z) of Z, where 2n = dimN . Suppose WP ×WQ is a parabolic subgroup of W × W . We show that the space of WP × WQ-invariants of H4n(Z) is eP QWeQ, where eP is the idempotent in group algebra of WP affording the trivial representation of WP and eQ is defined similarly. We also show that the space of WP × WQ-anti-invariants of H4n(Z) is ǫP QWǫQ, where ǫP is the idempotent in group algebra of WP affording the sign representation of WP and ǫQ is defined similarly.