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 W P × W Q of W × W the space of W P × W Q -invariants and the space of W P ×W Q -anti-invariants of H 4n (Z) are

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... hic to the top Borel-Moore homology groups of certain generalized Steinberg varieties introduced by Douglass and Röhrle (2004) . The rational group algebra of the Weyl group W of G is isomorphic to the opposite of the top Borel-Moore homology where e P is the idempotent in the group algebra of W P affording the trivial representation of W P and e Q is defined similarly. We also show that the space of W P × W Q -anti-invariants of H 4n (Z) is Q QW P , where P is the idempotent in the group algebra of W P affording the sign representation of W P and Q is defined similarly.