2005/05/26 by J. Matthew Douglass, J. M. Douglass, Gerhard Roehrle +3
Mathematics · #20G99 (Secondary) #22E46 (Primary) #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Representation Theory (math.RT) #math.KT #math.RT #msc:20G99 #msc:22E46
paper · pdf · doi:10.48550/arxiv.math/0505567
38 pages, to appear in Trans. Amer. Math. Soc
openalex publication_date 2005/05/26 · arxiv created 2007/06/12 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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, \mathcal 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 group W of G is isomorphic to the opposite of the top Borel-Moore homology H4n(Z) of Z, where 2n = dim \mathcal N. Suppose WP × WQ is a parabolic subgroup of W × W. We show that the space of WP × WQ-invariants of H4n(Z) is eQ\mathbb Q WeP, 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 εQ\mathbb Q WεP, where εP is the idempotent in group algebra of WP affording the sign representation of WP and εQ is defined similarly.