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Homology of generalized Steinberg varieties and Weyl group invariants

Authors: J. Matthew Douglass and Gerhard Röhrle
Journal: Trans. Amer. Math. Soc. 360 (2008), 5959-5998
MSC (2000): Primary 22E46; Secondary 20G99
Published electronically: July 10, 2008
Previous version: Originally posted June 26, 2008
MathSciNet review: 2425698
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Abstract: 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 $ \operatorname{Lie}(G)$ for the Steinberg variety $ Z$ of triples.

Using a general specialization argument we show that for a parabolic subgroup $ W_P \times W_Q$ of $ W \times W$ the space of $ W_P \times W_Q$-invariants and the space of $ W_P \times 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 Röhrle (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 \mathcal{N}$. Suppose $ W_P \times W_Q$ is a parabolic subgroup of $ W \times W$. We show that the space of $ W_P \times W_Q$-invariants of $ H_{4n}(Z)$ is $ e_Q\mathbb{Q} We_P$, 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 \times W_Q$-anti-invariants of $ H_{4n}(Z)$ is $ \epsilon_Q\mathbb{Q} W\epsilon_P$, where $ \epsilon_P$ is the idempotent in the group algebra of $ W_P$ affording the sign representation of $ W_P$ and $ \epsilon_Q$ is defined similarly.

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Additional Information

J. Matthew Douglass
Affiliation: Department of Mathematics, University of North Texas, Denton, Texas 76203

Gerhard Röhrle
Affiliation: Fakultät für Mathematik, Ruhr-Universität Bochum, D-44780 Bochum, Germany

Received by editor(s): October 25, 2006
Published electronically: July 10, 2008
Additional Notes: Part of the research for this paper was carried out while both authors were staying at the Mathematisches Forschungsinstitut Oberwolfach supported by the “Research in Pairs” program.
Part of this paper was written during visits of the first author to the University of Birmingham, where the second author was supported by an EPSRC grant.
Article copyright: © Copyright 2008 American Mathematical Society

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