Homology of generalized Steinberg varieties and Weyl group invariants
| dc.creator | Douglass, J. M. | |
| dc.creator | Roehrle, G. | |
| dc.date | 2005-05-26 | |
| dc.date | 2007-06-12 | |
| dc.date.accessioned | 2026-07-07T08:05:06Z | |
| dc.date.available | 2026-07-07T08:05:06Z | |
| dc.description | 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 in [5]. 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 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 $ε_Q{\mathbb Q} Wε_P$, where $ε_P$ is the idempotent in group algebra of $W_P$ affording the sign representation of $W_P$ and $ε_Q$ is defined similarly. | |
| dc.description | 38 pages, to appear in Trans. Amer. Math. Soc | |
| dc.identifier | https://arxiv.org/abs/math/0505567 | |
| dc.identifier | http://arxiv.org/abs/math/0505567 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/130173 | |
| dc.subject | Representation Theory | |
| dc.subject | K-Theory and Homology | |
| dc.subject | 22E46 (Primary); 20G99 (Secondary) | |
| dc.title | Homology of generalized Steinberg varieties and Weyl group invariants | |
| dc.type | text |