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Homology of generalized Steinberg varieties and Weyl group invariants |
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| Authors: | J Matthew Douglass Gerhard Rö hrle |
| Institution: | Department of Mathematics, University of North Texas, Denton, Texas 76203 ; Fakultät für Mathematik, Ruhr-Universität Bochum, D-44780 Bochum, Germany |
| Abstract: | Let  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,  , of  for the Steinberg variety  of triples. Using a general specialization argument we show that for a parabolic subgroup The rational group algebra of the Weyl group |
| Keywords: | |
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of 
the space of 
-invariants and the space of 
-anti-invariants of 
are isomorphic to the top Borel-Moore homology groups of certain generalized Steinberg varieties introduced by Douglass and Röhrle (2004). 
of 
is isomorphic to the opposite of the top Borel-Moore homology 
of 
, where 
. Suppose 
is a parabolic subgroup of 
. We show that the space of 
-invariants of 
is 
, where 
is the idempotent in the group algebra of 
affording the trivial representation of 
and 
is defined similarly. We also show that the space of 
-anti-invariants of 
is 
, where 
is the idempotent in the group algebra of 
affording the sign representation of 
and 
is defined similarly.