Exterior Powers of the Reflection Representation in the Cohomology of Springer Fibres
نویسنده
چکیده
Let H∗(Be) be the cohomology of the Springer fibre for the nilpotent element e in a simple Lie algebra g, on which the Weyl group W acts by the Springer representation. Let ΛV denote the ith exterior power of the reflection representation of W . We determine the degrees in which ΛV occurs in the graded representation H∗(Be), under the assumption that e is regular in a Levi subalgebra and satisfies a certain extra condition which holds automatically if g is of type A, B, or C. This partially verifies a conjecture of Lehrer–Shoji, and extends the results of Solomon in the e = 0 case and Lehrer–Shoji in the i = 1 case. The proof proceeds by showing that (H∗(Be)⊗Λ ∗ V ) is a free exterior algebra on its subspace (H∗(Be)⊗ V ) W .
منابع مشابه
Exterior Powers of the Reflection Representation in Springer Theory
Let H∗(Be) be the total Springer representation of W for the nilpotent element e in a simple Lie algebra g. Let ∧V denote the exterior powers of the reflection representation V of W . The focus of this paper is on the algebra of W -invariants in H∗(Be)⊗ ∧ ∗V and we show that it is an exterior algebra on the subspace (H∗(Be) ⊗ V ) in some new cases. This was known previously for e = 0 by a resul...
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