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Exterior powers of the reflection representation in Springer theory

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An Erratum to this article was published on 03 August 2011

Abstract

Let \( {H^*}\left( {{\mathcal{B}_e}} \right) \) be the total Springer representation of W for the nilpotent element e in a simple Lie algebra \( \mathfrak{g} \). Let Λi V denote the ith exterior power of the reflection representation V of W. The focus of this paper is on the algebra of W-invariants in

$$ {H^*}\left( {{\mathcal{B}_e}} \right) \otimes {\Lambda^*}V $$

and we show that it is an exterior algebra on the subspace \( {\left( {{H^*}\left( {{\mathcal{B}_e}} \right) \otimes V} \right)^W} \) in some cases that were not previously known. This result was established for e = 0 by Solomon [28] and was proved by Henderson [17] in types A, B, C when e is regular in a Levi subalgebra.

The above statement about the W-invariants implies a conjecture of Lehrer–Shoji [19] about the occurrences of Λi V in \( {H^*}\left( {{\mathcal{B}_e}} \right) \), which was originally stated when e is regular in a Levi subalgebra. In this paper we prove the Lehrer–Shoji conjecture in all types and its natural extension to any nilpotent e, not only those that are regular in a Levi subalgebra.

In the last part of the paper we make a connection to rational Cherednik algebras and this leads to an explanation for the appearance in Springer theory of the Orlik–Solomon exponents coming from hyperplane arrangements. This connection was established in the classical groups in [19], [30] after being observed empirically by Orlik, Solomon, and Spaltenstein in the exceptional groups.

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  1. Department of Mathematics and Statistics, University of Massachusetts—Amherst, Amherst, MA, 01003, USA

    Eric Sommers

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  1. Eric Sommers
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Correspondence to Eric Sommers.

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To Tonny Springer, on his 85th birthday

Supported by NSF grant DMS-0201826.

An erratum to this article can be found at http://dx.doi.org/10.1007/s00031-011-9160-7

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Sommers, E. Exterior powers of the reflection representation in Springer theory. Transformation Groups 16, 889–911 (2011). https://doi.org/10.1007/s00031-011-9148-3

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