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 result of Solomon [25] and was recently proved by Henderson [12] in typesA,B, C when e is regular in a Levi subalgebra. The above statement about the W -invariants implies a conjecture of Lehrer-Shoji [15] about the occurrences of ∧V inH∗(Be) (which was stated for 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 e (not only those with the regular condition). In the last part of the paper we make a connection to rational Cherednik algebras which implies a result about the appearance of the Orlik-Solomon exponents in Springer theory, a connection that was established in the classical groups in [15], [27] after being observed empirically by Orlik, Solomon, and Spaltenstein in the exceptional groups.