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Exterior powers of a parabolic Springer sheaf on a Lie algebra

Roman Bezrukavnikov, Kostiantyn Tolmachov

Abstract

We compute the exterior powers, with respect to the additive convolution on the general linear Lie algebra, of a parabolic Springer sheaf corresponding to a maximal parabolic subgroup of type (1, n -- 1). They turn out to be isomorphic to the semisimple perverse sheaves attached by the Springer correspondence to the exterior powers of the permutation representation of the symmetric group.

Exterior powers of a parabolic Springer sheaf on a Lie algebra

Abstract

We compute the exterior powers, with respect to the additive convolution on the general linear Lie algebra, of a parabolic Springer sheaf corresponding to a maximal parabolic subgroup of type (1, n -- 1). They turn out to be isomorphic to the semisimple perverse sheaves attached by the Springer correspondence to the exterior powers of the permutation representation of the symmetric group.
Paper Structure (6 sections, 3 theorems, 14 equations)

This paper contains 6 sections, 3 theorems, 14 equations.

Table of Contents

  1. Basic notations.
  2. Introduction.
  3. Springer theory.
  4. Fourier-Deligne transform.
  5. Étale neighbourhoods in $\mathfrak{g}$.
  6. Rank estimate.

Key Result

Theorem 1

Exterior powers of $\mathfrak{spr}(V)$ with respect to the symmetric monoidal structure $\star$ on $D^b(\mathfrak{gl}_n/G)$ satisfy In particular, $\wedge^{k}_+\mathfrak{spr}(V) = 0$ for $k > n$.

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Lemma 3: Proposition 2.11 in gunninghamGeneralizedSpringerTheory2018