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Linear structure on a finite Hecke category in type A

Kostiantyn Tolmachov

Abstract

For the group GL(n), we construct an action of the equivariant derived category of coherent sheaves on the Grothendieck-Springer resolution on a certain subcategory of a finite monodromic Hecke category. We use this to construct a partial categorification of the projection from the extened affine to the finite Hecke algebra of GL(n). As a crucial intermediate step, we compute the exterior powers, with respect to the perversely truncated multiplicative convolution, of a parabolic Springer sheaf corresponding to a maximal parabolic subgroup fixing a line in the defining n-dimensional representation of GL(n).

Linear structure on a finite Hecke category in type A

Abstract

For the group GL(n), we construct an action of the equivariant derived category of coherent sheaves on the Grothendieck-Springer resolution on a certain subcategory of a finite monodromic Hecke category. We use this to construct a partial categorification of the projection from the extened affine to the finite Hecke algebra of GL(n). As a crucial intermediate step, we compute the exterior powers, with respect to the perversely truncated multiplicative convolution, of a parabolic Springer sheaf corresponding to a maximal parabolic subgroup fixing a line in the defining n-dimensional representation of GL(n).
Paper Structure (29 sections, 30 theorems, 123 equations)

This paper contains 29 sections, 30 theorems, 123 equations.

Table of Contents

  1. Notations.
  2. Introduction.
  3. Linear structures.
  4. Tannakian formalism for the Steinberg variety.
  5. Homomorphism between affine and finite Hecke algebras in type A.
  6. Action of $\operatorname{Rep}(\mathbf{T})$.
  7. Action of $\operatorname{Rep}(\mathbf{G})$.
  8. Jucys--Murphy filtrations and monodromy.
  9. Related work.
  10. Acknowledgments.
  11. Finite monodromic Hecke category.
  12. Free-monodromic objects.
  13. Jucys--Murphy sheaves and the $\operatorname{Rep}(\mathbf{T})$-action.
  14. Construction of the $\operatorname{Rep}(\mathbf{G})$-action.
  15. Mellin transform.
  16. ...and 14 more sections

Key Result

Theorem 1.1.1

There is an exact monoidal functor where the monoidal structure on the perfect equivariant category $\mathrm{perf}_G(\mathbf{St}_\mathfrak{g})$ is given by convolution. The functor takes values in $\mathcal{H}_n^{\mathrm{perf}}$ and is compatible with the linear structures over $\tilde{\mathfrak{g}}/\mathbf{G}$.

Theorems & Definitions (59)

  • Theorem 1.1.1
  • Remark 1.2.1
  • Remark 1.3.1
  • Theorem 1.3.2
  • Corollary 1.3.3
  • Remark 1.3.4
  • Remark 1.3.5
  • Proposition 2.1.1: bezrukavnikovKoszulDualityKacMoody2013, zbMATH07610537
  • proof
  • Remark 2.1.2
  • ...and 49 more