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Cohomological boundedness of twisted coherent Springer sheaves

Oron Y. Propp

Abstract

We prove that the coherent Springer sheaf and its parabolic analogues are concentrated in cohomological degree $0$, as predicted by Ben-Zvi-Chen-Helm-Nadler, Zhu, Emerton-Gee-Hellmann, Hansen, and others. More generally, we show that the universal trace functor for a mixed partial affine Hecke category is right t-exact with respect to the exotic t-structure given by Bezrukavnikov-Mirković's noncommutative Springer resolution, and left t-exact with respect to the monoidally dual t-structure. To this end, we construct an explicit complex computing the universal trace functor for certain monoidal categories over quotient stacks.

Cohomological boundedness of twisted coherent Springer sheaves

Abstract

We prove that the coherent Springer sheaf and its parabolic analogues are concentrated in cohomological degree , as predicted by Ben-Zvi-Chen-Helm-Nadler, Zhu, Emerton-Gee-Hellmann, Hansen, and others. More generally, we show that the universal trace functor for a mixed partial affine Hecke category is right t-exact with respect to the exotic t-structure given by Bezrukavnikov-Mirković's noncommutative Springer resolution, and left t-exact with respect to the monoidally dual t-structure. To this end, we construct an explicit complex computing the universal trace functor for certain monoidal categories over quotient stacks.
Paper Structure (34 sections, 54 theorems, 323 equations)

This paper contains 34 sections, 54 theorems, 323 equations.

Table of Contents

  1. Introduction
  2. Origins of the problem
  3. Statement of results
  4. Strategy of the proof
  5. Overview
  6. Assumptions and notation
  7. Acknowledgments
  8. Partial affine Hecke categories
  9. Springer theory
  10. Convolution actions
  11. Categorical traces
  12. The noncommutative partial Springer resolution
  13. Recollections
  14. Generalization to connected reductive groups
  15. A covering group of the reductive centralizer
  16. ...and 19 more sections

Key Result

Theorem 1.2.2

Theorems & Definitions (118)

  • Theorem 1.2.2
  • Corollary 1.2.4
  • Proposition 1.3.2
  • Proposition 1.3.5
  • Corollary 2.3.2
  • proof
  • Lemma 2.3.4
  • proof
  • Corollary 2.3.5
  • Proposition 3.1.4
  • ...and 108 more