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Higher nearby cycles and central sheaves on affine flag varieties

Pramod N. Achar, Simon Riche

Abstract

In this paper we generalize and study a notion of (unipotent) nearby cycles over a higher dimensional base based on Beĭlinson's description of unipotent nearby cycles, following an idea of Gaitsgory. This generalization, in the setting of affine Grassmannians, is required in recent work of Bezrukavnikov-Braverman-Finkelberg-Kazhdan.

Higher nearby cycles and central sheaves on affine flag varieties

Abstract

In this paper we generalize and study a notion of (unipotent) nearby cycles over a higher dimensional base based on Beĭlinson's description of unipotent nearby cycles, following an idea of Gaitsgory. This generalization, in the setting of affine Grassmannians, is required in recent work of Bezrukavnikov-Braverman-Finkelberg-Kazhdan.
Paper Structure (32 sections, 22 theorems, 173 equations, 1 figure)

This paper contains 32 sections, 22 theorems, 173 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Beilinson's unipotent nearby cycles functor
  3. Higher dimensional version
  4. Composition of higher nearby cycles functors
  5. Compatibilities
  6. Further comments
  7. Acknowledgements
  8. Preliminaries
  9. Pointed maps and associated linear morphisms
  10. Scheme morphisms associated with pointed maps
  11. Jordan block local systems
  12. Z-closed polynomials and binomial identities
  13. Functions on the Tate module
  14. Definition of the local systems
  15. Higher nearby cycles
  16. ...and 17 more sections

Key Result

Lemma 2.2

Let $g \in \mathbb{Z}_\ell(1)$ be an element that generates $\mathbb{Z}_\ell(1)$ as a $\mathbb{Z}_\ell$-module, and for $a \geq 0$ let

Figures (1)

  • Figure 1: Diagram for the proof of Lemma \ref{['lem:conv-calc2']}

Theorems & Definitions (51)

  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Definition 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • ...and 41 more