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Braden-MacPherson sheaves on alcoves

Noriyuki Abe

TL;DR

The paper studies Braden–MacPherson sheaves on the alcove moment graph and constructs a monoidal action of Soergel bimodules on the BM-category, providing a categorification of the periodic Hecke module. It shows that BM-sheaves are preserved under the bimodule action via global sections and localization, and establishes a Hecke-algebra–module link on the level of split Grothendieck groups. A key contribution is the stability result: for indecomposable BM-sheaves $\mathcal{F},\mathcal{G}$, the Hom-space $\mathrm{Hom}(\mathcal{F}|_{O},\mathcal{G}|_{O})$ stabilizes for sufficiently large intervals $O$ in the alcove poset, yielding finite-dimensional Hom-spaces. The work leverages translation functors, KL combinatorics, and Lanini’s corollaries to connect the alcove BM-structure with parabolic and dominant-regime behavior, with implications for representation theory in positive characteristic and for the categorification of periodic modules.

Abstract

We study Braden-MacPherson sheaves on the moment graph associated to the set of of alcoves. We define an action of Soergel bimodules on the category of Braden-MacPherson sheaves. We also prove a certain stability of morphisms between Braden-MacPherson sheaves.

Braden-MacPherson sheaves on alcoves

TL;DR

The paper studies Braden–MacPherson sheaves on the alcove moment graph and constructs a monoidal action of Soergel bimodules on the BM-category, providing a categorification of the periodic Hecke module. It shows that BM-sheaves are preserved under the bimodule action via global sections and localization, and establishes a Hecke-algebra–module link on the level of split Grothendieck groups. A key contribution is the stability result: for indecomposable BM-sheaves $\mathcal{F},\mathcal{G}$, the Hom-space $\mathrm{Hom}(\mathcal{F}|_{O},\mathcal{G}|_{O})$ stabilizes for sufficiently large intervals $O$ in the alcove poset, yielding finite-dimensional Hom-spaces. The work leverages translation functors, KL combinatorics, and Lanini’s corollaries to connect the alcove BM-structure with parabolic and dominant-regime behavior, with implications for representation theory in positive characteristic and for the categorification of periodic modules.

Abstract

We study Braden-MacPherson sheaves on the moment graph associated to the set of of alcoves. We define an action of Soergel bimodules on the category of Braden-MacPherson sheaves. We also prove a certain stability of morphisms between Braden-MacPherson sheaves.

Paper Structure

This paper contains 17 sections, 61 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Moment graphs of alcoves
  3. Notation
  4. Moment graphs and sheaves
  5. Root datum
  6. Moment graphs
  7. Global sections and localization
  8. Translation
  9. The action of Soergel bimodules
  10. Definition
  11. Sheaves on $W_{\mathrm{aff}}$ and $W_{\mathrm{aff},\lambda}\backslash W_{\mathrm{aff}}$
  12. A result of Lanini
  13. Some calculation of graded ranks
  14. Stability of homomorphisms
  15. Some Kazhdan-Lusztig combinatorics
  16. ...and 2 more sections

Key Result

Theorem 1.1

If $\mathcal{F}$ is a Braden-MacPherson sheaf on $\mathcal{A}$, then $\mathcal{F}\star B$ is also a Braden-MacPherson sheaf. The operation $(\mathcal{F},B)\mapsto \mathcal{F}\star B$ defines an action of $\mathcal{S}$ on $\mathcal{BM}(\mathcal{A})$.

Theorems & Definitions (117)

  • Theorem 1.1: Theorem \ref{['thm:action of Soergel bimodule preserves BM sheaves']}
  • Theorem 1.2: Theorem \ref{['thm:finiteness']}
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • proof
  • ...and 107 more