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Microlocal characterization of Lusztig sheaves for affine quivers and $g$-loops quivers

Lucien Hennecart

TL;DR

The paper establishes a precise microlocal criterion for identifying Lusztig perverse sheaves across quivers of finite type, affine, cyclic, and those with loops. By linking the singular support of a simple perverse sheaf to appropriate nilpotent varieties via the moment map, it proves a converse to Lusztig’s nilpotency-based characterization in the affine case and extends these ideas to cyclic and loop quivers, including the $g$-loop setting with $g\ge 2$. The work introduces extended categories (e.g., extended Hall category) and uses AR stratifications, Fourier-Sato transforms, and smallness arguments to achieve a microlocal classification, culminating in explicit descriptions of simple objects via local systems on smooth strata and their Fourier transforms. These results deepen the geometric understanding of canonical bases in quantum groups and provide a robust framework for quivers with loops, with potential ties to Bozec–Schiffmann–Vasserot nilpotent varieties and related representation-theoretic structures. Overall, the paper advances a unified microlocal perspective on Lusztig’s perverse-sheaf theory across broad classes of quivers, clarifying when nilpotent-support conditions characterize Lusztig objects and enabling explicit descriptions in new settings.

Abstract

We prove that for extended Dynkin quivers, simple perverse sheaves in Lusztig category are characterized by the nilpotency of their singular support. This proves a conjecture of Lusztig in the case of affine quivers. For cyclic quivers, we prove a similar result for a larger nilpotent variety and a larger class of perverse sheaves. We formulate conjectures for similar results for quivers with loops, for which we have to use the appropriate notion of nilpotent variety, due to Bozec, Schiffmann and Vasserot. We prove our conjecture for $g$-loops quivers ($g\geq 2$).

Microlocal characterization of Lusztig sheaves for affine quivers and $g$-loops quivers

TL;DR

The paper establishes a precise microlocal criterion for identifying Lusztig perverse sheaves across quivers of finite type, affine, cyclic, and those with loops. By linking the singular support of a simple perverse sheaf to appropriate nilpotent varieties via the moment map, it proves a converse to Lusztig’s nilpotency-based characterization in the affine case and extends these ideas to cyclic and loop quivers, including the -loop setting with . The work introduces extended categories (e.g., extended Hall category) and uses AR stratifications, Fourier-Sato transforms, and smallness arguments to achieve a microlocal classification, culminating in explicit descriptions of simple objects via local systems on smooth strata and their Fourier transforms. These results deepen the geometric understanding of canonical bases in quantum groups and provide a robust framework for quivers with loops, with potential ties to Bozec–Schiffmann–Vasserot nilpotent varieties and related representation-theoretic structures. Overall, the paper advances a unified microlocal perspective on Lusztig’s perverse-sheaf theory across broad classes of quivers, clarifying when nilpotent-support conditions characterize Lusztig objects and enabling explicit descriptions in new settings.

Abstract

We prove that for extended Dynkin quivers, simple perverse sheaves in Lusztig category are characterized by the nilpotency of their singular support. This proves a conjecture of Lusztig in the case of affine quivers. For cyclic quivers, we prove a similar result for a larger nilpotent variety and a larger class of perverse sheaves. We formulate conjectures for similar results for quivers with loops, for which we have to use the appropriate notion of nilpotent variety, due to Bozec, Schiffmann and Vasserot. We prove our conjecture for -loops quivers ().

Paper Structure

This paper contains 89 sections, 77 theorems, 208 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The main results
  3. Analogy with character sheaves on Lie groups and Lie algebras
  4. Steps of the proof for affine quivers
  5. Steps of the proof for $g$-loops quivers
  6. Contents of the paper
  7. Remark
  8. Notations
  9. Terminology
  10. Some representation theory of quivers and stratification of the representation spaces
  11. Representation theory of finite type quivers
  12. Nilpotent representations of cyclic quivers
  13. Some Auslander-Reiten theory
  14. Quiver representation varieties
  15. Stratification of the representation spaces of acyclic quivers
  16. ...and 74 more sections

Key Result

Theorem 1.1

Let $Q$ be a finite type or affine quiverIn this paper, we include under the terminology affine quiver extended Dynkin quivers, Jordan and cyclic quivers.. Let $\mathscr{F}\in\mathop{\mathrm{\textup{Perv}}}\nolimits_{G_{\mathop{\mathrm{\mathbf{d}}}\nolimits}}(E_{\mathop{\mathrm{\mathbf{d}}}\nolimits

Figures (1)

  • Figure 1: Non-homogeneous tubes of affine quivers and their period MR3202707

Theorems & Definitions (141)

  • Theorem 1.1
  • Conjecture 1.2: Lusztig
  • Theorem 1.3
  • Theorem 2.1: Gabriel, MR332887
  • Theorem 2.2
  • Proposition 2.3
  • Corollary 2.4
  • proof
  • Theorem 2.5: Ringel, MR774589
  • Remark 2.6
  • ...and 131 more