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On singular supports of Lusztig's perverse sheaves

Jiepeng Fang

TL;DR

The paper proves a microlocal criterion for Lusztig's perverse sheaves on quiver representation varieties: a simple $G_\nu$-equivariant perverse sheaf has $SS(L)\subset \Lambda_\nu$ if and only if it is a Lusztig's perverse sheaf. The authors develop and leverage Lusztig's induction/restriction framework, a key inductive lemma, and a Fourier-Sato transform to establish the equivalence for all finite quivers without loops. This extends known results in finite-type and affine cases and strengthens the link between singular supports and categorification of the nilpotent quantum group part via $\Lambda_\nu$. The work provides a robust microlocal criterion with implications for the geometry of preprojective algebras and the structure of Lusztig's perverse sheaves in quiver theory.

Abstract

We prove a conjecture of Lusztig on a microlocal characterization of his perverse sheaves. For any finite quiver without loops, an equivariant simple perverse sheaf on the variety of quiver representations is a Lusztig's perverse sheaf if and only if its singular support is contained in Lusztig's Lagrangian variety, that is, the variety of nilpotent representations of the preprojective algebra of the quiver.

On singular supports of Lusztig's perverse sheaves

TL;DR

The paper proves a microlocal criterion for Lusztig's perverse sheaves on quiver representation varieties: a simple -equivariant perverse sheaf has if and only if it is a Lusztig's perverse sheaf. The authors develop and leverage Lusztig's induction/restriction framework, a key inductive lemma, and a Fourier-Sato transform to establish the equivalence for all finite quivers without loops. This extends known results in finite-type and affine cases and strengthens the link between singular supports and categorification of the nilpotent quantum group part via . The work provides a robust microlocal criterion with implications for the geometry of preprojective algebras and the structure of Lusztig's perverse sheaves in quiver theory.

Abstract

We prove a conjecture of Lusztig on a microlocal characterization of his perverse sheaves. For any finite quiver without loops, an equivariant simple perverse sheaf on the variety of quiver representations is a Lusztig's perverse sheaf if and only if its singular support is contained in Lusztig's Lagrangian variety, that is, the variety of nilpotent representations of the preprojective algebra of the quiver.
Paper Structure (9 sections, 10 theorems, 21 equations)

This paper contains 9 sections, 10 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quiver representations varieties and Lusztig's Lagrangian varieties
  4. Induction functors
  5. Restriction functors
  6. Lusztig's perverse sheaves
  7. A key inductive lemma
  8. Singular supports
  9. Main results

Key Result

Theorem 1.1

Let $G$ be a connected complex reductive group, then a $G$-equivariant simple perverse sheaf on $G$ is a character sheaf if and only if its singular support is contained in $G\times {\mathcal{N}}$, where ${\mathcal{N}}$ is the nilpotent cone in the Lie algebra of $G$.

Theorems & Definitions (13)

  • Theorem 1.1: Mirkovic-Vilonen-1988
  • Conjecture 1.2: Lusztig
  • Theorem 2.1: Lusztig-1991
  • Lemma 2.2: Lusztig-2000
  • Lemma 2.3: Lusztig-1991
  • Theorem 2.4: Lusztig-1991
  • Lemma 2.5: Lusztig-1991, Lusztig-1993
  • Proposition 2.6: Kashiwara-Schapira-1994
  • Theorem 3.1: Lusztig-1991
  • Lemma 3.2
  • ...and 3 more