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Lusztig sheaves, decomposition rule and restriction rule

Yixin Lan

TL;DR

This work geometrizes the decomposition and restriction rules for symmetric Kac–Moody algebras by realizing subquotient-based modules through Lusztig's perverse sheaves on multi-framed quivers and constructing their canonical bases. By developing a thick-subcategory localization framework and connecting with Nakajima quiver varieties via characteristic cycles, it shows that decomposition and restriction coefficients equal the dimensions of top Borel-Moore homology on certain locally closed loci. The authors establish subquotient realizations M[≥μ]/M[>μ] and their restricted analogues, and prove that simple perverse sheaves encode the canonical basis for these subquotients. The results reveal a tight link between categorical geometric constructions (via Ind/Res functors, Fourier-Sato transforms, and micro-local analysis) and the representation-theoretic data (decomposition, restriction, and coinvariants) of tensor products and Levi-restricted modules, with explicit ties to Nakajima’s quiver varieties and BM-homology through characteristic cycles.

Abstract

In this article, we realize the subquotient based modules of certain tensor products or restricted modules via Lusztig's perverse sheaves on multi-framed quivers, and provide a construction of their canonical bases. As an application, we prove that the decomposition and restriction coefficients of symmetric Kac-Moody algebras equal to the dimensions of top Borel-Moore homology groups for certain locally closed subsets of Nakajima's quiver varieties.

Lusztig sheaves, decomposition rule and restriction rule

TL;DR

This work geometrizes the decomposition and restriction rules for symmetric Kac–Moody algebras by realizing subquotient-based modules through Lusztig's perverse sheaves on multi-framed quivers and constructing their canonical bases. By developing a thick-subcategory localization framework and connecting with Nakajima quiver varieties via characteristic cycles, it shows that decomposition and restriction coefficients equal the dimensions of top Borel-Moore homology on certain locally closed loci. The authors establish subquotient realizations M[≥μ]/M[>μ] and their restricted analogues, and prove that simple perverse sheaves encode the canonical basis for these subquotients. The results reveal a tight link between categorical geometric constructions (via Ind/Res functors, Fourier-Sato transforms, and micro-local analysis) and the representation-theoretic data (decomposition, restriction, and coinvariants) of tensor products and Levi-restricted modules, with explicit ties to Nakajima’s quiver varieties and BM-homology through characteristic cycles.

Abstract

In this article, we realize the subquotient based modules of certain tensor products or restricted modules via Lusztig's perverse sheaves on multi-framed quivers, and provide a construction of their canonical bases. As an application, we prove that the decomposition and restriction coefficients of symmetric Kac-Moody algebras equal to the dimensions of top Borel-Moore homology groups for certain locally closed subsets of Nakajima's quiver varieties.

Paper Structure

This paper contains 34 sections, 41 theorems, 190 equations.

Table of Contents

  1. Introduction
  2. Decomposition rule and Restriction Rule
  3. Perverse sheaves and Canonical bases for quantum groups and their representations
  4. Nakajima's quiver varieties and their relations with canonical bases
  5. The structure of this article
  6. Preliminaries
  7. Canonical bases of tensor products
  8. Based modules and their subquotients
  9. Realization of integrable highest weight modules
  10. The $N$-framed quiver
  11. Lusztig's induction and restriction functors
  12. The thick subcategory $\mathcal{N}$ and localization
  13. The realization of integrable modules
  14. Filtration of based modules arising from tensor products
  15. The thick subcategory $\mathcal{D}^{\geqslant \mu}$ and $\mathcal{D}^{>\mu}$
  16. ...and 19 more sections

Key Result

Theorem 2.1

lusztig1992canonicalbao2016canonical Let $\mathcal{L}(M\otimes M')$ be the $\mathbf{A}$-submodule of $M\otimes M'$ generated by $B \otimes B'$, then the pair $(M\otimes M', B(M) \diamond B(M'))$ is a based module. More precisely, we have the following statements. (1) For any $(b,b') \in B(M)\times B Moreover, $p_{b,b',b,b'}=1$ and $p_{b,b',b_{2},b'_{2}} \neq 0$ implies that $(b_{2},b'_{2}) < (b,b'

Theorems & Definitions (74)

  • Theorem 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Proposition 3.6
  • ...and 64 more