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Lusztig sheaves, characteristic cycles and the Borel-Moore homology of Nakajima's quiver varieties

Jiepeng Fang, Yixin Lan

TL;DR

The paper develops a geometric, characteristic-cycle framework to connect Lusztig’s canonical basis with the top Borel-Moore homology of Nakajima quiver varieties and their tensor-product variants. By localizing Lusztig’s sheaves on framed and two-framed quivers and employing Fourier-Sato transforms, it constructs CC maps from twisted Grothendieck groups to BM-homology, yielding $_{\mathbb{Z}}\mathbf{U}(\mathfrak{g})$-equivariant isomorphisms after suitable base change and sign-twists. This provides an integer realization of Nakajima’s irreducible highest-weight modules and their tensor products and offers a new proof of Nakajima’s conjecture on canonical tensor-product isomorphisms, with upper-triangular transition control under the refine string order. The results harmonize perverse-sheaf-theoretic canonical bases with geometric fundamental classes, enabling canonical tensor-product identifications and clarifying how characteristic cycles govern the passage from categorical to homological realizations.

Abstract

By using characteristic cycles, we build a morphism from the canonical bases of integrable highest weight modules of quantum groups to the top Borel-Moore homology groups of Nakajima's quiver and tensor product varieties, and compare the canonical bases and the fundamental classes. As an application, we show that Nakajima's realization of irreducible highest weight modules and their tensor products can be defined over integers. We also give a new proof of Nakajima's conjecture on the canonical isomorphism of tensor product varieties.

Lusztig sheaves, characteristic cycles and the Borel-Moore homology of Nakajima's quiver varieties

TL;DR

The paper develops a geometric, characteristic-cycle framework to connect Lusztig’s canonical basis with the top Borel-Moore homology of Nakajima quiver varieties and their tensor-product variants. By localizing Lusztig’s sheaves on framed and two-framed quivers and employing Fourier-Sato transforms, it constructs CC maps from twisted Grothendieck groups to BM-homology, yielding -equivariant isomorphisms after suitable base change and sign-twists. This provides an integer realization of Nakajima’s irreducible highest-weight modules and their tensor products and offers a new proof of Nakajima’s conjecture on canonical tensor-product isomorphisms, with upper-triangular transition control under the refine string order. The results harmonize perverse-sheaf-theoretic canonical bases with geometric fundamental classes, enabling canonical tensor-product identifications and clarifying how characteristic cycles govern the passage from categorical to homological realizations.

Abstract

By using characteristic cycles, we build a morphism from the canonical bases of integrable highest weight modules of quantum groups to the top Borel-Moore homology groups of Nakajima's quiver and tensor product varieties, and compare the canonical bases and the fundamental classes. As an application, we show that Nakajima's realization of irreducible highest weight modules and their tensor products can be defined over integers. We also give a new proof of Nakajima's conjecture on the canonical isomorphism of tensor product varieties.
Paper Structure (44 sections, 50 theorems, 172 equations)

This paper contains 44 sections, 50 theorems, 172 equations.

Table of Contents

  1. Introduction
  2. Background
  3. The canonical basis of Lusztig's integral form
  4. Realization of integrable modules of universal enveloping algebras
  5. Realization of integrable modules of quantum groups
  6. The results of the present paper
  7. The structure of the present paper
  8. Preliminaries
  9. Equivariant characteristic cycle and induction
  10. Equivariant induction diagram
  11. Induction of equivariant complex
  12. Induction of (equivariant) Borel-Moore homology
  13. Compatibility of induction operations
  14. Geometric realization of $_{\mathcal{A}} \mathbf{U}^{-}_{v}(\mathfrak{g})$ and canonical bases
  15. Lusztig sheaves
  16. ...and 29 more sections

Key Result

Theorem 1.1

The Grothendieck group $\mathcal{L}(\omega^{1})$ of the localization of Lusztig sheaves on framed quiver is canonically isomorphic to the irreducible highest weight module ${_{\mathcal{A}}L}_{v}(\lambda_{1})$, Let $L_{0}$ be the constant sheaf on $\mathbf{E}_{\mathbf{V}\oplus\mathbf{W}^{1},\Omega^{(1)}}$ with $\mathbf{V}=0$, then $[L_{0}]$ is sent to the highest weight vector. Moreover, the image

Theorems & Definitions (80)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Definition 2.1
  • Theorem 2.2: hennecart2024geometric
  • Lemma 3.1: MR1088333
  • ...and 70 more