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Affine Laumon space and contragredient dual Verma module of $U_q(\widehat{\mathfrak{gl}_n})$

Che Shen

TL;DR

The paper develops a geometric realization of the contragredient dual Verma module for the quantum affine algebra $U_q(\widehat{\mathfrak{gl}}_n)$ on the localized equivariant K-theory of (affine) Laumon spaces. Building on PBW bases and a Maulik–Okounkov stable envelope framework, it proves that the dual PBW basis maps into the K-theory and spans it, both in the finite and affine settings, with the affine case requiring a stability-based triangularity argument and a rigidity technique. It extends prior work of Braverman–Finkelberg and Negu\c{t} by providing an explicit geometric realization of the dual Verma module via stable envelopes and by showing almost-integrality of the action away from the critical level. A key technical achievement is the control of denominators in the $U_q(\widehat{\mathfrak{gl}}_n)$ action, enabling specialization to generic highest weights and a clear identification with contragredient dual Verma modules, which has implications for the geometric representation theory of quantum affine algebras.

Abstract

We study the action of the quantum group $U_q(\widehat{\mathfrak{gl}_n})$ on the equivariant K-theory of affine Laumon spaces. We show that, at any highest weight away from the critical level, this can be identified with the contragredient dual Verma module of $U_q(\widehat{\mathfrak{gl}_n})$, improving earlier results of Braverman-Finkelberg and Negu{ţ}. The proof uses a variant of stable envelopes first introduced by Maulik-Okounkov in the study of Nakajima quiver varieties.

Affine Laumon space and contragredient dual Verma module of $U_q(\widehat{\mathfrak{gl}_n})$

TL;DR

The paper develops a geometric realization of the contragredient dual Verma module for the quantum affine algebra on the localized equivariant K-theory of (affine) Laumon spaces. Building on PBW bases and a Maulik–Okounkov stable envelope framework, it proves that the dual PBW basis maps into the K-theory and spans it, both in the finite and affine settings, with the affine case requiring a stability-based triangularity argument and a rigidity technique. It extends prior work of Braverman–Finkelberg and Negu\c{t} by providing an explicit geometric realization of the dual Verma module via stable envelopes and by showing almost-integrality of the action away from the critical level. A key technical achievement is the control of denominators in the action, enabling specialization to generic highest weights and a clear identification with contragredient dual Verma modules, which has implications for the geometric representation theory of quantum affine algebras.

Abstract

We study the action of the quantum group on the equivariant K-theory of affine Laumon spaces. We show that, at any highest weight away from the critical level, this can be identified with the contragredient dual Verma module of , improving earlier results of Braverman-Finkelberg and Negu{ţ}. The proof uses a variant of stable envelopes first introduced by Maulik-Okounkov in the study of Nakajima quiver varieties.
Paper Structure (25 sections, 22 theorems, 105 equations)

This paper contains 25 sections, 22 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Laumon space
  3. Affine Laumon space
  4. Strategy of proof
  5. Laumon spaces and quantum affine algebra
  6. Affine Laumon space
  7. Definition as a quiver variety
  8. Fixed points
  9. Local coordinates
  10. Tautological bundle
  11. Quantum group ${U_q(\widehat{{\mathfrak{gl}}}_n)}$
  12. Anti-involution
  13. Universal Verma module and Shapovalov form
  14. PBW basis
  15. Action on $K_{\mathsf{T}}({\mathcal{M}}_{\mathbf{d}} )$
  16. ...and 10 more sections

Key Result

Lemma 2.2

The maps $p^+, p^-, \overline{p}^+$ are proper.

Theorems & Definitions (43)

  • Remark 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Theorem 3.1
  • Remark 3.2
  • Corollary 3.3
  • Lemma 3.4
  • proof
  • Proposition 3.5
  • ...and 33 more