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Coherent IC-sheaves on type $A_{n}$ affine Grassmannians and dual canonical basis of affine type $A_{1}$

Michael Finkelberg, Ryo Fujita

Abstract

The convolution ring $K^{GL_n(\mathcal{O})\rtimes\mathbb{C}^\times}(\mathrm{Gr}_{GL_n})$ was identified with a quantum unipotent cell of the loop group $LSL_2$ in [Cautis-Williams, J. Amer. Math. Soc. 32 (2019), pp. 709-778]. We identify the basis formed by the classes of irreducible equivariant perverse coherent sheaves with the dual canonical basis of the quantum unipotent cell.

Coherent IC-sheaves on type $A_{n}$ affine Grassmannians and dual canonical basis of affine type $A_{1}$

Abstract

The convolution ring was identified with a quantum unipotent cell of the loop group in [Cautis-Williams, J. Amer. Math. Soc. 32 (2019), pp. 709-778]. We identify the basis formed by the classes of irreducible equivariant perverse coherent sheaves with the dual canonical basis of the quantum unipotent cell.

Paper Structure

This paper contains 23 sections, 24 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Overall convention
  4. Quantum unipotent cell of $LSL_{2}$
  5. Quantum algebras of type $\mathsf{A}_{1}^{(1)}$
  6. Quantum unipotent subgroup
  7. PBW and dual PBW bases
  8. Dual canonical basis
  9. Quantum cluster structure
  10. Berenstein-Zelevinsky's bar involution
  11. Localization
  12. Perverse coherent sheaves on type $A$ affine Grassmannian
  13. Affine Grassmannian
  14. Convolution product
  15. Perverse coherent sheaves
  16. ...and 8 more sections

Key Result

Theorem 2.1

For each $\beta \in \mathsf{Q}^{+}$, there exists a unique $\mathbb{Q}(q)$-basis $\mathscr{B}_{n}(\beta) = \{ B^{*}(\mathbf{a}) \mid \mathbf{a} \in \mathrm{KP}_{n}(\beta) \}$ of $(A_{n})_{\beta}$ characterized by the following properties: Moreover we have $\mathscr{B}_{n}(\beta) = \mathbf{B}^{*} \cap (A_{n})_{\beta}.$

Theorems & Definitions (42)

  • Theorem 2.1: Kim12
  • Theorem 2.2: GLS13, KKKO18
  • Lemma 2.3: cf. KO17
  • Corollary 2.4
  • Proposition 2.5: cf. KO17
  • Proposition 2.6: cf. KO17
  • Definition 3.1
  • Theorem 3.2: Cautis-Williams CW18
  • Theorem 3.3: CW18
  • Theorem 4.1
  • ...and 32 more