Coherent IC-sheaves on type $A_n$ affine Grassmannians and dual canonical basis of affine type $A_1$

Abstract

The convolution ring $K^{G{L}_n(\mathcal{O})⋊{\mathbb{C}}^{\times }}({\mathrm{Gr}}_{G{L}_n})$ was identified with a quantum unipotent cell of the loop group $LS{L}_2$ 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.

23 pages, v2: minor revision, v3: replaced the statements of Theorems 2.2, 3.2 and 4.1 with ones for the integral forms, improved some explanations slightly, v4: minor revision, v5: Erratum to the published version added in Section 5, correcting Lemma 4.14

References in corpus (3)

• Shifted quantum affine algebras: integral forms in type $A$ (with appendices by Alexander Tsymbaliuk and Alex Weekes)
• Twist automorphisms on quantum unipotent cells and dual canonical bases
• Towards a cluster structure on trigonometric zastava