Coherent IC-sheaves on type 𝐴_{𝑛} affine Grassmannians and dual canonical basis of affine type 𝐴₁

Abstract

The convolution ring K G L n ( O ) ⋊ C × ( G r G L n ) K^{GL_n(\mathcal {O})\rtimes \mathbb {C}^\times }(\mathrm {Gr}_{GL_n}) was identified with a quantum unipotent cell of the loop group L S L 2 LSL_2 in Cautis and 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.