  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.
