Quantum unipotent subgroup and dual canonical basis

Abstract

In a series of works, Geiss, Leclerc, and Schröer defined the cluster algebra structure on the coordinate ring ${\mathbb {C}}[N(w)]$ of the unipotent subgroup, associated with a Weyl group element $w$. And they proved that cluster monomials are contained in Lusztig’s dual semicanonical basis ${\mathcal {S}}^{*}$. We give a setup for the quantization of their results and propose a conjecture that relates the quantum cluster algebras in Berenstein and Zelevinsky’s work to the dual canonical basis ${\mathbf {B}}^{\operatorname{up}}$. In particular, we prove that the quantum analogue ${\mathcal {O}}_{q}[N(w)]$ of ${\mathbb {C}}[N(w)]$ has the induced basis from ${\mathbf {B}}^{\operatorname{up}}$, which contains quantum flag minors and satisfies a factorization property with respect to the “$q$-center” of ${\mathcal {O}}_{q}[N(w)]$. This generalizes Caldero’s results from finite type to an arbitrary symmetrizable Kac–Moody Lie algebra.