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 $\mathbb{C}\left[N\right(w\left)\right]$ 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 ${\mathcal{S}}^{\ast }$. 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 ${\mathbf{B}}^{up}$. In particular, we prove that the quantum analogue ${\mathcal{O}}_q\left[N\right(w\left)\right]$ of $\mathbb{C}\left[N\right(w\left)\right]$ has the induced basis from ${\mathbf{B}}^{up}$, which contains quantum flag minors and satisfies a factorization property with respect to the “$q$-center” of ${\mathcal{O}}_q\left[N\right(w\left)\right]$. This generalizes Caldero’s results from finite type to an arbitrary symmetrizable Kac–Moody Lie algebra.