Dual canonical bases of quantum groups and $\imath$quantum groups II: geometry

TL;DR

This paper proves the equivalence and positivity properties of dual canonical bases for $ extit{i}$quantum groups of type ADE, using geometric and algebraic realizations, and confirms their invariance under braid group actions.

Contribution

It establishes the coincidence of dual canonical bases constructed via perverse sheaves and Hall algebras for $ extit{i}$quantum groups of type ADE, demonstrating their invariance and positivity.

Findings

Dual canonical bases coincide for type ADE $ extit{i}$quantum groups.

Bases are invariant under braid group actions.

Structural constants are integral and positive.

Abstract

The $\imath$quantum groups admit two realizations: one via the $\imath$Hall algebras and the other via the quantum Grothendieck rings of quiver varieties, as developed by the first author and Wang. Based on these two realizations, we establish the dual canonical bases for $\imath$quantum groups of type ADE in two distinct ways, using perverse sheaves and Hall algebras respectively. In this paper, we prove that these two dual canonical bases coincide, thereby proving their invariance under braid group actions, and that their structural constants are integral and positive. Furthermore, we establish the positivity of the coefficients of the transition matrix from the Hall basis (and PBW basis) to the dual canonical basis.