Dual canonical bases of quantum groups and $\imath$quantum groups

Abstract: The $\imath$quantum groups have 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. Perverse sheaves provide the dual canonical bases for $\imath$quantum groups of type ADE with integral and positive structure constants. In this paper, we present a new construction of the dual canonical bases in the setting of $\imath$Hall algebras. We also introduce Fourier transforms for both $\imath$Hall algebras and quantum Grothendieck rings, and prove the invariance of the dual canonical bases under braid group actions and Fourier transforms. Additionally, we establish the positivity of the transition matrix coefficients from the Hall basis (and PBW basis) to the dual canonical basis. As quantum groups can be regarded as $\imath$quantum groups of diagonal type, we demonstrate that the dual canonical bases of Drinfeld double quantum groups coincide with the double canonical bases defined by Berenstein and Greenstein, resolving several conjectures therein.