Dual canonical bases of quantum groups and $\imath$quantum groups I: Hall algebras

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. The isoclasses of 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 $\imath$Hall algebras, and prove the invariance of the dual canonical bases under braid group actions and Fourier transforms. Since quantum groups are $\imath$quantum groups of diagonal type, all results also apply to this classical case.