Sheaf realization of Bridgeland's Hall algebra of Dynkin type

Abstract: As one of results in [6], Bridgeland realized the quantum group $\mathrm{U}_v(\mathfrak{g})$ via the localization of Ringel-Hall algebra for two-periodic projective complexes of quiver representations over a finite field. In the present paper, we generalize Lusztig's categorical construction and (dual) canonical basis for the nilpotent part $\mathrm{U}_v(\mathfrak{n}^+)$ to Bridgeland's Hall algebra of Dynkin type, and obtain a perverse sheaf realization of global basis for Bridgeland's localizing algebra. In particular, we prove that the dual of canonical basis elements are part of our basis up to powers of $v$.