PBW bases and KLR algebras

We generalize Lusztig’s geometric construction of the PBW bases of finite quantum groups of type $\mathsf{A}\mathsf{D}\mathsf{E}$ under the framework of [Varagnolo-Vasserot, J. reine angew. Math. 659 (2011)]. In particular, every PBW basis of such quantum groups is proven to yield a semi-orthogonal collection in the module category of the KLR-algebras. This enables us to prove Lusztig’s conjecture on the positivity of the canonical (lower global) bases in terms of the (lower) PBW bases in the $\mathsf{A}\mathsf{D}\mathsf{E}$ case. In addition, we verify Kashiwara’s problem on the finiteness of the global dimensions of the KLR-algebras of type $\mathsf{A}\mathsf{D}\mathsf{E}$.