Intersection K-theory of quotient singularities

For a class of quotient stacks V/G for G a reductive group and V a symmetric G-representation, we propose a definition of the intersection K-theory of the quotient V//G, which is a Q-vector space denoted IK(V//G). There is a Chern character from IK(V//G) to intersection cohomology IH(V//G) and IK(V//G) satisfies Kirwan surjectivity. We construct IK(V//G) as a direct summand of the Grothendieck group (tensor with Q) of a noncommutative resolution of singularities of V//G constructed by Spenko-Van den Bergh. We explain extensions of some of these results for stacks with good moduli spaces. We discuss an application of this construction to a PBW-type theorem for (Kontsevich-Soibelman) K-theoretic Hall algebras of quivers with potential.