A geometric construction of generalized quiver Hecke algebras

We provide a common generalization of the Springer map and quiver-graded Springer map due to Lusztig, called generalized quiver-graded Springer map associated to generalized quiver representations introduced by Derksen and Weyman. Following Chriss and Ginzburg for any equivariant projective map π : E → V , there is an algebra structure on the equivariant Borel-Moore homology of Z = E ×V E, we call it the Steinberg algebra (of π). For the classical Springer map, it is a skew group ring of the Weyl group operating on a polynomial ring (having the Weyl group ring as degree zero part) and for the quiver-graded Springer map, it is (one graded part of) the quiver Hecke algebra, this is due to Varagnolo and Vasserot. As our main result we calculate (under mild assumptions) generators and relations for Steinberg algebra of the generalized quiver-graded Springer map using the methods of Varagnolo and Vasserot. In the end, we discuss examples. As a third example we define the symplectic symmetric quiver-graded Springer map and give for quivers without loops the generators and relations of the associated algebra. We end with an overview table of known examples.