Toward AGT for Parabolic Sheaves

We construct explicit elements W-ij(k) in (a completion of) the shifted quantum toroidal algebra of type A and show that these elements act by 0 on the K-theory of moduli spaces of parabolic sheaves. We expect that the quotient of the shifted quantum toroidal algebra by the ideal generated by the elements W-ij(k) will be related to q-deformed W-algebras of type A for arbitrary nilpotent, which would imply a q-deformed version of the Alday-Gaiotto-Tachikawa (AGT) correspondence between gauge theory with surface operators and conformal field theory.