Affine nil-Hecke algebras and quantum cohomology

Let G be a compact, connected Lie group and T⊂G a maximal torus. Let (M,ω) be a monotone closed symplectic manifold equipped with a Hamiltonian action of G. We construct a module action of the affine nil-Hecke algebra Hˆ_⁎^S1×T(LG/T) on the S¹×T-equivariant quantum cohomology of M, QH_S¹×T^⁎(M). Our construction generalizes the theory of shift operators for Hamiltonian torus actions [46,40]. We show that, as in the abelian case, this action behaves well with respect to the quantum connection. As an application of our construction, we show that the G-equivariant quantum cohomology QH_G^⁎(M) defines a canonical holomorphic Lagrangian subvariety L_G(M)↪BFM(G_C^∨) in the BFM-space of the Langlands dual group, confirming an expectation of Teleman from [51].