Affine nil-Hecke algebras and Quantum cohomology

TL;DR

This paper constructs an affine nil-Hecke algebra action on the equivariant quantum cohomology of symplectic manifolds with Hamiltonian Lie group actions, revealing new geometric structures and confirming theoretical predictions in the Langlands dual setting.

Contribution

It introduces a module action of the affine nil-Hecke algebra on equivariant quantum cohomology, generalizing shift operators and linking to the BFM-space of the Langlands dual group.

Findings

The affine nil-Hecke algebra acts on equivariant quantum cohomology.

The action respects the quantum connection.

When G is semi-simple, quantum cohomology defines a holomorphic Lagrangian subvariety.

Abstract

Let $G$ be a compact, connected Lie group and $T \subset G$ a maximal torus. Let $(M,\omega)$ be a monotone closed symplectic manifold equipped with a Hamiltonian action of $G$. We construct a module action of the affine nil-Hecke algebra $\hat{H}_*^{S^1 \times T}(LG/T)$ on the $S^1 \times T$-equivariant quantum cohomology of $M$, $QH^*_{S^1 \times T}(M).$ Our construction generalizes the theory of shift operators for Hamiltonian torus actions [OP,LJ]. 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 when $G$ is semi-simple, the $G$-equivariant quantum cohomology $QH_G^*(M)$ defines a canonical holomorphic Lagrangian subvariety $\mathbb{L}_G(M) \hookrightarrow BFM(G_{\mathbb{C}}^{\vee})$ in the BFM-space of the Langlands dual group, confirming an expectation of Teleman from [T1].