On the center of the generic affine Hecke algebra

TL;DR

The paper identifies the center of the generic affine Hecke algebra ${\mathcal{H}}_{\mathbf q}$ with the $W_0$-invariants of a commutative subalgebra ${\mathcal{A}}_{\mathbf q}$, realized as the Rees algebra of the coweight lattice filtered by the length function. It proves a central isomorphism ${\mathcal{Z}}_{\mathbf q} \cong {\mathbb{C}}[{\mathbf q}][{\check X}^+]$, interpolating between the $q$-deformed center ${\mathcal{Z}}_{\mathbf q^{\pm 1}}$ and the $q=0$ limit ${\mathcal Z}_0$, and links this center to toric and Hessenberg geometry. The authors interpret ${\mathcal{A}}_0$ as equivariant cohomology $H_T^*(X_{\Sigma},\mathbb{C})$ of a toric variety associated to the Weyl fan, while ${\mathcal{A}}_{\mathbf q}$ encodes quantum cohomology with respect to the same Kähler class, unifying algebraic, geometric, and topological perspectives. They further show that ${\mathcal H}_{\mathbf q}$ is finite over its center and, under suitable assumptions, a Frobenius extension with explicit Nakayama automorphism. The toric-geometry viewpoint yields concrete identifications with Stanley-Reisner rings and clarifies how convex piecewise-linear deformations govern the quantum product structure.

Abstract

We identify the center of the generic affine Hecke algebra $H_q$ corresponding to some root datum with the semigroup algebra $\mathbb C[q][\check X^+]$ of the dominant chamber of its coweight lattice. This is done by first identifying a maximal commutative subalgebra $A_q \subset H_q$ with the Rees algebra associated to the semigroup algebra of the coweight lattice for the filtration induced by the length function. We explain how this subalgebra $A_q$ can also be identified with (quantum) cohomology of the toric variety given by the fan corresponding to the coroot lattice (a Hessenberg variety).

Theorems & Definitions (34)

• Lemma 2.1
• Remark 2.2
• Lemma 2.3
• proof
• Theorem 3.1
• Corollary 3.2
• Lemma 3.3
• proof
• Lemma 3.4
• proof