On the center of the generic affine Hecke algebra

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).