The center conjecture for cyclotomic Hecke algebras and cohomology of quiver varieties

For a quiver without edge loops, we construct an isomorphism from the cohomology of a Nakajima quiver variety with coefficients in an arbitrary field to the Jucys–Murphy center of the corresponding cyclotomic KLR algebra over that field. In affine type $A_{e-1}^{(1)}$, combining this with a dimension count shows that this Jucys–Murphy center coincides with the full center in characteristic $0$, when the characteristic does not divide $e$, and when the characteristic equals $e$. This follows from a geometric theorem showing that the sum of these cohomology rings is cocyclic as a module over an integral form of a loop algebra. As a consequence, we prove that the center of a cyclotomic Hecke algebra of type $G(r,1,n)$ with $q\mathrm{\ne }1$ over any field is generated by symmetric polynomials in its Jucys–Murphy elements, and the same holds over any commutative domain in which $q-1$ is invertible. We also prove a generalization to the integral case of the conjecture of Krylov, Paegelow and Shlykov identifying the equivariant $K$-theory of the core of a Gieseker variety with the cocenter of the generic cyclotomic Hecke algebra, including the natural extension of this conjecture to positive characteristic.