K-theory of Gieseker variety and type A cyclotomic Hecke algebra

TL;DR

This paper provides an algebraic description of the equivariant K-theory of Gieseker varieties, linking it to cyclotomic Hecke algebras and their centers, with implications at roots of unity and for affine type A quiver varieties.

Contribution

It identifies the equivariant K-theory of Gieseker varieties with the Jucys--Murphy center of cyclotomic Hecke algebras, extending previous conjectures and results.

Findings

K-theory of Gieseker varieties equals the Jucys--Murphy center of cyclotomic Hecke algebra.

Specialization at q=1 recovers the group algebra description.

At roots of unity, K-theory relates to centers of blocks of cyclotomic Hecke algebras.

Abstract

We give an algebraic description of the equivariant $K$-theory of Gieseker varieties. Our main result identifies the equivariant $K$-theory of the Gieseker space with the Jucys--Murphy center of the cyclotomic Hecke algebra, over the equivariant $K$-theory of a point. The construction is inspired by the proof of the Hikita--Nakajima conjecture for Gieseker spaces given by the first and third authors. We discuss consequences for the center of cyclotomic Hecke algebras. Under the specialization $q=1$, we recover the corresponding description in terms of the group algebra, while at roots of unity, assuming an identification between the equivariant $K$-theory of the Lagrangian subvariety and the cocenter, our result identifies the $K$-theory of affine type A quiver varieties with the centers of the corresponding blocks of specialized cyclotomic Hecke algebras. This last result strengthens the correspondences obtained by the second author in earlier work.