K-theory of Springer varieties

TL;DR

This paper provides a presentation of the topological K-ring of Springer varieties of type A, extending known cohomology descriptions to K-theory, and offering new algebraic insights into their structure.

Contribution

It offers a new generator-and-relations description of the topological K-ring of Springer varieties, paralleling existing cohomology and equivariant K-theory results.

Findings

Explicit generators and relations for the K-ring of Springer varieties.

Extension of cohomology ring descriptions to K-theory.

Connections to existing cohomology and equivariant theories.

Abstract

The aim of this paper is to describe the topological $K$-ring, in terms of generators and relations, of a Springer variety $\mathcal{F}_{\lambda}$ of type $A$ associated to a nilpotent operator having Jordan canonical form whose block sizes form a weakly decreasing sequence $\lambda=(\lambda_1,\ldots, \lambda_l)$. Our description parallels the description of the integral cohomology ring of $\mathcal{F}_{\lambda}$ due to Tanisaki and also the equivariant analogue due to Abe and Horiguchi.