Equivariant $K$-theory of Springer Varieties

TL;DR

This paper provides a detailed description of the equivariant K-theory ring of Springer varieties of type A, extending previous cohomology and K-theory results to a more general setting using generators and relations.

Contribution

It offers a new presentation of the equivariant K-ring of Springer varieties, generalizing earlier cohomology and K-theory descriptions to include equivariant K-theory.

Findings

Explicit generators and relations for the equivariant K-ring are derived.

The results extend known cohomology descriptions to the K-theoretic setting.

The work generalizes previous ordinary K-ring descriptions to the equivariant case.

Abstract

The aim of this paper is to describe the topological equivariant $K$-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 $\lambda=(\lambda_1,\ldots, \lambda_l)$. This parallels the description of the equivariant cohomology ring of $\mathcal{F}_{\lambda}$ due to Abe and Horiguchi and generalizes the description of ordinary topological $K$-ring of $\mathcal{F}_{\lambda}$ due to Sankaran and Uma \cite{su}.