K-theory of weight varieties

TL;DR

This paper provides a new proof of the $K$-theoretic Kirwan surjectivity theorem for Hamiltonian torus actions, computes the kernel of the equivariant Kirwan map, and applies these results to flag varieties and weight varieties.

Contribution

It introduces a novel proof technique for the $K$-theoretic Kirwan surjectivity and explicitly computes the kernel for flag and weight varieties.

Findings

Computed the kernel of the equivariant Kirwan map.

Provided explicit formulas for the $K$-theory of weight varieties.

Presented a new proof of the $K$-theoretic Kirwan surjectivity theorem.

Abstract

Let $T$ be a compact torus and $(M,\omega)$ a Hamiltonian $T$-space. We give a new proof of the $K$-theoretic analogue of the Kirwan surjectivity theorem in symplectic geometry by using the equivariant version of the Kirwan map introduced in one of R. Goldin's papers. We compute the kernel of this equivariant Kirwan map, and hence give a computation of the kernel of the Kirwan map. As an application, we find the presentation of the kernel of the Kirwan map for the $T$-equivariant $K$-theory of flag varieties $G/T$ where $G$ is a compact, connected and simply-connected Lie group. In the last section, we find explicit formulae for the $K$-theory of weight varieties.