Equivariant $K$-theory of flag varieties revisited and related results

TL;DR

This paper revisits the equivariant $K$-theory of flag varieties, providing new formulas for structure constants and extending results to the wonderful compactification of the adjoint group.

Contribution

It introduces a novel approach to compute multiplicative structure constants in equivariant $K$-theory by lifting classes to a tensor product and applies these results to the wonderful compactification.

Findings

Explicit formulas for structure constants in $K_T(G/B)$

Extension of results to the wonderful compactification

Enhanced understanding of multiplicative structures in equivariant $K$-theory

Abstract

In this article we obtain many results on the multiplicative structure constants of $T$-equivariant Grothendieck ring of the flag variety $G/B$. We do this by lifting the classes of the structure sheaves of Schubert varieties in $K_{T}(G/B)$ to $R(T)\otimes R(T)$, where $R(T)$ denotes the representation ring of the torus $T$. We further apply our results to describe the multiplicative structure constants of $K(X)_{\mathbb Q}$ where $X$ is the wonderful compactification of the adjoint group of $G$, in terms of the structure constants of Schubert varieties in the Grothendieck ring of $G/B$.