The convolution algebra structure on $K^G(\mathcal{B} \times \mathcal{B})$

TL;DR

This paper establishes an isomorphism between the convolution algebra of equivariant K-theory on the product of a flag variety and itself, and the based ring of a specific two-sided cell in the extended affine Weyl group, revealing deep algebraic structure.

Contribution

It proves a new isomorphism linking the convolution algebra on the flag variety to the based ring of a two-sided cell in the affine Weyl group, enhancing understanding of geometric representation theory.

Findings

Convolution algebra $K^G(B imes B)$ is isomorphic to the based ring of the lowest two-sided cell.

Provides a geometric realization of algebraic structures in affine Weyl groups.

Connects algebraic K-theory with the combinatorial structure of affine Weyl groups.

Abstract

We show that the convolution algebra $K^G(\mathcal{B} \times \mathcal{B})$ is isomorphic to the Based ring of the lowest two-sided cell of the extended affine Weyl group associated to $G$, where $G$ is a connected reductive algebraic group over the field $\mathbb{C}$ of complex numbers and $\mathcal{B}$ is the flag variety of $G$.