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

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 $G$.

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