Mirković–Vilonen basis in type $A_1$

Abstract:

Let $G$ be a connected reductive algebraic group over $\mathbb C$. Through the geometric Satake equivalence, the fundamental classes of the Mirković–Vilonen cycles define a basis in each tensor product $V(\lambda _1)\otimes \cdots \otimes V(\lambda _r)$ of irreducible representations of $G$. We compute this basis in the case $G=\mathrm {SL}_2(\mathbb C)$ and conclude that in this case it coincides with the dual canonical basis at $q=1$.