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 $q=1$.

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中文翻译：

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𝐴₁ 类型的 Mirković–Vilonen 基

摘要：令$G$是$\mathbb C$上的连通还原代数群。通过几何 Satake 等价，Mirković-Vilonen 循环的基本类定义了 $G$ 的不可约表示的每个张量积 $V(\lambda_1)\otimes\cdots\otimes V(\lambda_r)$ 的基础。我们在 $G=\mathrm {SL}_2(\mathbb C)$ 的情况下计算这个基，并得出结论，在这种情况下，它与 $q=1$ 处的双正则基重合。

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