We state a conjecture about the Weyl group action coming from Geometric Satake on zero-weight spaces in terms of equivariant multiplicities of Mirković-Vilonen cycles. We prove it for small coweights in type A. In this case, using work of Braverman, Gaitsgory and Vybornov, we show that the Mirković-Vilonen basis agrees with the Springer basis. We rephrase this in terms of equivariant multiplicities using work of Joseph and Hotta. We also have analogous results for Ginzburg's Lagrangian construction of ${\mathfrak{sl}}_n$ representations.

中文翻译：

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权重为零的Mirković-Vilonen和等变多重的Weyl小组动作

我们根据Mirković-Vilonen循环的等变多重性，对来自几何Satake的零权空间上的Weyl群动作提出一个猜想。我们用A型小重量秤证明了这一点。在这种情况下，使用Braverman，Gaitsgory和Vybornov的工作，我们证明了Mirković-Vilonen基础与Springer基础一致。我们用约瑟夫和霍塔的著作用等变多重性来重新表述这一点。对于金茨堡（Ginzburg）的拉格朗日（Lagrangian）构造，我们也有类似的结果${\mathfrak{sl}}_{ñ}$ 表示形式。