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

Weyl group action on weight zero Mirkovi\'c-Vilonen basis and equivariant multiplicities