Weyl group action on weight zero Mirković-Vilonen basis and equivariant multiplicities

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