In mathematics, the Satake isomorphism, introduced by , identifies the Hecke algebra of a reductive group over a local field with a ring of invariants of the Weyl group. The geometric Satake equivalence is a geometric version of the Satake isomorphism, proved by . Statement Classical Satake isomorphism. Let be a semisimple algebraic group, be a non-Archimedean local field and be its ring of integers. It's easy to see that is grassmannian. For simplicity, we can think that and , a prime number; in this case, is a infinite dimensional algebraic variety . One denotes the category of all compactly supported spherical functions on biinvariant under the action of as , the field of complex numbers, which is a Hecke algebra and can be also treated as a group scheme over . Let be the maximal torus of , be the Weyl group of . one can associate a cocharacter variety to . Let be the set of all cocharacters of , i.e. . The cocharacter variety is basically the group scheme created by adding the elements of as variables to , i.e. . There is a natural action of on the cocharacter variety , induced by the natural action of on . Then the Satake isomorphism is a algebra isomorphism from the category of spherical functions to the -invariant part of the aforementioned cocharacter variety. In formulas: . Geometric Satake isomorphism. As Ginzburg said , "geometric" stands for sheaf theoretic. In order to obtain the geometric version of Satake isomorphism, one has to change the left part of the isomorphism, using Grothendieck group of the category of perverse sheaves on to replace the category of spherical functions; the replacement is de facto an algebra isomorphism over . One has also to replace the right hand side of the isomorphism by the Grothendieck group of finite dimensional complex representations of the Langlands dual of ; the replacement is also an algebra isomorphism over . Let denote the category of perverse sheaf on . Then, the geometric Satake isomorphism is , where the in stands for the Grothendieck group. This can be obviously simplified to , which is a fortiori an equivalence of Tannakian categories . Notes References Representation theory