Satake isomorphism

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 formula_1 be a semisimple algebraic group, formula_2 be a non-Archimedean local field and formula_3 be its ring of integers. It's easy to see that formula_4 is grassmannian. For simplicity, we can think that formula_5 and formula_6, formula_7 a prime number; in this case, formula_8 is a infinite dimensional algebraic variety . One denotes the category of all compactly supported spherical functions on formula_9 biinvariant under the action of formula_10 as formula_11, formula_12 the field of complex numbers, which is a Hecke algebra and can be also treated as a group scheme over formula_12. Let formula_14 be the maximal torus of formula_15, formula_16 be the Weyl group of formula_17. one can associate a cocharacter variety formula_18 to formula_14. Let formula_20 be the set of all cocharacters of formula_14, i.e. formula_22. The cocharacter variety formula_18 is basically the group scheme created by adding the elements of formula_20 as variables to formula_12, i.e. formula_26. There is a natural action of formula_16 on the cocharacter variety formula_18, induced by the natural action of formula_16 on formula_30. Then the Satake isomorphism is a algebra isomorphism from the category of spherical functions to the formula_16-invariant part of the aforementioned cocharacter variety. In formulas:
formula_32.
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 formula_8 to replace the category of spherical functions; the replacement is de facto an algebra isomorphism over formula_34 . 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 formula_35 of formula_1; the replacement is also an algebra isomorphism over formula_12 . Let formula_38 denote the category of perverse sheaf on formula_8. Then, the geometric Satake isomorphism is
formula_40,
where the formula_41 in formula_42 stands for the Grothendieck group. This can be obviously simplified to
formula_43,
which is a fortiori an equivalence of tannakian categories .