Satake isomorphism

In mathematics, the Satake isomorphism, introduced by Ichirō Satake (1963), 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 Ivan Mirković and Kari Vilonen (2007).

Classical Satake isomorphism.

be a semisimple algebraic group,

be a non-Archimedean local field and

a prime number; in this case,

is an infinite dimensional algebraic variety (Ginzburg 2000).

One denotes the category of all compactly supported spherical functions on

the field of complex numbers, which is a Hecke algebra and can be also treated as a group scheme over

One can associate a cocharacter variety

be the set of all cocharacters of

The cocharacter variety

is basically the group scheme created by adding the elements of

on the cocharacter variety

, induced by the natural action of

Then the Satake isomorphism is an algebra isomorphism from the category of spherical functions to the

-invariant part of the aforementioned cocharacter variety.

Geometric Satake isomorphism.

As Ginzburg said (Ginzburg 2000), "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 the 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

; the replacement is also an algebra isomorphism over

{\displaystyle \mathrm {Perv} (Gr)}

denote the category of perverse sheaves on

Then, the geometric Satake isomorphism is

{\displaystyle K(\mathrm {Perv} (Gr))\otimes _{\mathbb {Z} }\mathbb {C} \quad \xrightarrow {\sim } \quad K(\mathrm {Rep} ({}^{L}G))\otimes _{\mathbb {Z} }\mathbb {C} }

stands for the Grothendieck group.

{\displaystyle \mathrm {Perv} (Gr)\quad \xrightarrow {\sim } \quad \mathrm {Rep} ({}^{L}G)}

, which is a fortiori an equivalence of Tannakian categories (Ginzburg 2000).