# Satake isomorphism

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{{Short description|Mathematics concept}}
In mathematics, the '''Satake isomorphism''', introduced by {{harvs|txt|authorlink=Ichirō Satake|last=Satake|first=Ichirō|year=1963}}, identifies the [Hecke algebra](/source/Hecke_algebra_of_a_locally_compact_group) of a [reductive group](/source/reductive_group) over a [local field](/source/local_field) with a ring of invariants of the [Weyl group](/source/Weyl_group).  The '''geometric Satake equivalence''' is a geometric version of the Satake isomorphism, proved by {{harvs|txt|last1=Mirković | first1=Ivan | last2=Vilonen | first2=Kari | author2-link=Kari Vilonen|year=2007}}.

==Statement==
'''Classical Satake isomorphism'''.
Let <math>G</math> be a [semisimple algebraic group](/source/semisimple_algebraic_group), <math>K</math> be a non-Archimedean local field and <math>O</math> be its ring of integers. It's easy to see that <math>Gr = G(K)/G(O)</math> is a [Grassmannian](/source/Grassmannian). For simplicity, we can think that <math> K = \Z/p\Z((x)) </math> and <math> O = \Z/p\Z[x](/source/x) </math>, for <math> p </math> a prime number; in this case, <math> Gr </math> is an infinite dimensional [algebraic variety](/source/algebraic_variety) {{harv | Ginzburg | 2000 }}. One denotes the category of all compactly supported [spherical functions](/source/spherical_functions) on <math> G(K) </math> bi-invariant under the action of <math> G(O) </math> as <math> \Complex_c[G(O) \backslash G(K)/G(O)] </math>, <math> \Complex </math> the field of complex numbers, which is a [Hecke algebra](/source/Hecke_algebra_of_a_locally_compact_group) and can be also treated as a [group scheme](/source/group_scheme) over <math> \Complex </math>. Let <math> T(\Complex) </math> be the [maximal torus](/source/maximal_torus) of <math> G(\Complex) </math>, <math> W </math> be the [Weyl group](/source/Weyl_group) of <math> G </math>. One can associate a cocharacter variety <math> \mathbb{X}_*(T(\Complex)) </math> to <math> T(\Complex) </math>. Let <math> X_*(T(\Complex)) </math> be the set of all cocharacters of <math> T(\Complex) </math>, i.e. <math> X_*(T(\Complex)) = \mathrm{Hom}(\Complex^*, T(\Complex)) </math>. The cocharacter variety <math> \mathbb{X}_*(T(\Complex)) </math> is basically the [group scheme](/source/group_scheme) created by adding the elements of <math> X_*(T(\Complex)) </math> as variables to <math> \Complex </math>, i.e. <math> \mathbb{X}_*(T(\Complex)) = \Complex[X_*(T(\Complex))] </math>. There is a natural action of <math> W </math> on the cocharacter variety <math> \mathbb{X}_*(T(\Complex)) </math>, induced by the natural action of <math> W </math> on <math> T </math>. Then the Satake isomorphism is an algebra isomorphism from the category of [spherical functions](/source/spherical_functions) to the <math> W </math>-invariant part of the aforementioned cocharacter variety. In formulas:
<div style = "text-align:center;">
<math> \Complex_c [G(O) \backslash G(K)/G(O)] \quad \xrightarrow{\sim} \quad \mathbb{X}_*(T(\Complex))^W </math>.
</div>

'''Geometric Satake isomorphism'''.
As Ginzburg said {{harv | 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](/source/perverse_sheaf) on <math> Gr </math> to replace the category of [spherical functions](/source/spherical_functions); the replacement is de facto an algebra isomorphism over <math>\Complex</math> {{harv | Ginzburg | 2000 }}. One has also to replace the right hand side of the isomorphism by the [Grothendieck group](/source/Grothendieck_group) of finite dimensional complex representations of the [Langlands dual](/source/Langlands_dual) <math>{}^L G</math> of <math>G</math>; the replacement is also an algebra isomorphism over <math> \Complex </math> {{harv | Ginzburg | 2000 }}. Let <math> \mathrm{Perv}(Gr) </math> denote the category of [perverse sheaves](/source/perverse_sheaves) on <math> Gr </math>. Then, the geometric Satake isomorphism is
<div style = "text-align:center;">
<math> K(\mathrm{Perv}(Gr)) \otimes_\Z \Complex \quad \xrightarrow{\sim} \quad K(\mathrm{Rep}({}^LG)) \otimes_\Z \Complex</math>,
</div>
where the <math> K </math> in <math> K(\mathrm{Rep}({}^LG)) </math> stands for the [Grothendieck group](/source/Grothendieck_group). This can be obviously simplified to
<div style = "text-align:center;">
<math> \mathrm{Perv}(Gr) \quad \xrightarrow{\sim} \quad \mathrm{Rep}({}^LG) </math>,
</div>
which is ''a fortiori'' an equivalence of [Tannakian categories](/source/Tannakian_formalism) {{harv | Ginzburg | 2000 }}.

==Notes==
{{Reflist}}

==References==
* {{Citation | last1=Gross | first1=Benedict H. |authorlink=Benedict Gross| title=Galois representations in arithmetic algebraic geometry (Durham, 1996) | publisher=[Cambridge University Press](/source/Cambridge_University_Press) | series=London Math. Soc. Lecture Note Ser. | doi=10.1017/CBO9780511662010.006 | mr=1696481 | year=1998 | volume=254 | chapter=On the Satake isomorphism | pages=223–237| isbn=9780521644198 }}
* {{Citation | last1=Mirković | first1=Ivan | last2=Vilonen | first2=Kari | author2-link=Kari Vilonen| title=Geometric Langlands duality and representations of algebraic groups over commutative rings | doi=10.4007/annals.2007.166.95 | mr=2342692 | year=2007 | journal=[Annals of Mathematics](/source/Annals_of_Mathematics) |series=Second Series | issn=0003-486X | volume=166 | issue=1 | pages=95–143 |arxiv=math/0401222| s2cid=14127684 }}
* {{Citation | last1=Satake | first1=Ichirō |authorlink=Ichirō Satake| title=Theory of spherical functions on reductive algebraic groups over p-adic fields | url=http://www.numdam.org/item?id=PMIHES_1963__18__5_0 | mr=0195863 | year=1963 | journal=[Publications Mathématiques de l'IHÉS](/source/Publications_Math%C3%A9matiques_de_l'IH%C3%89S) | volume=18 | issn=1618-1913 | issue=18 | pages=5–69| doi=10.1007/BF02684781 | s2cid=4666554 }}
*{{cite arXiv |last= Ginzburg |first= Victor | title = Perverse sheaves on a loop group and Langlands' duality |eprint= alg-geom/9511007 |date= 2000 }}

Category:Representation theory

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Adapted from the Wikipedia article [Satake isomorphism](https://en.wikipedia.org/wiki/Satake_isomorphism) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Satake_isomorphism?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.