Satake isomorphism

Classical Satake isomorphism.

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

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 on $Gr$ to replace the category of spherical functions; the replacement is de facto an algebra isomorphism over $\backslash Complex$ {{harv | Ginzburg | 2000 }}. 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 $\{\}^L\; G$ of $G$; the replacement is also an algebra isomorphism over $\backslash Complex$ {{harv | Ginzburg | 2000 }}. Let $\backslash mathrm\{Perv\}(Gr)$ denote the category of perverse sheaves on $Gr$. Then, the geometric Satake isomorphism is

where the $K$ in $K(\backslash mathrm\{Rep\}(\{\}^LG))$ stands for the Grothendieck group. This can be obviously simplified to

which is a fortiori an equivalence of Tannakian categories {{harv | Ginzburg | 2000 }}.

{{Reflist}}

• {{Citation | last1=Gross | first1=Benedict H. |authorlink=Benedict Gross| title=Galois representations in arithmetic algebraic geometry (Durham, 1996) | publisher=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 |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 | 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