0200現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE
2019/11/28(木) 23:58:19.75ID:QdpmOFrx>Satake equivalence
下記かな〜?(^^;
”The geometric Satake equivalence is a geometric version of the Satake isomorphism, proved by Ivan Mirkovi? and Kari Vilonen (2007).”
”which is a fortiori an equivalence of tannakian categories (Ginzburg 2000).”
https://en.wikipedia.org/wiki/Satake_isomorphism
Satake isomorphism
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In mathematics, the Satake isomorphism, introduced by Ichir? 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 Mirkovi? and Kari Vilonen (2007).
Statement
Classical Satake isomorphism Let {\displaystyle G}G be a semisimple algebraic group, {\displaystyle K}K be a non-Archimedean local field and {\displaystyle O}O be its ring of integers. It's easy to see that {\displaystyle Gr=G(K)/G(O)}{\displaystyle Gr=G(K)/G(O)} is grassmannian.
Then, the geometric Satake isomorphism is
{\displaystyle K(Perv(Gr))\otimes _{\mathbb {Z} }\mathbb {C} \quad {\xrightarrow {\sim }}\quad K(Rep({}^{L}G))\otimes _{\mathbb {Z} }\mathbb {C} }{\displaystyle K(Perv(Gr))\otimes _{\mathbb {Z} }\mathbb {C} \quad {\xrightarrow {\sim }}\quad K(Rep({}^{L}G))\otimes _{\mathbb {Z} }\mathbb {C} },
which can be obviously simplified to
{\displaystyle Perv(Gr)\quad {\xrightarrow {\sim }}\quad Rep({}^{L}G)}{\displaystyle Perv(Gr)\quad {\xrightarrow {\sim }}\quad Rep({}^{L}G)},
which is a fortiori an equivalence of tannakian categories (Ginzburg 2000).