Satake isomorphism

In mathematics, the Satake isomorphism, introduced by 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, introduced by Mirković & Vilonen (2007).

Let G be a Chevalley group, K be a non-Archimedean local field and O be its ring of integers. Then the Satake isomorphism identifies the Grothendieck group of complex representations of the Langlands dual of G, with the ring of G(O) invariant compactly supported functions on the affine Grassmannian. In formulas:

Here G(O) acts on G(K) / G(O) by multiplication from the left.

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