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).

Statement

Let G be a Chevalley group, K be a non-Archimedian local field and O 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 Grassmanian. In formulas:

$\mathbb C_c [ G(K) / G(O) ]^{G(O)} \cong K_0(G^L-Rep)$

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

References

Gross, Benedict H. (1998), "On the Satake isomorphism", Galois representations in arithmetic algebraic geometry (Durham, 1996), London Math. Soc. Lecture Note Ser., 254, Cambridge University Press, pp. 223–237, doi:10.1017/CBO9780511662010.006, MR 1696481
Mirković, I.; Vilonen, K. (2007), "Geometric Langlands duality and representations of algebraic groups over commutative rings", Annals of Mathematics. Second Series 166 (1): 95–143, doi:10.4007/annals.2007.166.95, ISSN 0003-486X, MR arXiv:math/0401222 2342692[[arXiv]]:[[arXiv:math/0401222|math/0401222]]
Satake, Ichirô (1963), "Theory of spherical functions on reductive algebraic groups over p-adic fields", Publications Mathématiques de l'IHÉS (18): 5–69, ISSN 1618-1913, MR 0195863, http://www.numdam.org/item?id=PMIHES_1963__18__5_0