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-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:

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

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

Notes

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 2342692 arXiv:math/0401222
• Satake, Ichirō (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