Spherical Hecke algebra and classical Satake isomorphism
The Satake isomorphism is a way to identify the spherical Hecke algebra of a reductive group G with the invariant part (under the action of the Weyl group) of cocharacters (also called one-parameter subgroups) and both are isomorphic to the Grothendieck ring of the category of the Langlands dual group G^{\vee}. This fits perfectly with the so-called Langlands duality philosophy, suggesting that algebraic objects associated with G^{\vee} should be captured by topological objects associated with G. More concretely, let F be a non-archimedian local field, and O the ring of integers, and let G be a split reductive group over \mathcal{O}. Let T \subset G be a maximal torus and X_{\ast}(T) = \operatorname{Hom}(\mathbb{G}_m,T) the group of cocharacters. The group ring \mathbb{Z}[X_{\ast}(T)] is endowed with an action of the Weyl group W. Let q be the cardinality of the residue field of K, the Satake isomorphism reads $$\mathbb{Z}_c[G(\mathcal{O}) \setminus G(K)/G(\...