Geometry and Topology without figures

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(\...