Spherical Hecke Algebra and Satake Isomorphism

Let $k$ be a nonarchimedean local field (e.g. $\mathbb{Q}_p$), $G/k$ reductive, and $K\subset G(k)$ a hyperspecial maximal compact (see hyperspecial $K$ ).

The spherical Hecke algebra is $\mathcal H(G(k),K):=C_c(K\backslash G(k)/K)$ with convolution (compactly supported, $K$-bi-invariant functions).

The Satake isomorphism identifies $\mathcal H(G(k),K)$ with a commutative algebra built from the dual torus of $\\widehat G$ (e.g. $\mathbb{C}[X_*(\widehat T)]^{W}$); in the letter this is described as invariants in a group algebra of a dual lattice $cL$.

Key use: a Hecke eigencharacter $\chi:\mathcal H\to\mathbb{C}$ corresponds to a semisimple conjugacy class (a Satake parameter).