Maximal Compact and Hyperspecial Subgroup

Let $k$ be a nonarchimedean local field with ring of integers $\mathcal O_k$.

A maximal compact subgroup of $G(k)$ is a compact open subgroup maximal under inclusion.

A subgroup $K$ is hyperspecial if $G$ extends to a reductive group scheme $\mathcal G/\mathcal O_k$ and $K=\mathcal G(\mathcal O_k)$.

Key property (for the letter):

• For hyperspecial $K$, the spherical Hecke algebra has the clean Satake description; the letter’s $G_{\mathbb Z_p}$ plays this role for almost all $p$.

Example: $\mathrm{GL}_n(\mathcal O_k)\subset \mathrm{GL}_n(k)$ is hyperspecial.