Maximal compact and hyperspecial subgroups

A maximal compact subgroup of a locally compact group is a compact subgroup maximal under inclusion. For a connected reductive group over a nonarchimedean local field $F$ , maximal compact subgroups are open.

Let $\mathcal O_F$ be the valuation ring. A compact open subgroup $K\subset G(F)$ is hyperspecial if there is a smooth affine reductive group scheme $\mathcal G/\mathcal O_F$ , with generic fiber $G$ , such that

K=\mathcal G(\mathcal O_F).

Requiring the special fiber of $\mathcal G$ to remain reductive distinguishes a hyperspecial subgroup from a general parahoric subgroup.

Existence

A connected reductive $F$ -group admits a hyperspecial subgroup exactly when it is unramified: it is quasi-split over $F$ and becomes split over an unramified extension. Every hyperspecial subgroup is maximal compact, but a maximal compact subgroup need not be hyperspecial.

For $\operatorname{GL}_n(F)$ , the subgroup $\operatorname{GL}_n(\mathcal O_F)$ is hyperspecial.

Archimedean distinction

For a real reductive group, a maximal compact subgroup is an archimedean Lie subgroup, such as $\operatorname O(n)\subset\operatorname{GL}_n(\mathbb R)$ . There is no hyperspecial notion at an archimedean place. Bundling these phrases in one historical page should not blur the local-field distinction.

Spherical role

If $K$ is hyperspecial, an irreducible representation with $\pi^K\neq0$ is unramified, and the spherical Hecke algebra acts on its one-dimensional fixed line. The groups denoted $G_{\mathbb Z_p}$ in the letter have this role at almost all places.

References

François Bruhat and Jacques Tits, “Groupes réductifs sur un corps local II,” Publications Mathématiques de l'IHÉS 60 (1984), 5–184. Numdam.
Jacques Tits, “Reductive groups over local fields,” Proc. Sympos. Pure Math. 33, part 1, 1979.