Unramified reductive group

Definition

Let $F$ be a nonarchimedean local field. A connected reductive $F$ -group $G$ is unramified if it is quasi-split over $F$ and becomes split over a finite unramified extension of $F$ .

This is a property of the algebraic group $G/F$ , not of a representation of the locally compact group $G(F)$ .

Integral model and hyperspecial subgroup

Equivalently, $G$ extends to a reductive group scheme $\mathcal G$ over the valuation ring $\mathcal O_F$ . Then $\mathcal G(\mathcal O_F)$ is a hyperspecial maximal compact subgroup of $G(F)$ . Conversely, the existence of a hyperspecial vertex in the Bruhat–Tits building characterizes unramified $G$ .

The choice of integral model or hyperspecial subgroup is generally not unique, even though unramifiedness is intrinsic.

Langlands use

For an unramified group and a chosen hyperspecial subgroup, the spherical Hecke algebra has a normalized Satake isomorphism. Its characters correspond to semisimple conjugacy classes in the unramified part of the L-group.

References

François Bruhat and Jacques Tits, “Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d'une donnée radicielle valuée,” Publications Mathématiques de l'IHÉS 60 (1984), 5–184. Numdam.
Jacques Tits, “Reductive groups over local fields,” in Automorphic Forms, Representations and L-Functions, Proceedings of Symposia in Pure Mathematics 33, part 1, 1979.