Unramified representation of a p-adic group

Definition

Let $\mathbf G$ be an unramified connected reductive group over a nonarchimedean local field $F$ , and let $K\leq G=\mathbf G(F)$ be a hyperspecial maximal compact subgroup. An irreducible admissible smooth representation $\pi$ of $G$ is unramified with respect to $K$ if

\pi^K\ne0.

The pair $(G,K)$ is spherical, so $\dim\pi^K=1$ . A nonzero vector in this line is an unramified, or spherical, vector.

Satake classification

The spherical Hecke algebra acts on $\pi^K$ through a character. The normalized Satake isomorphism turns this character into the Satake parameter of $\pi$ , a semisimple conjugacy class in the appropriate Frobenius coset of the $L$ -group.

Choice and terminology

The adjective depends on the chosen integral model, hence on $K$ , although hyperspecial subgroups are suitably conjugate in the standard unramified setting. “Spherical representation” is also used more broadly for any representation with fixed vectors under a chosen maximal compact subgroup; the present definition is specifically nonarchimedean and hyperspecial.

References

Armand Borel, “Automorphic $L$ -functions,” in Automorphic Forms, Representations and $L$ -Functions, Proceedings of Symposia in Pure Mathematics 33, part 2, 1979, §§3–4.
Jayce R. Getz, An Introduction to Automorphic Representations, §9. Author notes.