Why are spherical representations subquotients of unramified principal series?

Question

I'm trying to learn the basics of the representation theory of $p$-adic groups and I'm stuck on a few things:

Let $G$ is a connected split reductive group over a non-archimedean local field $F$, and $K=G(\mathcal{O})$, a hyperspecial maximal compact subgroup of $G(F)$. Then there are two facts that apparently go hand in hand:

1. The Iwahori-Hecke algebra $H(G//K)$ is commutative.

2. If $\pi$ is any irreducible admissible (complex) representation of $G(F)$, then dim$(\pi^K) \leq 1$.

For the first statement there is a trick: one uses the Cartan decomposition to show that there is an anti-automorphism on the Hecke algebra that is the identity. But I don't see how to prove the second fact -- does it follow quickly from the first fact?

There is also this statement: For any irreducible admissible $\pi$, $\pi^K \neq 0$ if and only if $\pi$ is a Jordan-Holder component of an unramified principal series representation of $T(F)$, where $T$ is a maximal split torus in $G$.

I would appreciate any help on how these statements are proven. Part of the reason I'm confused is that the source I'm looking at seems to me to suggest that these statements follow from the first statement, but after looking on the internet, it seems that the statement about unramified principal series is a deep result of Satake, reproven by Casselman.

Answer 1

Statement 2. comes from the following classical fact whose proof can be found in e.g. Bushnell and Kutzko, "The admissible dual of ${\rm GL}(N)$ via compact open subgroups". This is a particular case of Proposition (4.2.3), page 147 of loc. cit. I state it in your case:

The following sets are in natural bijection:

(i) equivalence classes of irreducible representations $\pi$ such that $\pi^K \not= 0$;

(ii) Isomorphism classes of simple ${\mathcal H}(G//K)$-modules.

Of course, in your case, since ${\mathcal H }(G//K)$ is commutative, its simple left modules are $1$-dimensional.

For your last statement. There are several good references for the unramified principal series. Among them:

Casselman's lecture notes : Introduction to the theory of admissible representations of $p$-adic reductive groups (available on his web page).

Casselman, W. The unramified principal series of 𝔭-adic groups. I. The spherical function. Compositio Math. 40 (1980), no. 3, 387–406.

Important remark. Your last statement, as it is stated is false, or, to be true, it depends on what you mean by "unramified principal series". If you take as a definition that an unramified principal series is with no restriction a representation parabolically induced from an unramified character of a maximal split torus, then for your statement to be true you have to replace $K$ by an Iwahori subgroup of $G$. In that case the corresponding theorem is due to A. Borel (Invent. Math. 35, 1976, 233--259). For the case of your maximal compact subgroup $K$, only the direct implication holds. A classical counter-example is the steinberg representation. It is an irreducible (sub)quotient of some unramified principal series, but it is not $K$-spherical (but of course it is Iwahori-spherical).