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:

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

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.

sphericalHecke algebra, not "Iwahori-Hecke". For the Iwahori-Hecke algebra, I'd mean the left-and-right $J$-invariant compactly supported functions, with $J$ an Iwahori subgroup... $\endgroup$