Are local representations of an automorphic representation always infinite dimensional?

Question

We know an irreducible admissible automorphic representation $\pi$ of $GL_2$ decomposes to tensor product of local representations, and almost every local representation is spherical. To define the L function of $\pi$, I need to know every local spherical representation is in spherical principle series rather than $1$-dimensional. How to show that is true? Maybe this follows from existence of Whittaker model, but I don't know how to rule out the $1$-dimension possibility .

Answer 1

Three things:

1. If $\pi$ is an irreducible finite dimensional smooth representation of $G(\mathbb{Q}_p)$ with $G$ reductive and split (note that reductive implies that irreducible+smooth implies admissible in general, but we're finite dimensional here so this is a non-issue) then $\pi$ is a character.
2. It is certainly not true that irreducible spherical representations are $1$-dimensional. I think what you mean is that if $K_0\subseteq G(\mathbb{Q}_p)$ is hyperspecial, and $\pi$ is a $K_0$-spherical representation of $G(\mathbb{Q}_p)$ then $\pi^{K_0}$ is $1$-dimensional. This follows from the fact (deduced from the Satake isomorphism) that the spherical Hecke algebra is commutative.
3. This last fact (that $K_0$-spherical representations are unramified prinicipal series) is true. See Theorem 3.8 of Cartier's Corvallis article, or Theorem 1.2.12 of Blasius and Rogawski's article in Motives Vol. II. I think in the Cartier article he gives a reference to the original general proof due to Casselman.

EDIT: To address the question in the comments if you look at this nice note you can find that if $f$ is a cuspform of level $\Gamma_1(N)$ with associated automorphic representation $\pi_f$ then $(\pi_f)_v$ is an unramified principal series for $p\nmid N$

Answer 2

Let $G$ is a connected reductive group over a number field $F.$ Suppose that $v$ is a place for which $G(F_v)$ is not compact modulo center. Let $\pi$ be an automorphic representation of $G.$ If $\pi_v$ is one-dimensional then $\pi$ is one-dimensional. This statement appears in the appendix in this paper: https://arxiv.org/abs/1610.07567. I don't totally understand the proof yet.