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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 .

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    $\begingroup$ Yes 1-dimensional representations are not generic, so if you are globally generic, you cannot have a 1-dimensional local component. On the other hand, you can get 1-dimensional local components working on a quaternion division algebra. $\endgroup$
    – Kimball
    Apr 27, 2019 at 12:33

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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 $p\nmid N$

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  • $\begingroup$ Thank you, I mean there are two possibilities for irreducible spherical representations: 1-dim or infinite dimensional. An unramified 1-dim rep is spherical by definition. How do we know local reps of global automorphic one are not 1-dim? $\endgroup$
    – user395911
    Apr 26, 2019 at 19:05
  • $\begingroup$ @zzy See the edit. Does that answer your question? $\endgroup$ Apr 26, 2019 at 21:12
  • $\begingroup$ @zzy You asked for an example, didn't you? I gave an example for where it's not necessarily true that the local representation is $1$-dimensional. It's also true that they can be one dimension by taking an adelic character. I'm not sure what you're asking (also ping me so I know you responded). $\endgroup$ Apr 26, 2019 at 21:19
  • $\begingroup$ Thank you, I believe that for any irreducible cuspidal automorphic representation of $GL_2(\mathbb A)$, every local representation is infinite dimensional. $\endgroup$
    – user395911
    Apr 26, 2019 at 21:21
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    $\begingroup$ Maybe this was resolved in chat, but I read the question as asking how do you know you don't get 1-dimensional reps as local components. $\endgroup$
    – Kimball
    Apr 27, 2019 at 12:31
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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.

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  • $\begingroup$ Thanks for pointing this out! This is indeed a simpler proof to the question that the OP originally asked (and I misinterpreted)--it was also suggested by Kimball in the comments of my answer.. What about the proof are you unsure of? $\endgroup$ Aug 28, 2019 at 23:56

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