# Are local representations of an automorphic representation always infinite dimensional?

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 .

• 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. Apr 27, 2019 at 12:33

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 $$p\nmid N$$

• 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?
– user395911
Apr 26, 2019 at 19:05
• @zzy See the edit. Does that answer your question? Apr 26, 2019 at 21:12
• @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). Apr 26, 2019 at 21:19
• Thank you, I believe that for any irreducible cuspidal automorphic representation of $GL_2(\mathbb A)$, every local representation is infinite dimensional.
– user395911
Apr 26, 2019 at 21:21
• 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. Apr 27, 2019 at 12:31

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.

• 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? Aug 28, 2019 at 23:56