Can characters occur in automorphic representation

Question

Let $\pi$ be an irreducible cuspidal automorphic representation of $GL(2)$ over a global field with factorisation $\pi = \otimes \pi_v$.

Then at most finitely many $\pi_v$ are not spherical.

Questions:

• Can it happen that $\pi_v(g) = \chi_v(\det g)$ for a unitary character of $F_v^\times$? Can it happen infinitely often?

• Is the Steinberg spherical? Can it occur as $\pi_v$? Can it appear infinitely often?

Answer 1

A cuspform for GL(2) has a Fourier-Whittaker expansion, which is to say that it has a global Whittaker model. Thus, all the local repns have Whittaker models.

Anything factoring through $\det$ does not have a Whittaker model, so cannot appear.

Steinberg repns do have Whittaker models, but are not spherical, so only finitely-many places can have them appearing, and they certainly can appear.