# Can characters occur in automorphic representation

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?

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