Take the tour ×  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.     • 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?  share|improve
this question add comment  1 Answer  up vote 4 down vote
accepted  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.  share|improve this
answer All I wanted to know. Thank you! –  Marc Palm Apr 13 '12
at 7:44 add comment  Your Answer