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Oct 24, 2011 at 10:01 comment added David Loeffler You can always distinguish unram twists of Steinberg from tamely ramified principal series using $a_p$, because $|a_p|$ will be $p^{(k-1)/2}$ in the former case and $p^{(k-2)/2}$ in the latter. But you can't recover the inducing characters from $a_p$ in the PS case without knowing the central character as well.
Oct 24, 2011 at 6:47 history answered Olivier CC BY-SA 3.0