I am sorry in advance if this question is not "research level".
Let $F$ be a p-adic field.
I saw, in Bumps book, a proof which I liked, showing which principal series representations of $GL(2,F)$ are irreducible. I want to modify it to use for $SL(2,F)$, but run into trouble.
The proof in Bump, shows that the twisted Jacquet ( = Whittaker) functor of our principal series rep. $V$ is at most 1-dimensional. Then assuming that $V$ has some non-trivial sub-object, we get that this sub-object (or the quotient) has trivial twisted Jacquet functor.
Then, we realize that a representation which has trivial twisted Jacquet functor, "coincides" with its Jacquet functor. This is because a representation of $N$ is the same as a sheaf on $N^*$ (the Pontryagin dual of $N$), and the twisted Jacquet functor is the fiber of this sheaf at a non-zero point (all fibers are "the same" since $A$ acts transitively - this is the crucial point).
My question is: If we deal with $SL(2,F)$, then there are "two" twisted Jacquet functors (two orbits of $A$ on $N^* - 0$). So one can not do the proof exactly as above. How does one modify the proof? Is there a reference?
Thank you, Sasha