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

  • $\begingroup$ I would guess that you see this already at the level of finite fields. I am not so sure if the result will be independent of the residue characteristic as it is for Gl(2). Did you study what happens when restricting from Gl(2). Wheter or not it decomposes can be studied in the unitary cases via Mackey induction restriction formula. $\endgroup$
    – Marc Palm
    Jul 10 '13 at 19:44
  • $\begingroup$ Oh sorry... Mackey will not suffice here. $\endgroup$
    – Marc Palm
    Jul 10 '13 at 19:57

I dont think all of them are irreducible: when you restrict from GL(2) to SL(2) a representation can split into two. I think all the necessary information is in the following paper:

M. Tadic, Notes on representations of non-Archimedean SL(n). Pacific J. Math. 152 (1992), no. 2, 375--396.

  • $\begingroup$ Thanks! While this does answer the first question of which principal series are irreducible, I still wonder if there is a method that does not go up to $GL_2$ and then back down. $\endgroup$ Jul 12 '13 at 14:22

Paul Sally explains this in these notes:


I am afraid he excludes residue characteristic two there:(

But this seems to be more general: http://www.jstor.org/stable/2373536?seq=1

  • $\begingroup$ This also adresses Dror Speiser's follow-up question, because it works with SL(2) directly. The approach is via special functions on the $p$-adics, so fairly elementary. $\endgroup$
    – Marc Palm
    Jul 17 '13 at 11:57
  • $\begingroup$ Well, it's not really my follow up question... Sasha's final paragraph, beginning with the words "My question is", explicitly asks about the proof using Jacquet functors, which really are what the question is about (I think). Thank you for your answer, it is very helpful in piecing the ideas together. $\endgroup$ Sep 13 '13 at 15:06

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