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I was trying to understand the paper "Forms of GL(2) from the analytic point of view", by Gelbart and Jacquet.

On Page 226 in Remark (4.13) they mention that the kernel of the local intertwining operator $M(\eta_{\nu})$ has codimension one.

However, I cannot immediately see this. Any ideas?

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    $\begingroup$ I think this is nontrivial, e.g. one has to look at the Jacquet modules of the reducible principal series representations $\mathbf{H}(\eta_v)$ and $\mathbf{H}(\tilde\eta_v)$. See Theorem 8.12.22 in Goldfeld-Hundley's Volume I, and also Exercise 4.5.5 in Bump: Automorphic forms and representations. $\endgroup$
    – GH from MO
    May 28, 2015 at 11:41
  • $\begingroup$ @GHfromMO Thanks for the precise reference. However, I cannot figure out the relation between Jacquet modules and the kernel of the intertwining operator. $\endgroup$
    – Vishal Gupta
    Jun 1, 2015 at 8:27

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This question is from a while ago, but for what it's worth, I agree with GH from MO: the fact in question is well-known but not particularly obvious. We didn't do any intertwining operators in the book (regrettably!) but you can see what you wanted fairly easily from the Bruhat pairing in corollary 8.10.11. The point is: you have this pairing between a principal series representation and its contragredient (which is another principal series representation). At the reducibility point you're interested in, the contragredient has a one dimensional invariant subspace corresponding to a scalar composed with det. The statement is that the kernel of the intertwining operator is precisely the functions orthogonal to this subspace of the contragredient. (This perp space is clearly codim 1.) Now look at the Bruhat pairing and assume that $f_2$ factors through det. Vanishing of the pairing should be equivalent to vanishing of the int. op. (You'll probably need to use invariance of the pairing.)

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  • $\begingroup$ Its a little late, but thanks for taking the time to answer. $\endgroup$
    – Vishal Gupta
    Jan 19, 2016 at 10:24
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Another point is that (by Borel-Casselman-Matsumoto) in general subrepns and quotients of unramified principal series (for p-adic fields) are generated by their Iwahori-fixed vectors. For $GL_2$, there are only two of these. As in Casselman's 1980 Compositio paper, one can track the spherical vector, and see that for "typical" parameters it generates the whole, and the whole principal series is irreducible. The only parameter values for which the spherical vector fails to generate the whole turn out to be such that the spherical vector is determinant composed with a character, and this is certainly a one-dimensional subrepn.

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