TITLE: Analytic continuation of intertwining operator
QUESTION [7 upvotes]: 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?

REPLY [2 votes]: 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.