TITLE: When are the principal series of GL_2(F) square integrable?
QUESTION [9 upvotes]: Let $F$ be a local field, $G=GL_2(F)$, $\chi_1,\chi_2$ quasicharacters, and $B(\chi_1,\chi_2)$ be the principal series (we are assuming $\chi_1,\chi_2$ to be such that the representation is irreducible).
Is $B(\chi_1,\chi_2)$ a constituent of $L^2(Z\backslash G)$?
I'm pretty sure it never is, but can't prove it. Am familiar with Bump up to 4.5.
On the other hand, say $\chi_1\chi_2^{-1} = |\cdot |^{\pm 1}$ (so not a principal series). Same question?
This time I think it is...

REPLY [2 votes]: If the principal series would be square integrable, they would be discrete points in the Fell topology. It can be seen that the $B(\chi_1, \chi_2)$ varies continuously with the pair $(\chi_1, \chi_2)$ with the usual topology of local characters as long as it is irreducible.
Now, the point $\chi_1 \chi_2^{-1}$ being the norm or its inverse, the representation contains the Steinberg representation (times a one-dim'l twist) as a submodule or -quotient, being not irreducible anymore. Otherwise, it is irreducible.
The Steinberg (times a one-dim'l twist) and the supercuspidal representations are square-intgerable, if the central character is unitary.