Take $F$ a local field and $\chi_1, \chi_2$ two characters, write $M(\chi_1, \chi_2)$ for the standard intertwining integral $$M(\chi_1. \chi_2).f(g) := \int_{F} f\left( \begin{pmatrix} 0&-1\\ 1& 0 \end{pmatrix} \begin{pmatrix} 1& x \\ 0 & 1 \end{pmatrix}g \right) \, dx.$$ (Which varies meromorphically as a function of $\chi_1,$ and $\chi_2$.) I would expect that if $L(\eta)$ denotes the usual local $L$ function attached to a character $\eta,$ then $L(\frac{\chi_1}{\chi_2})^{-1} M(\chi_1, \chi_2)$ is holomorphic and never the zero operator. That is, if I've formulated correctly this is a pretty easy check when $F$ is nonarchimedean and immediate from the explicit formula given in (6.15) of Bump's Automorphic forms and representations if $F=\mathbb R.$ So, I imagine it holds and is well known for $\mathbb C$ as well, but does anyone have a reference?
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$\begingroup$ Did you consult Arthur's early papers on the trace formula? $\endgroup$– user1688Commented Jan 8, 2016 at 16:59
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$\begingroup$ A little. I didn't come across anything GL(2) (or SL(2)) specific. It seemed like this simplest case would probably be simpler and older. $\endgroup$– Joseph HundleyCommented Jan 9, 2016 at 14:37
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$\begingroup$ Why do you expect the normalized operator, as you have defined, to be nonzero? Shouldn't it have a zero at the special representations, i.e. $\chi_1\chi_2^{-1}=p^{\pm1}$? Most probably to get a non-vanishing holomorphic operator one normalizes by $\epsilon(0,\chi_1\chi_2^{-1})\frac{L(1,\chi_2\chi_1^{-1})}{L(0,\chi_1\chi_2^{-1})}.$ $\endgroup$– Subhajit JanaCommented Dec 6, 2016 at 17:30
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1 Answer
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This is on page 57-58 of M. Duflo, "Representations irreductibles des groupes semisimples complexes", in "Analyse harmonique sure les groupes de Lie", SLN 497, 1975, 26-88. V. Drinfeld also noted in email to me (2014) that Zelobenko wrote up SL(2,C) in 1963, in Izv. Akad. Nauk SSSR Ser. Mat. 27, 1343-1394, but I did not corroborate the latter. Wallach states the outcome for SL(2,C) in his article in Corvallis.
(A more recent working of this exercise is at http://www.math.umn.edu/~garrett/m/v/intertwinings_SL2C.pdf )
answered Jan 8, 2016 at 17:20