vanishing of spectral term in Arthur-Selberg trace formula for GL(2)? Hi,
In the Arthur-Selberg trace formula for $G = GL(2)/\mathbf Q$ (as seen for example in
Gelbart's "Lectures on the Trace Formula"), the spectral side includes terms
like:
$$
\int_{-\infty}^\infty tr (\rho(\mu,it)(f))dt
$$
where $f$ is the test function, $\mu$ is a Hecke character, and $\rho(\mu,s)$
is the induced representation
$$
Ind_{B(\mathbf A)}^{G(\mathbf A)}\mu(a_1/a_2)|a_1/a_2|^s_{\mathbf A}
$$
where we write elements of $B(\mathbf A)$ (= the standard Borel of $G$) as having $a_1$, $a_2$ in the diagonal. This
term appears both in the hyperbolic and unipotent spectral terms.
I would like to understand under which condition this term vanishes. In particular, is 
there a condition on $f_{\mathbf R}$ (the infinite component of my test function $f$) that would imply the vanishing of this term?
Thanks!
 A: The space of regular semi-simple elements in the infinite component $GL_2({\mathbb R})$ which are elliptic is an open set. If you take a compactly supported smooth function $f$ whose support in the reals lies in this open set, the contribution of your term vanishes. This is used, for example,in the article by Jacquet and Gelbart in the Corvallis volume. 
Of course, this is not the only condition that ensures this. Since I am not very familiar with these questions, I think someone more "automorphic" may be able to tell you weaker conditions.  
A: Sorry, my original answer was adressing vanishing of the contribution of the continuous spectrum. You are asking for something different.
If the $\infty$ component is a pseudo coefficient of the discrete series representations, the contribution you ask for vanish by the definition of a pseudo coefficient and the fact that
$$ tr \  Ind_B^G (\mu) (\phi)  = \prod\limits_v tr \ Ind_{B_v}^{G_v} (\mu_v) (\phi_v).$$
Another natural thing to choose is a pseudo coefficient of square-integrable reps at one or several $p$-adic places. Then as well your distribution vanishes.
The existence of pseudo-coefficients was proved by Delorme, Arthur, Kahzdan etc. I have constructed them explicitly for GL(2) in my PhD thesis, but the idea is essentially already in Hejhal's book.
