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!



vanishing of spectral term in Arthur-Selberg trace formula for GL(2)?
*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.


