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!