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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?


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2 Answers 2

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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.

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My PhD thesis contains a lot of similar computations. You can find it on my homepage. Best, MP. –  Marc Palm Feb 11 '13 at 13:19
the whole spectral side does not vanish. It picks out those automorphic representations whose infinity component is discrete series. These correspond to holomorphic (and anti holomorphic) modular forms on a congruence subgroup. –  Venkataramana Feb 11 '13 at 13:27
No, these traces vanish as well. The trace of a discrete series rep is essentially that of a principal series representations, except that you take away certain $K$-types in the $K$-expansion. The trace of a principal series representation equals an integral over the Levisubgroup after a $K$-expansion. –  Marc Palm Feb 11 '13 at 13:37
There is no nice analogy between conjugacy classes and irrep here:) –  Marc Palm Feb 11 '13 at 13:39
I do not have access to the library right now (late in the evening here). I think in Lang's book, traces of functions like $f$ on discrete series reps are computed; the only time these traces are non-zero are on the space I described. By the Selberg trace formula (geometric side) for such functions, all terms except identity vanish. This proves in fact that discrete series multplicities for congruence subgroups $\Gamma$ grow like the volume of $\Gamma$. I suggest you see the arguments of Speh-Rohlfs or Degeorgi-Wallach where more general things like these are proved. –  Venkataramana Feb 11 '13 at 13:59

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.

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