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One can see the following two equations,

  • Theorem 6.1 (Selberg Trace formula) on page 26 of these notes.

  • Equation 3.19 and 3.20 on page 11 of this paper.

I vaguely feel that these two are the same statements but I can't completely get the second from the first. It would be great if someone can help connect the dots..

I guess that the equation in the second reference is the special case of the first with the Fuschian group $\Gamma$ being set to just the identity element but still there are some gaps - like how does one get the correct "h" function?

Also 3.20 is bit more tricky..

In the first reference there are a few steps in the derivation towards this theorem 6.1 that I am not clear about and it would be great if someone can help fill the gaps,

  • Like how does equation 6.7 really arise? (..I don't understand that $y^r$ factor there..) I mean when given the integral kernel $K$ (as in equation 6.1) from there how does one derive the function $h$ as defined in 6.7?

    In my second attached reference I guess the equivalent $K$ is the Laplacian?

  • In equation 6.8 in the first reference can one help understand what is the last sum $\sigma \in \Gamma/\langle g \rangle$

  • I am also confused about the second set of equation at the top of page 26 - how is that integral involving square-roots written down?

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The physics paper regularizes the volume and I don't expect a straight forward translation between the Selberg trace formula setting for finite volume Riemann surface and the regularized upper halfplane setting (not involving a Fuchsian group at all). The measure $\lambda \tanh(\lambda)$ is closely related to the Plancherel measure of $\mathbb{H}=SL_2(\mathbb{R})/SO(2)$, which naturally turns up in the spectral analysis of the Casimir operator of $\mathfrak{sl}_2$ from which you can derive the Laplacian.

So no, $\Gamma= \{1\}$ is not allowed in the context of the Selberg trace formula because $\Gamma \backslash \mathbb{H}$ needs to be finite-volume with respect to the measure $\frac{dxdy}{y^2}$.

For your remainig questions, I suggest additional references: Iwaniec- "Spectral theory of automorphic forms" Chapter 10 or Deitmar-Echterhoff - "Principles of Harmonic Analysis" Chapter 11 or Hejhal "The Selberg trace formula Volume I for $PSL(2, \mathbb{R})$".

  • 6.7 is a definition. Usually, $K$ denotes kernel.

  • 6.8 is a partition into conjugacy classes. The group $\langle g \rangle$ should denote the centralizer. $g = \gamma_0$ is more common. The centralizer is a cyclic group and $g$ is its generator (we have $g^n =\gamma$ for some $n$). Conjugating by elements from the centralizer doesn't add anything new, so should be moded out;)

  • For the last equation, look at the suggested reference.

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  • $\begingroup$ So can you explain how the equations 3.19 and 3.20 can be derived? (independently of the Selberg trace formula?) I have seen a derivation of the spectral measure of the upper half plane being $\pi \lambda tanh(\pi \lambda)$ but $3.19$ is not really that and hence I am wondering where the difference comes from... $\endgroup$
    – user6818
    Commented Aug 13, 2013 at 13:02
  • $\begingroup$ The article suggest further references. I has nothing to do with the trace formula for sure, but only with the Plancherel measure which enters into the trace formula only as one ingredient. The translation between irreducible representations goes as follows. You only restrict to unitary irreducible representation of SL(2,R) with $SO(2)$-invariant vectors. Those are called spherical. You obtain principal series representation. The complementary series representation don't turn up. Their $SO(2)$-invariant vectors correspond to eigenfunctions of the Laplacian. This is the representation... $\endgroup$
    – Marc Palm
    Commented Aug 13, 2013 at 13:06
  • $\begingroup$ theoretic perspective. For a purely analytic proof, I am the wrong person to ask. $\endgroup$
    – Marc Palm
    Commented Aug 13, 2013 at 13:11
  • $\begingroup$ Even in their references I never see this curious Vol(H^2)/(4\pi) factor - that is totally mysterious. [...thats why I remembered the Selberg trace formula since it produces similar prefactors..] $\endgroup$
    – user6818
    Commented Aug 13, 2013 at 13:27
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    $\begingroup$ No, this factor in the physics paper is not a volume. To call it volume is confusing. The volume of $\mathbb{H}$ is infinite. The authors simply encode the chosen normalization of the Plancherel measure in "vol(H^2)". $\endgroup$
    – Marc Palm
    Commented Aug 13, 2013 at 13:42

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