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I'm reading Daniel Bump's "Automorphic forms and representations" chapter 2, and in the book they define an integral over $\Gamma\backslash\mathcal{H}$ (here, $\Gamma$ is a discontinuous subgroup of $\mathrm{SL}(2,\mathbb{R})$, and $\mathcal{H}$ is the upper half plane.)

But how to define it? what is the definition of that integral? (or, is $\Gamma\backslash\mathcal{H}$ a manifold? how to prove it?)

If possible, please recommend a book or chapter of a book for elementary theory about $\Gamma\backslash\mathcal{H}$ when $\Gamma$ is a discrete subgroup of $\mathrm{SL}(2,\mathbb{R})$.

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  • $\begingroup$ When you ask "what is the definition of that integral", could you describe in your question what integral you mean? It is not clear whether the measure on $\Gamma\setminus{\mathcal H}$ is intended to be invariant or quasi-invariant with respect to the natural group action ... $\endgroup$
    – Yemon Choi
    Commented Oct 2, 2020 at 17:15
  • $\begingroup$ That said, I think Deitmar's Introduction to Harmonic Analysis has a treatment of a similar situation for the Heisenberg group (lattices inside the real Heisenberg group) and it might also have some discussion of the Selberg trace formula for the situation you describe. Unfortunately I don't have access to a copy right now to check $\endgroup$
    – Yemon Choi
    Commented Oct 2, 2020 at 17:16
  • $\begingroup$ From the answers it should be clear that it is sometimes better to read Chapter 1 first, and only then Chapter 2. Shimura's book "Introduction to the arithmetic theory of automorphic functions" explains in a detailed way that $\Gamma\backslash\mathcal{H}$ is a Riemann surface. Iwaniec's book "Introduction to the spectral theory of automorphic forms" is another good source. Finally, you might want to learn about the quotient measure in the context of locally compact groups, see e.g. www-users.math.umn.edu/~garrett/m/mfms/notes_2013-14/… $\endgroup$
    – GH from MO
    Commented Oct 3, 2020 at 21:17

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In Section 1.2 of Bump’s book ‘The Modular Group’, you can see that Bump identifies $\mathcal{H}$ with the coset $SL_2(\mathbb{R})/ SO(2)$ by means of its Iwasawa decomposition.

In exercise 1.2.6 he gives away the Haar measure on $SL_2(\mathbb{R})$, and then pushing it forward to the coset, we get the measure on $\mathcal{H}$. This measure is invariant under $SL_2(\mathbb{R})$, so it is also invariant under any of its discontinuous subgroup ‘$\Gamma$’ , and so we can define the integral over $\Gamma \backslash \mathcal{H}$ as a quotient measure. (Mostly we are only concerned with $\Gamma$ such that $\Gamma \backslash \mathcal{H}$ has a finite volume)

Note that although I’m saying define, it’s more so using a few simple theorems from harmonic analysis to say that a such a measure exists and is unique up to scalar multiplication. Things are much simpler if $\Gamma$ is a subgroup of $SL_2(\mathbb{Z})$, where one can explicitly make use of the translates of the fundamental domain to construct $\Gamma \backslash \mathcal{H}$.

As for $\Gamma \backslash \mathcal{H}$ being a manifold, the same section 1.2 has the answer, that it can be made into a compact Riemann surface by adding a finite number of points (here also we require the finite volume of $\Gamma \backslash \mathcal{H}$), one point for each ‘cusps’ of $\Gamma$.

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