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Harmonic analysis is a generalisation of Fourier analysis that studies the properties of functions. Check out this tag for abstract harmonic analysis (on abelian locally compact groups), or Euclidean harmonic analysis (eg, Littlewood-Paley theory, singular integrals). It also covers harmonic analysis on tube domains, as well as the study of eigenvalues and eigenvectors of the Laplacian on domains, manifolds and graphs.

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Quotient space by discrete group and $L^2(\Gamma\backslash \mathcal{H})$

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 g …
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