# Decomposition of $L^2(\Gamma \backslash H)$ into irreducible representations using the spectral theorem

I'm reading the introduction of An Introduction to the Trace Formula by James Arthur and wanted to understand something in the introduction.

Let $$H$$ be a unimodular locally compact Hausdorff group, and $$\Gamma$$ a discrete subgroup of $$H$$. Let $$\mathscr H = L^2(\Gamma \backslash H)$$ be the Hilbert space of measurable functions $$\phi: \Gamma \backslash H \rightarrow \mathbb C$$ satisfying $$||\phi||^2 = \int\limits_{\Gamma \backslash H} |\phi(h)|dh < \infty$$.

Assume $$\Gamma \backslash H$$ is compact. Fix $$f \in C_c^{\infty}(H)$$, and let $$K \in L^2(\Gamma \backslash H \times \Gamma \backslash H)$$ be the function defined by

$$K(x,y) = \sum\limits_{\gamma \in \Gamma} f(x^{-1}\gamma y)$$ which is a finite sum. To this kernel we can associate a compact operator $$R(f)$$ on $$\mathscr H$$ defined by

$$[R(f)\phi](x) = \int\limits_{\Gamma \backslash H} K(x,y)\phi(y)dy$$

It can be shown that this integral is equal to just $$\int\limits_H f(y)\phi(xy)dy$$.

Arthur arranges that $$R(f)$$ is a compact self adjoint operator, and claims the decomposition of $$\mathscr H$$ into a Hilbert space direct sum of irreducible subrepresentations (under the action of $$H$$ by right translation) follows from the spectral theorem for self adjoint operators. How does this follow?

• Are you familiar with what the spectral theorem for compact self-adjoint operators says? – Yemon Choi Feb 10 at 3:22
• Yes, you can find an orthonormal basis of eigenvectors for real eigenvalues which tend to zero. – D_S Feb 10 at 3:57
• OK, I must confess I've never checked how the proof of the result mentioned by Arthur goes (something like this is in an introductory book on harmonic analysis by Deitmar, IIRC?) but I think this is the same idea as in the usual proof of the Peter-Weyl theorem, see e.g. Prop. 7 in terrytao.wordpress.com/2011/01/23/… – Yemon Choi Feb 10 at 4:10
• I haven't checked carefully, but I think this is essentially in Tamagawa's paper "On Selberg's trace formula" (see Theorem 2). – Kimball Feb 11 at 19:28