Let \(\mathcal{P}\) be the primal manifold, defined as the compact abelian topological group
\[ \mathcal{P} := \ker(|\cdot| : C_{\mathbb{Q}} \to \mathbb{R}_{>0}), \]where \(C_{\mathbb{Q}} = \mathbb{A}_{\mathbb{Q}}^\times / \mathbb{Q}^\times\) is the idele class group over \(\mathbb{Q}\), and \(|\cdot|\) is the idele norm. As established, \(\mathcal{P} \cong \hat{\mathbb{Z}}^\times = \prod_p \mathbb{Z}_p^\times\), where the product runs over all primes \(p\), and \(\mathbb{Z}_p^\times\) denotes the group of units in the \(p\)-adic integers.
Let \(\mu\) be the normalized Haar measure on \(\mathcal{P}\), so that \(\mu(\mathcal{P}) = 1\). The Hilbert space \(L^2(\mathcal{P})\) consists of all measurable functions \(f : \mathcal{P} \to \mathbb{C}\) such that
\[ \|f\|_2^2 = \int_{\mathcal{P}} |f(x)|^2 \, d\mu(x) < \infty, \]equipped with the inner product
\[ \langle f, g \rangle = \int_{\mathcal{P}} f(x) \overline{g(x)} \, d\mu(x). \]The Pontryagin dual of \(\mathcal{P}\), denoted \(\widehat{\mathcal{P}}\), is the discrete abelian group of all continuous homomorphisms \(\omega : \mathcal{P} \to S^1\), where \(S^1 = \{ z \in \mathbb{C} : |z| = 1 \}\) is the circle group. These \(\omega\) are precisely the unitary characters of \(\mathcal{P}\).
The set \(\{ \omega : \omega \in \widehat{\mathcal{P}} \}\) forms an orthonormal basis for \(L^2(\mathcal{P})\). For any \(f \in L^2(\mathcal{P})\),
\[ f = \sum_{\omega \in \widehat{\mathcal{P}}} \hat{f}(\omega) \, \omega, \]where the Fourier coefficients are
\[ \hat{f}(\omega) = \langle f, \omega \rangle = \int_{\mathcal{P}} f(x) \overline{\omega(x)} \, d\mu(x), \]and the series converges in the \(L^2\)-norm. Moreover, Parseval's identity holds:
\[ \|f\|_2^2 = \sum_{\omega \in \widehat{\mathcal{P}}} |\hat{f}(\omega)|^2. \]Since \(\mathcal{P}\) is a compact abelian topological group, the result follows from the Peter-Weyl theorem applied to abelian groups (or directly from Pontryagin duality and Fourier analysis on locally compact abelian groups).
Orthogonality of Characters: For distinct \(\omega_1, \omega_2 \in \widehat{\mathcal{P}}\),
\[ \langle \omega_1, \omega_2 \rangle = \int_{\mathcal{P}} \omega_1(x) \overline{\omega_2(x)} \, d\mu(x) = \int_{\mathcal{P}} (\omega_1 \omega_2^{-1})(x) \, d\mu(x). \]Since \(\omega_1 \omega_2^{-1} \neq 1\) (the trivial character) and is continuous, the integral vanishes by the uniqueness of the Haar measure and the fact that non-trivial characters have mean zero over compact groups. If \(\omega_1 = \omega_2\), then \(\omega_1 \omega_2^{-1} = 1\), and the integral equals \(\mu(\mathcal{P}) = 1\).
Density and Completeness: The linear span of \(\{ \omega : \omega \in \widehat{\mathcal{P}} \}\) is dense in the space of continuous functions on \(\mathcal{P}\) (by the Stone-Weierstrass theorem, as the characters separate points and form an algebra). Since continuous functions are dense in \(L^2(\mathcal{P})\) (as \(\mathcal{P}\) is compact), and the characters are orthonormal, they form a complete orthonormal basis by the Hilbert space theory of orthonormal sets.
Fourier Expansion and Parseval: For any \(f \in L^2(\mathcal{P})\), the partial sums of the series \(\sum \hat{f}(\omega) \omega\) converge to \(f\) in \(L^2\)-norm, and Parseval follows from the completeness of the orthonormal basis.
The characters \(\omega \in \widehat{\mathcal{P}}\) are precisely the "arithmetic" or "unitary" components of the Hecke Grössencharaktere (grossencharacters) on \(C_{\mathbb{Q}}\). Specifically, since \(C_{\mathbb{Q}} \cong \mathbb{R}_{>0} \times \mathcal{P}\), every continuous homomorphism \(\chi : C_{\mathbb{Q}} \to \mathbb{C}^\times\) (a Hecke character) decomposes as
\[ \chi(j) = \omega(j) \cdot |j|^s, \]where \(s \in \mathbb{C}\), \(|j|^s\) is the continuous character on the \(\mathbb{R}_{>0}\) factor (extended trivially to \(\mathcal{P}\)), and \(\omega \in \widehat{\mathcal{P}}\) is the restriction of \(\chi\) to \(\mathcal{P}\) (trivial on \(\mathbb{R}_{>0}\)).
This decomposition bridges to the Riemann zeta function \(\zeta(s)\) and its generalizations via Tate's thesis. The zeta function arises as a Mellin transform or integral over the idele class group:
\[ \zeta(s) = \int_{C_{\mathbb{Q}}} |j|^{s-1} \, d^\times j, \](up to normalization and local factors), where the principal (trivial) Hecke character corresponds to \(\omega = 1\). More generally, for a Hecke character \(\chi = \omega \cdot |\cdot|^s\), the associated L-function is
\[ L(s, \chi) = \sum_{n=1}^\infty \frac{\chi(n)}{n^s} = \prod_p (1 - \chi(p) p^{-s})^{-1}, \](analytically continued), encoding arithmetic information through \(\omega\).
The decomposition of \(L^2(\mathcal{P})\) via \(\{ \omega \}\) thus provides the spectral foundation for harmonic analysis on \(\mathcal{P}\), linking the "arithmetic spectrum" (characters \(\omega\)) to the analytic properties of \(\zeta(s)\) and L-functions, including the functional equation and the distribution of zeros. This step formalizes the transition from the group-theoretic structure of \(\mathcal{P}\) to the analytic continuation and pole residues in the zeta function, as the Fourier transform on \(\mathcal{P}\) (via Pontryagin duality) mirrors the decomposition of automorphic forms or theta functions in the adelic setting.