I have the following situation: $G$ is a real unimodular locally compact semisimple Lie group. Then it is known that the regular representation $H:=L(G,\mu_H)$ decomposes as
\begin{equation}
\int^{\oplus}_{\hat{G}} H_\lambda^*\otimes H_\lambda \, d\mu_P.
\end{equation}
On the other hand, by the nuclear spectral theorem (btw, do you know a nice reference with rigorous proof?), for $A:D(A) \subseteq H \to H$ being a densely defined self-adjoint operator, there exists a dense nuclear subspace $\Phi(A) \subseteq H$, s.t. $H$ decomposes as a direct integral over generalized eigenspaces $\tilde{H}_\lambda$ of $A$, s.t. we get maps $\Phi(A) \to \tilde{H}_\lambda$ for every $\lambda \in \sigma(A)$. Now, I'm looking for such an operator $A$, such that the support of the above Plancherel measure coincides with the generalized spectrum of $A$. I know that the Laplacian is the obvious choice here, but I'm not sure if this is always possible (self-adjointness, etc.). Also, do you know a good reference where harmonic analysis and the theory of generalized functions are combined?