I am currently working through the details of theorem 2.10 in this paper. There are a few things which don't make sense to me, and I'm hoping someone could help me work through the details of the proof.
First, why does the Hilbert subspace $H_{R} := \overline{Lin(\pi (R) \xi)}$, which I am guessing is just the Hilbert space closure of the linear span of $\{\pi (g) \xi : g \in R \}$, decompose as $H_{R} = H_1 \bigoplus H_2$, where $H_1$ and $H_2$ are $R$-invariant subspaces with $\pi (R) | H_1$ being the identity representation and $\pi (R) | H_2$ being the regular representation? It obviously has something to do with the fact that the only indecomposable/extreme characters on the subgroup $R$ are the identity character and regular, but I can't quite make the connection yet.
Second, what exactly does it mean to say that $\pi (R)|H_2$ is the regular representation. I guess that $\pi (R)|H_1$ being the identity representation means that for each $g \in R$, $\pi (g)$ acts like the identity operator on $H_1$, but this seems like strange notation. As for what exactly it means to say $\pi (R)|H_2$ is the regular representation, I can't quite figure that out. To me, the (left) regular representation of a group $G$ is the unitary representation $\lambda : G \to U(\ell^2 G)$ defined by $\lambda (g) \delta_{x} = \delta_{gx}$, where $\{\delta_{x}\}_{x \in G}$ forms an orthonormal basis for the Hilbert space $\ell^2(G)$ of square summable functions on $G$. I am sure it have something to do with that, but I am not exactly sure.