I would like to understand the following statement taken from this paper, dealing with the so called Transverse Index Theory or in other words with the index theory for diffeomorphism invariant geometry. The authors construct second order signature operator $Q$ being a part of spectral triple which encodes the diffeomorphism invariant geometry of a given manifold. The task is to compute the Chern character of the corresponding spectral triple. Here is the statement which I would like to understand:
The operator $Q$ is in fact the image under the right regular representation of the affine group $G_1$ (which is the semidirect product of $\mathbb{R}^n$ and $GL^+(n,\mathbb{R})$ ) of a (matrix valued) hypoelliptic symmetric element in $\mathcal{U}(\mathfrak{g}_1)$
(here $\mathfrak{g}_1$ is the Lie algebra of $G_1$ and $\mathcal{U}$ stands for universal enveloping algebra). In another paper dealing with the same topic there is a similar statement which reads as follows:
Note that $Q$ is affiliated with the universal enveloping algebra of the group of affine motions of $\mathbb{R}^n$ in the sense that it is of the form $Q=R(Q_{{alg}})$ with $$Q_{alg} \in \Big(\mathfrak{A}\big(\mathbb{R}^n \rtimes \mathfrak{gl}(n,\mathbb{R})\big) \otimes End(E) \Big)^{SO(n)}$$ where $R$ is right regular representation of $\mathbb{R}^n \rtimes GL(n,\mathbb{R})$ and $E$ is unitary $SO(n)$-module.
(here I suspect that $\mathfrak{A}$ should mean teh same as $\mathcal{U}$ above). I would like to understand it in detail, where each term lives, how the right regular representation is understood here, how $E$ is defined etc. As it is written it is complete mystery for me.
