
In our classification, the main family of uniserial $\mathfrak{g}_m$modules is actually constructed for any $\mathfrak{g}=\mathfrak{s}\ltimes V(\mu)$, where $\mathfrak{s}$ is a semisimple Lie algebra and $V(\mu)$ is the irreducible $\mathfrak{s}$module with highest weight $\mu\neq 0$. It turns out that the members of this family are, but for a few exceptions of lengths 2, 3 and 4, the only uniserial $\mathfrak{g}_m$modules.
One major step towards this classification is the determination of all admissible sequences of length 3, these are sequences $V(a),V(b),V(c)$ for which there is a uniserial $\mathfrak{g}_m$module with these composition factors. This step depends in an essential manner on the determination of certain nontrivial zeros of RacahWigner $6j$symbol.
\begin{thebibliography}{RBMW}
\bibitem[DR]{DR} A. Douglas, J. Repka, \emph{Embedding of the Euclidean algebra e(3) into sl(4,C) and restriction of irreducible representations of sl(4,C)}, Journal of Mathematical Physics \textbf{52} 013504 (2011).
\bibitem[Pi]{Pi} A. Piard, \emph{Sur des repr\'esentations ind\'ecomposables de dimension finie de $SL(2).R^2$}, Journal of Geometry and Physics, Volume \textbf{3}, Issue 1, 1986, 153.
\end{thebibliography}
This is joint work with J. P. Cossey.
Several times these groups afford Bruhatlike presentations. This is the case when $A$ is an artinian simple involutive ring, and when $A$ admits a weak $*$analogue of the euclidean algorithm for coprime elements. A very general Weil representation can be constructed in this case, from abstract core data, (recovering as particular cases, the Weil representations of the symplectic groups $Sp(2n,k)$ for $k$ a finite field and the generalized Weil representation of a nonclassical case of an involutive base ring having a nilpotent radical).
When a presentation is not at hand a different but also elementary approach to the construction of Weil representations, which is more geometric in nature, can be applied.