Weil's representation of the Lie algebra of type \(G(2)\), and representations of \(\mathrm{SL}(3)\) connected with it (Q1145200)

For a semisimple Lie algebra \(G\) its representation \(T\) of differentiations in some associative algebra \(A\) is called a multiplicative model if (1) \(T\) splits into the direct sum of irreducible finite-dimensional representations \(\rho\), (2) the multiplicity of the occurrence of \(\rho\) in \(T\) is equal to the maximal multiplicity of the weight in \(\rho\), (3) there exist \(G\)-invariant subspaces \(V\) in \(A\) such that the restriction of \(T\) onto \(V_\rho\) is equivalent to \(\rho\) and the restriction of \(T\) onto \(V_\rho V_\tau\) (the space generated by products of elements from \(V_\rho\) and \(V_\tau\)) is equivalent to \(\rho\otimes\tau\) for any irreducible finite-dimensional representations \(\rho\) and \(\tau\). Using the representation of the algebra of type \(G_2\) the multiplicative model is constructed for \(G = \mathrm{sl}_3(\mathbb R)\). A verification of properties (1)--(3) is left to the reader. This paper continues the paper of \textit{L. Biedenharn} and \textit{J. Louck} [Ann. Phys. 63, 459--475 (1971; \url{doi:10.1016/0003-4916(71)90022-4})] where a model for \(\mathrm{sl}_2(\mathbb R)\) is given. \{Remark. In the English text the translation of the last sentence from the Russian original is not correct: Read ``It most probably exists \dots'' instead of ``It clearly exists \dots''\}.