Diamond representations of rank two semisimple Lie algebras

The present work is a part of a larger program to construct explicit combinatorial models for the (indecomposable) regular representation of the nilpotent factor \(N\) in the Iwasawa decomposition of a semi-simple Lie algebra \(\mathfrak g\), using the restrictions to \(N\) of the simple finite dimensional modules of \(\mathfrak g\). Such a description is given in \cite{[ABW]}, for the cas \(\mathfrak g=\mathfrak{sl}(n)\). Here, we give the analog for the rank 2 semi simple Lie algebras (of type \(A_1\times A_1\), \(A_2\), \(C_2\) and \(G_2\)). The algebra \(\mathbb C[N]\) of polynomial functions on \(N\) is a quotient, called reduced shape algebra of the shape algebra for \(\mathfrak g\). Basis for the shape algebra are known, for instance the so called semi standard Young tableaux (see \cite{[ADLMPPrW]}). We select among the semi standard tableaux, the so called quasi standard ones which define a kind basis for the reduced shape algebra.