Diamond representations of rank two semisimple Lie algebras

Abstract

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.