Multi-Component NLS Models on Symmetric Spaces: Spectral Properties versus Representations Theory

The algebraic structure and the spectral properties of a special class of multi-component NLS equations, related to the symmetric spaces of $\mathbf{BD.I}$-type are analyzed. The focus of the study is on the spectral theory of the relevant Lax operators for different fundamental representations of the underlying simple Lie algebra $\mathfrak g$. Special attention is paid to the structure of the dressing factors in spinor representation of the orthogonal simple Lie algebras of $\mathbf B_r\simeq so(2r+1,\mathbb C)$ type.