Complete Nevanlinna-Pick property of $\mathbb K$-Invariant Reproducing Kernels

Let $\Omega$ be a Cartan domain and $K = \sum_{\underline s}a_{\underline s}K_{\underline s}$ be a $\mathbb K$-invariant kernel on $\Omega$. In this article, we first obtain a necessary condition on $K$ to have the complete Nevanlinna-Pick property in terms of the sequence $\{a_{\underline s}\}_{\underline s}$ with the assumption that each $a_{\underline s}$ is non-zero and $K$ is non-vanishing. This generalizes the well-known Kaluza's Lemma in the context of $\mathbb K$-invariant kernels. The notion of the characteristic function of the classical Sz.-Nagy--Foias Theory is extended to a commuting tuple of $\frac{1}{K}$-contraction where $K$ is an irreducible $\mathbb K$-invariant kernel. An explicit construction of the characteristic function of a $\frac{1}{K}$-contraction is provided. A characterization of a $\mathbb K$-invariant kernel with the complete Nevanlinna-Pick property is obtained via the existence of characteristic functions associated with $\frac{1}{K}$-contractions.