Journal of Lie Theory

\newcommand{\fn}{\mathfrak n} In prior work an orbit method, due to Pukanszky and Lipsman, was used to produce an injective mapping $\Psi\colon \Delta(K,N)\rightarrow \fn^*/K$ from the space of bounded $K$-spherical functions for a nilpotent Gelfand pair $(K,N)$ into the space of $K$-orbits in the dual for the Lie algebra $\fn$ of $N$. We have conjectured that $\Psi$ is a topological embedding. This has been proved for all pairs $(K,N)$ with $N$ a Heisenberg group. A nilpotent Gelfand pair $(K,N)$ is said to be {\em irreducible} if $K$ acts irreducibly on $\fn/[\fn,\fn]$. In this paper and its sequel we will prove that $\Psi$ is an embedding for all such irreducible pairs. Our proof involves careful study of the non-Heisenberg entries in Vinberg's classification of irreducible nilpotent Gelfand pairs. Part I concerns generalities and six related families of examples from Vinberg's list in which the center for $\fn$ can have arbitrarily large dimension.