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If $Q$ is a non degenerate quadratic form on ${\bb C}^n$, it is well knownthat the differential operators $X=Q(x)$, $Y=Q(\partial)$, and$H=E+\frac{n}{2}$, where $E$ is the Euler operator, generate a Lie algebraisomorphic to ${\go sl}_{2}$. Therefore the associative algebra they generateis a quotient of the universal enveloping algebra ${\cal U}({\go sl}_{2})$.This fact is in some sense the foundation of the metaplectic representation.The present paper is devoted to the study of the case where $Q(x)$ is replacedby $\Delta_{0}(x)$, where $\Delta_{0}(x)$ is the relative invariant of aprehomogeneous vector space of commutative parabolic type ($ {\go g},V $), orequivalently where $\Delta_{0}$ is the "determinant" function of a simpleJordan algebra $V$ over ${\bb C}$. In this Part I we show several structureresults for the associative algebra generated by $X=\Delta_{0}(x)$,$Y=\Delta_{0}(\partial)$. Our main result shows that if we consider thisalgebra as an algebra over a certain commutative ring ${\bf A}$ of invariantdifferential operators it is isomorphic to the quotient of what we call ageneralized Smith algebra $S(f, {\bf A}, n)$ where $f\in {\bf A}[t]$. The Smithalgebras (over ${\bb C}$) were introduced by P. Smith as "natural"generalizations of ${\cal U}({\go sl}_{2})$.

Invariant differential operators and an infinite dimensional Howe-type  correspondence. Part I: Structure of the associated algebras of differential  operators