Résumé : If $Q$ is a non degenerate quadratic form on ${\bb C}^n$, it is well known that the differential operators $X=Q(x)$, $Y=Q(\partial)$, and $H=E+\frac{n}{2}$, where $E$ is the Euler operator, generate a Lie algebra isomorphic to ${\go sl}_{2}$. Therefore the associative algebra they generate is 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 a generalization where $Q(x)$ is replaced by $\Delta_{0}(x)$ which is a relative invariant of a multiplicity free representation $(G,V)$ with a one dimensional quotient (see definition below). Over these spaces we study various algebras of differential operators. In particular if $G'=[G,G]$ is the derived group of the reductive group $G$, we prove that the algebra $D(V)^{G'}$ of $G'$-invariant differential operators with polynomial coefficients on $V$, is a quotient of a Smith algebra over its center. Over ${\bb C}$ this class of algebras was introduced by S.P. Smith as a class of algebras similar to ${\cal U}({\go s}{\go l}_{2})$. This allows us to describe by generators and relations the structure of $D(V)^{G'}$. As a corollary we obtain that various "algebras of radial components" are quotients of ordinary Smith algebras over ${\bb C}$. We also give the complete classification of the multiplicity free spaces $(G,V)$ with a one dimensional quotient, and pay particular attention to the subclass of prehomogeneous vector spaces of commutative parabolic type, for which further results are obtained.