  We study the algebras of differential operators invariant with respect to the
scalar slash actions of real Jacobi groups of arbitrary rank. These algebras
are non-commutative and are generated by their elements of orders 2 and 3. We
prove that their centers are polynomial in one variable and are generated by
the Casimir operator. For slash actions with invertible indices we also compute
the characters of the IDO algebras: in rank exceeding 1 there are two, and in
rank 1 there are in general five. In rank 1 we compute in addition all
irreducible admissible representations of the IDO algebras.
