Centers and characters of Jacobi group-invariant differential operator algebras

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.