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. We consider only slash actions with invertible indices. The corresponding 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. We also compute their characters: in rank exceeding 1 there are two, and in rank 1 there are in general five. In rank 1 we compute in addition all of their irreducible admissible representations.