Unitary representations of the Lie superalgebra osp(1|2n) and parabosons

It is known that there is a close connection between the Fock space of n pairs of boson operators B± i (i = 1, 2,..., n) and the so-called meta-plectic representation V (1) of the Lie superalgebra osp(1|2n) with lowest weight (1/2, 1/2,..., 1/2). On the other hand, the defining relations of osp(1|2n) are equivalent to the defining relations of n pairs of paraboson operators b± i. In particular, with the usual star conditions, this implies that the “parabosons of order p ” correspond to a unitary irreducible (infinite-dimensional) lowest weight representation V (p) of osp(1|2n) with lowest weight (p/2, p/2,..., p/2). Apart from the simple cases p = 1 or n = 1, these representations had never been constructed due to computational dif-ficulties, despite their importance. We have now managed to give an explicit and elegant construction of these representations V (p), and can present explicit actions or matrix elements of the osp(1|2n) generators. Essentially, V (p) is constructed as a quotient module of an induced module. In all steps of the construction and for the chosen basis vectors, the subalgebra u(n) of osp(1|2n) plays a crucial role.