Harmonic analysis of compact Lie supergroups

We realize the irreducible representations of a compact Lie supergroup G, with a contragredient simple Lie superalgebra, in the space of square integrable (in the sense of Berezin) holomorphic sections on X=GA, A is the real torus in the complexification of G. We give an explicit realization of unitary representations when G=SU(1|1).