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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).
Chuah et al. (Mon,) studied this question.