Projective representations of the current group \(\text{SU} (1,1)^ X\) (Q1337915)

Let \(G\) be the pseudo-unitary group SU(1,1) and let \(X\) be a topological space with continuous measure \(dx\). Then we can construct the group \(G^X\) of all measurable maps \(f : X \to G\) with pointwise multiplication. In a previous paper the authors constructed families of irreducible unitary representations of \(G^X\), associated to representations of the group \(G\) by motions of a real Hilbert space. In this paper, instead of representations of \(G\) by motions of a real Hilbert space, the authors consider representations of \(G\) by motions of a complex Hilbert space. There are two series of irreducible representations of \(G\) of this kind. Starting with these representations, two new series of irreducible unitary representations of the group \(G^X\) are constructed. Unlike the representations constructed by means of real Hilbert spaces, these representations are projective. They can be considered as square roots of the faithful representations constructed before. It is indicated that similar series of representations for the current group \(\text{SU} (n,1)^X\), \(n > 1\), can be constructed.