A note on the multipliers and and projective representations of semi-simple Lie groups

Abstract

We show that, for any connected semi-simple Lie group G, there is a natural isomorphism between the Galois cohomology H²(G, T) (with respect to the trivial action of G on the circle group T) and the Pontryagin dual of the homology group H₁(G) (with integer coefficients) of G as a manifold. As an application, we find that there is a natural correspondence between the projective representations of any such group and a class of ordinary representations of its universal cover. We illustrate these ideas with the example of the group of bi-holomorphic automorphisms of the unit disc.