Sankhya: The Indian Journal of Statistics

2000, Volume 62, Series A, Pt. 3, pp. 425--432

A NOTE ON THE MULTIPLIERS AND PROJECTIVE REPRESENTATIONS OF SEMI-SIMPLE LIE GROUPS

By

BHASKAR BAGCHI and GADADHAR MISRA, Indian Statistical Institute, Bangalore

SUMMARY. 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.

AMS (1991) subject classification. 20C 25, 22 E 46.

Key words and phrases. Semi-simple Lie groups, projective representations, multipliers.

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