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 *H*^{2}(*G*,**T**) (with respect to the trivial
action of *G* on the circle group *T*) and the Pontryagin dual of
the homology group *H*_{1}(*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.