Inventiones math. 59, 1-11 (1980) Invelltiolle$ mathematicae 9 by Springer-Verlag 1980 On the Canonical k-Types in the Irreducible Unitary g-Modules with Non-Zero Relative Cohomology S. Kumaresan School of Mathematics, Tata Institute of Fundamental Research, Bombay 400005, India 1. Introduction Let G be a connected real semi-simple Lie group with finite center, and K a maximal compact subgroup of G. We denote by go and ko the Lie algebras of G and K respectively, g and [ stand for their respective complexifications. Let g =fGP be the corresponding Cartan decomposition and 0 the corresponding Cartan involution. Let (zt,/~) be an irreducible unitary representation of G on H. Let H denote the space of K-finite vectors in H. It is of current interest to study H(g, [:H) the relative (g, [)-cohomology of the g-module H. For this cohomology to be non-zero there is a necessary condition that the infinitesimal character of H coincides with the infinitesimal character of the trivial one dimensional representation of g. If this condition is satisfied then one knows Hi(g,t:H) =Homr(Aip, H). In particular an irreducible t-component of Aip occurs in H. The precise qualitative nature of such irreducible components is not known in general (see Remark below). The same problem arises in another context also. Let F be a discrete subgroup of G such that FIG/K is a compact manifold. Then one knows that G acts on I-?(FIG) via right regular representation and that as G-representation spaces, L2(FIG)= @Nr(r0n, where Nr(r0=0 ~EG except for countably many reed and Nr(n ) is finite for all ned. Here d is the set of equivalence classes of irreducible unitary representations of G. Matsushima's formula is then given as follows: dim Hi(FIG~K)= ~. Nr(~) dim Hom~(A~p,H) ~EG gr== go Here Xo stands for the infinitesimal character of the trivial one dimensional representation of G. Thus we are again led to consider Homt(Aip, H), for H an irreducible g-unitary module with ZH=Zo. 0020-9910/80/0059/0001/$02.20