The geometric realization of the irreducible square integrable representations for semisimple Lie groups (cf. [3], [6] ) and also for nilpotent Lie groups [5] suggests that, as a general phenomenon, such representations should appear as L -kernels of invariant elliptic operators. One basic problem in this respect is to decide when such a kernel is nonzero. In the compact case the basic tool for this, used in the Borel-Weil-Bott approach, is the Hirzebruch-Riemann-Roch theorem. In the noncompact case one needs an analogue of the index theorem of Atiyah-Singer [2] for noncompact manifolds. When G possesses a discrete cocompact subgroup, the L-index theorem for covering spaces of [1] and [7] provides the required analogue. Our purpose here is to give a general index theorem for homogeneous spaces of arbitrary connected unimodular Lie groups, essentially based on the index theorem for foliations [4]. So let G be a connected unimodular Lie group, and let H be a closed subgroup of G which contains the center Z of G and such that H/Z is compact. Let x be a character of Z, and let E, F be finite-dimensional unitary representations of H whose restrictions to Z are given by xDenote by E, F the corresponding (invariant) induced bundles on the homogeneous space M = G/H, and let D be an invariant elliptic differential operator from E to F. The representation of G in the kernel of D in L(M, E) is square integrable modulo the center of G (see [4]), though not necessarily irreducible. Its formal degree deg(Ker D) (as defined in [4]) is always finite, so that the analytical index of D can be defined as