The L 2 -Index Theorem for Homogeneous Spaces of Lie Groups

Abstract

The L2-index theorem for covering spaces of Atiyah [3] and Singer [20] asserts that, given a discrete group F acting smoothly and freely on a manifold M with compact quotient M = F M, and an elliptic differential operator D on M which is F -invariant and thus descends to an elliptic differential operator D on the compact manifold M, then the P-index indrD = dimrKerD dimrKerD* of D coincides with the ordinary index ind D = dim Ker D dim Ker D* of D. Here Ker D is the space of L2-solutions of Du = 0, and dim r denotes the dimension function corresponding to the trace (on the commutant of F acting on the Hilbert space of L2-sections over M) naturally associated to P. The importance of this theorem lies in the fact that indD > 0 implies the existence of nontrivial L2-solutions for the equation Du = 0, and as such it was used in a crucial way by Atiyah and Schmid [5] to construct explicit realizations of the discrete series representations for semisimple Lie groups. Indeed, if G is a Lie group which possesses a discrete, torsion-free, cocompact subgroup F, and H is a compact subgroup of G, then the L2-index theorem applied to the covering space M = G/H of M =F G/H, combined with the index formula of Atiyah-Singer [6], yields existence results for L2-solutions of G-invariant elliptic equations on the homogeneous space G/H. It is relevant to note that, with D denoting this time a G-invariant elliptic differential operator on G/H, the ratio between the F-index of D and the covolume of F gives a number independent of F. This real number, indG D, can be in fact intrinsically defined as the difference of the two formal degrees corresponding to the representations of G on Ker D and Ker D* respectively, and hence makes sense for any unimodular Lie group G, even when it has no discrete, cocompact subgroups (which, outside the semisimple case, is the generic