2 Eta Invariants of Homogeneous Spaces

We derive a formula for the η-invariants of equivariant Dirac operators on quotients of compact Lie groups, and for their infinitesimally equivariant extension. As an example, we give some computations for spheres. Quotients M = G/H of compact Lie groups form a very special class of manifolds, but yet they provide many important examples of Riemannian manifolds with non-negative sectional curvature. The η-invariant has been introduced by Atiyah, Patodi and Singer in [APS] as the boundary contribution in an index theorem for manifolds with boundary. It is an important tool for the construction of subtle differential-topological invariants suitable to distinguish homeomorphic or PL-equivalent manifolds that are not diffeomorphic, see e.g. [APS], [D1], [KS]. It is therefore worthwhile to have a formula for η-invariants of homogeneous spaces that yields itself to explicit computations. First steps in this direction have been made in [G1]–[G3], however, one complicated (though local) term remained. The goal of this paper is to present a formula for the non-equivariant eta invariant η(D) that is more tractable. More generally, we also obtain a formula for the infinitesimally equivariant η-invariant ηg(D ); this invariant is the universal η-form for all families with fibrewise Dirac operator D and compact structure group G, see [G3]. Our main result is stated in Theorem 2.33 for twisted Dirac operators, and in Corollary 2.34 for the odd signature operator. As an application, we compute the infinitesimally equivariant η-invariants of the untwisted Dirac operator and the signature operator for round spheres in Theorem 1.27. In a forthcoming joint paper [GKS] with N. Kitchloo and K. Shankar, we use our formula to calculate the Eells-Kuiper invariant of the Berger space SO(5)/SO(3), which is then used to determine the diffeomorphism type. Let us sketch our method. In [G1], [G2], we presented a formula for the equivariant η-invariant of Slebarski’s deformed Dirac operator D̃ = D 1 3 ,κ ([Sl], this operator has been generalised and made popular by Kostant in [Ko]). The difference between ηG(D ) and ηG(D̃ ) is comparatively easy to control on the subset G0 ⊂ G of elements that act freely on M . In [G3], we showed how to obtain the infinitesimally equivariant η-invariant ηg(D ), which is a power series on g, from ηG(D )|G0 . In particular, one obtains the classical η-invariant as the value of ηg(D ) at 0 ∈ g. However, the formula for the difference ηX(D )−ηe−X (D) stated in [G3] unfortunately involves the integration of an equivariant differential form on M . This causes problems, because the integrand is in general not invariant under G, so that one cannot reduce the problem to a calculation at a single point in M . The new contribution in this paper is an evaluation of this integral in terms of representation theoretical data of the groups G and H and the relative position of their maximal tori, up to an equivariant Chern-Simons term, see Theorem 2.30. This is an improvement compared with respect to [G1]–[G3] for two reasons. First, our formula for η(D) involves no more 2000 Mathematics Subject Classification. Primary 58J28; Secondary 53C30.