Equivariant $\eta$-invariants on homogeneous spaces

Abstract.

Let D be a homogeneous Dirac operator on the quotient M = G/H of two compact connected Lie groups. We construct a deformation $\tilde D$ ofD and calculate its equivariant $\eta$-invariant $\eta_G(\tilde D)$ explicitly on the dense subset $G_0$ of G that acts freely onM. On $G_0$, $\eta_G(\tilde D)$ and $\eta_G(D)$ differ only by a virtual character of $G$. Moreover, if $G\supset H$ is a symmetric pair or if D is the untwisted Dirac operator, then $\eta_G(D)=\eta_G(\tilde D)$ on $G_0$. We also sketch some applications of $\eta_G(\tilde D)$.