Index theorem for homogenous spaces of Lie groups

We study index theory on homogeneous spaces associated to an almost connected Lie group in terms of the topological aspect and the analytic aspect. On the topological aspect, we obtain a topological formula as a result of the Riemann-Roch formula for proper cocompact actions of the Lie group, inspired by the work of Paradan and Vergne. On the analytic aspect, we apply heat kernel methods to obtain a local index formula representing the higher indices of equivariant elliptic operators with respect to a proper cocompact actions of the Lie group.