The Sixties was a time of great ferment in topology and one of its crowning glories was the Atiyah-Singer index theorem. Independently of Atiyah and Singer's work, Bott's paper  on homogeneous differential operators analyzes an interesting example where the analytical difficulties can be avoided by representation theory.
Suppose is a compact connected Lie group and a closed connected subgroups. As in our earlier discussion of homogeneous vector bundles, a representation of gives rise to a vector bundle over the homogeneous space . Now suppose and are two vector bundles over arising from representations of . Since acts on the left on both and , it also acts on their spaces of sections, and . We say that a differential operator is homogeneous if it commutes with the actions of on and . If is elliptic, then its index
Atiyah and Singer had given a formula for the index of an elliptic operator on a manifold in terms of the topological data of the situation: the characteristic classes of , , the tangent bundle of the base manifold, and the symbol of the operator. In  Raoul Bott verified the Atiyah-Singer index theorem for a homogeneous operator by introducing a refined index, which is not a number, but a character of the group . The usual index may be obtained from the refined index by evaluating at the identity. A similar theorem in the infinite-dimensional case has recently been proven in the context of physics-inspired mathematics.