The index theorem for homogeneous differential operators

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 [37] on homogeneous differential operators analyzes an interesting example where the analytical difficulties can be avoided by representation theory.

Suppose $G$ is a compact connected Lie group and $H$ a closed connected subgroups. As in our earlier discussion of homogeneous vector bundles, a representation $\rho$ of $H$ gives rise to a vector bundle $G \times _{\rho} H$ over the homogeneous space $X=G/H$ . Now suppose $E$ and $F$ are two vector bundles over $G/H$ arising from representations of $H$ . Since $G$ acts on the left on both $E$ and $F$ , it also acts on their spaces of sections, $\Gamma (E)$ and $\Gamma (F)$ . We say that a differential operator $D: \Gamma (E) \to \Gamma (F)$ is homogeneous if it commutes with the actions of $G$ on $\Gamma (E)$ and $\Gamma (F)$ . If $D$ is elliptic, then its index

$$\operatorname{index}(D) = \dim \ker D - \dim \operatorname{coker}D$$

is defined.

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 $E$ , $F$ , the tangent bundle of the base manifold, and the symbol of the operator. In [37] 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 $G$ . 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.