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
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

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 , , 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 . 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.