Equivariant index theorem

Let a compact Lie group $G$ act on a closed manifold and on the vector bundles of a $G$-equivariant elliptic differential operator $D$. Its finite-dimensional kernel and cokernel define the virtual representationAn equivariant index theorem computes this character of a representation from local principal symbol and curvature data on the fixed-point set of $g$. At the identity element it recovers the ordinary Atiyah-Singer index theorem; for isolated fixed points the fixed-set integral becomes a finite sum.

For example, the holomorphic fixed-point formula for the trivial line bundle on a compact complex manifold giveswhen the fixed points are nondegenerate. On the Riemann sphere, $f(z)=az$ with $|a|=1$ and $a\ne 1$ has fixed points zero and infinity, with tangent multipliers $a$ and $a^{-1}$. Their contributions areThe global answer is also one: only the constant holomorphic functions contribute. Local fixed-point terms can thus be nonintegral complex numbers even though their sum is the character of an actual virtual representation.^[ref]