= 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 <representation>
$$
\operatorname{ind}_G D=[\ker D]-[\operatorname{coker}D],
\qquad
\chi_D(g)=\operatorname{tr}(g|\ker D)-\operatorname{tr}(g|\operatorname{coker}D).
$$
An 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> gives
$$
\sum_q(-1)^q\operatorname{tr}(f^*|H^{0,q})
=\sum_{f(p)=p}\frac1{\det_{\mathbb C}(I-df_p)}
$$
when the fixed <points> are nondegenerate. On the <Riemann sphere>, $f(z)=az$ with $|a|=1$ and $a\ne1$ has fixed <points> zero and infinity, with tangent multipliers $a$ and $a^{-1}$. Their contributions are
$$
\frac1{1-a}+\frac1{1-a^{-1}}=1.
$$
The 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>.<A Lefschetz Fixed Point Formula for Elliptic Complexes: II. Applications by M. F. Atiyah and R. Bott>[Theorem 4.12, printed pp.458–459]{ref}
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