---
tag: T866256Y1W
kind: statement
role: theorem
name: Equivariant index theorem
msc: [58J20, 19L47, 57S15]
uses: [44GBFZ16SG, QDCD33CS21, MT9P2X85V8, 7JBRGCVJRB, 7JA7WH92K0, RMWMYBY8TC, G44NQXCWT9, PKS8JE1PW1]
about: [44GBFZ16SG, QDCD33CS21]
sources:
  - url: https://en.wikipedia.org/wiki/Atiyah–Singer_index_theorem
---

## Statement

Let $G$ be a compact [[Lie group|7JBRGCVJRB]] acting smoothly on a closed smooth manifold $M$.
Let $E$ and $F$ be $G$-equivariant complex vector bundles over $M$, and let $P \colon C^\infty(M; E) \to C^\infty(M; F)$ be a $G$-invariant [[elliptic operator|MT9P2X85V8]].
Then $\ker P$ and $\operatorname{coker} P$ are finite-dimensional [[representations|7JA7WH92K0]] of $G$, and the equivariant analytic index
$$\operatorname{ind}_G P = [\ker P] - [\operatorname{coker} P] \in R(G)$$
lies in the representation ring $R(G)$.
The equivariant analytic index equals the equivariant topological index:
$$\operatorname{ind}_G P = \operatorname{t-ind}_G P.$$
Here $\operatorname{t-ind}_G P$ is defined from the class of the principal symbol in the equivariant K-theory $K_G(T^*M)$.
The definition uses a $G$-equivariant embedding of $M$ into a real representation of $G$.

## Remarks

The proof follows the K-theoretic proof of the non-equivariant theorem.
Both sides are homomorphisms $K_G(T^*M) \to R(G)$ with the same formal properties, and they agree on the basic case of a representation space.
Taking $G$ trivial gives the Atiyah–Singer index theorem, since $R(\{1\}) = \Z$ and the class of a vector space is its dimension.
Evaluating the [[character|RMWMYBY8TC]] of $\operatorname{ind}_G P$ at $g \in G$ gives the Lefschetz number $\operatorname{tr}(g \mid \ker P) - \operatorname{tr}(g \mid \operatorname{coker} P)$.
Atiyah, Segal and Singer computed this number by localization to the fixed point set $M^g$; when $g$ has no fixed points the number is $0$.
Example: the $G$-signature theorem is the case of the signature operator.

## Explanations

### L1 human

When a group acts by symmetries on a manifold and on an operator, the spaces of solutions are not only vector spaces but representations of the group.
The equivariant index records the difference of these representations, not just of their dimensions.
The theorem says that this difference is still computed by topology.

### L2 human

The main use is the fixed point formula: the trace of a symmetry $g$ on the index is a sum of contributions of the fixed points of $g$.
For the de Rham complex this recovers the [[Lefschetz fixed-point theorem|G44NQXCWT9]].
For the Dolbeault complex it gives the holomorphic Lefschetz formula of [[Atiyah and Bott|PKS8JE1PW1]], from which they derived the Weyl character formula for compact connected Lie groups.