trace formula

Let $\Gamma$ be a compact Lie group acting on $V$ and let $\Sigma\subset\Gamma$ be a Lie subgroup. Then

\operatorname{dim}\;\operatorname{Fix}(\Sigma)=\int_{\sigma\in\Sigma}% \operatorname{trace}(\sigma)

where $\int$ denotes the normalized Haar integral on $\Sigma$ and $\operatorname{Fix}(\Sigma)$ is the fixed-point subspace of $\Sigma$ .

References

GSS Golubitsky, Martin. Stewart, Ian. Schaeffer, G. David: Singularities and Groups in Bifurcation Theory (Volume II). Springer-Verlag, New York, 1988.