trace formula

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

$$\dim \mathrm{Fix}(\mathrm{\Sigma })={\int }_{\sigma \in \mathrm{\Sigma }}\mathrm{trace}(\sigma )$$

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