fixed-point subspace

Let $\mathrm{\Sigma }\subset \mathrm{\Gamma }$ be a subgroup where $\mathrm{\Gamma }$ is a compact Lie Group acting on a vector space $V$. The fixed-point subspace of $\mathrm{\Sigma }$ is

$$\mathrm{Fix}(\mathrm{\Sigma })=\{x\in V\mid \sigma x=x,\forall \sigma \in \mathrm{\Sigma }\}$$

$\mathrm{Fix}(\mathrm{\Sigma })$ is a linear subspace of $V$ since

$$\mathrm{Fix}(\mathrm{\Sigma })=\bigcap _{\sigma \in \mathrm{\Sigma }}\ker (\sigma -\mathrm{I})$$

where $I$ is the identity. If it is important to specify the space $V$ we use the following notation ${\mathrm{Fix}}_V(\mathrm{\Sigma })$.