$\mathrm{\Gamma }$-equivariant

Let $\mathrm{\Gamma }$ be a compact Lie group acting linearly on $V$ and let $g$ be a mapping defined as $g:V\to V$. Then $g$ is $\mathrm{\Gamma }$-equivariant if

$$g(\gamma v)=\gamma g(v)$$

for all $\gamma \in \mathrm{\Gamma }$, and all $v\in V$.
Therefore if $g$ commutes with $\mathrm{\Gamma }$ then $g$ is $\mathrm{\Gamma }$-equivariant.

[GSS]