I have difficulties seeing why is the following true:
Let $G$ be a lie group and $H$ a closed subgroup, with $\tilde{g}$ and $\tilde{h}$ their lie algebras. The adjoint action of $g\in G$ is given by the differential at the identity of $(g,g')\mapsto gg'g^{-1}$. The adjoint action of $H$ leaves $\tilde{h}$ stable: $Ad(H)(\tilde{h})\subset \tilde{h}$. Thus the action can be defined on the quotient: $Ad_{\tilde{g}/\tilde{h}}(h) (x+\tilde{h})=Ad(h)(x)+\tilde{h}$. On the other hand we define the linear isotropy representation by the differential of the natural action of $H$ on $G/H$ at $eH$ (the tangent space can be identifed with $\tilde{g}/\tilde{h}$). The Two representations are then identical.
In other words, why does the adjoint representation of $H$ coincides on $\tilde{g}/\tilde{h}$ with the linear isotropic representation?
Thanks for your answers