$\alpha(z \cdot m_0)=\alpha(m_0)$ for all $\alpha \in M^*$. Then $z \cdot m_0=m_0$ and then $z \in H$. Hence $H=\{z \in G: z \cdot f = f \text{ for all } f \in {}^H\Bbbk[G]\}$. It is well known from the general theory of affine algebraic groups, that the above characterization of $H$ as the stabilizer of ${}^H\Bbbk[G]$ guarantees that $[{}^H\Bbbk[G]]={}^H[\Bbbk[G]]$. Along the proof of Theorem \[theo:geocarac\] we proved that the condition $[{}^H\Bbbk[G]]={}^H[\Bbbk[G]]$ guarantees the observability of $H$ on $G$.

[Problem 2. Descent of the finite generation condition.]{}

Assume that $H \subseteq G$ is a closed inclusion of affine algebraic groups and that $A$ is a rational $G$–module algebra we say that the *the finite generation condition descends from $G$ to $H$* if for all $A$ as above, in the inclusion ${}^G A \subseteq {}^HA$ the finite generation of the smallest $\Bbbk$–algebra implies the finite generation of the larger.

It is natural to search for conditions for $G$ and $H$ for which the finite generation of invariants descends from $G$ to $H$.

The first thing to notice is that having $H''$ the same invariants than $H$ we can assume without loss of generality that $H$ is observable in $G$ as ${}^GA \subseteq 