some partial relevant results that were later rediscovered by Weitzenböck and guaranteed a positive answer for the case of the invariants of $(\mathbb C, +)$ and $(\mathbb C^*,
\times)$. Later Weyl and Schiffer gave a complete positive answer for semisimple groups over $\mathbb C$. More recently –based on the platform established by Mumford in [@kn:Mumford]–, Nagata’s school contributions (see [@kn:nagataicm] and [@kn:nagatata] together with Haboush’s results ([@kn:haboushred]) settled the question affirmatively for reductive groups over fields of arbitrary characteristic.

In the case of non reductive groups, positive answers are more scarce. It is worth mentioning –besides the contributions by Maurer and Weitzenböck for the case of the additive group of the field of complex numbers– a result by Hochschild and Mostow (valid in characteristic zero): if $U$ is the unipotent radical of a subgroup $H$ of $G$ that contains a maximal unipotent subgroup of $G$ then the $U$–invariants of a finitely generated commutative $G$–module algebra are finitely generated ([@kn:homounip]).

Around the same time of the publication of the paper just mentioned, Grosshans’ published the above mentioned papers that provide more general insights into the problem of the finite generation of invariants for a non reductive group in arbitrary characteristic. For 