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Assume we work over an algebraically closed field of characteristic zero. I know that for a connected semisimple algebraic group there is an upper bound for the number of isomorphism classes of representations with free algebra of invariants (due to Popov, 1983: http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=im&paperid=1619&option_lang=eng ).

In Algebraic Geometry IV (2010) it is stated that the connected semisimple linear groups admitting such a representation have not yet been determined. Does anyone know if there has been some recent work in this direction?

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    $\begingroup$ As far as I know, there is no further progress on the classification; this doesn't answer your question, however. I'd suggest you provide more explicit references, e.g., to Popov's paper, which I guess is this one: mathscinet.ams.org/mathscinet-getitem?mr=651651 (it's freely available online in Russian but not English). Also, Lie group theory probably has little to do with the question asked. Anyway, the connected semisimple Lie groups coincide with the connected semisimple algebraic groups over an algebraically closed field of characteristic 0 (Chevalley). $\endgroup$
    – Jim Humphreys
    Commented Mar 19, 2018 at 0:37
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    $\begingroup$ P.S. Here is the link to the Russian original: mathnet.ru/php/… $\endgroup$
    – Jim Humphreys
    Commented Mar 20, 2018 at 20:07
  • $\begingroup$ @JimHumphreys thank you very much. I chose the tag "Lie groups" because the question is very close to "which are the semisimple lie group actions with an affine space as a quotient" $\endgroup$
    – svelaz
    Commented Mar 20, 2018 at 20:34
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    $\begingroup$ This question is slightly confusing - it took me some time to track down the passage in Algebraic Geometry IV which says that "the connected ... admitting ... have not yet been determined". The point is that any semisimple algebraic group admits at least one representation for which the invariants are free: the adjoint representation. When Popov & Vinberg talk about "connected semisimple linear groups satisfying (FA)" they mean: pairs $(G,V)$ where $G$ is semisimple and $V$ is a representation. So the question really is, for a fixed $G$, what is the number $\geq 1$ of such representations? $\endgroup$
    – Paul Levy
    Commented Mar 30, 2018 at 14:01

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