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?
