Equidimensional representations of \(2\)-simple groups
Let \(G\) be a complex reduction algebraic group, \(\rho:G\to\text{GL}(V)\) a finite dimensional complex representation and \(\pi:V\to V//G\) the categorical quotient of the action of \(G\) on \(V\) induced by \(\rho\). There is the conjecture in invariant theory that if \(G\) is a connected semisimple group and \(\pi\) an equidimensional morphism, then \(V//G\) is smooth (and hence isomorphic to an affine space). This conjecture was formulated twenty years ago and since then has been verified for simple groups, for semisimple groups and their irreducible representations \(\rho\) (the verification followed a posteriori from the corresponding classification results) and for tori. In the present paper this conjecture is proved for a large class of the semisimple groups having exactly two simple factors (also in an a posteriori way, via classification of the equidimensional representations of these groups).
scientific article; zbMATH DE number 168592