On the orbit space of an irreducible representation of the special unitary group

Let $V$ be a real vector space and $G\subset\mathrm{GL}(V)$ a compact linear Lie group. The author considers the question whether the orbit space (topological quotient) $V/G$ is a smooth manifold. The case in which $G$ is either abelian or locally isomorphic to $SU_2$ has been studied in a previous work of the author. In this article, $G$ is a compact group locally isomorphic to $SU_n$ and $V$ is an irreducible representation of $G$. The output of the author’s case-by-case computations is that if $G$ is connected then the quotient is never a smooth manifold.