Linear preservers and representations with a 1-dimensional ring of invariants

Abstract

We determine the group of linear transformations preserving a polynomial function $f$ on a vector space $V$ for several interesting pairs $(V,f)$ by introducing a subgroup $G$ of $\mathrm {GL}(V)$ and applying the theory of semisimple algebraic groups. Along the way, we give an explicit description of the normalizer $N_{\mathrm {GL}(V)}(G)$ and prove that, under a mild technical assumption, the normalizer agrees with the stabilizer in $\mathrm {GL}(V)$ of the orbit of the highest weight vector in $\mathbb {P}(V)$.