LINEAR PRESERVERS AND REPRESENTATIONS WITH A 1-DIMENSIONAL RING OF INVARIANTS

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 GL(V) and applying the theory of semisimple algebraic groups. Along the way, we give an explicit description of the normalizer N-GL(V)(G) and prove that, under a mild technical assumption, the normalizer agrees with the stabilizer in GL(V) of the orbit of the highest weight vector in P(V).