Invariants and Slices for Reductive and Biparabolic Coadjoint Actions

1.1 Let V be a finite dimensional vector space and G a group of linear automorphisms of V , Then G acts by transposition on the dual V ∗ of V and hence on the space of regular functions R[V ∗] on V ∗, which may be identified with the symmetric algebra S(V ) of V , that is to say, the polynomial algebra generated by a basis of V . A basic problem of algebraic invariant theory is to determine S(V )G the algebra of G-invariant functions on V ∗.