Orbits and invariants for coisotropy representations

For a subgroup H of a reductive group G, let m g^* be the cotangent space of eH G/H. The linear action (H: m) is the coisotropy representation. It is known that the complexity and rank of G/H (denoted c and r, respectively) are encoded in properties of (H: m). We complement existing results on c, r, and (H: m), especially for quasiaffine varieties G/H. If the algebra of invariants k mH is finitely generated, then we establish a connection between the nullcones in m and g^*. Two other topics considered are (i) a relationship between varieties G/H of complexity at most 1 and the homological dimension of the algebra of invariants k mH and (ii) the Poisson structure of k mH and Poisson-commutative subalgebras in k mH with maximal transcendence degree.