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[ m]H 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[ m]H and (ii) the Poisson structure of k[ m]H and Poisson-commutative subalgebras in k[ m]H with maximal transcendence degree.