Orbits and invariants for coisotropy representations

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