Orbits and invariants for coisotropy representations

Abstract

For a subgroup $H$ of a reductive group $G$, let $\mathfrak m\subset \mathfrak g^*$ be the cotangent space of $eH\in G/H$. The linear action $(H:\mathfrak m)$ 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 $(H:\mathfrak m)$. We complement existing results on $c$, $r$, and $(H:\mathfrak m)$, especially for quasiaffine varieties $G/H$. If the algebra of invariants $k[\mathfrak m]^H$ is finitely generated, then we establish a connection between the nullcones in $\mathfrak m$ and $\mathfrak g^*$. 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 $k[\mathfrak m]^H$ and (ii) the Poisson structure of $k[\mathfrak m]^H$ and Poisson-commutative subalgebras in $k[\mathfrak m]^H$ with maximal transcendence degree.