Orbits and invariants for coisotropy representations

TL;DR

This paper explores the properties of coisotropy representations for subgroups of reductive groups, linking invariants, nullcones, and Poisson structures to the geometric complexity of homogeneous spaces.

Contribution

It extends existing results on invariants, nullcones, and Poisson structures for coisotropy representations, especially in the context of quasiaffine varieties G/H.

Findings

Established a connection between nullcones in al m and al g^* when invariants are finitely generated.

Analyzed the relationship between complexity al and homological dimension of invariants.

Investigated the Poisson structure and Poisson-commutative subalgebras in the invariant algebra.

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.