Orbits and invariants for coisotropy representations

For a subgroup H of a reductive group G, let \(\mathfrak {m}=\mathfrak {h}^\perp \subset \mathfrak {g}^*\) $\mathfrak{m}={\mathfrak{h}}^{\perp }\subset {\mathfrak{g}}^{\ast }$ be the cotangent space of \(\{H\}\in G/H\) $\{H\}\in G/H$ . The linear action \((H:\mathfrak {m})\) $(H:\mathfrak{m})$ 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 \((H:\mathfrak {m})\) $(H:\mathfrak{m})$ . We complement existing results on c, r, and \((H:\mathfrak {m})\) $(H:\mathfrak{m})$ , especially for quasiaffine varieties G/H. For instance, if the algebra of invariants \(\Bbbk [\mathfrak {m}]^H\) $\mathbb{k}{[\mathfrak{m}]}^H$ is finitely generated, then \(\mathfrak {N}_H(\mathfrak {m})\subset \mathfrak {m}\cap \mathfrak {N}_G(\mathfrak {g}^*)\) ${\mathfrak{N}}_H(\mathfrak{m})\subset \mathfrak{m}\cap {\mathfrak{N}}_G({\mathfrak{g}}^{\ast })$ . Moreover, if G/H is affine, then \(\mathfrak {N}_H(\mathfrak {m})=\mathfrak {m}\cap \mathfrak {N}_G(\mathfrak {g}^*)\) ${\mathfrak{N}}_H(\mathfrak{m})=\mathfrak{m}\cap {\mathfrak{N}}_G({\mathfrak{g}}^{\ast })$ if and only if \(c=0\) $c=0$ . We also prove that the variety \(\mathfrak {m}\cap \mathfrak {N}_G(\mathfrak {g}^*)\) $\mathfrak{m}\cap {\mathfrak{N}}_G({\mathfrak{g}}^{\ast })$ is pure, of dimension \(\dim \mathfrak {m}-r\) $dim\mathfrak{m}-r$ . Two other topics considered are (i) a relationship between varieties G/H of complexity at most 1 and the homological dimension of the algebra \(\Bbbk [\mathfrak {m}]^H\) $\mathbb{k}{[\mathfrak{m}]}^H$ and (ii) the Poisson structure of \(\Bbbk [\mathfrak {m}]^H\) $\mathbb{k}{[\mathfrak{m}]}^H$ and Poisson-commutative subalgebras \({\mathcal {A}}\subset \Bbbk [\mathfrak {m}]^H\) $\mathcal{A}\subset \mathbb{k}{[\mathfrak{m}]}^H$ such that \({\mathrm {trdeg\,}}{\mathcal {A}}\) $\mathrm{trdeg}\mathcal{A}$ is maximal.