Invariants of a maximal unipotent subgroup and equidimensionality
Let \(G\) be a semisimple algebraic group over an algebraically closed field \(\mathbb{K}\) on characteristic zero. Fix a maximal unipotent subgroup \(U\subseteq G\) and a maximal torus \(T\subseteq B=N_G(U)\). Let \(G\) act on an affine variety~\(X\). As was shown by Hadžiev, the algebra of \(U\)-invariants, \(\mathbb{K}[X]^U\), is finitely generated. Moreover, there is a natural isomorphism \(\mathbb{K}[X]^U\cong (\mathbb{K}[X]\otimes\mathbb{K}[G/U])^G\). Denote by \(X/\!/U\) the affine variety \(\text{Spec}\,\mathbb{K}[X]^U\) and consider the quotient morphism \(\pi_U\,:\, X \to X/\!/U\) induced by the inclusion \(\mathbb{K}[X]^U\subseteq \mathbb{K}[X]\). The first result of the paper establishes a relation between the fibres of the quotient morphisms \[ \pi_U\,:\, X \to X/\!/U \quad \text{and} \quad \pi_G\,:\, X\times\text{Spec}\,\mathbb{K}[G/U] \to (X\times\text{Spec}\,\mathbb{K}[G/U])/\!/G \] that contain \(T\)-fixed points. For \(X\) conical, this implies that \(\pi_U\) is equidimensional if and only if \(\pi_G\) is. Recall that an S-variety is an affine \(G\)-variety with a dense \(G\)-orbit such that the stabilizer of any point contains a maximal unipotent subgroup of \(G\). S-varieties were introduced and studied in [\textit{V. L. Popov} and \textit{E. B. Vinberg}, ``On a class of quasihomogeneous affine varieties'', Math. USSR, Izv. 6(1972), 743--758 (1973; Zbl 0255.14016)]. The author gives a criterion of equidimensionality of \(\pi_U\) in the case when \(X\) is an S-variety.
scientific article; zbMATH DE number 6017078
scientific article