Orbits, rings of invariants and Weyl groups for classical \(\Theta\)-groups
The present paper is devoted to the invariant theory of Vinberg's \(\varTheta\)-groups. Namely, let \(G\) be a complex reductive algebraic group with Lie algebra \(\mathfrak g\) and \(\theta: G\to G\) an automorphism of order \(m\). We also denote by \(\theta:\mathfrak g\to \mathfrak g\) the Lie algebra automorphism defined by \(\theta\). Let \(\mathfrak g_1\) be the eigenspace of \(\theta\) with eigenvalue \(e^{2\pi\sqrt{-1}/m}\). Then the isotropy subgroup \(G_0:=G^\theta\) acts on \(\mathfrak g_1\) by the adjoint action. We call \((G,\theta)\) a \(\varTheta\)-group of order \(m\) and \((G_0,\mathfrak g_1)\) the \(\varTheta\)-representation defined by \((G,\theta)\). If \(G\) is \(\text{GL}(V)\), \(\text{O}(V)\) or \(\text{Sp}(V)\) and \(\theta:G\to G\) is an automorphism of classical type, we call \((G_0,\mathfrak g_1)\) a classical \(\varTheta\)-representation. Here we say that \(\theta\) is of classical type if \(\theta\) is an inner automorphism of \(G\) or an outer automorphism of \(G=\text{GL}(V)\). One of the main problems in this theory is whether the restriction map \(\text{rest}: \mathbb C[\mathfrak g]^G\to \mathbb C[\mathfrak g_1]^{G_0}\) is surjective or not. For example, \textit{S. Helgason} [Invariant differential operators and Weyl group invariants, Harmonic analysis on reductive groups, Proc. Conf., Brunswick/ME (USA), 1989, Prog. Math. 101, 193--200 (1991; Zbl 0760.43002)] showed that the restriction is surjective for a \(\varTheta\)-representation of order \(2\), and \textit{D. I. Panyushev} [On invariant theory of \(\theta\)-groups, J. Algebra 283, No. 2, 655--670 (2005; Zbl 1071.17005)] gave similar results for \(N\)-regular \(\varTheta\)-representations (i.e., when the regular nilpotent \(G\)-orbit in \(\mathfrak g\) meets \(\mathfrak g_1\)). The present author extends these results further with the following Theorem: For a classical \(\varTheta\)-representation \((G_0,\mathfrak g_1)\) naturally included in \((\text{GL}(V),\mathfrak{gl}(V))\), the restriction map \(\text{rest}: \mathbb C[\mathfrak{gl}(V)]^{\text{GL}(V)}\to \mathbb C[\mathfrak g_1]^{G_0}\), \(f\mapsto f|_{\mathfrak g_1}\) is surjective. As an immediate corollary of this Theorem, the author also obtains that: par For any classical \(\varTheta\)-representation \((G_0,\mathfrak g_1)\), the restriction map r\(\text{rest}: \mathbb C[\mathfrak g]^G\to \mathbb C[\mathfrak g_1]^{G_0}\) is surjective. In order to prove the main results, the author gives in Sections 2 and 3 a complete classification of general and semisimple orbits of three types of \(\varTheta\)-representations. This is achieved by means of \(\mathbb Z_m\)-labeled Young diagrams with eigenvalues. The results from Sections 2 and 3 are also applied by the author in Section 5 to describe the Weyl groups of \(\varTheta\)-representations. The notions of Cartan subspaces and Weyl groups were introduced by \textit{E. B. Vinberg} [The Weyl group of a graded Lie algebra, Math. USSR, Izv. 10(1976), 463--495 (1977); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 40, 488--526 (1976; Zbl 0363.20035)]. Ohta extends the description of Weyl groups given by Vinberg for further \(\varTheta\)-representations.
scientific article; zbMATH DE number 5856429
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