Orbits, rings of invariants and Weyl groups for classical \(\Theta\)-groups
type: Item
instance of: scholarly article
title: Orbits, rings of invariants and Weyl groups for classical \(\Theta\)-groups
zbMATH Open document ID: 1242.17024
author: Takuya Ohta
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2011-02-23T00:00:00Z
review text: 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 \((\text{GL}(V),\mathfrak{gl}(V))\),…
reviewed by: Ivo M. Michailov
Mathematics Subject Classification ID: 17B70
Mathematics Subject Classification ID: 13A50
Mathematics Subject Classification ID: 14R20
zbMATH DE Number: 5856429
zbMATH Keywords: Vinberg theta-groups
zbMATH Keywords: rings of invariants
zbMATH Keywords: Weyl groups
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1294170345
OpenAlex ID: W2092431448
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DOI: 10.2748/TMJ/1294170345
description: scientific article; zbMATH DE number 5856429