The Kempf-Ness theorem and invariant theory for real reductive representations
type: Item
instance of: scholarly article
title: The Kempf-Ness theorem and invariant theory for real reductive representations
zbMATH Open document ID: 1490.14081
author: L. Biliotti
published in: São Paulo Journal of Mathematical Sciences
publication date: 2021-07-28T00:00:00Z
review text: The paper under review provides new proofs of two well-known results in the invariant theory of real reductive representations, namely appropriate analogues of the Kempf-Ness Theorem and the Hilbert-Mumford criterion. As originally formulated, the Kempf-Ness theorem [\textit{G. Kempf} and \textit{L. Ness}, Lect. Notes Math. 732, 233--243 (1979; Zbl 0407.22012)] tells us that a vector in a linear representation of a complex reductive group is stable if and only if an invariant norm function achieves its minimum value on that orbit. The most important consequence of this is the following, which is sometimes itself called the Kempf-Ness theorem: when a complex reductive group \(G\) acting linearly on a smooth complex projective variety \(X \subset \mathbb{P}^n\), the geometric invariant theory quotient of \(X\) by the action of \(G\) is homeomorphic to the symplectic reduction of \(X\) by the maximal compact subgroup \(K \leq G\). A related result is the Hilbert-Mumford criterion in Geometric Invariant Theory, the original version of which applies to any reductive group \(G\) acting linearly on a projective variety over an algebraically closed field of characteristic zero. This result states that stability (respectively, semistability) of a point \(x\in X\), is equivalently stability (resp. semistability) with respect to all 1-parameter subgroups \(\lambda : \mathbb{G}_m \rightarrow G\). This allows one to formulate stability and semi-stability as purely combinatorial notions, involving the convex hulls of subsets of the weights appearing in the representation. This theorem is crucial to the utility of GIT in practice. The original proof of the Hilbert-Mumford criterion makes use of Iwahori's theorem on double cosets of semisimple linear algebraic groups, and from this deduces a statement to the effect that any point in the closure of a \(G\)-orbit can be reached by taking a limit under an appropriate 1-parameter subgroup. It is this statement that goes by the name `Hi…
reviewed by: Joshua Jackson
Mathematics Subject Classification ID: 14L24
zbMATH DE Number: 7375677
zbMATH Keywords: gradient map
zbMATH Keywords: geometric invariant theory
zbMATH Keywords: stability
zbMATH Keywords: polystability
zbMATH Keywords: semi-stability
MaRDI profile type: Publication
full work available at URL: s40863-019-00151-6
OpenAlex ID: W2979374665
Wikidata QID: Q107168554
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DOI: 10.1007/S40863-019-00151-6
description: scientific article; zbMATH DE number 7375677