The Kempf-Ness theorem and invariant theory for real reductive representations
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 `Hilbert-Mumford criterion' in the real reductive setting. Instead of Iwahori's theorem, this paper uses the slice theorem of \textit{P. Heinzner} and \textit{H. Stötzel} [in: Global aspects of complex geometry. Berlin: Springer. 211--226 (2006; Zbl 1117.32017)], itself a translation of a well-known GIT result into the real reductive setting. The version of the Kempf-Ness theorem proved in this paper is for representations of real reductive groups, of which several different definitions exist in the literature. This theorem was originally proved by \textit{R. W. Richardson} and \textit{P. J. Slodowy} [J. Lond. Math. Soc., II. Ser. 42, No. 3, 409--429 (1990; Zbl 0675.14020)] around 1990, and since has been re-proved by \textit{C. Böhm} and \textit{R. A. Lafuente} [``Real geometric invariant theory'', Preprint, \url{arXiv:1701.00643}] using less algebraic methods. The proof in this paper makes use of the notion of a Kempf-Ness function and introduced by the author in the joint paper [\textit{L. Biliotti} and \textit{M. Zedda}, Ann. Mat. Pura Appl. (4) 196, No. 6, 2185--2211 (2017; Zbl 1397.14061)] with Zedda, and results from the author's prior work, especially [\textit{L. Biliotti} and \textit{M. Zedda}, Ann. Mat. Pura Appl. (4) 196, No. 6, 2185--2211 (2017; Zbl 1397.14061); \textit{L. Biliotti}, J. Geom. Phys. 151, Article ID 103621, 15 p. (2020; Zbl 1443.22015)]. The results and proofs of the first half of the paper are mostly taken from these papers.
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