A connected closed subgroup \(G \subset \mathrm{GL}(n,\mathbb{R})\) that is invariant under transposition is called a real reductive Lie group. The Lie algebra \(\mathfrak{g}\) of \(G\) can be considered as a subalgebra of the linear maps \(\mathrm{gl}(n,\mathbb{R})\) on \(\mathbb{R}^n\), with the exponential map \(\exp\) given by the regular exponential map of matrices. In that case, there exists a scalar product \(\langle \cdot, \cdot \rangle\) on \(V = \mathbb{R}^n\) such that \(G\) can be written as \[ G = K \exp(\mathfrak{p})\] with \(K = G \cap O(V)\) and \(\mathfrak{p} = \mathfrak{g} \cap \mathrm{Sym}(V)\). Here we denote by \(O(V)\) the orthogonal group with respect to the scalar product \(\langle \cdot, \cdot \rangle\) and by \(\mathrm{Sym}(V)\) the set of symmetric endomorphisms on \(V\). If \(\mathfrak{k}\) is the Lie algebra of the group \(K\), which is moreover the maximal compact subgroup of \(G\), then \(\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}\) is the Cartan decomposition of \(\mathfrak{g}\). We can define the map \(\psi: G \times V \to \mathbb{R}\) via \[\psi(g,x) = \frac{1}{2}\left( \langle g x, gx \rangle - \langle x, x \rangle \right)\] and this is a Kempf-Ness function as defined in the third section of the paper. The study of real reductive groups via the Kempf-Ness function and corresponding gradient map, in particular their convexity properties, is very active and has for example lead to a better understanding of metrics with negative Ricci curvature on solvmanifolds, see [J. Deré and J. Lauret, Math. Nachr. 292, No. 7, 1462–1481 (2019; Zbl 1425.53042)].
In the paper under review, the author gives numerous different results about the image of the gradient map. For example, the image of the gradient map for abelian subalgebras of \(\mathfrak{p}\) is computed, which generalizes a result of Kac and Peterson from the complex numbers to the reals [V. G. Kac and D. H. Peterson, Invent. Math. 76, 1–14 (1984; Zbl 0534.17008)]. On the other hand, a new proof of the Hilbert-Mumford criterion for real reductive groups is given in the last section.