The Moment-Weight Inequality and the Hilbert–Mumford Criterion

Abstract

This series reports on new developments in all areas of mathematics and their applications -quickly, informally and at a high level. Mathematical texts analysing new developments in modelling and numerical simulation are welcome. The type of material considered for publication includes:

Convexity properties of gradient maps associated to real reductive representations

“…The proof of the following Theorem is based on the Lojasiewicz gradient inequality, which holds in general for analytic gradient flows. A proof for the case of an action of a complex reductive group is given in [14].…”

“…We recall some result from Riemannian geometry. We refer the reader to Apendix A in [14] for further details. Suppose M is an Hadamard manifold, i.e., connected, complete, simplyconnected with non-positive curvature.…”

“…The following result asserts that any critical points of f in the same G-orbit in fact belong to the same K-orbit. We use original ideas from [14] in a different context. Theorem 4.6.…”

“…Therefore, the limit of the negative gradient flow exists and it is unique and any G-orbit collapses to a single K-orbit (Theorem 4.9). We use the original ideas from [14] in a different context. By the stratification theorem, we have…”

“…Applying results proved in [10,11,12], we completely describe the convex hull of the image of the gradient map, with respect to G, restricted on the closure of G orbits (Theorems 26, 34). Finally, using in a different context original ideas from [25], which are themselves of some interest, we give a probably new proof of the Hilbert-Mumford criterion for real reductive groups (Theorem 47) avoiding any algebraic result. This paper is organized as follows.…”