On distinguished orbits of reductive representations

Abstract

Abstract. Let G be a real reductive Lie group and let τ ∶ G → GL(V ) be a real reductive representation of G with (restricted) moment map mg ∶ V ∖ {0} → g. In this work, we introduce the notion of nice space of a real reductive representation to study the problem of how to determine if a G-orbit is distinguished (i.e. it contains a critical point of the norm squared of mg). We give an elementary proof of the well-known convexity theorem of Atiyah-Guillemin-Sternberg in our particular case and we use it to give an easy-to-check sufficient condition for a G-orbit of a element in a nice space to be distinguished. In the case where G is algebraic and τ is a rational representation, the above condition is also necessary (making heavy use of recent results of M. Jablonski), obtaining a generalization of Nikolayevsky's nice basis criterium. We also provide useful characterizations of nice spaces in terms of the weights of τ . Finally, some applications to ternary forms and minimal metrics on nilmanifolds are presented.

Classification of Nilsoliton metrics in dimension seven

“…It is easy to see that Nikolayevsky's nice basis criterium is also true in the complex case (see [F3,Remark 3.2]); i.e. given a complex nilpotent Lie algebra (C n , µ) which is written in a nice basis for the action of .3 has at least one solution x with positive coordinates where m = #(R(µ)) and U is the Gram matrix of (R(µ), ⟨⟨⋅, ⋅⟩⟩) (here, ⟨⟨⋅, ⋅⟩⟩ is the usual Hermitian inner product on gl n (C)).…”

“…We refer the reader to [J2,Corollary 3.4] or [F3,Section 3] for further information on results related with Theorem 2.3. Another application from geometric invariant theory in the study of nilsoliton metrics is the following result, which was proven independently Michael Jablonski in [J2,Theorem 6.5] and Yuri Nikolayevsky in [Nk2, Theorem 6].…”

Soliton Almost Kähler Structures on 6-Dimensional Nilmanifolds for the Symplectic Curvature Flow

“…By using strong results from real geometric invariant theory (real GIT for short), the properties that make a minimal metric "special" are given in [Lau1]: a minimal metric is unique (up to isometry and scaling) when it exists, and it can be characterized as a soliton solution of the invariant Ricci flow ([Lau1, Theorem 4.4]). By using results given in [Fer2], we introduce the notion of nice basis (Definition 2.18) in the context of minimal metrics and give the corresponding criterion to know when a geometric structure γ on a nilpotent Lie algebra admitting a γ-nice basis has a minimal compatible metric.…”

“…In general, it is difficult to know when a pair (N, γ) admits a γ-nice basis, even if γ = 0 (nilsoliton case). In [Fer2,Section 4] we study this problem in the general case of real reductive representations and some results obtained will be very useful in the study of minimal metrics.…”

“…we must determine when a orbit contains a critical point of the norm-square of the moment map m sp associated to the action. In [Fer2], by using convexity properties of the moment map and recent results of Michael Jablonski in [Jab1], we gave a criterion to know when a nice element of a real reductive representation has a distinguished orbit. Such result can be considered as a generalization of Nikolayevsky's nice basis criterium (see [Nik2,Theorem 3]).…”