On distinguished orbits of reductive representations

EDISON ALBERTO FERNÁNDEZ CULMA

JOURNAL OF ALGEBRA

ACADEMIC PRESS INC ELSEVIER SCIENCE

Lugar: Amsterdam; Año: 2013 vol. 396 p. 61 - 81

0021-8693

Let $G$ be a real reductive Lie group and let ${tau}:{G} longrightarrow {GL}({V})$ be a real reductive representation of $G$ with (restricted) moment map $mm_{g}:V --> g$. 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 $mm_{g}$). 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 $tau$ is a rational representation, the above condition is also necessary (making heavy use of recent results of Michael Jablonski), obtaining a generalization of Nikolayevsky´s nice basis criterium. We also provide useful characterizations of nice spaces in terms of the weights of $tau$. Finally, some applications to ternary forms are presented.