On distinguished orbits of representations of Real reductive Lie groups

EDISON ALBERTO FERNÁNDEZ CULMA

Cartagena

Congreso; Algebraic Structures, their representation and applications in geometry and non-associative models; 2012

Centre International de Mathématiques Pures et Appliquées

Let $G = Kexp(p)$ be a real reductive Lie group acting linearly on a finite dimensional inner product space $(V ; ip )$, where $ip$ is a $K$-$p$-invariant inner product ($K$ acts by isometries and $p$ acts by symmetric operators). Motivated by Nikolayevsky´s nice basis criterium ([Nk, Theorem 3]) and by using Jablonski´s detection theorem ([J, Corollary 3.4]), we extend this criterium to any representation as above and we give an easy-to-check necessary and sufficient condition for a "nice element" to have a $G$-distinguished orbit. Also, we give some important applications, with an emphasis on the study of Ricci flow on nilmanifolds ([L],[F1]) ............................................................................................................................................................. References .................................................................................................................................................................. [F1] E. A. Fernández Culma, Classification of 7-dimensional Einstein Nilradicals, arXiv.org (2011). Under review. .................................................................................................................................................................. [F3] E. A. Fernández Culma, On distinguished orbits of representations of Real reductive Lie groups, (2011). Paper in preparation. ............................................................................................................................................ [J] M. Jablonski, Detecting orbits along subvarieties via the moment map, Münster J. of Math. 3 (2010), 67-88 .................................................................................................................................................................. [L] J. Lauret, Einstein solvmanifolds and nilsolitons, Contemporary Mathematics Volume 491 (2009), 35 p. .................................................................................................................................................................. [Nk] Y. Nikolayevsky Einstein solvmanifolds and the pre-Einstein derivation, Trans. Amer. Math. Soc. 363 (2011), 3935-3958.