Stratifications with respect to actions of real reductive groups

• Published in: Compositio mathematica Vol. 144; no. 1; pp. 163 - 185
• Main Authors: Heinzner, Peter, Schwarz, Gerald W., Stötzel, Henrik
• Format: Journal Article
• Language: English
• Published: London, UK London Mathematical Society 01.01.2008; Cambridge University Press
• Subjects: moment map; Hamiltonian action; Cartan decomposition; stratification; 32M05
• ISSN: 0010-437X; 1570-5846
• DOI: 10.1112/S0010437X07003259

We study the action of a real reductive group G on a real submanifold X of a Kähler manifold Z. We suppose that the action of G extends holomorphically to an action of the complexified group $G^{\mathbb {C}}$ and that with respect to a compatible maximal compact subgroup U of $G^{\mathbb {C}}$ the action on Z is Hamiltonian. There is a corresponding gradient map $\mu _{\mathfrak {p}}\colon X\to \mathfrak {p}^*$ where $\mathfrak {g}=\mathfrak {k}\oplus \mathfrak {p}$ is a Cartan decomposition of $\mathfrak {g}$. We obtain a Morse-like function $\eta _{\mathfrak {p}}:=\Vert \mu _{\mathfrak {p}}\Vert ^2$ on X. Associated with critical points of $\eta _{\mathfrak {p}}$ are various sets of semistable points which we study in great detail. In particular, we have G-stable submanifolds Sβ of X which are called pre-strata. In cases where $\mu _{\mathfrak {p}}$ is proper, the pre-strata form a decomposition of X and in cases where X is compact they are the strata of a Morse-type stratification of X. Our results are generalizations of results of Kirwan obtained in the case where $G=U^{\mathbb {C}}$ and X=Z is compact.