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Stratifications with respect to actions of real reductive groups
| Content Provider | Scilit |
|---|---|
| Author | Heinzner, Peter Schwarz, Gerald W. Stötzel, Henrik |
| Copyright Year | 2008 |
| Description | Journal: Compositio Mathematica 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 $S_{β}$ 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. |
| Ending Page | 185 |
| Starting Page | 163 |
| ISSN | 00221295 |
| e-ISSN | 15705846 |
| DOI | 10.1112/s0010437x07003259 |
| Journal | Compositio Mathematica |
| Issue Number | 1 |
| Volume Number | 144 |
| Language | English |
| Publisher | Wiley-Blackwell |
| Publisher Date | 2008-01-01 |
| Access Restriction | Open |
| Subject Keyword | Journal: Compositio Mathematica |
| Content Type | Text |
| Resource Type | Article |
| Subject | Algebra and Number Theory |
