We consider the action of a real semisimple Lie group G on the complexification G(C)/H-C of a semisimple symmetric space G/H and we present a refinement of Matsuki's results (Matsuki, 1997 [1]) in this case. We exhibit a finite set of points in G(C)/H-C, sitting on closed G-orbits of locally minimal dimension, whose slice representation determines the G-orbit structure of G(C)/H-C. Every such point (p) over bar lies on a compact torus and occurs at specific values of the restricted roots of the symmetric pair (g. h). The slice representation at (p) over bar is equivalent to the isotropy representation of a real reductive symmetric space, namely Z(G)(p(4))/G((p) over bar). In principle, this gives the possibility to explicitly parametrize all G-orbits in G(C)/H-C. (C) 2012 Elsevier B.V. All rights reserved.
Geatti, L. (2012). A remark on the orbit structure of the complexification of a semisimple symmetric space. DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS, 30(2), 195-205 [10.1016/j.difgeo.2012.01.001].
A remark on the orbit structure of the complexification of a semisimple symmetric space
GEATTI, LAURA
2012-01-01
Abstract
We consider the action of a real semisimple Lie group G on the complexification G(C)/H-C of a semisimple symmetric space G/H and we present a refinement of Matsuki's results (Matsuki, 1997 [1]) in this case. We exhibit a finite set of points in G(C)/H-C, sitting on closed G-orbits of locally minimal dimension, whose slice representation determines the G-orbit structure of G(C)/H-C. Every such point (p) over bar lies on a compact torus and occurs at specific values of the restricted roots of the symmetric pair (g. h). The slice representation at (p) over bar is equivalent to the isotropy representation of a real reductive symmetric space, namely Z(G)(p(4))/G((p) over bar). In principle, this gives the possibility to explicitly parametrize all G-orbits in G(C)/H-C. (C) 2012 Elsevier B.V. All rights reserved.File | Dimensione | Formato | |
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