Closure ordering and the Kostant-Sekiguchi correspondence

Barbasch, Dan; Sepanski, Mark

doi:10.1090/S0002-9939-98-04090-8

Closure ordering and the Kostant-Sekiguchi correspondence
HTML articles powered by AMS MathViewer

by Dan Barbasch and Mark R. Sepanski PDF

Proc. Amer. Math. Soc. 126 (1998), 311-317 Request permission

Abstract:

Let $S$ be a real semisimple Lie group with Lie algebra $\mathfrak {s}=\mathfrak {k} +\mathfrak {p}$. The Kostant–Sekiguchi correspondence is a bijection between nilpotent $S$ orbits on $\mathfrak {s}$ and nilpotent $K_{\mathbb {C}}$ orbits on $\mathfrak {p}_{\mathbb {C}}$. In this note we prove that the closure relations among nilpotent orbits are preserved under the Kostant–Sekiguchi correspondence. The techniques rely on work of M. Vergne and P. Kronheimer.

References

S. K. Donaldson, Nahm’s equations and the classification of monopoles, Comm. Math. Phys. 96 (1984), no. 3, 387–407. MR 769355, DOI 10.1007/BF01214583
P. B. Kronheimer, A hyper-Kählerian structure on coadjoint orbits of a semisimple complex group, J. London Math. Soc. (2) 42 (1990), no. 2, 193–208. MR 1083440, DOI 10.1112/jlms/s2-42.2.193
P. B. Kronheimer, Instantons and the geometry of the nilpotent variety, J. Differential Geom. 32 (1990), no. 2, 473–490. MR 1072915, DOI 10.4310/jdg/1214445316
Takuya Ohta, The closures of nilpotent orbits in the classical symmetric pairs and their singularities, Tohoku Math. J. (2) 43 (1991), no. 2, 161–211. MR 1104427, DOI 10.2748/tmj/1178227492
Jir\B{o} Sekiguchi, Remarks on real nilpotent orbits of a symmetric pair, J. Math. Soc. Japan 39 (1987), no. 1, 127–138. MR 867991, DOI 10.2969/jmsj/03910127
Michèle Vergne, Instantons et correspondance de Kostant-Sekiguchi, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 8, 901–906 (French, with English and French summaries). MR 1328708