Abstract
We consider the complex ind-group \(G=\mathrm {SL}(\infty ,\mathbb {C})\) and its real forms \(G^0={\mathrm {SU}}(\infty ,\infty )\), \({\mathrm {SU}}(p,\infty )\), \(\mathrm {SL}(\infty ,\mathbb {R})\), \(\mathrm {SL}(\infty ,\mathbb {H})\). Our main object of study are the \(G^0\)-orbits on an ind-variety G / P for an arbitrary splitting parabolic ind-subgroup \(P\subset G\), under the assumption that the subgroups \(G^0\subset G\) and \(P\subset G\) are aligned in a natural way. We prove that the intersection of any \(G^0\)-orbit on G / P with a finite-dimensional flag variety \(G_n/P_n\) from a given exhaustion of G / P via \(G_n/P_n\) for \(n\rightarrow \infty \), is a single \((G^0\cap G_n)\)-orbit. We also characterize all ind-varieties G / P on which there are finitely many \(G^0\)-orbits, and provide criteria for the existence of open and closed \(G^0\)-orbits on G / P in the case of infinitely many \(G^0\)-orbits.
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Notes
- 1.
In what follows we use the terms isotropic and totally isotropic as synonyms.
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Acknowledgements
The first author was supported in part by the Russian Foundation for Basic Research through grants no. 14–01–97017 and 16–01–00154, by the Dynasty Foundation and by the Ministry of Science and Education of the Russian Federation, project no. 204. A part of this work was done at the Oberwolfach Research Institute for Mathematics (program “Oberwolfach Leibniz Fellows”) and at Jacobs University Bremen, and the first author thanks these institutions for their hospitality. The second and third authors thank Professor V.K. Dobrev for the invitation to speak at the XI International Workshop “Lie Theory and its Applications in Physics” in Varna, 15–21 June 2015. The second author acknowledges continued partial support by the DFG through Priority Program SPP 1388 and grant PE 980/6–1. The third author acknowledges partial support from the Dickson Emeriti Professorship at the University of California and from a Simons Foundation Collaboration Grant for Mathematicians.
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Ignatyev, M.V., Penkov, I., Wolf, J.A. (2016). Real Group Orbits on Flag Ind-Varieties of \(\mathrm {SL}(\infty ,\mathbb {C})\) . In: Dobrev, V. (eds) Lie Theory and Its Applications in Physics. LT 2015. Springer Proceedings in Mathematics & Statistics, vol 191. Springer, Singapore. https://doi.org/10.1007/978-981-10-2636-2_8
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