On the Dual Topology of the Groups \(\mathbf {U(n)\ltimes \mathbb H_n}\)

  • Mounir Elloumi
  • Janne-Kathrin Günther
  • Jean Ludwig
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 207)

Abstract

Let \(G_n=U(n)\ltimes H_n \) be the semi-direct product of the unitary group acting by automorphisms on the Heisenberg group \(\mathbb H_n\). According to Lipsman, the unitary dual \(\widehat{G_n} \) of \(G_n \) is in one to one correspondence with the space of admissible coadjoint orbits \(\mathfrak g_n^\ddagger /G_n \) of \(G_n \). In this paper, we determine the topology of the space \(\mathfrak g_n^\ddagger /G_n \) and we show that the correspondence with \(\widehat{G_{n}} \) is a homeomorphism.

Keywords

Unitary group Semi-direct product Dual topology Admissible coadjoint orbit space 

2000 Mathematics Subject Classification

Primary 43A40 Secondary 22E45 

Notes

Acknowledgements

The authors would like to thank the referee for his / her careful reading of our paper and many valuable suggestions. Janne-Kathrin Günther was supported for this work by the Fonds National de la Recherche, Luxembourg (Project Code 3964572).

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Mounir Elloumi
    • 1
  • Janne-Kathrin Günther
    • 2
    • 3
  • Jean Ludwig
    • 2
  1. 1.Mathematics DepartmentCollege of Science, King Faisal UniversityAhsaaKingdom of Saudi Arabia
  2. 2.Université de Lorraine, Institut Elie Cartan de Lorraine, UMR 7502MetzFrance
  3. 3.University of LuxembourgMathematical Research UnitLuxembourgLuxembourg

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