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On Involutions in the Weyl Group and B-Orbit Closures in the Orthogonal Case

  • Mikhail V. Ignatyev
Chapter
Part of the Progress in Mathematics book series (PM, volume 330)

Abstract

We study coadjoint B-orbits on \(\mathfrak {n}^*\), where B is a Borel subgroup of a complex orthogonal group G, and \(\mathfrak {n}\) is the Lie algebra of the unipotent radical of B. To each basis involution w in the Weyl group W of G one can assign the associated B-orbit Ωw. We prove that, given basis involutions σ, τ in W, if the orbit Ωσ is contained in the closure of the orbit Ωτ then σ is less than or equal to τ with respect to the Bruhat order on W. For a basis involution w, we also compute the dimension of Ωw and present a conjectural description of the closure of Ωw.

AMS Subject Classification:

17B22 17B08 17B30 20F55 

Keywords

Involution in the Weyl group Bruhat order Coadjoint orbit Orthogonal group 

Notes

Acknowledgements

A part of this work (Sect. 3) was done during my stay at University of Haifa. I would like to express my gratitude to Prof. Dr. Anna Melnikov for her hospitality and fruitful discussions.

The work on Sect. 2 was performed at the NRU HSE with the support from the Russian Science Foundation, grant no. 16-41-01013. The work on Sect. 3 has been supported by RFBR grant no. 16-01-00154a and by ISF grant no. 797/14.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Mikhail V. Ignatyev
    • 1
  1. 1.Samara National Research UniversitySamaraRussia

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