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.
To Anthony Joseph in the occasion of his 75th birthday
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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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Ignatyev, M.V. (2019). On Involutions in the Weyl Group and B-Orbit Closures in the Orthogonal Case. In: Gorelik, M., Hinich, V., Melnikov, A. (eds) Representations and Nilpotent Orbits of Lie Algebraic Systems. Progress in Mathematics, vol 330. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-23531-4_8
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DOI: https://doi.org/10.1007/978-3-030-23531-4_8
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