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

## 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 Ω_{w}. We prove that, given basis involutions *σ*, *τ* in *W*, if the orbit Ω_{σ} is contained in the closure of the orbit Ω_{τ} 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 Ω_{w} and present a conjectural description of the closure of Ω_{w}.

## 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.

## References

- 1.E. Bagno, Y. Cherniavsky. Congruence
*B*-orbits and the Bruhat poset of involutions of the symmetric group. Discrete Math.**312**(2012), no. 6, 1289–1299.MathSciNetCrossRefGoogle Scholar - 2.S. Billey, V. Lakshmibai. Singular loci of Schubert varieties. Progr. in Math.
**182**, Birkhäuser, 2000.Google Scholar - 3.A. Bjorner, F. Brenti. Combinatorics of Coxeter groups. Graduate Texts in Mathematics
**231**, Springer, 2005.Google Scholar - 4.N. Bourbaki. Lie groups and Lie algebras. Chapters 4–6. Springer, 2002.Google Scholar
- 5.M.V. Ignat’ev. Orthogonal subsets of classical root systems and coadjoint orbits of unipotent groups (in Russian). Mat. Zametki
**86**(2009), no. 1, 65–80. English translation: Math. Notes**86**(2009), no. 1, 65–80; arXiv: math.RT/0904.2841.MathSciNetCrossRefGoogle Scholar - 6.M.V. Ignatev. Orthogonal subsets of root systems and the orbit method (in Russian). Algebra i Analiz
**22**(2010), no. 5, 104–130. English translation: St. Petersburg Math. J.**22**(2011), no. 5, 777–794; arXiv: math.RT/1007.5220.Google Scholar - 7.M.V. Ignatyev. Combinatorics of
*B*-orbits and the Bruhat–Chevalley order on involutions. Transformation Groups**17**(2012), no. 3, 747–780; arXiv: math.RT/1101.2189.MathSciNetCrossRefGoogle Scholar - 8.M.V. Ignatyev. The Bruhat-Chevalley order on involutions of the hyperoctahedral group and combinatorics of
*B*-orbit closures (in Russian). Zapiski Nauchnykh Seminarov POMI**400**(2012), 166–188. English translation: J. Math. Sci.**192**(2013), no. 2, 220–231; arXiv: math.RT/1112.2624.Google Scholar - 9.M.V. Ignatyev, A.S. Vasyukhin. Rook placements in
*A*_{n}and combinatorics of*B*-orbit closures. J. Lie Theory**24**(2014), no. 4, 931–956; arXiv: math.RT/1310.3164.Google Scholar - 10.F. Incitti. Bruhat order on the involutions of classical Weyl groups. Ph.D. thesis. Dipartimento di Matematika “Guido Castelnuovo”, Università di Roma “La Sapienza”, 2003.Google Scholar
- 11.F. Incitti. The Bruhat order on the involutions of the symmetric groups. J. Alg. Combin.
**20**(2004), no. 3, 243–261.MathSciNetCrossRefGoogle Scholar - 12.F. Incitti. Bruhat order on classical Weyl groups: minimal chains and covering relation. European J. Combinatorics
**26**(2005), 729–753.MathSciNetCrossRefGoogle Scholar - 13.A.A. Kirillov. Unitary representations of nilpotent Lie groups. Russian Math. Surveys
**17**(1962), 53–110.MathSciNetCrossRefGoogle Scholar - 14.A.A. Kirillov. Lectures on the orbit method, Grad. Studies in Math.
**64**, AMS, 2004.Google Scholar - 15.A. Melnikov.
*B*-orbit in solution to the equation*X*^{2}= 0 in triangular matrices. J. Algebra**223**(2000), 101–108.MathSciNetCrossRefGoogle Scholar - 16.A. Melnikov. Description of
*B*-orbit closures of order 2 in upper-triangular matrices. Transformation Groups**11**(2006), no. 2, 217–247.MathSciNetCrossRefGoogle Scholar - 17.A. Melnikov.
*B*-orbits of nilpotent order 2 and link patterns. Indagationes Mathematicae**24**(2013), no.2, 443–473.MathSciNetCrossRefGoogle Scholar - 18.A.N. Panov. Involutions in
*S*_{n}and associated coadjoint orbits. J. Math. Sci.**151**(2008), no. 3, 3018–3031.MathSciNetCrossRefGoogle Scholar - 19.R.W. Richardson, T.A. Springer. The Bruhat order on symmetric varieties. Geom. Dedicata
**35**(1990), no. 1–3, 389–436.MathSciNetzbMATHGoogle Scholar - 20.T.A. Springer. Some remarks on involutions in Coxeter groups. Comm. Algebra
**10**(1982), no. 6, 631–636.MathSciNetCrossRefGoogle Scholar