## Parametrization, structure and Bruhat order of certain spherical quotients

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- by Pierre-Emmanuel Chaput, Lucas Fresse and Thomas Gobet
- Represent. Theory
**25**(2021), 935-974 - DOI: https://doi.org/10.1090/ert/584
- Published electronically: October 21, 2021
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## Abstract:

Let $G$ be a reductive algebraic group and let $Z$ be the stabilizer of a nilpotent element $e$ of the Lie algebra of $G$. We consider the action of $Z$ on the flag variety of $G$, and we focus on the case where this action has a finite number of orbits (i.e., $Z$ is a spherical subgroup). This holds for instance if $e$ has height $2$. In this case we give a parametrization of the $Z$-orbits and we show that each $Z$-orbit has a structure of algebraic affine bundle. In particular, in type $A$, we deduce that each orbit has a natural cell decomposition. In the aim to study the (strong) Bruhat order of the orbits, we define an abstract partial order on certain quotients associated to a Coxeter system. In type $A$, we show that the Bruhat order of the $Z$-orbits can be described in this way.## References

- M. Bender and N. Perrin,
*Singularities of closures of $B$-conjugacy classes of nilpotent elements of height 2*, Transform. Groups**24**(2019), no. 3, 741–768. MR**3989689**, DOI 10.1007/s00031-018-9505-6 - A. Białynicki-Birula,
*Some theorems on actions of algebraic groups*, Ann. of Math. (2)**98**(1973), 480–497. MR**366940**, DOI 10.2307/1970915 - Anders Björner and Francesco Brenti,
*An improved tableau criterion for Bruhat order*, Electron. J. Combin.**3**(1996), no. 1, Research Paper 22, approx. 5. MR**1399399**, DOI 10.37236/1246 - Magdalena Boos, Giovanni Cerulli Irelli, and Francesco Esposito,
*Parabolic orbits of 2-nilpotent elements for classical groups*, J. Lie Theory**29**(2019), no. 4, 969–996. MR**4000994** - Magdalena Boos and Markus Reineke,
*$B$-orbits of 2-nilpotent matrices and generalizations*, Highlights in Lie algebraic methods, Progr. Math., vol. 295, Birkhäuser/Springer, New York, 2012, pp. 147–166. MR**2866850**, DOI 10.1007/978-0-8176-8274-3_{6} - Paolo Bravi and Stéphanie Cupit-Foutou,
*Classification of strict wonderful varieties*, Ann. Inst. Fourier (Grenoble)**60**(2010), no. 2, 641–681 (English, with English and French summaries). MR**2667789**, DOI 10.5802/aif.2535 - M. Brion,
*Rational smoothness and fixed points of torus actions*, Transform. Groups**4**(1999), no. 2-3, 127–156. Dedicated to the memory of Claude Chevalley. MR**1712861**, DOI 10.1007/BF01237356 - Michel Brion,
*On orbit closures of spherical subgroups in flag varieties*, Comment. Math. Helv.**76**(2001), no. 2, 263–299. MR**1839347**, DOI 10.1007/PL00000379 - David H. Collingwood and William M. McGovern,
*Nilpotent orbits in semisimple Lie algebras*, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993. MR**1251060** - Vinay V. Deodhar,
*Some characterizations of Bruhat ordering on a Coxeter group and determination of the relative Möbius function*, Invent. Math.**39**(1977), no. 2, 187–198. MR**435249**, DOI 10.1007/BF01390109 - Matthew Dyer,
*On the “Bruhat graph” of a Coxeter system*, Compositio Math.**78**(1991), no. 2, 185–191. MR**1104786** - Jacopo Gandini, Pierluigi Möseneder Frajria, and Paolo Papi,
*Spherical nilpotent orbits and abelian subalgebras in isotropy representations*, J. Lond. Math. Soc. (2)**95**(2017), no. 1, 323–352. MR**3653095**, DOI 10.1112/jlms.12022 - Jacopo Gandini, Pierluigi Möseneder Frajria, and Paolo Papi,
*Nilpotent orbits of height 2 and involutions in the affine Weyl group*, Indag. Math. (N.S.)**31**(2020), no. 4, 568–594. MR**4126755**, DOI 10.1016/j.indag.2020.04.006 - James E. Humphreys,
*Linear algebraic groups*, Graduate Texts in Mathematics, No. 21, Springer-Verlag, New York-Heidelberg, 1975. MR**0396773**, DOI 10.1007/978-1-4684-9443-3 - Victor G. Kac,
*Infinite-dimensional Lie algebras*, Progress in Mathematics, vol. 44, Birkhäuser Boston, Inc., Boston, MA, 1983. An introduction. MR**739850**, DOI 10.1007/978-1-4757-1382-4 - Bertram Kostant,
*The cascade of orthogonal roots and the coadjoint structure of the nilradical of a Borel subgroup of a semisimple Lie group*, Mosc. Math. J.**12**(2012), no. 3, 605–620, 669 (English, with English and Russian summaries). MR**3024825**, DOI 10.17323/1609-4514-2012-12-3-605-620 - B. Mühlherr,
*Coxeter groups in Coxeter groups*, Finite geometry and combinatorics (Deinze, 1992) London Math. Soc. Lecture Note Ser., vol. 191, Cambridge Univ. Press, Cambridge, 1993, pp. 277–287. MR**1256283**, DOI 10.1017/CBO9780511526336.027 - Dmitrii I. Panyushev,
*Complexity and nilpotent orbits*, Manuscripta Math.**83**(1994), no. 3-4, 223–237. MR**1277527**, DOI 10.1007/BF02567611 - Dmitri I. Panyushev,
*On spherical nilpotent orbits and beyond*, Ann. Inst. Fourier (Grenoble)**49**(1999), no. 5, 1453–1476 (English, with English and French summaries). MR**1723823**, DOI 10.5802/aif.1726 - R. W. Richardson and T. A. Springer,
*The Bruhat order on symmetric varieties*, Geom. Dedicata**35**(1990), no. 1-3, 389–436. MR**1066573**, DOI 10.1007/BF00147354 - R. W. Richardson and T. A. Springer,
*Combinatorics and geometry of $K$-orbits on the flag manifold*, Linear algebraic groups and their representations (Los Angeles, CA, 1992) Contemp. Math., vol. 153, Amer. Math. Soc., Providence, RI, 1993, pp. 109–142. MR**1247501**, DOI 10.1090/conm/153/01309 - Brian David Rothbach,
*Borel orbits of X2 = 0 in gl(n)*, ProQuest LLC, Ann Arbor, MI, 2009. Thesis (Ph.D.)–University of California, Berkeley. MR**2714006**

## Bibliographic Information

**Pierre-Emmanuel Chaput**- Affiliation: Université de Lorraine, CNRS, Institut Élie Cartan de Lorraine, UMR 7502, Vandoeuvre-lès-Nancy F-54506, France
- MR Author ID: 697366
- Email: pierre-emmanuel.chaput@univ-lorraine.fr
**Lucas Fresse**- Affiliation: Université de Lorraine, CNRS, Institut Élie Cartan de Lorraine, UMR 7502, Vandoeuvre-lès-Nancy F-54506, France
- MR Author ID: 875745
- Email: lucas.fresse@univ-lorraine.fr
**Thomas Gobet**- Affiliation: Institut Denis Poisson, CNRS UMR 7350, Faculté des Sciences et Techniques, Université de Tours, Parc de Grandmont, 37200 Tours, France
- MR Author ID: 1076592
- Email: thomas.gobet@lmpt.univ-tours.fr
- Received by editor(s): September 18, 2020
- Received by editor(s) in revised form: June 2, 2021
- Published electronically: October 21, 2021
- Additional Notes: The first two authors were supported in part by the ANR project GeoLie ANR-15-CE40-0012. The third author was partially supported by the same project and by the ARC project DP170101579.
- © Copyright 2021 American Mathematical Society
- Journal: Represent. Theory
**25**(2021), 935-974 - MSC (2020): Primary 20G05, 17B08
- DOI: https://doi.org/10.1090/ert/584
- MathSciNet review: 4329194