Glasnik Matematicki, Vol. 56, No. 2 (2021), 287-327. \( \)

APPROXIMATION OF NILPOTENT ORBITS FOR SIMPLE LIE GROUPS

Lucas Fresse and Salah Mehdi

Institut Elie Cartan de Lorraine, CNRS - UMR 7502, Université de Lorraine, France
e-mail:lucas.fresse@univ-lorraine.fr
e-mail:salah.mehdi@univ-lorraine.fr

Abstract.   We propose a systematic and topological study of limits \(\lim_{\nu\to 0^+}G_\mathbb{R}\cdot(\nu x)\) of continuous families of adjoint orbits for a non-compact simple real Lie group \(G_\mathbb{R}\). This limit is always a finite union of nilpotent orbits. We describe explicitly these nilpotent orbits in terms of Richardson orbits in the case of hyperbolic semisimple elements. We also show that one can approximate minimal nilpotent orbits or even nilpotent orbits by elliptic semisimple orbits. The special cases of \(\mathrm{SL}_n(\mathbb{R})\) and \(\mathrm{SU}(p,q)\) are computed in detail.

2020 Mathematics Subject Classification.   17B08, 22E15

Key words and phrases.   Lie groups, semisimple and nilpotent orbits, approximation, asymptotic cones

https://doi.org/10.3336/gm.56.2.06

References:

1. D. Barbasch and M. R. Sepanski, Closure ordering and the Kostant–Sekiguchi correspondence, Proc. Amer. Math. Soc. 126 (1998), 311–317.
   MathSciNet    CrossRef

2. D. Barbasch and D. A. Vogan Jr, The local structure of characters, J. Functional Analysis 37 (1980), 27–55.
   MathSciNet    CrossRef

3. D. Barbasch and D. A. Vogan Jr, Weyl group representations and nilpotent orbits, in: Representation theory of reductive groups (Park City, Utah, 1982), Birkhäuser Boston, Boston, 1983, 21–33.
   MathSciNet

4. W. Borho and H. Kraft, Über Bahnen und deren Deformationen bei linearen Aktionen reduktiver Gruppen, Comment. Math. Helv. 54 (1979), 61–104.
   MathSciNet    CrossRef

5. M. Božičević, A limit formula for even nilpotent orbits, Internat. J. Math. 19 (2008), 223–236.
   MathSciNet    CrossRef

6. D. H. Collingwood and W.M. McGovern, Nilpotent orbits in semisimple Lie algebras, van Nostrand Reinhold Co., New York, 1993.
   MathSciNet

7. D. Ž. Djoković, Closures of conjugacy classes in classical real linear Lie groups. II, Trans. Amer. Math. Soc. 270 (1982), 217–252.
   MathSciNet    CrossRef

8. B. Harris, Tempered representations and nilpotent orbits, Represent. Theory 16 (2012), 610–619.
   MathSciNet    CrossRef

9. B. Harris, H. He and G. Ólafsson, Wave front sets of reductive Lie group representations, Duke Math. J. 165 (2016), 793–846.
   MathSciNet    CrossRef

10. B. Harris and Y. Oshima, Irreducible characters and semisimple coadjoint orbits, J. Lie Theory 30 (2020), 715–765.
    MathSciNet

11. A. W. Knapp, Lie groups beyond an introduction, second ed., Birkhäuser Boston Inc., Boston, 2002.
    MathSciNet

12. T. Kobayashi and B. Ørsted, Conformal geometry and branching laws for unitary representations attached to minimal nilpotent orbits, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), 925–930.
    MathSciNet    CrossRef

13. T. Kobayashi and B. Ørsted, Analysis on the minimal representation of \(O(p,q)\). I. Realization via conformal geometry, Adv. Math. 180 (2003), 486–512.
    MathSciNet    CrossRef

14. K. Nishiyama, Asymptotic cone of semisimple orbits for symmetric pairs, Adv. Math. 226 (2011), 4338–4351.
    MathSciNet    CrossRef

15. T. Ohta, The closures of nilpotent orbits in the classical symmetric pairs and their singularities, Tohoku Math. J. (2) 43 (1991), 161–211.
    MathSciNet    CrossRef

16. T. Okuda, Smallest complex nilpotent orbits with real points, J. Lie Theory 25 (2015), 507–533.
    MathSciNet

17. W. Rossmann, Limit characters of reductive Lie groups, Invent. Math. 61 (1980), 53–66.
    MathSciNet    CrossRef

18. W. Rossmann, Limit orbits in reductive Lie algebras, Duke Math. J. 49 (1982), 215–229.
    MathSciNet    Link

19. L. P. Rothschild, Orbits in a real reductive Lie algebra, Trans. Amer. Math. Soc. 168 (1972), 403–421.
    MathSciNet    CrossRef

20. W. Schmid and K. Vilonen, Characteristic cycles and wave front cycles of representations of reductive Lie groups, Ann. of Math. (2) 151 (2000), 1071–1118.
    MathSciNet    CrossRef

21. J.-M. Souriau, Structure des systèmes dynamiques, Dunod, Paris, 1970.
    MathSciNet

22. P. E. Trapa, Annihilators and associated varieties of \(A_{{\mathfrak q}(\lambda)}\) modules for \(\mathrm{U}(p,q)\), Compositio Math. 129 (2001), 1–45.
    MathSciNet    CrossRef

23. P. E. Trapa, Richardson orbits for real classical groups, J. Algebra 286 (2005), 361–385.
    MathSciNet    CrossRef

24. D. A. Vogan Jr., Associated varieties and unipotent representations, in: Harmonic analysis on reductive groups (Brunswick, ME, 1989), Birkhäuser Boston, Boston, 1991, 315–388.
    MathSciNet

25. D. A. Vogan Jr., The method of coadjoint orbits for real reductive groups, in: Representation theory of Lie groups (Park City, UT, 1998), Amer. Math. Soc., Providence, 2000, 179–238.
    MathSciNet    CrossRef