## Kyoto Journal of Mathematics

### Regular functions on spherical nilpotent orbits in complex symmetric pairs: Classical non-Hermitian cases

#### Abstract

Given a classical semisimple complex algebraic group $G$ and a symmetric pair $(G,K)$ of non-Hermitian type, we study the closures of the spherical nilpotent $K$-orbits in the isotropy representation of $K$. For all such orbit closures, we study the normality, and we describe the $K$-module structure of the ring of regular functions of the normalizations.

#### Article information

Source
Kyoto J. Math., Volume 57, Number 4 (2017), 717-787.

Dates
Accepted: 9 June 2016
First available in Project Euclid: 30 August 2017

https://projecteuclid.org/euclid.kjm/1504080147

Digital Object Identifier
doi:10.1215/21562261-2017-0013

Mathematical Reviews number (MathSciNet)
MR3725259

Zentralblatt MATH identifier
06825576

Subjects
Primary: 14M27: Compactifications; symmetric and spherical varieties
Secondary: 20G05: Representation theory

#### Citation

Bravi, Paolo; Chirivî, Rocco; Gandini, Jacopo. Regular functions on spherical nilpotent orbits in complex symmetric pairs: Classical non-Hermitian cases. Kyoto J. Math. 57 (2017), no. 4, 717--787. doi:10.1215/21562261-2017-0013. https://projecteuclid.org/euclid.kjm/1504080147

#### References

• [1] J. Adams, J.-S. Huang, and D. A. Vogan, Jr.,Functions on the model orbit in $\mathsf{E}_{8}$, Represent. Theory2(1998), 224–263.
• [2] B. Binegar,On a class of multiplicity-free nilpotent $K_{\mathbb{C}}$-orbits, J. Math. Kyoto Univ.47(2007), 735–766.
• [3] P. Bravi,Primitive spherical systems, Trans. Amer. Math. Soc.365, no. 1 (2013), 361–407.
• [4] P. Bravi, J. Gandini, and A. Maffei,Projective normality of model varieties and related results, Represent. Theory20(2016), 39–93.
• [5] P. Bravi and D. Luna,An introduction to wonderful varieties with many examples of type ${\sf F}_{4}$, J. Algebra329(2011), 4–51.
• [6] P. Bravi and G. Pezzini,Wonderful varieties of type ${\sf D}$, Represent. Theory9(2005), 578–637.
• [7] P. Bravi and G. Pezzini,Wonderful subgroups of reductive groups and spherical systems, J. Algebra409(2014), 101–147.
• [8] P. Bravi and G. Pezzini,The spherical systems of the wonderful reductive subgroups, J. Lie Theory25(2015), 105–123.
• [9] P. Bravi and G. Pezzini,Primitive wonderful varieties, Math. Z.282(2016), 1067–1096.
• [10] R. Chirivì, P. Littelmann, and A. Maffei,Equations defining symmetric varieties and affine Grassmannians, Int. Math. Res. Not. IMRN2009, no. 2, 291–347.
• [11] R. Chirivî and A. Maffei,Projective normality of complete symmetric varieties, Duke Math. J.122(2004), 93–123.
• [12] D. H. Collingwood and W. M. McGovern,Nilpotent Orbits in Semisimple Lie Algebras, Van Nostrand Reinhold, New York, 1993.
• [13] S. Cupit-Foutou,Wonderful varieties: a geometrical realization, preprint,arXiv:0907.2852v4[math.AG].
• [14] D. Ž. \DJoković,Proof of a conjecture of Kostant, Trans. Amer. Math. Soc.302, no. 2 (1987), 577–585.
• [15] J. Gandini,Spherical orbit closures in simple projective spaces and their normalizations, Transform. Groups16(2011), 109–136.
• [16] W. Hesselink,The normality of closures of orbits in a Lie algebra, Comment. Math. Helv.54(1979), 105–110.
• [17] J.-S. Huang and J.-S. Li,Unipotent representations attached to spherical nilpotent orbits, Amer. J. Math.121(1999), 497–517.
• [18] D. R. King,Spherical nilpotent orbits and the Kostant-Sekiguchi correspondence, Trans. Amer. Math. Soc.354, no. 12 (2002), 4909–4920.
• [19] D. R. King,Classification of spherical nilpotent orbits in complex symmetric space, J. Lie Theory14(2004), 339–370.
• [20] D. R. King,Small spherical nilpotent orbits and $K$-types of Harish Chandra modules, preprint,arXiv:math/0701034v1[math.RT].
• [21] B. Kostant and S. Rallis,Orbits and representations associated with symmetric spaces, Amer. J. Math.93(1971), 753–809.
• [22] H. Kraft and C. Procesi,On the geometry of conjugacy classes in classical groups, Comment. Math. Helv.57(1982), 539–602.
• [23] D. Luna,Variétés sphériques de type ${\sf A}$, Publ. Math. Inst. Hautes Études Sci.94(2001), 161–226.
• [24] K. Nishiyama,Multiplicity-free actions and the geometry of nilpotent orbits, Math. Ann.318(2000), 777–793.
• [25] K. Nishiyama,Classification of spherical nilpotent orbits for $\mathrm{U}(p,p)$, J. Math. Kyoto Univ.44(2004), 203–215.
• [26] K. Nishiyama, H. Ochiai, and C.-B. Zhu,Theta lifting of nilpotent orbits for symmetric pairs, Trans. Amer. Math. Soc.358, no. 6 (2006), 2713–2734.
• [27] D. I. Panyushev,On spherical nilpotent orbits and beyond, Ann. Inst. Fourier (Grenoble)49(1999), 1453–1476.
• [28] D. I. Panyushev,Some amazing properties of spherical nilpotent orbits, Math. Z.245(2003), 557–580.
• [29] H. Sabourin,Orbites nilpotentes sphériques et représentations unipotentes associées: le cas $\mathfrak{sl}_{n}$, Represent. Theory9(2005), 468–506.
• [30] J. Sekiguchi,Remarks on nilpotent orbits of a symmetric pair, J. Math. Soc. Japan39(1987), 127–138.
• [31] D.A. Timashev,Homogeneous spaces and equivariant embeddings, Encyclopaedia Math. Sci.138, Springer, Heidelberg, 2011.
• [32] K. D. Wong,Regular functions of symplectic spherical nilpotent orbits and their quantizations, Represent. Theory19(2015), 333–346.
• [33] L. Yang,On the quantization of spherical nilpotent orbits, Trans. Amer. Math. Soc.365, no. 12 (2013), 6499–6515.