Advertisement

Branching Rules Related to Spherical Actions on Flag Varieties

  • Roman AvdeevEmail author
  • Alexey Petukhov
Article
  • 3 Downloads

Abstract

Let G be a connected semisimple algebraic group and let HG be a connected reductive subgroup. Given a flag variety X of G, a result of Vinberg and Kimelfeld asserts that H acts spherically on X if and only if for every irreducible representation R of G realized in the space of sections of a homogeneous line bundle on X the restriction of R to H is multiplicity free. In this case, the information on restrictions to H of all such irreducible representations of G is encoded in a monoid, which we call the restricted branching monoid. In this paper, we review the cases of spherical actions on flag varieties of simple groups for which the restricted branching monoids are known (this includes the case where H is a Levi subgroup of G) and compute the restricted branching monoids for all spherical actions on flag varieties that correspond to triples (G, H, X) satisfying one of the following two conditions: (1) G is simple and H is a symmetric subgroup of G; (2) G = SLn.

Keywords

Algebraic group Representation Flag variety Spherical variety Branching rule 

Mathematics Subject Classification (2010)

20G05 22E46 14M15 14M27 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The authors are grateful to Dmitry Timashev for useful discussions. The first author thanks the Institute for Fundamental Science in Moscow for providing excellent working conditions.

The results of Sections 6.5–6.12, 7.4–7.7 are obtained by the first author supported by the grant RSF–DFG 16-41-01013. The results of Sections 6.2–6.4, 7.2–7.3 are obtained by the second author supported by the RFBR grant no. 16-01-00818 and by the DFG grant PE 980/6-1.

References

  1. 1.
    Akhiezer, D., Panyushev, D.: Multiplicities in the branching rules and the complexity of homogeneous spaces. Mosc. Math. J. 2(1), 17–33 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arzhantsev, I., Derenthal, U., Hausen, J., Laface, A.: Cox Rings. Cambridge Studies in Advanced Mathematics, vol. 144. Cambridge University Press, Cambridge (2015)zbMATHGoogle Scholar
  3. 3.
    Avdeev, R.S., Petukhov, A.V.: Spherical actions on flag varieties. Sb. Math. 205(9), 1223–1263 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Benson, C., Ratcliff, G.: A classification of multiplicity free actions. J. Algebra 181(1), 152–186 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bourbaki, N.: Éléments de mathématique. Fasc. XXXIV. Groupes et Algèbres de Lie. Chapitre IV: Groupes de Coxeter et Systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: Systèmes de racines. Actualités Scientifiques et Industrielles, No. 1337. Hermann, Paris (1968)Google Scholar
  6. 6.
    Brion, M.: The total coordinate ring of a wonderful variety. J. Algebra 313, 61–99 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Elashvili, A.G.: Invariant Algebras. In: Lie Groups, Their Discrete Subgroups, and Invariant Theory, Adv. Soviet Math., vol. 8, pp. 57–64. Amer. Math. Soc, Providence (1992)CrossRefGoogle Scholar
  8. 8.
    Gelfand, I.M., Tsetlin, M.L.: Finite-dimensional representations of the group of unimodular matrices. Dokl. Akad. Nauk SSSR 71(5), 825–828 (1950). (in Russian)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Gelfand, I.M., Tsetlin, M.L.: Finite-dimensional representations of the group of orthogonal matrices. Dokl. Akad. Nauk SSSR 71(6), 1017–1020 (1950). (in Russian)MathSciNetGoogle Scholar
  10. 10.
    Goodman, R., Wallach, N.R.: Symmetry, Representations, and Invariants, Grad. Texts in Math., vol. 255. Springer, Dordrecht (2009)CrossRefGoogle Scholar
  11. 11.
    He, X., Nishiyama, K., Ochiai, H., Oshima, Y.: On orbits in double flag varieties for symmetric pairs. Transform Groups 18(4), 1091–1136 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Howe, R., Tan, E.-C., Willenbring, J.F.: Reciprocity Algebras and Branching for Classical Symmetric Pairs. In: Groups and Analysis, London Math. Soc. Lecture Note Ser., vol. 354, pp. 191–231. Cambridge Univ. Press, Cambridge (2008)Google Scholar
  13. 13.
    Howe, R., Umeda, T.: The Capelli identity, the double commutant theorem, and multiplicity-free actions. Math. Ann. 290(3), 565–619 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kac, V.G.: Some remarks on nilpotent orbits. J. Algebra 64(1), 190–213 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Knop, F.: Some remarks on multiplicity free spaces. In: Representation Theories and Algebraic Geometry, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 514, pp. 301–317. Springer Netherlands, Dordrecht (1998)Google Scholar
  16. 16.
    Kraft, H.: Geometrische Methoden in der Invariantentheorie. Aspects of Math. D1. Friedr. Vieweg & Sohn, Braunschweig (1984)CrossRefzbMATHGoogle Scholar
  17. 17.
    Krämer, M: Multiplicity-free subgroups of compact connected Lie groups. Arch. Math. 27(1), 28–36 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Leahy, A.S.: A classification of multiplicity free representations. J. Lie Theory 8 (2), 367–391 (1998)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Lee, C.Y.: On the branching theorem of the symplectic groups. Canad. Math. Bull. 17, 535–545 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lepowsky, J.: Multiplicity formulas for certain semisimple Lie groups. Bull. Amer. Math. Soc. 77, 601–605 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    LiE, A computer algebra package for Lie group computations, see http://wwwmathlabo.univ-poitiers.fr/~maavl/LiE
  22. 22.
    Littelmann, P.: On spherical double cones. J. Algebra 166(1), 142–157 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Magyar, P., Weyman, J., Zelevinsky, A.: Multiple flag varieties of finite type. Adv. Math. 141, 97–118 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Magyar, P., Weyman, J., Zelevinsky, A.: Symplectic multiple flag varieties of finite type. J. Algebra 230(1), 245–265 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Niemann, B.: Spherical affine cones in exceptional cases and related branching rules, preprint (2011), see arXiv:1111.3823 [math.RT]
  26. 26.
    Onishchik, A.L., Vinberg, E.B.: Lie Groups and Algebraic Groups, Springer Ser. Soviet Math. Springer, Berlin (1990)CrossRefGoogle Scholar
  27. 27.
    Panyushev, D.I.: On the conormal bundle of a G-stable subvariety. Manuscripta Math. 99(2), 185–202 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Petukhov, A.V.: Bounded reductive subalgebras of \(\mathfrak {sl}_n\). Transform Groups 16(4), 1173–1182 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Ponomareva, E.V.: Classification of double flag varieties of complexity 0 and 1. Izv. Math. 77(5), 998–1020 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Ponomareva, E.V.: Invariants of the Cox rings of low-complexity double flag varieties for classical groups. Trans. Moscow Math. Soc. 2015, 71–133 (2015)Google Scholar
  31. 31.
    Ponomareva, E.V.: Invariants of the Cox rings of double flag varieties of low complexity for exceptional groups. Sb. Math. 208(5), 707–742 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Popov, V.L.: Picard groups of homogeneous spaces of linear algebraic groups and one-dimensional homogeneous vector bundles. Math. USSR-Izv. 8(2), 301–327 (1974)CrossRefzbMATHGoogle Scholar
  33. 33.
    Steinberg, R.: Endomorphisms of Linear Algebraic Groups. Memoirs of the Americal Mathematical Society, vol. 80. American Mathematical Society, Providence (1968)Google Scholar
  34. 34.
    Stembridge, J.R.: Multiplicity-free products and restrictions of Weyl characters. Represent. Theory 7, 404–439 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Timashev, D.A.: Homogeneous Spaces and Equivariant Embeddings, Encycl. Math. Sci., vol. 138. Springer, Berlin (2011)CrossRefGoogle Scholar
  36. 36.
    Vinberg, E.B., Kimel’fel’d, B.N.: Homogeneous domains on flag manifolds and spherical subgroups of semisimple Lie groups. Funct. Anal. Appl. 12(3), 168–174 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Wallach, N., Yacobi, O.: A Multiplicity Formula for Tensor Products of S L 2 Modules and an Explicit S p 2n to S p 2n− 2 × S p 2 Branching Formula. In: Symmetry in Mathematics and Physics. Contemp. Math., vol. 490, pp. 151–155. Amer. Math. Soc., Providence (2009)Google Scholar
  38. 38.
    Yacobi, O.: An analysis of the multiplicity spaces in branching of symplectic groups. Selecta Math. (N.S.) 16(4), 819–855 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Zhelobenko, D.P.: Compact Lie Groups and Their Representations. Translations of Mathematical Monographs, vol. 40. American Mathematical Society, Providence (1973)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.National Research University “Higher School of Economics”MoscowRussia
  2. 2.Institute for Information Transmission ProblemsMoscowRussia

Personalised recommendations