Journal of the Mathematical Society of Japan

Multiplicity-free branching rules for outer automorphisms of simple Lie algebras

Hidehisa ALIKAWA

Full-text: Open access

Abstract

We find explicit multiplicity-free branching rules of some series of irreducible finite dimensional representations of simple Lie algebras 𝔤 to the fixed point subalgebras 𝔤 σ of outer automorphisms σ . The representations have highest weights which are scalar multiples of fundamental weights or linear combinations of two scalar ones. Our list of pairs of Lie algebras ( 𝔤 , 𝔤 σ ) includes an exceptional symmetric pair ( E 6 , F 4 ) and also a non-symmetric pair ( D 4 , G 2 ) as well as a number of classical symmetric pairs. Some of the branching rules were known and others are new, but all the rules in this paper are proved by a unified method. Our key lemma is a characterization of the ``middle'' cosets of the Weyl group of 𝔤 in terms of the subalgebras 𝔤 σ on one hand, and the length function on the other hand.

Article information

Source
J. Math. Soc. Japan, Volume 59, Number 1 (2007), 151-177.

Dates
First available in Project Euclid: 25 May 2007

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1180135505

Digital Object Identifier
doi:10.2969/jmsj/1180135505

Mathematical Reviews number (MathSciNet)
MR2302667

Zentralblatt MATH identifier
1136.17005

Subjects
Primary: 17B10: Representations, algebraic theory (weights)
Secondary: 05E15: Combinatorial aspects of groups and algebras [See also 14Nxx, 22E45, 33C80]

Keywords
branching rule multiplicity free outer automorphism graph automorphism semisimple Lie groups characters of Lie groups

Citation

ALIKAWA, Hidehisa. Multiplicity-free branching rules for outer automorphisms of simple Lie algebras. J. Math. Soc. Japan 59 (2007), no. 1, 151--177. doi:10.2969/jmsj/1180135505. https://projecteuclid.org/euclid.jmsj/1180135505


Export citation

References

  • H. Alikawa, Multiplicity-free branching rules for symmetric pair $({E}_6,{F}_4)$, Master's thesis, Graduate School of Mathematical Sciences, University of Tokyo, March, 2001.
  • N. Bourbaki, Éléments de Mathématique, Groupes et Algèbres de Lie, chaptietres 4 à 6, Hermann, 1968.
  • É. Cartan, Le principe de dualité et la théorie des groupes simples et semi-simples, Bull. des Sci. Math., 49 (1925), 361–374.
  • J. Fuchs, U. Ray and C. Schweigert, Some automorphisms of generalized Kac-Moody algebras, J. Algebra, 191 (1997), 518–540.
  • S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Grad. Stud. Math., Amer. Math. Soc., 34 (2001), Corrected reprint of the 1978 original.
  • J. E. Humphreys, Reflection groups and Coxeter groups, Cambridge Stud. Adv. Math., 29 (1990).
  • M. Ishikawa and M. Wakayama, Minor summation formula of Pfaffians, Linear Multilinear algebra, 39 (1995), 285–305.
  • V. G. Kac, Infinite-dimensional Lie Algebras, third ed., Cambridge Univ. Press, Cambridge, 1990.
  • A. W. Knapp, Lie Groups Beyond an Introduction, Progr. math., Birkhäuser, 1996.
  • T. Kobayashi, Discrete decomposability of the restriction of ${A}_{\mathfrak{q}}(\lambda)$ with respect to reductive subgroups and its applications, Invent. math., 117 (1994), 181–205.
  • T. Kobayashi, Multiplicity-free theorem in branching problems of unitary highest modules, Proceedings of the Symposium on Representation Theory, Saga, Japan (K. Mimachi, ed.), 1997, pp.,9–17.
  • T. Kobayashi, Multiplicity-free theorems of the restrictions of unitary heighest weight modules with respect to reductive symmetric pairs, to appear in Progr. Math., Birkhäuser.
  • T. Kobayashi, Geometry of multiplicity-free representations of ${GL}(n)$, visible actions on flag varieties, and triunity, Acta Appl. Math., 81 (2004), 129–146.
  • T. Kobayashi, Multiplicity-free representations and visible actions on complex manifolds, Publ. RIMS, Kyoto Univ., 41 (2005), 497–549, Special Issue of Publications of RIMS commemorating the fortieth anniversary of the founding of the Research Institute for Mathematical Sciences.
  • K. Koike and I. Terada, Young diagrammatic methods for the representation theory of the classical groups of type ${B}_n,{C}_n,{D}_n$, J. Algebra, 107 (1987), 466–511.
  • C. Krattenthaler, Identities for classical group characters of nearly rectangular shape, J. Algebra, 209 (1998), 1–64.
  • I. G. Macdonald, Symmetric Functions and Hall Polynomials, Oxford Univ. Press, 1997.
  • S. Okada, Applications of minor summation formulas to rectangular-shaped representations of classical groups, J. Algebra, 205 (1998), 337–367.
  • R. A. Proctor, Shifted plane partition of trapezoidal shape, Proc. Amer. Math. Soc., 89 (1983), 553–559.
  • I. Satake, On representations and compactifications of symmetric Riemannian spaces, Ann. of Math., 71 (1960), 77–110.
  • W. Schmid, Die Randwerte holomorpher Funktionen auf hermitesch symmetrischen Räumen, Invent. Math., 9 (1969/1970), 61–80.
  • J. R. Stembridge, Hall-Littlewood functions, plane partitions, and Roger-Ramanujan identities, Trans. Amer. Math., 319 (1990), 469–498.
  • D. P. Želobenko, Compact Lie Groups and Their Representations, Translations of Mathematical Monographs, Amer. Math. Soc., 40 (1973).