Journal of the Mathematical Society of Japan

Visible actions on flag varieties of type D and a generalization of the Cartan decomposition

Yuichiro TANAKA

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Abstract

We give a generalization of the Cartan decomposition for connected compact Lie groups motivated by the work on visible actions of T. Kobayashi [J. Math. Soc. Japan, 2007] for type A group. This paper extends his results to type D group. First, we classify a pair of Levi subgroups $(L,H)$ of a simple compact Lie group $G$ of type D such that $G=LG^{\sigma}H$ where $\sigma$ is a Chevalley–Weyl involution. This gives the visibility of the $L$-action on the generalized flag variety $G/H$ as well as that of the $H$-action on $G/L$ and of the $G$-action on $(G\times G)/(L\times H)$. Second, we find a generalized Cartan decomposition $G=LBH$ with $B$ in $G^{\sigma}$ by using the herringbone stitch method which was introduced by Kobayashi in his 2007 paper. Applications to multiplicity-free theorems of representations are also discussed.

Article information

Source
J. Math. Soc. Japan, Volume 65, Number 3 (2013), 931-965.

Dates
First available in Project Euclid: 23 July 2013

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1374586630

Digital Object Identifier
doi:10.2969/jmsj/06530931

Mathematical Reviews number (MathSciNet)
MR3079290

Zentralblatt MATH identifier
1296.22015

Subjects
Primary: 22E46: Semisimple Lie groups and their representations
Secondary: 32A37: Other spaces of holomorphic functions (e.g. bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA)) [See also 46Exx] 53C30: Homogeneous manifolds [See also 14M15, 14M17, 32M10, 57T15]

Keywords
Cartan decomposition multiplicity-free representation semisimple Lie group flag variety visible action herringbone stitch

Citation

TANAKA, Yuichiro. Visible actions on flag varieties of type D and a generalization of the Cartan decomposition. J. Math. Soc. Japan 65 (2013), no. 3, 931--965. doi:10.2969/jmsj/06530931. https://projecteuclid.org/euclid.jmsj/1374586630


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References

  • S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Grad. Stud. Math., 34, Amer. Math. Soc., Providence, RI, 2001.
  • B. Hoogenboom, Intertwining Functions on Compact Lie Groups, CWI Tract, 5, Math. Centrum, Centrum Wisk. Inform., Amsterdam, 1984.
  • A. W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Progr. Math., 140, Birkhäuser, Boston, 2002.
  • T. Kobayashi and T. Oshima, Lie Groups and Representation Theory (Japanese), Iwanami, 2005.
  • 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. Res. Inst. Math. Sci., 41 (2005), 497–549.
  • T. Kobayashi, Propagation of multiplicity-freeness property for holomorphic vector bundles, In: Lie Groups: Structure, Actions, and Representations: in Honor of Joseph A. Wolf on the Occasion of His 75th Birthday, Progr. Math., 306, Birkhäuser, Boston, 2013, arXiv:0607004v2.
  • T. Kobayashi, A generalized Cartan decomposition for the double coset space $({\rm U}(n_{1}) \times {\rm U}(n_{2}) \times {\rm U}(n_{3})) \backslash {\rm U}(n)/({\rm U}(p) \times {\rm U}(q))$, J. Math. Soc. Japan, 59 (2007), 669–691.
  • T. Kobayashi, Visible actions on symmetric spaces, Transform. Groups, 12 (2007), 671–694.
  • T. Kobayashi, Multiplicity-free theorems of the restriction of unitary highest weight modules with respect to reductive symmetric pairs, In: Representation Theory and Automorphic Forms, Progr. Math., 255, Birkhäuser, Boston, 2008, pp.,45–109.
  • 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.
  • K. Koike and I. Terada, Young Diagrammatic Methods for the Restriction of Representations of Complex Classical Lie Groups to Reductive Subgroups of Maximal Rank, Adv. Math., 79 (1990), 104–135.
  • P. Littelmann, A generalization of the Littlewood-Richardson rule, J. Algebra, 130 (1990), 328–368.
  • P. Littelmann, On spherical double cones, J. Algebra, 166 (1994), 142–157.
  • T. Matsuki, Double coset decompositions of algebraic groups arising from two involutions. II (Japanese), In: Non-Commutative Analysis on Homogeneous Spaces, Kyoto, 1994, Sūrikaisekikenkyūsho Kōkyūroku, 895 (1995), 98–113.
  • T. Matsuki, Double coset decompositions of algebraic groups arising from two involutions. I, J. Algebra, 175 (1995), 865–925.
  • T. Matsuki, Double coset decompositions of reductive Lie groups arising from two involutions, J. Algebra, 197 (1997), 49–91.
  • A. Sasaki, Visible actions on irreducible multiplicity-free spaces, Int. Math. Res. Not. IMRN, 2009 (2009), 3445–3466.
  • A. Sasaki, A characterization of non-tube type Hermitian symmetric spaces by visible actions, Geom. Dedicata, 145 (2010), 151–158.
  • A. Sasaki, A generalized Cartan decomposition for the double coset space $SU(2n+1)$ $\backslash SL(2n+1,{\mathbb C}) / Sp(n,{\mathbb C})$, J. Math. Sci. Univ. Tokyo, 17 (2010), 201–215.
  • J. R. Stembridge, Multiplicity-free products of Schur functions, Ann. Comb., 5 (2001), 113–121.
  • J. R. Stembridge, Multiplicity-free products and restrictions of Weyl characters, Represent. Theory, 7 (2003), 404–439.
  • J. A. Wolf, Harmonic Analysis on Commutative Spaces, Math. Surveys Monogr., 142, Amer. Math. Soc., Providence, RI, 2007.