previous :: next

### Classification of visible actions on flag varieties

Yuichiro Tanaka
Source: Proc. Japan Acad. Ser. A Math. Sci. Volume 88, Number 6 (2012), 91-96.

#### Abstract

We give a complete classification of the pairs $(L,H)$ of Levi subgroups of compact simple Lie groups $G$ such that the $L$-action on a generalized flag variety $G/H$ is strongly visible (or equivalently, the $H$-action on $G/L$ or the diagonal $G$-action on $(G\times G)/(L\times H)$). The notion of visible actions on complex manifolds was introduced by T. Kobayashi, and a classification was accomplished by himself for the type A groups [J. Math. Soc. Japan, 2007]. A key step is to classify the pairs $(L,H)$ for which the multiplication mapping $L\times G^{\sigma}\times H\to G$ is surjective, where $\sigma$ is a Chevalley–Weyl involution of $G$. We then see that strongly visible actions, multiplicity-free restrictions of representations (c.f. Littelmann, Stembridge), the decomposition $G=LG^{\sigma}H$ and spherical actions are all equivalent in our setting.

First Page:
Primary Subjects: 22E46
Secondary Subjects: 32A37, 53C30
Full-text: Open access

Permanent link to this document: http://projecteuclid.org/euclid.pja/1338900735
Digital Object Identifier: doi:10.3792/pjaa.88.91
Zentralblatt MATH identifier: 06071047
Mathematical Reviews number (MathSciNet): MR2928897

### References

N. Bourbaki, Lie groups and Lie algebras. Chapters 4–6, translated from the 1968 French original by Andrew Pressley, Elements of Mathematics (Berlin), Springer, Berlin, 2002.
Mathematical Reviews (MathSciNet): MR1890629
M. Flensted-Jensen, Spherical functions of a real semisimple Lie group. A method of reduction to the complex case, J. Funct. Anal. 30 (1978), no. 1, 106–146.
Mathematical Reviews (MathSciNet): MR513481
Zentralblatt MATH: 0419.22019
Digital Object Identifier: doi:10.1016/0022-1236(78)90058-7
B. Hoogenboom, Intertwining functions on compact Lie groups, CWI Tract, 5, Math. Centrum, Centrum Wisk. Inform., Amsterdam, 1984.
Mathematical Reviews (MathSciNet): MR751783
Zentralblatt MATH: 0553.43005
T. Kobayashi, Geometry of multiplicity-free representations of $\mathrm{GL}(n)$, visible actions on flag varieties, and triunity, Acta Appl. Math. 81 (2004), no. 1–3, 129–146.
Mathematical Reviews (MathSciNet): MR2069335
Digital Object Identifier: doi:10.1023/B:ACAP.0000024198.46928.0c
T. Kobayashi, Multiplicity-free representations and visible actions on complex manifolds, Publ. Res. Inst. Math. Sci. 41 (2005), no. 3, 497–549.
Mathematical Reviews (MathSciNet): MR2153533
Digital Object Identifier: doi:10.2977/prims/1145475221
T. Kobayashi, Propagation of multiplicity-freeness property for holomorphic vector bundles, Progr. Math., Birkhäuser, Boston, 2012 (in press). arXiv:0607004v2.
Mathematical Reviews (MathSciNet): MR2369496
Digital Object Identifier: doi:10.1007/978-0-8176-4646-2_3
T. Kobayashi, A generalized Cartan decomposition for the double coset space $(\mathrm{U}(n_{1})\times\mathrm{U}(n_{2})\times\mathrm{U}(n_{3}))\backslash\mathrm{U}(n)/(\mathrm{U}(p)\times\mathrm{U}(q))$, J. Math. Soc. Japan 59 (2007), no. 3, 669–691.
Mathematical Reviews (MathSciNet): MR2344822
Zentralblatt MATH: 1124.22003
Digital Object Identifier: doi:10.2969/jmsj/05930669
Project Euclid: euclid.jmsj/1191591852
T. Kobayashi, Visible actions on symmetric spaces, Transform. Groups 12 (2007), no. 4, 671–694.
Mathematical Reviews (MathSciNet): MR2365440
Zentralblatt MATH: 1147.53041
Digital Object Identifier: doi:10.1007/s00031-007-0057-4
T. Kobayashi, Multiplicity-free theorems of the restrictions of unitary highest weight modules with respect to reductive symmetric pairs, in Representation theory and automorphic forms, Progr. Math., 255, Birkhäuser, Boston, MA, 2008, pp. 45–109.
Mathematical Reviews (MathSciNet): MR2369496
Digital Object Identifier: doi:10.1007/978-0-8176-4646-2_3
P. Littelmann, On spherical double cones, J. Algebra 166 (1994), no. 1, 142–157.
Mathematical Reviews (MathSciNet): MR1276821
Zentralblatt MATH: 0823.20040
Digital Object Identifier: doi:10.1006/jabr.1994.1145
T. Matsuki, Double coset decompositions of reductive Lie groups arising from two involutions, J. Algebra 197 (1997), no. 1, 49–91.
Mathematical Reviews (MathSciNet): MR1480777
Zentralblatt MATH: 0887.22009
Digital Object Identifier: doi:10.1006/jabr.1997.7123
A. Sasaki, A characterization of non-tube type Hermitian symmetric spaces by visible actions, Geom. Dedicata 145 (2010), 151–158.
Mathematical Reviews (MathSciNet): MR2600951
Zentralblatt MATH: 1190.32017
Digital Object Identifier: doi:10.1007/s10711-009-9412-z
J. R. Stembridge, Multiplicity-free products and restrictions of Weyl characters, Represent. Theory 7 (2003), 404–439 (electronic).
Mathematical Reviews (MathSciNet): MR2017064
Digital Object Identifier: doi:10.1090/S1088-4165-03-00150-X
É. B. Vinberg and B. N. Kimel'fel'd, Homogeneous domains on flag manifolds and spherical subgroups of semisimple Lie groups, Funct. Anal. Appl. 12 (1978), no. 3, 168–174.
previous :: next