Visible actions on flag varieties of type D and a generalization of the Cartan decomposition. (English) Zbl 1296.22015
Let \(G\) be a simple connected compact Lie group, \(L, H\) be two compact Lie subgroups containing a same maximal connected abelian subgroup \(T\) of \(G\) such that their root systems are subsystems of the root systems of \(G\). Following T. Kobayashi [Transform. Groups 12, No. 4, 671–694 (2007; Zbl 1147.53041)], the author defines the action of \(L\) on the generalized flag manifold \(M=G/H\) (which is also a compact complex manifold and is projective and rational) to be strongly visible if (i) there is a real submanifold \(S\) such that \(O=LS\) is an open subset of \(M\); (ii) there is an anti-holomorphic involution \(\tau\) on \(O\) such that \(S\) is in the fixed point set of \(\tau\).
Now let \(\tau\) be an involution of \(G\) such that \(\tau (t)=t^{ -1}\) for all \(t\) in \(T\). Define \(N\) to be the fixed point set of \(G\) with respect to \(\tau\), then \(NH\) defines a real submanifold \(S\) in \(G/H\). The author classifies all the triples \((G,H,L)\) such that \(G=LNH\) with \(G=SO(2n)\).
The author calls this a generalized Cartan decomposition of the given matrix group \(G\) with respect to the triple \((H,L,\tau )\). We notice that \(L\) acts on \(G/H\) strongly visible with \(S=NH/H\). This also has something to do with multiplicity-free representations of \(G\) induced from \(G/H\).
The cases in which \(G=SU(n)\) was first treated by T. Kobayashi [J. Math. Soc. Japan 59, No. 3, 669–691 (2007; Zbl 1124.22003)]. The author finished the classification program started therein, for all the simple Lie groups \(G\). The announcement appeared in [Y. Tanaka, Proc. Japan Acad., Ser. A 88, No. 6, 91–96 (2012; Zbl 1250.32021)]. The details for the cases in which \(G=SO(2n+1)\) appeared in [Bull. Aust. Math. Soc. 88, No. 1, 81–97 (2013; Zbl 1347.22012)]. The details for the cases in which \(G=Sp(2n)\) appeared in [Tohoku Math. J. (2) 65, No. 2, 281–295 (2013; Zbl 1277.22014)]. The details in which \(G\) are exceptional appeared in [J. Algebra 399, 170–189 (2014; Zbl 1305.22013)]. Possibly, the author should mention these related works.
Now let \(\tau\) be an involution of \(G\) such that \(\tau (t)=t^{ -1}\) for all \(t\) in \(T\). Define \(N\) to be the fixed point set of \(G\) with respect to \(\tau\), then \(NH\) defines a real submanifold \(S\) in \(G/H\). The author classifies all the triples \((G,H,L)\) such that \(G=LNH\) with \(G=SO(2n)\).
The author calls this a generalized Cartan decomposition of the given matrix group \(G\) with respect to the triple \((H,L,\tau )\). We notice that \(L\) acts on \(G/H\) strongly visible with \(S=NH/H\). This also has something to do with multiplicity-free representations of \(G\) induced from \(G/H\).
The cases in which \(G=SU(n)\) was first treated by T. Kobayashi [J. Math. Soc. Japan 59, No. 3, 669–691 (2007; Zbl 1124.22003)]. The author finished the classification program started therein, for all the simple Lie groups \(G\). The announcement appeared in [Y. Tanaka, Proc. Japan Acad., Ser. A 88, No. 6, 91–96 (2012; Zbl 1250.32021)]. The details for the cases in which \(G=SO(2n+1)\) appeared in [Bull. Aust. Math. Soc. 88, No. 1, 81–97 (2013; Zbl 1347.22012)]. The details for the cases in which \(G=Sp(2n)\) appeared in [Tohoku Math. J. (2) 65, No. 2, 281–295 (2013; Zbl 1277.22014)]. The details in which \(G\) are exceptional appeared in [J. Algebra 399, 170–189 (2014; Zbl 1305.22013)]. Possibly, the author should mention these related works.
Reviewer: Daniel Guan (Riverside)
MSC:
| 22E46 | Semisimple Lie groups and their representations |
| 32M10 | Homogeneous complex manifolds |
| 53C30 | Differential geometry of homogeneous manifolds |
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
Keywords:
visible actions; generalized Cartan decompositions; semisimple Lie groups; generalized flag manifolds; multiplicity-free representations; anti-holomorphic involutions.Citations:
Zbl 1147.53041; Zbl 1124.22003; Zbl 1250.32021; Zbl 1277.22014; Zbl 1347.22012; Zbl 1305.22013References:
| [1] | S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Grad. Stud. Math., 34 , Amer. Math. Soc., Providence, RI, 2001. · Zbl 0993.53002 |
| [2] | B. Hoogenboom, Intertwining Functions on Compact Lie Groups, CWI Tract, 5 , Math. Centrum, Centrum Wisk. Inform., Amsterdam, 1984. · Zbl 0553.43005 |
| [3] | A. W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Progr. Math., 140 , Birkhäuser, Boston, 2002. · Zbl 1075.22501 |
| [4] | T. Kobayashi and T. Oshima, Lie Groups and Representation Theory (Japanese), Iwanami, 2005. |
| [5] | T. Kobayashi, Geometry of multiplicity-free representations of GL\((n)\), visible actions on flag varieties, and triunity, Acta Appl. Math., 81 (2004), 129-146. · Zbl 1050.22018 · doi:10.1023/B:ACAP.0000024198.46928.0c |
| [6] | T. Kobayashi, Multiplicity-free representations and visible actions on complex manifolds, Publ. Res. Inst. Math. Sci., 41 (2005), 497-549. · Zbl 1085.22010 · doi:10.2977/prims/1145475221 |
| [7] | 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, · Zbl 1304.22013 · doi:10.1007/978-0-8176-4646-2_3 |
| [8] | 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), 669-691. · Zbl 1124.22003 · doi:10.2969/jmsj/05930669 |
| [9] | T. Kobayashi, Visible actions on symmetric spaces, Transform. Groups, 12 (2007), 671-694. · Zbl 1147.53041 · doi:10.1007/s00031-007-0057-4 |
| [10] | 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. · Zbl 1304.22013 · doi:10.1007/978-0-8176-4646-2_3 |
| [11] | 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. · Zbl 0622.20033 · doi:10.1016/0021-8693(87)90099-8 |
| [12] | 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. · Zbl 0698.22013 · doi:10.1016/0001-8708(90)90059-V |
| [13] | P. Littelmann, A generalization of the Littlewood-Richardson rule, J. Algebra, 130 (1990), 328-368. · Zbl 0704.20033 · doi:10.1016/0021-8693(90)90086-4 |
| [14] | P. Littelmann, On spherical double cones, J. Algebra, 166 (1994), 142-157. · Zbl 0823.20040 · doi:10.1006/jabr.1994.1145 |
| [15] | 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. · Zbl 0900.22010 |
| [16] | T. Matsuki, Double coset decompositions of algebraic groups arising from two involutions. I, J. Algebra, 175 (1995), 865-925. · Zbl 0831.22002 · doi:10.1006/jabr.1995.1218 |
| [17] | T. Matsuki, Double coset decompositions of reductive Lie groups arising from two involutions, J. Algebra, 197 (1997), 49-91. · Zbl 0887.22009 · doi:10.1006/jabr.1997.7123 |
| [18] | A. Sasaki, Visible actions on irreducible multiplicity-free spaces, Int. Math. Res. Not. IMRN, 2009 (2009), 3445-3466. · Zbl 1180.32006 · doi:10.1093/imrn/rnp060 |
| [19] | A. Sasaki, A characterization of non-tube type Hermitian symmetric spaces by visible actions, Geom. Dedicata, 145 (2010), 151-158. · Zbl 1190.32017 · doi:10.1007/s10711-009-9412-z |
| [20] | 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. · Zbl 1251.53031 |
| [21] | J. R. Stembridge, Multiplicity-free products of Schur functions, Ann. Comb., 5 (2001), 113-121. · Zbl 0990.05130 · doi:10.1007/s00026-001-8008-6 |
| [22] | J. R. Stembridge, Multiplicity-free products and restrictions of Weyl characters, Represent. Theory, 7 (2003), 404-439. · Zbl 1060.17001 · doi:10.1090/S1088-4165-03-00150-X |
| [23] | J. A. Wolf, Harmonic Analysis on Commutative Spaces, Math. Surveys Monogr., 142 , Amer. Math. Soc., Providence, RI, 2007. · Zbl 1156.22010 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
