A characterization of non-tube type Hermitian symmetric spaces by visible actions. (English) Zbl 1190.32017
A holomorphic action of a Lie group \(H\) on a connected complex manifold \(D\) is called strongly visible in the sense of T. Kobayashi [Publ. Res. Inst. Math. Sci 41, No. 3, 497–549 (2005; Zbl 1085.22010)] if there exists a real submanifold \(S\) (called a slice) in \(D\) and an anti-holomorphic diffeomorphism \(\sigma\) of \(D\) such that \(D=H\cdot S\), \(\sigma|_S = \text{id}_S\), and \(\sigma\) preserves each \(H\)-orbit in \(D\). Let \(G/K\) be an irreducible Hermitian symmetric space of non-compact type, \(G_{\mathbb C}/K_{\mathbb C}\) its complexification by forgetting the original complex structure. Then \(D:= G_{\mathbb C}/[K_{\mathbb C},K_{\mathbb C}]\) is a non-symmetric Stein manifold.
In the paper, the author proves that a maximal compact subgroup of \(G_{\mathbb C}\) acts on \(D\) in a strongly visible fashion if and only if \(G/K\) is of non-tube type. The proof uses the theory of multiplicity-free representations and the construction of a slice and an anti-holomorphic involution on \(D\).
In the paper, the author proves that a maximal compact subgroup of \(G_{\mathbb C}\) acts on \(D\) in a strongly visible fashion if and only if \(G/K\) is of non-tube type. The proof uses the theory of multiplicity-free representations and the construction of a slice and an anti-holomorphic involution on \(D\).
Reviewer: Manfred Stoll (Columbia)
MSC:
| 32M15 | Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) |
| 22E46 | Semisimple Lie groups and their representations |
| 32A07 | Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010) |
| 20G05 | Representation theory for linear algebraic groups |
Citations:
Zbl 1085.22010References:
| [1] | Flensted-Jensen M.: Spherical functions on a simply connected semisimple Lie group. Am. J. Math. 99, 341–361 (1977) · Zbl 0372.43005 · doi:10.2307/2373823 |
| [2] | Flensted-Jensen M.: Spherical functions of a real semisimple Lie group. A method of reduction to the complex case. J. Funct. Anal. 30, 106–146 (1978) · Zbl 0419.22019 · doi:10.1016/0022-1236(78)90058-7 |
| [3] | Knapp, A.W.: Lie groups beyond an introduction, 2nd edition. Progress in Mathematics 140, Birkhäuser, Boston (2002) · Zbl 1075.22501 |
| [4] | Kobayashi T.: Discrete decomposability of the restriction of \({A_{\mathfrak{q}}(\lambda )}\) with respect to reductive subgroups. Part I. Invent. Math. 117, 181–205 (1994) · Zbl 0826.22015 · doi:10.1007/BF01232239 |
| [5] | Kobayashi T.: Discrete decomposability of the restriction of \({A_{\mathfrak{q}}(\lambda )}\) with respect to reductive subgroups. Part II. Ann. Math. 147, 709–729 (1998) · Zbl 0910.22016 |
| [6] | Kobayashi T.: Discrete decomposability of the restriction of \({A_{\mathfrak{q}}(\lambda )}\) with respect to reductive subgroups. Part III. Invent. Math. 131, 229–256 (1998) · Zbl 0907.22016 · doi:10.1007/s002220050203 |
| [7] | Kobayashi T.: Geometry of multiplicity-free representations of GL(n), visible actions on flag varieties, and triunity. Acta. Appl. Math. 81, 129–146 (2004) · Zbl 1050.22018 · doi:10.1023/B:ACAP.0000024198.46928.0c |
| [8] | Kobayashi, T.: Multiplicity-free representations and visible actions on complex manifolds. Publ. Res. Inst. Math. Sci. 41, 497–549 (2005), special issue commemorating the fortieth anniversary of the founding of RIMS · Zbl 1085.22010 |
| [9] | Kobayashi T.: A generalized Cartan decomposition for the double coset space (U(n 1) {\(\times\)} U(n 2) {\(\times\)} U(n 3))\(\backslash\) U(n)/(U(p) {\(\times\)} U(q)). J. Math. Soc. Japan 59, 669–691 (2007) · Zbl 1124.22003 · doi:10.2969/jmsj/05930669 |
| [10] | Kobayashi T.: Visible actions on symmetric spaces. Transform. Group 12, 671–694 (2007) · Zbl 1147.53041 · doi:10.1007/s00031-007-0057-4 |
| [11] | Kobayashi, T.: Propagation of multiplicity-free property for holomorphic vector bundles. math.RT/0607004 · Zbl 1284.32011 |
| [12] | Korányi A., Wolf J.A.: Realization of Hermitian symmetric spaces as generalized half-planes. Ann. Math. 81, 265–288 (1965) · Zbl 0137.27402 · doi:10.2307/1970616 |
| [13] | Krämer M.: Sphärische Untergruppen in kompakten zusammenhängenden Liegruppen. Compositio Math. 38, 129–153 (1979) · Zbl 0402.22006 |
| [14] | Matsushima Y.: Espaces homogènes de Stein des groupes de Lie complexes. Nagoya Math. J. 16, 205–218 (1960) · Zbl 0094.28201 |
| [15] | Sasaki, A.: Visible actions on irreducible multiplicity-free spaces. Int. Math. Res. Not. (2009). doi: 10.1093/imrn/rnp060 · Zbl 1180.32006 |
| [16] | Vinberg É.B., Kimelfeld B.N.: Homogeneous domains on flag manifolds and spherical subgroups of semisimple Lie groups. Funct. Anal. Appl. 12, 168–174 (1978) · Zbl 0439.53055 · doi:10.1007/BF01681428 |
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