Some Remarks on Visible Actions on Multiplicity-free Spaces
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Abstract
This paper provides a summary of the recent work on visible actions, based on Sasaki [1-3]. A holomorphic action of a Lie group on a complex manifold is called stronglyvisible if a real submanifold meeting every -orbit in and an anti-holomorphic diffeomorphism preserving each -orbit such that exists. In case of complex linear spaces, a deep relationship between visible actions and multiplicity-freerepresentations is found. A holomorphic representation of a complex reductive it’s Lie group has a strongly visible action if and only if its polynomial ring is multiplicity-free as it’s representation. Furthermore, the dimension of our choice of a slice coincides with the rank of the semigroup of highest weights occurring in the polynomial ring.
Keywords: Visible actions, slice, multiplicity-free representation, rank
*Corresponding author: E-mail: atsumu@tokai-u.jp
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References
[2] A. Sasaki, 2011. Visible actions on reducible multiplicity-free spaces, International Mathematics Research Notices, 885-929.
[3] A. Sasaki, 2012. Compatible automorphisms for strongly visible linear actions, preprint.
[4] T. Kobayashi, 2005. Multiplicity-free representations and visible actions on complex manifolds, Publications of the Research Institute for Mathematical Sciences 41, 497-549, special issue commemorating the fortieth anniversary of the founding of RIMS.
[5] T. Kobayashi, Propagation of multiplicity-free property for holomorphic vector bundles, preprint, arXiv: math.RT/0607004.
[6] A. Sasaki, 2010. A characterization of non-tube type Hermitian symmetric spaces by visible actions, GeometriaeDedicata 145, 151-158.
[7] A. Sasaki, 2010. A generalized Cartan decomposition for the double coset space SU(2n+1)∖SL(2n+1,C)/Sp(n,C), Journal of Mathematical Sciences, the University of Tokyo 17, 201-215.
[8] A. Sasaki, 2011. Visible actions on the non-symmetric homogeneous space SO(8,C)/G_2 (C), Advances in Pure and Applied Mathematics 2, 437-450.
[9] C, Benson, G. Ratcliff, 1996. A classification of multiplicity-free actions, Journal of Algebra 181, 152-186
[10] C, Benson, G. Ratcliff, 2004. On multiplicity-free actions, Representations of Real and p-Adic Groups, 221-304, Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore 2, Singapore University Press, Singapore.
[11] V. Kac, 1980. Some remarks on nilpotent orbits, Journal of Algebra 64, 190-213.
[12] A. Leahy, 1998. A classification of multiplicity free representations, Journal of Lie Theory 8, 367-391.
[13] R. Howe, T. Umeda, 1991. The Capelli identity, the double commutant theorem, and multiplicity-free actions, MathematischeAnnalen 290, 565-619.
[14] F. Knop, 1998. Some remarks on multiplicity-free spaces, Representation Theories and Algebraic Geometry, 301-317, NATO Advances Science Institute Series C: Mathematical and Physocal Sciences 514, Kluwer Academic Publishers, Dordrecht.
[15] A. Sasaki, 2012. Visible actions on spherical nilpotent orbits, preprint.
