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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