Some Remarks on Visible Actions on Multiplicity-Free Spaces

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