Baklouti A, Sasaki A (2021)
Publication Type: Journal article
Publication year: 2021
Book Volume: 31
Pages Range: 719-750
Journal Issue: 3
A visible action on a complex manifold is a holomorphic action that admits a J -transversal totally real submanifold S . It is said to be strongly visible if there exists an orbitpreserving anti-holomorphic diffeomorphism σ such that σjS = idS . Let G be the Heisenberg group and H a non-trivial connected closed subgroup of G. We prove that any complex homogeneous space D = GC/HC admits a strongly visible L-action, where L stands for a connected closed subgroup of G explicitly constructed through a co-exponential basis of H in G. This leads in turn that G itself acts strongly visibly on D. The proof is carried out by finding explicitly an orbit-preserving anti-holomorphic diffeomorphism and a totally real submanifold S , for which the dimension depends upon the dimensions of G and H. As a direct application, our geometric results provide a proof of various multiplicity-free theorems on continuous representations on the space of holomorphic sections on D. Moreover, we also generate as a consequence, a geometric criterion for a quasi-regular representation of G to be multiplicity-free.
Baklouti, A., & Sasaki, A. (2021). Visible actions and criteria for multiplicity-freeness of representations of heisenberg groups. Journal of Lie Theory, 31(3), 719-750.
Baklouti, Ali, and Atsumu Sasaki. "Visible actions and criteria for multiplicity-freeness of representations of heisenberg groups." Journal of Lie Theory 31.3 (2021): 719-750.