## Visible actions on symmetric spaces.(English)Zbl 1147.53041

Consider a Lie group $$H$$ acting holomorphically on a complex manifold $$D$$. This action is said to be strongly visible if there exists a real submanifold $$S$$ and an antiholomorphic involution $$\sigma$$ of $$D$$ such that
(i) $$S$$ meets every $$H$$-orbit in $$D$$, (ii) $$\sigma (z)=z$$ for every $$z\in S$$, (iii) $$\sigma$$ preserves each $$H$$-orbit in $$D$$.
This notion is useful in representation theory for proving various multiplicity free results.
If $$D=G/K$$ is a semi-simple Hermitian symmetric space, then the action on $$D$$ of every symmetric subgroup $$H$$ of $$G$$ is strongly visible. Further, the action of a maximal unipotent subgroup $$N$$ of $$G$$ on $$D$$ is strongly visible as well.
As an application, if $$(\pi ,{\mathcal H})$$ is an irreducible unitary highest weight representation of $$G$$ of scalar type, then the restriction of $$\pi$$ to every symmetric subgroup $$H$$ of $$G$$ is multiplicity free. As a special case it follows that the tensor product $$\pi _1\otimes \pi _2$$ of two unitary highest weight representations $$\pi _1$$ and $$\pi _2$$ of scalar type is multiplicity free. The representation $$\pi _1\otimes \pi _2^*$$ is multiplicity free as well. Further, the restriction to a maximal unipotent subgroup $$N$$ of $$G$$ of an irreducible unitary highest weight representation of $$G$$ of scalar type is multiplicity free.

### MSC:

 53C35 Differential geometry of symmetric spaces
Full Text: