Visible actions on symmetric spaces.

*(English)*Zbl 1147.53041Consider 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.

(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.

Reviewer: Jacques Faraut (Paris)

##### MSC:

53C35 | Differential geometry of symmetric spaces |