Classical limit of the tensor product of holomorphic discrete series representations

Abstract

For three coadjoint orbits $\mathcal {O}_1, \mathcal {O}_2$ and $\mathcal {O}_3$ in $\mathfrak {g}^*$, the Corwin–Greenleaf function $n(\mathcal {O}_1 \times \mathcal {O}_2, \mathcal {O}_3)$ is given by the number of $G$-orbits in $\{(\lambda , \mu ) \in \mathcal {O}_1 \times \mathcal {O}_2 \, : \, \lambda + \mu \in \mathcal {O}_3 \}$ under the diagonal action. In the case where $G$ is a simple Lie group of Hermitian type, we give an explicit formula of $n(\mathcal {O}_1 \times \mathcal {O}_2, \mathcal {O}_3)$ for coadjoint orbits $\mathcal {O}_1$ and $\mathcal {O}_2$ that meet $\left( [\mathfrak {k}, \mathfrak {k}] + \mathfrak {p}\right) ^{\perp }$, and show that the formula is regarded as the ‘classical limit’ of a special case of Kobayashi’s multiplicity-free theorem (Progr. Math. 2007) in the branching law to symmetric pairs.