Multiplicity free subgroups of semi-direct products

Recently, Sun and Zhu have shown that the pairs of groups of the list (O(p,q), O(p - l,q)), (U(p,q), U(p - l,q)), (GL(n, ℝ), GL(n - 1, ℝ)) and their complex counterparts, namely the pairs (O(n, ℂ), O(n - 1, ℂ)) and (GL(n, ℂ), GL(n - 1, ℂ)), are so-called multiplicity free pairs. In this note we consider two types of semi-direct products. We show, in particular, that (U(p, q) ⋉ℂp+q, U(p, q)) and (U(p, q) ⋉ Hp+q,U(p,q)) are strong multiplicity free pairs. Here Hp+q denotes the Heisenberg group of real dimension 2(p+q) + 1.