\((\mathrm{GL}(n+1,\mathbb R),\mathrm{GL}(n,\mathbb R))\) is a generalized Gelfand pair (Q1033812)

Let \(G\) be a Lie group, and \(H\) a closed subgroup. The pair \((G,H)\) is said to be a generalized Gelfand pair if every \(G\)-invariant Hilbert subspace of \({\mathcal D}'(G/H)\), the space of distributions on the homogeneous space \(G/H\), decomposes multiplicity free. It is known that the symmetric pair \((\text{GL}(n+1,\mathbb R), \text{GL}(1,\mathbb R)\times \text{GL}(n,\mathbb R))\) is a generalized Gelfand pair. In this note it is shown that \((\text{GL}(n+1,\mathbb R), \text{GL}(n,\mathbb R))\) is a generalized Gelfand pair as well.