The explicit Plancherel formula on the line budle of $\mathrm{S}\mathrm{L}(n+1,\mathbb{R})\mathrm{/}\mathrm{S}(\mathrm{G}\mathrm{L}(1,\mathbb{R})\times \mathrm{G}\mathrm{L}(n,\mathbb{R}))$

The purpose of this paper is to study the Plancherel formula for the spaces of $L^2$-sections of the line bundles over the pseudo-Riemannian space $G\mathrm{/}H$, where $G=\mathrm{S}\mathrm{L}(n+1,\mathbb{R})$ and $H={\rm S}({\rm GL}(1, {\mathbb R})\times {\rm GL}(n, {\mathbb R}))$. The formula is given in an explicit form by means of spherical distributions associated with the character ${\chi }_{{}_{\lambda }}$ of the subgroup $H$.