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

TL;DR

This paper derives an explicit Plancherel formula for the space of square-integrable sections of line bundles over a pseudo-Riemannian space formed by SL(n+1,R) mod a subgroup, using spherical distributions linked to specific characters.

Contribution

It provides the first explicit formulation of the Plancherel formula for these line bundle spaces over the given pseudo-Riemannian homogeneous space.

Findings

Explicit Plancherel formula derived for the space of L^2-sections.

Use of spherical distributions associated with subgroup characters.

Enhanced understanding of harmonic analysis on pseudo-Riemannian spaces.

Abstract

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 $G/H$, where $G={\rm SL}(n+1, {\mathbb R})$ 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 $H$.