Scattering and a Plancherel formula of spherical varieties for real reductive split groups

TL;DR

This paper extends the scattering theorem to real spherical varieties, providing a Plancherel formula for real reductive split groups using Harish-Chandra homomorphism and spectral analysis.

Contribution

It establishes the analog of Sakellaridis and Venkatesh's scattering theorem for real spherical varieties, a significant extension from p-adic cases.

Findings

Proves a Plancherel formula for real spherical varieties.

Uses properties of the Harish-Chandra homomorphism for invariant differential operators.

Employs spectral projections and special coverings of the variety.

Abstract

We establish the analog for real spherical varieties of the Scattering Theorem of Sakellaridis and Venkatesh (\cite{SV}, Theorem 7.3.1) for p-adic spherical varieties. We use properties of the Harish-Chandra homomorphism of Knop for invariant differential operators of the variety, special coverings of the variety and spectral projections.