Scattering and a Plancherel formula for real reductive spherical spaces
arXiv:2305.13867
The paper proves a scattering theorem and a Plancherel formula for real homogeneous spherical varieties, extending results of Sakellaridis–Venkatesh to the real case using Harish‑Chandra homomorphisms and spectral projections, assuming an analog of their discrete series conjecture.
Abstract
We establish the analog for real homogeneous spherical varieties of the Scattering Theorem of Sakellaridis and Venkatesh (Periods and harmonic analysis on spherical varieties, Asterisque 396, (2017), Theorem 7.3.1) for p-adic wavefront 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. Our main result depend on an analog of the Discrete Series Conjecture of Sakellaridis and Venkatesh (\cite{SV}, Conjecture 9.4.6). Their result quoted above depends on this Discrete series conjecture.
A misunderstanding of the coverages makes section 8 completely wrong. The defition of scatering operators seems correct but their unitarity is not proved