The tempered spectrum of a real spherical space

Let G/H be a unimodular real spherical space which is either absolutely spherical or wave-front. It is shown that every tempered representation of G/H embeds into a relative discrete series of a boundary degeneration of G/H. If in addition G/H is of wave-front type it follows that the tempered representation is parabolically induced from a discrete series representation of a lower dimensional real spherical space. Final version. To appear in Acta Math.Comment: 64 page

The local structure theorem for real spherical varieties

Let $G$ be an algebraic real reductive group and $Z$ a real spherical $G$-variety, that is, it admits an open orbit for a minimal parabolic subgroup $P$. We prove a local structure theorem for $Z$. In the simplest case where $Z$ is homogeneous, the theorem provides an isomorphism of the open $P$-orbit with a bundle $Q \times_L S$. Here $Q$ is a parabolic subgroup with Levi decomposition $LU$, and $S$ is a homogeneous space for a quotient $D=L/L_n$ of $L$, where $L_n$ is normal in $L$, such that $D$ is compact modulo center.Comment: v1: 18 pages, no figures; v2: 19 pages, revised, final versio

Classification of reductive real spherical pairs II. The semisimple case

If ${\mathfrak g}$ is a real reductive Lie algebra and ${\mathfrak h} < {\mathfrak g}$ is a subalgebra, then $({\mathfrak g}, {\mathfrak h})$ is called real spherical provided that ${\mathfrak g} = {\mathfrak h} + {\mathfrak p}$ for some choice of a minimal parabolic subalgebra ${\mathfrak p} \subset {\mathfrak g}$. In this paper we classify all real spherical pairs $({\mathfrak g}, {\mathfrak h})$ where ${\mathfrak g}$ is semi-simple but not simple and ${\mathfrak h}$ is a reductive real algebraic subalgebra. The paper is based on the classification of the case where ${\mathfrak g}$ is simple (see arXiv:1609.00963) and generalizes the results of Brion and Mikityuk in the (complex) spherical case.Comment: Extended revised version. Section 6 and Appendix B are new. To appear in Transformation Groups. 40

Volume growth, temperedness and integrability of matrix coefficients on a real spherical space

We apply the local structure theorem and the polar decomposition to a real spherical space Z=G/H and control the volume growth on Z. We define the Harish-Chandra Schwartz space on Z. We give a geometric criterion to ensure $L^p$-integrability of matrix coefficients on Z.Comment: Additional material of 4 pages added. To appear in J. Funct. Analysi

Comment on "Material Evidence of a 38 MeV Boson"

In the recent preprint 1202.1739 it was claimed that preliminary data presented by COMPASS at recent conferences confirm the existence of a resonant state of mass 38 MeV decaying to two photons. This claim was made based on structures observed in two-photon mass distributions which however were shown only to demonstrate the purity and mass resolution of the {\pi}0 and {\eta} signals. The additional structures are understood as remnants of secondary interactions inside the COMPASS spectrometer. Therefore, the COMPASS data do not confirm the existence of this state.Comment: 2 pages, 7 figure