Volume growth, temperedness and integrability of matrix coefficients on a real spherical space (Q281507)

First, the authors consider a real algebraic homogeneous space \(Z=G/H\), attached to an algebraic real reductive group \(G\) and assumed to carry a \(G\)-invariant measure, where they define the volume weight \(v\) and study the growth of \(v\) as a function of \(z\in Z\). Next, they study the Harish-Chandra Schwartz space on \(Z\) by means of a function space \( \mathcal C (Z)\) and show several interesting properties (e.g., \( \mathcal C (Z)\) is nuclear). Finally, they define the integrability property \((I)\) by means of unitary irreducible representations \((\pi,\mathcal H)\) of an \(H\)-invariant distribution vector \(\eta\) and of the \(L^{p}\)-integrability of the matrix coefficient \(m_{v,\eta}\). They discuss different cases where \(Z\) has the property \((I)\) (e.g. if \(Z=G/H\) is a wavefront real spherical space with \(H\) reductive). The investigation of several related phenomena is also very interesting.