Spherical spaces for which the Hasse principle and weak approximation fail (Q1363138)

The main object of this paper is a spherical homogeneous \(k\)-space \(X=H \backslash G\) of a connected semisimple simply connected group \(G\) with connected stabilizer \(H\). If the ground field \(k\) is algebraically closed, this means that there exists a Borel subgroup \(B\subset G\) acting on \(X\) with an open orbit; in general, this means that there is a transitive action \(X\times G\to X\) such that \(\overline X= X\times_k \overline k\) is spherical in the above sense. In many cases spherical spaces enjoy good arithmetic properties such as weak approximation and the Hasse principle; it is true for all projective homogeneous spaces (= flag varieties) [\textit{G. Harder}, Jahresber. Dtsch. Math.-Verf. 70, 182-216 (1968; Zbl 0194.05701)] and all affine spherical spaces [\textit{B. È. Kunyavskij} in: Séminaire Théorie Nombres, Paris 1991-92, Prog. Math. 116, 63-71 (1993; Zbl 0829.11022)]. In the paper under review it is shown that for a general spherical space this is not true. Counter-examples are found among horospherical spaces that is such that \(H\) contains a maximal unipotent subgroup of \(G\).