On cohomogeneity one linear actions on pseudo-Euclidean space \(\mathbb{R}^{p , q} \) (Q2304646)

In this article, the authors study the natural actions of some (noncompact) subgroups of \( SO(p,q) \) (matrices preserving the quadratic form of signature \( p,q \ (p\geq q)\)) on \( \mathbb{R}^{p+q} \). The case \( q=1 \) has been investigated in [\textit{J. Berndt} et al., Monatsh. Math. 184, No. 2, 185--200 (2017; Zbl 1379.53091)]. In this article, similar results are obtained for \( q>1 \). All orbits of this cohomogeniety one action are found. A decomposition of the subgroup \( Q \) leaving a certain subspace invariant is determined. Restricted actions to a certain class of subgroups of \( Q \) are studied in detail. The authors prove that the actions of these subgroups are also of cohomogeniety one and all their orbits are determined. It is shown that, when \( p>q+1\geq2 \), the orbits outside a certain subspace (\( \mathbb{W} \) of dimension \( p \)) are independent of the acting group (in this class) but the orbits of points in \( \mathbb{W} \) depend on the acting group as in [loc. cit.].