Inhomogeneity of Isoparametric Hypersurfaces of OT-FKM-type in the Pseudo-Sphere

We study isoparametric hypersurfaces, whose principal curvatures are all constant, in the pseudo-Riemannian space forms. In this paper, we investigate two topics. Firstly, according to representations of Clifford algebras, we give a construction of Clifford systems of signature $(m,r)$ for any $(m,r)$ explicitly. Secondly, we show that a (connected) isoparametric hypersurface of OT-FKM-type whose focal variety is $M_{+}$ in the pseudo-sphere is inhomogeneous if the signature $(m,r)$ of its Clifford system on ${\mathbb{R}}_s^{2l}$ satisfies $m\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}4)$, $r\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}2)$ and $l>m$, showing that each connected component of $M_{+}$ is inhomogeneous.