Degenerate principal series representations of $$\mathrm{SO}(p+1,p)$$ SO ( p + 1 , p )

For \(p\) odd, the Lie group \(G^\sharp =\mathrm{SO}_0(p+1,p+1)\) has a family of complementary series representations realized on the space of real \((p+1)\times (p+1)\) skew symmetric matrices, similar to the Stein’s complementary series for \(\mathrm{SL}(2n, {\mathbb C})\). We consider their restriction on the subgroup \(G_0=\mathrm{SO}_0(p+1,p)\) and prove that they are still irreducible and is equivalent to (a unitarization of) the principal series representation of \(G=\mathrm{SO}(p+1, p)\), and also irreducible under a maximal parabolic subgroup of \(G\)