Equivariant vector bundles over quantum spheres

We quantize homogeneous vector bundles over an even complex sphere ${\mathbb{S}}^{2n}$ as one-sided projective modules over its quantized coordinate ring. We realize them in two different ways: as locally finite $\mathbb{C}$-homs between pseudo-parabolic Verma modules and as induced modules of the quantum orthogonal group. Based on this alternative, we study representations of a quantum symmetric pair related to ${\mathbb{S}}_q^{2n}$ and prove their complete reducibility.