Extremal twist and quantization of homogeneous vector bundles
Andrey Mudrov (University of Leicester)
Wednesday 29th November, 2017 16:00-17:00 Maths 311B
We will discuss vector bundles over quantum homogeneous spaces. Our basic examples are even dimensional spheres regarded as conjugacy classes of orthogonal groups. This case is particularly interesting because the stabilizer subgroup is of pseudo-Levi type and has no natural quantum counterpart in the total quantum group, in the standard Drinfeld's context. Yet the polynomial algebra on the quantum sphere admits two different realizations: by invariants of a coideal subalgebra in the total quantum group, and by linear operators on a certain (base) module of highest weight. Although the quantum stabilizer is present in the first picture, it is not compatible with the canonical polarization of the total quantum group, which makes its representation theory extremely difficult. Our alternative allows to address it through a pseudo-parabolic subcategory in the category O. It is a module category over finite dimensional representations of the quantum orthogonal group, generated by the base module. We prove that it is semi-simple, as well as the representation category of the coideal subalgebra. As a result, we alternatively realize the homogeneous vector bundles as either induced modules of the quantum symmetric pair or by linear maps between simple objects of the pseudo-parabolic category. Our analysis relies on an extremal twist operator cooked up of the inverse Shapovalov form of the base module.