Picard groups of homogeneous spaces of linear algebraic groups and one-dimensional homogeneous vector bundles

We construct models of finite-dimensional linear and projective irreducible representations of a connected semisimple group $G$ in linear systems on the variety $G$. We establish an algebro-geometric criterion for the linearizability of an irreducible projective representation of $G$. We explain the algebro-geometric meaning of the numerical characteristic of an arbitrary rational character of a maximal torus of $G$. Using these results we compute the Picard group of an arbitrary homogeneous space of any connected linear algebraic group $H$, prove the homogeneity of an arbitrary one-dimensional algebraic vector bundle over such a space relative to some covering group of $H$, and compute the Chern class of such a bundle.