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.