A common method of construction of representations of groups in representation theory is to consider the invariant subspaces of the induced representation (set-theoretic or -version). The induced representation is too big and Frobenius reciprocity indicates that they are normally not irreducible. Given a subgroup and a -module , with the induced module can be represented as a space of (set-theoretic or ‑) sections of the associated bundle to the principal fiber bundle , at least when these words make sense. In geometric quantization, the method to single out a sufficiently small space of sections is to look at sections which are horizontal in the sense of some polarization, or equivalently horizontal for an appropriate choice of connection on the bundle.
The first instance is the theorem of Borel–Weil, (J-P. Serre, Bourbaki Seminar 100, 1953/54) which asserts that if is the Borel subgroup of the complex semisimple group (which can be considered as the complexification? of a compact Lie group with the maximal torus? ), then all unitary irreducible representations can be obtained as the spaces of (anti)-holomorphic line bundles associated to the principal fibration over the generalized flag variety with the fiber , which is the -dimensional representation corresponding to a dominant integral character ; and viceversa, all such spaces of sections are irreducible. The inner product is inherited from the hermitean structure on the line bundle.
There is an extension to higher cohomologies instead of spaces of sections, so-called Borel–Weil–Bott theorem and numerous extensions, e.g. to Harish–Chandra sheaves to construct the infinite-dimensional representations. The original proof is by geometric and analytic methods; some of the modern extensions of the method use the algebraic D-module theory and are based on the Beilinson–Bernstein localization theorem. There are even extensions to quantum groups.