nLab Borel-Weil theorem

Idea

The Borel-Weil-Bott theorem characterizes representations of suitable Lie groups $G$ as spaces of holomorphic sections of complex line bundles over flag varieties $G/B$, for $B$ a Borel subgroup.

With suitable qualifiers added this is Kirillov’s orbit method, and the construction may be interpreted as sending a symplectic manifold equipped with $G$-Hamiltonian action to its geometric quantization. As discussed there, this is equivalently given by fiber integration in K-theory and accordingly the Borel-Weil-Bott theorem is naturally regarded in the context of generalized Schubert calculus (e.g. Kumar 12).

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 $L^2$-version). The induced representation is too big and Frobenius reciprocity indicates that they are normally not irreducible. Given a subgroup $B\subset G$ and a $B$-module $V$, with $\rho\colon B\to Aut(V)$ the induced module can be represented as a space of (set-theoretic or $L^2$‑) sections of the associated bundle $G\times_{G/B} V$ to the principal fiber bundle $G\to G/B$, 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 $B$ is the Borel subgroup of the complex semisimple group $G^{\mathbb{C}}$ (which can be considered as the complexification of a compact Lie group $G$ with the maximal torus $T=G\cap B\subset G$), then all unitary irreducible representations can be obtained as the spaces of (anti)-holomorphic line bundles associated to the principal fibration $G\to G/B$ over the generalized flag variety $G^{\mathbb{C}}/B\cong G/T$ with the fiber $\mathbb{C}_\chi$, which is the $1$-dimensional representation corresponding to a dominant integral character $\chi$; 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, called the 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.

Related concepts

• orbit method

• Schubert calculus

References

• Neil Chriss, Victor Ginzburg, Representation theory and complex geometry, Birkhäuser 1994

• Jean-Pierre Serre, Représentations linéaires et espaces homogènes kählériens des groupes de Lie compacts (d’après Armand Borel et André Weil)“, Séminaire Bourbaki 100: 447–454, Paris: Soc. Math. France, 1953/54, numdam

• Bertram Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. 74, 2, 1961, pp. 329-387 pdf

• Pierre Cartier, Remarks on “Lie algebra cohomology and the generalized Borel-Weil theorem” by B. Kostant, Ann. of Math. 74, 2, 1961, pp. 388-390 pdf

• Jacob Lurie, A proof of the Borel-Weil-Bott theorem (pdf)

Lecture notes include

• Wilfried Schmid (notes by Matvei Libine), Geometric methods in representation theory (pdf)

• Shrawan Kumar, Borel-Weil-Bott theorem and geometry of Schubert varieties (pdf)

• P. Woit, Topics in representation theory: the Borel-Weil theorem, (pdf), Quantum field theory and representation theory: A sketch (pdf)

• Lisa Jeffrey, Remarks on geometric quantization and representation theory pdf

See also

• eom: Bott-Borel-Weil theorem

• wikipedia: Borel–Weil–Bott theorem, Borel–Weil theorem