Английская Википедия:Borel–Weil–Bott theorem
Шаблон:Short description In mathematics, the Borel–Weil–Bott theorem is a basic result in the representation theory of Lie groups, showing how a family of representations can be obtained from holomorphic sections of certain complex vector bundles, and, more generally, from higher sheaf cohomology groups associated to such bundles. It is built on the earlier Borel–Weil theorem of Armand Borel and André Weil, dealing just with the space of sections (the zeroth cohomology group), the extension to higher cohomology groups being provided by Raoul Bott. One can equivalently, through Serre's GAGA, view this as a result in complex algebraic geometry in the Zariski topology.
Formulation
Let Шаблон:Mvar be a semisimple Lie group or algebraic group over <math>\mathbb C</math>, and fix a maximal torus Шаблон:Mvar along with a Borel subgroup Шаблон:Mvar which contains Шаблон:Mvar. Let Шаблон:Mvar be an integral weight of Шаблон:Mvar; Шаблон:Mvar defines in a natural way a one-dimensional representation Шаблон:Math of Шаблон:Mvar, by pulling back the representation on Шаблон:Math, where Шаблон:Mvar is the unipotent radical of Шаблон:Mvar. Since we can think of the projection map Шаблон:Math as a [[Principal bundle|principal Шаблон:Mvar-bundle]], for each Шаблон:Math we get an associated fiber bundle Шаблон:Math on Шаблон:Math (note the sign), which is obviously a line bundle. Identifying Шаблон:Math with its sheaf of holomorphic sections, we consider the sheaf cohomology groups <math>H^i( G/B, \, L_\lambda )</math>. Since Шаблон:Mvar acts on the total space of the bundle <math>L_\lambda</math> by bundle automorphisms, this action naturally gives a Шаблон:Mvar-module structure on these groups; and the Borel–Weil–Bott theorem gives an explicit description of these groups as Шаблон:Mvar-modules.
We first need to describe the Weyl group action centered at <math> - \rho </math>. For any integral weight Шаблон:Mvar and Шаблон:Mvar in the Weyl group Шаблон:Mvar, we set <math>w*\lambda := w( \lambda + \rho ) - \rho \,</math>, where Шаблон:Mvar denotes the half-sum of positive roots of Шаблон:Mvar. It is straightforward to check that this defines a group action, although this action is not linear, unlike the usual Weyl group action. Also, a weight Шаблон:Mvar is said to be dominant if <math>\mu( \alpha^\vee ) \geq 0</math> for all simple roots Шаблон:Mvar. Let Шаблон:Mvar denote the length function on Шаблон:Mvar.
Given an integral weight Шаблон:Mvar, one of two cases occur:
- There is no <math>w \in W</math> such that <math>w*\lambda</math> is dominant, equivalently, there exists a nonidentity <math>w \in W</math> such that <math>w * \lambda = \lambda</math>; or
- There is a unique <math>w \in W</math> such that <math>w * \lambda</math> is dominant.
The theorem states that in the first case, we have
- <math>H^i( G/B, \, L_\lambda ) = 0</math> for all Шаблон:Mvar;
and in the second case, we have
- <math>H^i( G/B, \, L_\lambda ) = 0</math> for all <math>i \neq \ell(w)</math>, while
- <math>H^{ \ell(w) }( G/B, \, L_\lambda )</math> is the dual of the irreducible highest-weight representation of Шаблон:Mvar with highest weight <math> w * \lambda</math>.
It is worth noting that case (1) above occurs if and only if <math>(\lambda+\rho)( \beta^\vee ) = 0</math> for some positive root Шаблон:Mvar. Also, we obtain the classical Borel–Weil theorem as a special case of this theorem by taking Шаблон:Mvar to be dominant and Шаблон:Mvar to be the identity element <math>e \in W</math>.
Example
For example, consider Шаблон:Math, for which Шаблон:Math is the Riemann sphere, an integral weight is specified simply by an integer Шаблон:Mvar, and Шаблон:Math. The line bundle Шаблон:Math is <math>{\mathcal O}(n)</math>, whose sections are the homogeneous polynomials of degree Шаблон:Mvar (i.e. the binary forms). As a representation of Шаблон:Mvar, the sections can be written as Шаблон:Math, and is canonically isomorphic to Шаблон:Math.
This gives us at a stroke the representation theory of <math>\mathfrak{sl}_2(\mathbf{C})</math>: <math>\Gamma({\mathcal O}(1))</math> is the standard representation, and <math>\Gamma({\mathcal O}(n))</math> is its Шаблон:Mvarth symmetric power. We even have a unified description of the action of the Lie algebra, derived from its realization as vector fields on the Riemann sphere: if Шаблон:Mvar, Шаблон:Mvar, Шаблон:Mvar are the standard generators of <math>\mathfrak{sl}_2(\mathbf{C})</math>, then
- <math>
\begin{align} H & = x\frac{\partial}{\partial x}-y\frac{\partial}{\partial y}, \\[5pt] X & = x\frac{\partial}{\partial y}, \\[5pt] Y & = y\frac{\partial}{\partial x}. \end{align} </math>
Positive characteristic
One also has a weaker form of this theorem in positive characteristic. Namely, let Шаблон:Mvar be a semisimple algebraic group over an algebraically closed field of characteristic <math>p > 0</math>. Then it remains true that <math>H^i( G/B, \, L_\lambda ) = 0</math> for all Шаблон:Mvar if Шаблон:Mvar is a weight such that <math>w*\lambda</math> is non-dominant for all <math>w \in W</math> as long as Шаблон:Mvar is "close to zero".[1] This is known as the Kempf vanishing theorem. However, the other statements of the theorem do not remain valid in this setting.
More explicitly, let Шаблон:Mvar be a dominant integral weight; then it is still true that <math>H^i( G/B, \, L_\lambda ) = 0</math> for all <math>i > 0</math>, but it is no longer true that this Шаблон:Mvar-module is simple in general, although it does contain the unique highest weight module of highest weight Шаблон:Mvar as a Шаблон:Mvar-submodule. If Шаблон:Mvar is an arbitrary integral weight, it is in fact a large unsolved problem in representation theory to describe the cohomology modules <math>H^i( G/B, \, L_\lambda )</math> in general. Unlike over <math>\mathbb{C}</math>, Mumford gave an example showing that it need not be the case for a fixed Шаблон:Mvar that these modules are all zero except in a single degree Шаблон:Mvar.
Borel–Weil theorem
The Borel–Weil theorem provides a concrete model for irreducible representations of compact Lie groups and irreducible holomorphic representations of complex semisimple Lie groups. These representations are realized in the spaces of global sections of holomorphic line bundles on the flag manifold of the group. The Borel–Weil–Bott theorem is its generalization to higher cohomology spaces. The theorem dates back to the early 1950s and can be found in Шаблон:Harvtxt and Шаблон:Harvtxt.
Statement of the theorem
The theorem can be stated either for a complex semisimple Lie group Шаблон:Math or for its compact form Шаблон:Math. Let Шаблон:Math be a connected complex semisimple Lie group, Шаблон:Math a Borel subgroup of Шаблон:Math, and Шаблон:Math the flag variety. In this scenario, Шаблон:Math is a complex manifold and a nonsingular algebraic Шаблон:Nowrap. The flag variety can also be described as a compact homogeneous space Шаблон:Math, where Шаблон:Math is a (compact) Cartan subgroup of Шаблон:Math. An integral weight Шаблон:Math determines a Шаблон:Nowrap holomorphic line bundle Шаблон:Math on Шаблон:Math and the group Шаблон:Math acts on its space of global sections,
- <math>\Gamma(G/B,L_\lambda).\ </math>
The Borel–Weil theorem states that if Шаблон:Math is a dominant integral weight then this representation is a holomorphic irreducible highest weight representation of Шаблон:Math with highest weight Шаблон:Math. Its restriction to Шаблон:Math is an irreducible unitary representation of Шаблон:Math with highest weight Шаблон:Math, and each irreducible unitary representations of Шаблон:Math is obtained in this way for a unique value of Шаблон:Math. (A holomorphic representation of a complex Lie group is one for which the corresponding Lie algebra representation is complex linear.)
Concrete description
The weight Шаблон:Math gives rise to a character (one-dimensional representation) of the Borel subgroup Шаблон:Math, which is denoted Шаблон:Math. Holomorphic sections of the holomorphic line bundle Шаблон:Math over Шаблон:Math may be described more concretely as holomorphic maps
- <math> f: G\to \mathbb{C}_{\lambda}: f(gb)=\chi_{\lambda}(b^{-1})f(g)</math>
for all Шаблон:Math and Шаблон:Math.
The action of Шаблон:Math on these sections is given by
- <math>g\cdot f(h)=f(g^{-1}h)</math>
for Шаблон:Math.
Example
Let Шаблон:Math be the complex special linear group Шаблон:Math, with a Borel subgroup consisting of upper triangular matrices with determinant one. Integral weights for Шаблон:Math may be identified with integers, with dominant weights corresponding to nonnegative integers, and the corresponding characters Шаблон:Math of Шаблон:Math have the form
- <math> \chi_n
\begin{pmatrix} a & b\\ 0 & a^{-1} \end{pmatrix}=a^n. </math>
The flag variety Шаблон:Math may be identified with the complex projective line Шаблон:Math with homogeneous coordinates Шаблон:Math and the space of the global sections of the line bundle Шаблон:Math is identified with the space of homogeneous polynomials of degree Шаблон:Math on Шаблон:Math. For Шаблон:Math, this space has dimension Шаблон:Math and forms an irreducible representation under the standard action of Шаблон:Math on the polynomial algebra Шаблон:Math. Weight vectors are given by monomials
- <math> X^i Y^{n-i}, \quad 0\leq i\leq n </math>
of weights Шаблон:Math, and the highest weight vector Шаблон:Math has weight Шаблон:Math.
See also
Notes
References
- Шаблон:Fulton-Harris.
- Шаблон:Citation. (reprinted by Dover)
- Шаблон:Springer
- A Proof of the Borel–Weil–Bott Theorem, by Jacob Lurie. Retrieved on Jul. 13, 2014.
- Шаблон:Citation.
- Шаблон:Citation.
- Шаблон:Citation.
- Шаблон:Citation. Reprint of the 1986 original.
Further reading
