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Bott-Borel-Weil theorem

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A holomorphic action of a complex Lie group on a holomorphic vector bundle \pi : E \rightarrow M is a left holomorphic action, G \times E \rightarrow E, which projects onto M and which sends \mathbf{C}-linearly each vector space fibre E _ { m } = \pi ^ { - 1 } ( m ) onto g E _ { m } = \pi ^ { - 1 } ( g m ). In this situation E is conveniently said to be G-equivariant. If E _ { 1 } and E _ { 2 } are equivariant bundles over M and f : E _ { 1 } \rightarrow E _ { 2 } is a mapping of bundles, it is easy to see what is meant by "f is G-equivariant" and also what is meant by "E1 and E2 are equivalent" , as G-equivariant vector bundles.

When G, \pi : E \rightarrow M, etc. are as above, one sees that, by restriction, the given action defines a complex linear representation of the stabilizer \operatorname { stab}_{G} (m) of a point m \in M on the \mathbf{C}-vectorial fibre E _ { m }. The equivalence class of this representation depends only on the G-equivariant holomorphism class of E. If M is a homogeneous G-space, this correspondence between equivalence classes is bijective. This may be explained as follows: If M = G / H is a complex homogeneous space and \varrho : H \rightarrow F is a holomorphic complex linear representation, one considers the following equivalence relation on G \times F:

\begin{equation*} ( g , \mathbf{f} ) \sim ( g h ^ { - 1 } , \varrho ( h ) \mathbf{f} ), \end{equation*}

where g \in G, h \in H, \mathbf{f} \in F. The quotient space G \times F / \sim will be denoted by G \times^{\varrho} F, and the equivalence class of ( g , \mathbf{f} ) will be denoted by g \times ^ { \varrho } {\bf f} \in G \times ^ { \varrho } F. The formula \pi ( g \times ^ { \varrho } \mathbf f ) = g H makes G \times^{\varrho} F into a vector bundle of fibre type F via \pi : G \times^\varrho F \rightarrow G / H. This fibration is naturally holomorphically G-equivariant via the action g ( g ^ { \prime } \times ^ { \varrho } \mathbf{f} ) = g g ^ { \prime } \times ^ { \varrho } \mathbf{f} and one checks that the stabilizer, H, of the "neutral element" of G / H acts (see above) on the "neutral fibre" exactly by the representation \varrho .

Below, the case F = \mathbf C will be regarded in some detail. Thus, the representation \varrho may be interpreted as a multiplicative character \varrho : H \rightarrow \mathbf{C} ^ { * } and \xi = G \times ^ { \varrho } \bf C will be a complex line bundle.

Background.

Let G be a semi-simple complex Lie group with Lie algebra \frak g (cf. also Lie group, semi-simple; Lie algebra), \mathfrak { h } \subset \mathfrak { g } a Cartan subalgebra of \frak g, S ^ { + } a system of positive roots (cf. Root system), S^{-} the corresponding system of opposite roots (termed negative), and S = S ^ { + } \cup S ^ { - } \subset \mathfrak{h} ^ { * } the set of all roots. Let \mathfrak { g } _ { \alpha } be the root space associated to \alpha \in S ^ { + }. Then \mathfrak{n} ^ { + } = \oplus _ { \alpha \in S ^{ + }} \mathfrak { g } _ { \alpha } is a nilpotent Lie subalgebra and one defines the maximal solvable subalgebra (the Borel subalgebra) \mathfrak{b} by \mathfrak { b } = \mathfrak { h } \oplus \mathfrak { n } ^ { + }. This is the Lie algebra of a closed complex Lie subgroup B \subset G such that G / B is compact. Finally, \mathfrak { n } ^ { + } = [ \mathfrak { b } , \mathfrak { b } ].

Note that there is a subspace \mathfrak{h}_R of \mathfrak h that is, in the vector spaces sense, a real form (that is, \mathfrak { h } = \mathfrak { h } _ { R } \oplus i \mathfrak { h } _ { R } and i \mathfrak{h} _ { R } is the Lie subalgebra of a compact connected group). It follows that the restriction to \mathfrak{h}_R of the Killing form of the complex algebra, denoted by \langle \, .\, ,\, . \, \rangle, is a real scalar product. From this one deduces an isomorphism \mathfrak { h } _ { R } \rightarrow \mathfrak { h } _ { R } ^ { * } : = \operatorname { hom } _ { \mathbf{R} } ( \mathfrak { h } _ { R } , \mathbf{R} ) and thus a scalar product \langle \, .\, ,\, . \, \rangle on \mathfrak{h} _ { R } ^ { * }. Notice that the evaluation of the weights of representations (and also of the roots) on \mathfrak{h}_R are real numbers. Recall that the closed Weyl chamber C ^ { + } \subset \mathfrak { h } _ { R } ^ { * } is the set of h ^ { * } for which \langle \alpha , h ^ { * } \rangle \geq 0 for all \alpha \in S ^ { + }. The Weyl group acts on \mathfrak{h} _ { R } ^ { * }, with C ^ { + } as "fundamental domain" . It is worth noting that while the transformation h ^ { * } \mapsto - h ^ { * } is not necessarily in the Weyl group, the opposite C ^ { - } = - C ^ { + } is the transformation w C ^ { + } of C ^ { + } by an element of the Weyl group (in fact, by the longest element). Now consider an irreducible representation R : G \rightarrow V. The theory of H. Weyl classically characterizes such a representation by its dominant weight (cf. also Representation of a Lie algebra). Contrary to tradition, it is perhaps wiser to characterize a representation by its dominated weight. This is the unique weight p \in \mathfrak { h } ^ { * } of the representation R such that the other weights of R may be obtained from p by the addition of an \mathbf{N}-linear combination of positive roots. In general, the dominated weight of a representation R is not the opposite of the dominant weight of R, but the opposite of the dominant weight of the contragredient representation \check{R} : G \rightarrow V ^ { * }. This dominated weight is always in the opposite of the Weyl chamber.

Bott–Borel–Weil theorem.

In the above context, consider the hyperplane H _ { R } \subset V that is the sum of all the proper spaces associated to the weights different from the dominated weight p of the representation R. By the definition of dominated weight, one sees that R _{*} ( \mathfrak{b} ) H _ { R } \subset H _ { R }. Now consider the holomorphically trivial bundle G / B \times V, and make it equivariant by the action g ^ { \prime } ( g B , v ) = ( g ^ { \prime } g B , R ( g ^ { \prime } ) v ). This G-equivariant bundle is exactly G \times ^ { R } V, which leads to the equivariant exact sequence of holomorphic bundles:

\begin{equation*} 0 \rightarrow G \times ^ { R } H _ { R } \rightarrow G \times ^ { R } V \rightarrow \xi \rightarrow 0. \end{equation*}

In fact, the weight p \in C^{-} extends to a character p : \mathfrak { b } \rightarrow \mathbf{C}, which can be integrated to give a character \varrho = e ^ { p } : B \rightarrow \mathbf C ^ { * }. One easily sees that \xi = G \times ^ { \varrho } \bf C and that the natural action of G on H ^ { 0 } ( G / B , G \times ^ { R } V ) is exactly the representation R.

In this context, the Borel–Weil theorem states:

a) The arrow V \rightarrow H ^ { 0 } ( G / B , \xi ) is a G-equivariant isomorphism;

b) H ^ { k } ( G / B , \xi ) = 0 for k \neq 0. These results are not unexpected (in case b), at least for those who are familiar with the idea of a sufficiently ample line bundle). This is not at all the case for the generalization to representations of G in H ^ { i } ( G / B , \xi ) when the line bundle is given by a representation \varrho : B \rightarrow \mathbf{C} ^ { * } such that the restriction to \mathfrak h of its derivative p : \mathfrak { b } \rightarrow \mathbf{C} is not the dominated weight of a holomorphic representation of G. Indeed, this generalization is the very unexpected Bott–Borel–Weil theorem: Let p \in \mathfrak{h} _ { R } ^ { * } \subset \mathfrak{h} ^ { * }, \varrho : B \rightarrow \mathbf{C} ^ { * } and \xi = G \times ^ { \varrho } \bf C be as above, and let also \bf W be the Weyl group relative to the Cartan algebra \mathfrak h and \delta : = ( 1 / 2 ) \sum _ { \alpha \in S ^ { + } } \alpha \in {\frak h} _ {\bf R } ^ { * }. Then:

i) If, for all w \in \operatorname{W}, the quantity w ( p - \delta ) + \delta is never the dominated weight of a representation, then all the cohomology groups H ^ { i } ( G / B , \xi ) are zero.

ii) If there exists an element w \in \operatorname{W}, hence unique with this property, such that w ( p - \delta ) + \delta \in C^- is the dominated weight of a representation R, then:

A) For i \neq \text{l} ( w ) (the length of w), the cohomology group H ^ { i } ( G / B , \xi ) is zero.

B) For i = \operatorname{l} ( w ), the natural representation of G on the cohomology group H ^ { i } ( G / B , \xi ) is exactly the representation R.

The proof is essentially a very beautiful application of the relative cohomology of Lie algebras, initiated by C. Chevalley and S. Eilenberg.

References

[a1] R. Bott, "Homogeneous vector bundles" Ann. of Math. , 66 (1957) pp. 203–248
[a2] N.R. Wallach, "Harmonic analysis on homogeneous spaces" , M. Dekker (1973)
[a3] M. Demazure, "A very simple proof of Bott's theorem" Invent. Math. , 33 (1976)
How to Cite This Entry:
Bott-Borel-Weil theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bott-Borel-Weil_theorem&oldid=50095
This article was adapted from an original article by F. Lescure (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article