Bott-Borel-Weil theorem

A holomorphic action of a complex Lie group $G$ 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)