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| In [[mathematics]], the '''Borel–Weil-Bott theorem''' is a basic result in the [[representation theory]] of [[Lie group]]s, showing how a family of representations can be obtained from holomorphic sections of certain complex [[vector bundle]]s, 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]].
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| ==Formulation==
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| Let ''G'' be a [[Semisimple algebraic group|semisimple]] Lie group or [[Linear algebraic group|algebraic group]] over <math>\mathbb C</math>, and fix a [[maximal torus]] ''T'' along with a [[Borel subgroup]] ''B'' which contains ''T''. Let λ be an [[weight (representation theory)|integral weight]] of ''T''; λ defines in a natural way a one-dimensional representation ''C''<sub>λ</sub> of ''B'', by pulling back the representation on ''T'' = ''B''/''U'', where ''U'' is the [[unipotent radical]] of ''B''. Since we can think of the projection map ''G'' → ''G/B'' as a [[Principal bundle|principal ''B''-bundle]], for each ''C''<sub>λ</sub> we get an [[associated fiber bundle]] ''L''<sub>-λ</sub> on ''G/B'' (note the sign), which is obviously a [[line bundle]]. Identifying ''L''<sub>λ</sub> with its [[sheaf (mathematics)|sheaf]] of holomorphic sections, we consider the [[sheaf cohomology]] groups <math>H^i( G/B, \, L_\lambda )</math>. Since ''G'' acts on the total space of the bundle <math>L_\lambda</math> by bundle automorphisms, this action naturally gives a ''G''-module structure on these groups; and the '''Borel–Weil–Bott theorem''' gives an explicit description of these groups as ''G''-modules.
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| We first need to describe the [[Weyl group]] action centered at <math>\rho</math>. For any integral weight <math>\lambda</math> and <math>w</math> in the Weyl group W, we set <math>w*\lambda := w( \lambda + \rho ) - \rho \,</math>, where <math>\rho</math> denotes the half-sum of positive roots of ''G''. 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 <math>\mu</math> is said to be ''dominant'' if <math>\mu( \alpha^\vee ) \geq 0</math> for all simple roots <math>\alpha</math>. Let <math>\ell</math> denote the [[Weyl group#Coxeter group structure|length function]] on ''W''.
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| Given an integral weight <math>\lambda</math>, one of two cases occur: (1) 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 (2) 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
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| :<math>H^i( G/B, \, L_\lambda ) = 0</math> for all i;
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| and in the second case, we have
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| :<math>H^i( G/B, \, L_\lambda ) = 0</math> for all <math>i \neq \ell(w)</math>, while
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| :<math>H^{ \ell(w) }( G/B, \, L_\lambda )</math> is the dual of the irreducible highest-weight representation of ''G'' with highest weight <math>\lambda</math>. | |
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| It is worth noting that case (1) above occurs if and only if <math>\lambda( \beta^\vee ) = 0</math> for some positive root <math>\beta</math>. Also, we obtain the classical [[Borel–Weil theorem]] as a special case of this theorem by taking <math>\lambda</math> to be dominant and <math>w</math> to be the identity element <math>e \in W</math>.
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| ==Example== | |
| For example, consider ''G'' = ''SL''<sub>2</sub>('''C'''), for which ''G/B'' is the [[Riemann sphere]], an integral weight is specified simply by an integer ''n'', and ρ = 1. The line bundle ''L<sub>n''</sub> is O(''n''), whose sections are the homogeneous polynomials of degree ''n'' (i.e. the [[binary form]]s). As a representation of ''G'', the sections can be written as Sym<sup>n</sup>('''C'''<sup>2</sup>)<sup>*</sup>, and is canonically isomorphic to Sym<sup>n</sup>('''C'''<sup>2</sup>). This gives us at a stroke the representation theory of <math>\mathfrak{sl}_2(\mathbf{C})</math>: Γ(O(1)) is the standard representation, and Γ(O(''n'')) is its ''n''-th [[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 ''H'', ''X'', ''Y'' are the standard generators of <math>\mathfrak{sl}_2(\mathbf{C})</math>, then we can write
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| :<math>H = x\frac{d}{dx}-y\frac{d}{dy}</math>
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| :<math>X = x\frac{d}{dy}</math>
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| :<math>Y = y\frac{d}{dx}.</math> | |
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| ==Positive characteristic==
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| One also has a weaker form of this theorem in positive characteristic. Namely, let ''G'' 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 i if <math>\lambda</math> is a weight such that <math>w*\lambda</math> is non-dominant for all <math>w \in W</math> as long as <math>\lambda</math> is "close to zero".<ref name="Jantzen">{{cite book|last=Jantzen|first=Jens Carsten|title=Representations of algebraic groups|edition=second|year=2003|publisher=American Mathematical Society|isbn=0-8218-3527-0}}</ref> This is known as the [[Kempf vanishing theorem]]. However, the other statements of the theorem do not remain valid in this setting.
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| More explicitly, let <math>\lambda</math> 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 ''G''-module is simple in general, although it does contain the unique highest weight module of highest weight <math>\lambda</math> as a ''G''-submodule. If <math>\lambda</math> 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 <math>\lambda</math> that these modules are all zero except in a single degree i.
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| ==Notes==
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| <references />
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| ==References==
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| * {{Fulton-Harris}}.
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| * {{citation|first1=Robert J.|last1=Baston|first2=Michael G.|last2=Eastwood|authorlink2=Michael Eastwood|title=The Penrose Transform: its Interaction with Representation Theory|publisher=Oxford University Press|year=1989}}.
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| * {{Springer|id=b/b120400|title=Bott–Borel–Weil theorem}}
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| *[http://www-math.mit.edu/~lurie/papers/bwb.pdf A Proof of the Borel–Weil–Bott Theorem], by Jacob Lurie. Retrieved on Dec. 14, 2007.
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| {{PlanetMath attribution|id=4585|title=Borel–Bott–Weil theorem}}
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| {{DEFAULTSORT:Borel-Weil-Bott theorem}}
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| [[Category:Representation theory of Lie groups]]
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| [[Category:Theorems in representation theory]]
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