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In [[mathematics]], in the field of [[representation theory]], the '''Borel–Weil theorem''', named after [[Armand Borel]] and [[André Weil]], provides a concrete model for [[irreducible representation]]s of [[compact Lie group]]s and [[complex numbers|complex]] [[semisimple Lie group]]s. These representations are realized in the spaces of global [[Section (fiber bundle)|sections]] of [[holomorphic line bundle]]s on the [[flag manifold]] of the group. Its generalization to higher cohomology spaces is called the [[Borel–Weil–Bott theorem]].
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== Statement of the theorem ==
The theorem can be stated either for a complex semisimple Lie group ''G'' or for its compact form ''K''. Let ''G'' be a [[connected space|connected]] complex semisimple Lie group, ''B'' a [[Borel subgroup]] of ''G'', and ''X''=''G''/''B'' the [[flag variety]]. In this picture, ''X'' is a complex manifold and a nonsingular algebraic ''G''-variety. The flag variety can also be described as a compact [[homogeneous space]] ''K''/''T'', where ''T''=''K''&cap;''B'' is a (compact) [[Cartan subgroup]] of ''K''. An [[Weight_(representation_theory)#Integral_weight|integral weight]] ''λ''
determines a ''G''-equivariant holomorphic line bundle ''L''<sub>''λ''</sub> on ''X'' and the group ''G'' acts on its space of global sections,
 
:<math>\Gamma(G/B,L_\lambda).\ </math>
 
The Borel–Weil theorem states that if ''λ'' is a ''dominant'' integral weight then this representation is an irreducible [[highest weight representation]] of ''G'' with highest weight ''λ''. Its restriction to ''K'' is an irreducible unitary representation of ''K'' with highest weight ''λ'', and each irreducible unitary representations of ''K'' is obtained in this way for a unique value of ''λ''.
 
== Concrete description ==
The weight ''λ'' gives rise to a character (one-dimensional representation) of the Borel subgroup ''B'', which is denoted ''&chi;''<sub>''λ''</sub>. Holomorphic sections of the holomorphic line bundle ''L''<sub>''λ''</sub> over ''G''/''B'' may be described more concretely as [[holomorphic map]]s
 
:<math> f: G\to \mathbb{C}_{\lambda}: f(gb)=\chi_{\lambda}(b)f(g)</math>
 
for all ''g''&isin;''G'' and ''b''&isin;''B''.  
 
The action of ''G'' on these sections is given by
: <math>g\cdot f(h)=f(g^{-1}h)</math>
 
for ''g'',''h''&isin;''G''.
 
== Example ==
Let ''G'' be the complex [[special linear group]] ''SL''(2,'''C'''), with a Borel subgroup consisting of upper triangular matrices with determinant one. Integral weights for ''G'' may be identified with [[integer]]s, with dominant weights corresponding to nonnegative integers, and the corresponding characters ''&chi;''<sub>''n''</sub> of ''B'' have the form
 
: <math> \chi_n
\begin{pmatrix}
a & b\\
0 & a^{-1}
\end{pmatrix}=a^n.
</math>
 
The flag variety ''G''/''B'' may be identified with the complex projective line '''P'''<sup>1</sup> with homogeneous coordinates ''X'', ''Y'' and the space of the global sections of the line bundle ''L''<sub>''n''</sub> is identified with the space of homogeneous polynomials of degree ''n'' on '''C'''<sup>''2''</sup>. For ''n''&ge;0, this space has dimension ''n''+1 and forms an irreducible representation under the standard action of ''G'' on the polynomial algebra '''C'''[''X'',''Y'']. Weight vectors are given by monomials
 
: <math> X^i Y^{n-i}, \quad 0\leq i\leq n </math>
 
of weights 2''i''&minus;''n'', and the highest weight vector ''X''<sup>''n''</sup> has weight ''n''.
 
==History==
The theorem dates back to the early 1950s and can be found in {{harvtxt|Serre|1995}} and {{harvtxt|Tits|1955}}.
 
==References==
*{{citation
| last = Serre | first = Jean-Pierre | authorlink = Jean-Pierre Serre
| title = Représentations linéaires et espaces homogènes kählériens des groupes de Lie compacts (d'après Armand Borel et André Weil)
| journal = Séminaire Bourbaki
| volume = 2 | issue = 100 | pages = 447–454 | publisher = Soc. Math. France | location = Paris | year = 1995}}. In French; translated title: “Linear representations and Kähler homogeneous spaces of compact Lie groups (after Armand Borel and André Weil.”
 
*{{citation
| last = Tits | first = Jacques | authorlink = Jacques Tits
| title = Sur certaines classes d'espaces homogènes de groupes de Lie
| series = Acad. Roy. Belg. Cl. Sci. Mém. Coll.
| volume = 29 | year = 1955}} In French.
 
*{{citation
| last = Sepanski | first = Mark R.
| title = Compact Lie groups.
| series = Graduate Texts in Mathematics | volume = 235
| publisher = Springer | location = New York | year = 2007}}.
 
*{{citation
| last = Knapp | first = Anthony W.
| title = Representation theory of semisimple groups: An overview based on examples
| series = Princeton Landmarks in Mathematics
| publisher = Princeton University Press | location = Princeton, NJ | year = 2001}}. Reprint of the 1986 original.
 
{{DEFAULTSORT:Borel-Weil theorem}}
[[Category:Representation theory of Lie groups]]
[[Category:Theorems in representation theory]]

Revision as of 00:47, 9 February 2014

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