|
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 representations of compact Lie groups and 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. Its generalization to higher cohomology spaces is called the Borel–Weil–Bott theorem. == 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 complex semisimple Lie group, ''B'' a Borel subgroup of ''G'', and ''X''=''G''/''B'' the flag variety. In this scenario, ''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''∩''B'' is a (compact) Cartan subgroup of ''K''. An integral weight ''λ'' determines a ''G''-equivariant holomorphic line bundle ''L''''λ'' on ''X'' and the group ''G'' acts on its space of global sections, : 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''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Borel–Weil theorem」の詳細全文を読む スポンサード リンク
|