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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Borel–Weil–Bott theorem ： ウィキペディア英語版 Borel–Weil–Bott theorem In mathematics, the Borel–Weil–Bott theorem is a basic result in the representation theory of Lie groups, showing how a family of representations can be obtained from holomorphic sections of certain complex vector bundles, 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. ==Formulation== Let be a semisimple Lie group or algebraic group over $\backslash mathbb\; C$, and fix a maximal torus along with a Borel subgroup which contains . Let be an integral weight of ; defines in a natural way a one-dimensional representation of , by pulling back the representation on , where is the unipotent radical of . Since we can think of the projection map as a , for each we get an associated fiber bundle on (note the sign), which is obviously a line bundle. Identifying with its sheaf of holomorphic sections, we consider the sheaf cohomology groups $H^i(\; G/B,\; \backslash ,\; L\_\backslash lambda\; )$. Since acts on the total space of the bundle $L\_\backslash lambda$ by bundle automorphisms, this action naturally gives a -module structure on these groups; and the Borel–Weil–Bott theorem gives an explicit description of these groups as -modules. We first need to describe the Weyl group action centered at . For any integral weight and in the Weyl group , we set $w$ *\lambda := w( \lambda + \rho ) - \rho \,, where denotes the half-sum of positive roots of . 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 is said to be ''dominant'' if $\backslash mu(\; \backslash alpha^\backslash vee\; )\; \backslash geq\; 0$ for all simple roots . Let denote the length function on . Given an integral weight , one of two cases occur: # There is no $w\; \backslash in\; W$ such that $w$ *\lambda is dominant, equivalently, there exists a nonidentity $w\; \backslash in\; W$ such that $w$ * \lambda = \lambda; or # There is a ''unique'' $w\; \backslash in\; W$ such that $w$ * \lambda is dominant. The theorem states that in the first case, we have :$H^i(\; G/B,\; \backslash ,\; L\_\backslash lambda\; )\; =\; 0$ for all ; and in the second case, we have :$H^i(\; G/B,\; \backslash ,\; L\_\backslash lambda\; )\; =\; 0$ for all $i\; \backslash neq\; \backslash ell(w)$, while :$H^(\; G/B,\; \backslash ,\; L\_\backslash lambda\; )$ is the dual of the irreducible highest-weight representation of with highest weight $w$ * \lambda. It is worth noting that case (1) above occurs if and only if $\backslash lambda(\; \backslash beta^\backslash vee\; )\; =\; 0$ for some positive root . Also, we obtain the classical Borel–Weil theorem as a special case of this theorem by taking to be dominant and to be the identity element $e\; \backslash in\; W$. 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–Bott theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.