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− | The structure of a real [[Lie group|Lie group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b1202101.png" /> can be studied by considering representations of the complexification <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b1202102.png" /> of its [[Lie algebra|Lie algebra]] (cf. also [[Representation of a Lie algebra|Representation of a Lie algebra]]). These are viewed as left modules over the [[Universal enveloping algebra|universal enveloping algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b1202103.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b1202104.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b1202105.png" />-modules. The Lie algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b1202106.png" /> considered here are the complexifications of real semi-simple Lie algebras corresponding to real, connected, semi-simple Lie groups. A Cartan subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b1202107.png" />, that is, a maximal Abelian subalgebra with the property that its adjoint representation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b1202108.png" /> is semi-simple, is chosen (cf. also [[Cartan subalgebra|Cartan subalgebra]]). A [[Root system|root system]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b1202109.png" />, corresponding to the resulting decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021010.png" />, is obtained. A further choice of a positive root system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021011.png" /> determines subalgebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021013.png" /> corresponding to the positive and negative root spaces, respectively. The building blocks in the study of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021014.png" /> are the finite-dimensional irreducible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021015.png" />-modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021016.png" />. They are indexed by the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021017.png" /> of dominant integral weights <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021018.png" /> relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021019.png" />.
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− | For any ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021020.png" /> with unity, a resolution of a left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021021.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021022.png" /> is an exact chain complex of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021023.png" />-modules:
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− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021024.png" /></td> </tr></table>
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| + | The structure of a real [[Lie group|Lie group]] $G$ can be studied by considering representations of the complexification $\frak g$ of its [[Lie algebra|Lie algebra]] (cf. also [[Representation of a Lie algebra|Representation of a Lie algebra]]). These are viewed as left modules over the [[Universal enveloping algebra|universal enveloping algebra]] $U ( \mathfrak { g } )$ of $\frak g$, or $\frak g$-modules. The Lie algebras $\frak g$ considered here are the complexifications of real semi-simple Lie algebras corresponding to real, connected, semi-simple Lie groups. A Cartan subalgebra $\mathfrak h $, that is, a maximal Abelian subalgebra with the property that its adjoint representation on $\frak g$ is semi-simple, is chosen (cf. also [[Cartan subalgebra|Cartan subalgebra]]). A [[Root system|root system]] $\Delta \subset \mathfrak { h } ^ { * }$, corresponding to the resulting decomposition of $\frak g$, is obtained. A further choice of a positive root system $\Delta ^ { + } \subset \Delta$ determines subalgebras $\mathfrak n$ and $\mathfrak{n}^{-}$ corresponding to the positive and negative root spaces, respectively. The building blocks in the study of $G$ are the finite-dimensional irreducible $\frak g$-modules $L ( \lambda )$. They are indexed by the set $P ^ { + } \subset \mathfrak { h } ^ { * }$ of dominant integral weights $\lambda$ relative to $\Delta ^ { + }$. |
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− | For example, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021025.png" /> be a complex Lie algebra, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021026.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021027.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021028.png" />th exterior power of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021030.png" />. Let | + | For any ring $A$ with unity, a resolution of a left $A$-module $M$ is an exact chain complex of $A$-modules: |
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− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021031.png" /></td> </tr></table>
| + | \begin{equation*} \ldots \rightarrow D _ { 2 } \stackrel { \delta _ { 2 } } { \rightarrow } D _ { 1 } \stackrel { \delta _ { 1 } } { \rightarrow } D _ { 0 } \stackrel { \delta _ { 0 } } { \rightarrow } M \rightarrow 0. \end{equation*} |
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− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021032.png" /></td> </tr></table>
| + | For example, let $\frak a$ be a complex Lie algebra, and let $D _ { k } = U ( \mathfrak{a} ) \otimes _ { \mathbf{C} } \wedge ^ { k } ( \mathfrak{a} )$, where $\wedge ^ { k } (\mathfrak{a} )$ is the $k$th exterior power of $\frak a$, $k = 0 , \ldots , n = \operatorname { dim } \mathfrak{a}$. Let |
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− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021033.png" /></td> </tr></table>
| + | \begin{equation*} \delta _ { k } ( X \bigotimes X _ { 1 } \bigwedge \ldots \bigwedge X _ { k } ) = \end{equation*} |
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− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021034.png" /></td> </tr></table>
| + | \begin{equation*} = \sum _ { i = 1 } ^ { k } ( - 1 ) ^ { i + 1 } X X _ { i } \bigotimes X _ { 1 } \bigwedge \ldots \bigwedge \hat{X} _ { i } \bigwedge \ldots \bigwedge X _ { k } + \end{equation*} |
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− | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021037.png" /> means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021038.png" /> has been omitted. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021039.png" /> be the constant term of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021040.png" />. Then
| + | \begin{equation*} + \sum _ { 1 \leq i < j \leq k } ( - 1 ) ^ { i + j } X \bigotimes [ X , X _ { j } ] \bigwedge \end{equation*} |
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− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021041.png" /></td> </tr></table> | + | <table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021034.png"/></td> </tr></table> |
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− | is the standard resolution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021042.png" /> of the trivial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021043.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021044.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021045.png" /> is a subalgebra, one considers the relative version <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021046.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021047.png" /> by setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021048.png" />. One observes that the obvious modification of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021049.png" /> produces mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021051.png" />, and that the resulting complex is similarly exact.
| + | where $X \in U ( \mathfrak{a} )$, $X _ { i } \in \mathfrak{a}$ and $\hat{X}_i$ means that $X_i$ has been omitted. Let $\delta _ { 0 } ( X )$ be the constant term of $X \in U ( \mathfrak{a} )$. Then |
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− | In [[#References|[a3]]] two constructions of a resolution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021053.png" />, were obtained. They are described below. | + | \begin{equation*} 0 \rightarrow D _ { n } \stackrel { \delta _ { n } } { \rightarrow } \ldots \stackrel { \delta _ { 1 } } { \rightarrow } D _ { 0 } \stackrel { \delta _ { 0 } } { \rightarrow } \mathbf{C} \rightarrow 0 \end{equation*} |
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| + | is the standard resolution $V ( \mathfrak{a} )$ of the trivial $\frak a$-module $\mathbf{C}$. If $\mathfrak{p} \subset \mathfrak{a}$ is a subalgebra, one considers the relative version $V ( \mathfrak{a} , \mathfrak{p} )$ of $V ( \mathfrak{a} )$ by setting $\overline { D } _ { k } = U ( {\frak a} ) \otimes_{U ( {\frak p} )} \wedge ^ { k } ( {\frak a}/ \frak{p} )$. One observes that the obvious modification of the $\delta _ { k }$ produces mappings $\overline { \delta } _{k} : \overline { D } _ { k } \rightarrow \overline { D } _ { k - 1 }$, $k = 1 , \ldots , r = \operatorname { dim } \mathfrak{a} / \mathfrak{p}$, and that the resulting complex is similarly exact. |
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| + | In [[#References|[a3]]] two constructions of a resolution of $L = L ( \lambda )$, $\lambda \in P ^ { + }$, were obtained. They are described below. |
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| ==Weak BGG resolution.== | | ==Weak BGG resolution.== |
− | Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021054.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021055.png" /> be the category of finitely-generated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021056.png" />-diagonalizable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021057.png" />-finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021058.png" />-modules ([[#References|[a2]]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021059.png" /> denote the centre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021060.png" />. If M is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021061.png" />-module, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021062.png" /> denote the set of eigenvalues of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021063.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021064.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021065.png" /> denote the eigenspace associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021066.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021067.png" /> consists of only one element, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021068.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021069.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021070.png" /> defines an [[Exact functor|exact functor]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021071.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021072.png" />, let | + | Let $\mathfrak { b } = \mathfrak { h } \oplus \mathfrak { n } \subset \mathfrak { g }$ and let $\mathcal{O}$ be the category of finitely-generated $\mathfrak h $-diagonalizable $U ( {\frak n} )$-finite $\frak g$-modules ([[#References|[a2]]]). Let $Z ( {\frak g} )$ denote the centre of $U ( \mathfrak { g } )$. If M is a $\frak g$-module, let $\Theta( M ) \subset Z ( \mathfrak { g } ) ^ { * }$ denote the set of eigenvalues of $M$. For $\theta \in \Theta ( M )$, let $M _ { \theta }$ denote the eigenspace associated to $\theta$. The set $\Theta( L ( \lambda ) )$ consists of only one element, denoted by $\theta _ { \lambda }$. For $M \in \mathcal{O}$, $\mathfrak { F } _ { \lambda } ( M ) = ( M \otimes L ( \lambda ) ) _ { \theta _ { \lambda } }$ defines an [[Exact functor|exact functor]] in $\mathcal{O}$. If $r = \operatorname { dim } \mathfrak{n}^-$, let |
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− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021073.png" /></td> </tr></table>
| + | \begin{equation*} ( B , \delta ) : 0 \rightarrow B _ { r } \stackrel { \delta _ { r } } { \rightarrow } \ldots \stackrel { \delta _ { 1 } } { \rightarrow } B _ { 1 } \stackrel { \delta _ { 0 } } { \rightarrow } L ( \lambda ) \rightarrow 0 \end{equation*} |
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− | be the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021074.png" /> under the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021075.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021076.png" /> is known as the weak BGG resolution. Its importance lies in the property of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021077.png" /> explained below. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021078.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021079.png" /> denotes the trivially extended action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021080.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021081.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021082.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021083.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021084.png" /> is the Verma module associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021085.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021086.png" /> denote the set of simple (i.e. indecomposable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021087.png" />, positive roots. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021088.png" /> be the group of automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021089.png" /> generated by the reflections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021090.png" /> relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021091.png" /> (cf. also [[Weyl group|Weyl group]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021092.png" /> be the set of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021093.png" /> that are minimally expressed as a product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021094.png" /> reflections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021095.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021096.png" />. One writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021097.png" />. Each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021098.png" /> has a filtration (cf. also [[Filtered algebra|Filtered algebra]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b12021099.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210100.png" />-modules such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210101.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210102.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210103.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210104.png" />. | + | be the image of $V ( \mathfrak { g } , \mathfrak { b } )$ under the functor $\mathfrak { F } _ { \lambda }$. $( B , \delta )$ is known as the weak BGG resolution. Its importance lies in the property of the $B _ { k }$ explained below. For $\mu \in \mathfrak { h } ^ { * }$, ${\bf C} ( \mu )$ denotes the trivially extended action of $\mu$ from $\mathfrak h $ to $\mathfrak{b}$. The $\frak g$-module $M ( \mu ) = U ( \mathfrak { g } ) \otimes_{ U ( \mathfrak { b } )} \mathbf{C} ( \mu )$ is the Verma module associated to $\mu$. Let $\Pi \subset \Delta ^ { + }$ denote the set of simple (i.e. indecomposable in $\Delta ^ { + }$, positive roots. Let $W$ be the group of automorphisms of $\mathfrak{h} ^ { * }$ generated by the reflections $\sigma _ { \alpha }$ relative to $\alpha \in \Pi$ (cf. also [[Weyl group|Weyl group]]). Let $W ^ { ( i ) }$ be the set of elements $w \in W$ that are minimally expressed as a product of $i$ reflections $\sigma _ { \alpha }$, $\alpha \in \Pi$. One writes $W ^ { ( i ) } = \{ w \in W : l ( w ) = i \}$. Each $B _ { k }$ has a filtration (cf. also [[Filtered algebra|Filtered algebra]]) $B _ { k } = M _ { 1 } \supset \ldots \supset M _ { s } = 0$ of $\frak g$-modules such that $M _ { i } / M _ { i - 1 } \simeq M ( \mu _ { i } )$ and $\{ \mu _ { i } \} _ { i = 1 } ^ { s - 1 } = \{ w \cdot \lambda \} _ { w \in W ^ { ( k ) } }$, where $w.\mu = w ( \mu + \rho ) - \rho$ and $\rho = ( 1 / 2 ) \sum _ { \alpha \in \Delta ^ { + } } \alpha$. |
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− | If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210105.png" /> is a Lie algebra and | + | If $\frak a$ is a Lie algebra and |
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− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210106.png" /></td> </tr></table> | + | <table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210106.png"/></td> </tr></table> |
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− | is a resolution of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210107.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210108.png" /> by projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210109.png" />-modules, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210110.png" /> is the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210111.png" /> under the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210112.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210113.png" />. The cohomology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210114.png" /> are defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210115.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210116.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210117.png" />, the weak BGG resolution implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210118.png" />. | + | is a resolution of the $\frak a$-module $M$ by projective $\frak a$-modules, and $( \operatorname{Hom} _ {\frak a } ( D , N ) , \delta ^ { \prime } )$ is the image of $( D , \delta )$ under the functor $\operatorname { Hom } _ { a }( - , N ) : N ^ { \prime } \rightarrow \operatorname { Hom } _ { a } ( N ^ { \prime } , N )$, then $\operatorname { Ext } _ { a } ^ { i } ( M , N ) = \operatorname { Ker } \delta _ { i + 1 } ^ { \prime } / \operatorname { Im } \delta _ { i } ^ { \prime }$. The cohomology groups $H ^ { i } ( \mathfrak{a} , M )$ are defined as $\operatorname{Ext}_{\mathfrak{a}}^i( \mathbf{C} , M)$. If $L = L ( \lambda )$, and $\lambda \in P ^ { + }$, the weak BGG resolution implies that $\operatorname{dim} H ^ { i } ( \mathfrak { n } ^ { - } , L ) = \# W ^ { ( i ) }$. |
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| ==Strong BGG resolution.== | | ==Strong BGG resolution.== |
− | For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210119.png" /> one writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210120.png" /> if there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210121.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210122.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210123.png" />. This relation induces a partial ordering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210124.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210125.png" />, by setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210126.png" /> whenever there are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210127.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210128.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210129.png" />. It was shown in [[#References|[a1]]] that | + | For $w _ { 1 } , w _ { 2 } \in W$ one writes $w _ { 1 } \leftarrow w _ { 2 }$ if there exists a $\gamma \in \Delta _ { + }$ such that $w _ { 1 } = \sigma _ { \gamma } w _ { 2 }$ and $l ( w _ { 1 } ) = l ( w _ { 2 } ) + 1$. This relation induces a partial ordering $\leq$ on $W$, by setting $w \leq w ^ { \prime }$ whenever there are $w _ { 1 } , \dots , w _ { k }$ in $W$ such that $w = w _ { 1 } \leftarrow \ldots \leftarrow w _ { k } = w ^ { \prime }$. It was shown in [[#References|[a1]]] that |
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− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210130.png" /></td> </tr></table> | + | <table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210130.png"/></td> </tr></table> |
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− | if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210131.png" />. Furthermore, every such homomorphism is zero or injective. One fixes, for each pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210132.png" />, one such injection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210133.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210134.png" />. Therefore, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210135.png" />-homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210136.png" /> is determined by a complex matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210137.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210138.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210139.png" />. It is shown in [[#References|[a3]]] that there exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210140.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210141.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210142.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210143.png" />, such that | + | if and only if $w _ { 1 } \leq w _ { 2 }$. Furthermore, every such homomorphism is zero or injective. One fixes, for each pair $( w _ { 1 } , w _ { 2 } )$, one such injection $i_{w _ { 1 } , w_ { 2 }}$. Let $C _ { k } = \oplus _ { w \in W ^ { ( i ) } } M ( w . \lambda )$. Therefore, a $\frak g$-homomorphism $d _ { k } : C _ { k } \rightarrow C _ { k - 1 }$ is determined by a complex matrix $( c _ { w _ { 1 } , w _ { 2 }} )$ with $w _ { 1 } \in W ^ { ( k ) }$ and $w _ { 2 } \in W ^ { ( k - 1 ) }$. It is shown in [[#References|[a3]]] that there exist $c _{w_{ 1 } , w _ { 2 } } \in \{ \pm 1 \}$, $w _ { 1 } \in W ^ { ( k ) }$, $w _ { 1 } \leftarrow w _ { 2 }$, for $k = 1 , \dots , r = \operatorname { dim } \mathfrak{n} ^ { - }$, such that |
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− | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210144.png" /></td> </tr></table> | + | <table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210144.png"/></td> </tr></table> |
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− | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210145.png" /> is the canonical surjection, is exact. This strong BGG resolution refines the weak BGG resolution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210146.png" /> and, in particular, calculates the cohomology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210147.png" />. In [[#References|[a4]]] it was proved that the weak and the strong BGG resolutions are isomorphic. The results of [[#References|[a4]]] apply to the more general situation of parabolic subalgebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210148.png" />. They imply the existence of a complex in terms of the degenerate principal series representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210149.png" /> that has the same cohomology as the de Rham complex [[#References|[a4]]]. The BGG resolution has been extended to Kac–Moody algebras (see [[#References|[a5]]] and also [[Kac–Moody algebra|Kac–Moody algebra]]) and to the Lie algebra of vector fields on the circle [[#References|[a6]]]. | + | where $d _ { 0 } : M ( \lambda ) \rightarrow L ( \lambda )$ is the canonical surjection, is exact. This strong BGG resolution refines the weak BGG resolution $( B , \delta )$ and, in particular, calculates the cohomology groups $H ^ { i } ( \mathfrak { h } ^ { - } , L )$. In [[#References|[a4]]] it was proved that the weak and the strong BGG resolutions are isomorphic. The results of [[#References|[a4]]] apply to the more general situation of parabolic subalgebras $\mathfrak { p } \supset \mathfrak{b}$. They imply the existence of a complex in terms of the degenerate principal series representations of $G$ that has the same cohomology as the de Rham complex [[#References|[a4]]]. The BGG resolution has been extended to Kac–Moody algebras (see [[#References|[a5]]] and also [[Kac–Moody algebra|Kac–Moody algebra]]) and to the Lie algebra of vector fields on the circle [[#References|[a6]]]. |
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| ====References==== | | ====References==== |
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> I.N. Bernstein, I.M. Gelfand, S.I. Gelfand, "Structure of representations generated by vectors of highest weight" ''Funkts. Anal. Prilozh.'' , '''5''' : 1 (1971) pp. 1–9</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> I.N. Bernstein, I.M. Gelfand, S.I. Gelfand, "A certain category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210150.png" />-modules" ''Funkts. Anal. Prilozh.'' , '''10''' : 2 (1976) pp. 1–8</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> I.N. Bernstein, I.M. Gelfand, S.I. Gelfand, "Differential operators on the base affine space and a study of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210151.png" />-modules" I.M. Gelfand (ed.) , ''Lie groups and their representations, Proc. Summer School on Group Representations'' , Janos Bolyai Math. Soc.&Wiley (1975) pp. 39–64</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> A. Rocha-Caridi, "Splitting criteria for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210152.png" />-modules induced from a parabolic and the Bernstein–Gelfand–Gelfand resolution of a finite dimensional, irreducible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210153.png" />-module" ''Trans. Amer. Math. Soc.'' , '''262''' : 2 (1980) pp. 335–366</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> A. Rocha-Caridi, N.R. Wallach, "Projective modules over graded Lie algebras" ''Math. Z.'' , '''180''' (1982) pp. 151–177</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> A. Rocha-Caridi, N.R. Wallach, "Highest weight modules over graded Lie algebras: Resolutions, filtrations and character formulas" ''Trans. Amer. Math. Soc.'' , '''277''' : 1 (1983) pp. 133–162</TD></TR></table> | + | <table><tr><td valign="top">[a1]</td> <td valign="top"> I.N. Bernstein, I.M. Gelfand, S.I. Gelfand, "Structure of representations generated by vectors of highest weight" ''Funkts. Anal. Prilozh.'' , '''5''' : 1 (1971) pp. 1–9</td></tr><tr><td valign="top">[a2]</td> <td valign="top"> I.N. Bernstein, I.M. Gelfand, S.I. Gelfand, "A certain category of $\frak g$-modules" ''Funkts. Anal. Prilozh.'' , '''10''' : 2 (1976) pp. 1–8</td></tr><tr><td valign="top">[a3]</td> <td valign="top"> I.N. Bernstein, I.M. Gelfand, S.I. Gelfand, "Differential operators on the base affine space and a study of $\frak g$-modules" I.M. Gelfand (ed.) , ''Lie groups and their representations, Proc. Summer School on Group Representations'' , Janos Bolyai Math. Soc.&Wiley (1975) pp. 39–64</td></tr><tr><td valign="top">[a4]</td> <td valign="top"> A. Rocha-Caridi, "Splitting criteria for $\frak g$-modules induced from a parabolic and the Bernstein–Gelfand–Gelfand resolution of a finite dimensional, irreducible $\frak g$-module" ''Trans. Amer. Math. Soc.'' , '''262''' : 2 (1980) pp. 335–366</td></tr><tr><td valign="top">[a5]</td> <td valign="top"> A. Rocha-Caridi, N.R. Wallach, "Projective modules over graded Lie algebras" ''Math. Z.'' , '''180''' (1982) pp. 151–177</td></tr><tr><td valign="top">[a6]</td> <td valign="top"> A. Rocha-Caridi, N.R. Wallach, "Highest weight modules over graded Lie algebras: Resolutions, filtrations and character formulas" ''Trans. Amer. Math. Soc.'' , '''277''' : 1 (1983) pp. 133–162</td></tr></table> |
The structure of a real Lie group $G$ can be studied by considering representations of the complexification $\frak g$ of its Lie algebra (cf. also Representation of a Lie algebra). These are viewed as left modules over the universal enveloping algebra $U ( \mathfrak { g } )$ of $\frak g$, or $\frak g$-modules. The Lie algebras $\frak g$ considered here are the complexifications of real semi-simple Lie algebras corresponding to real, connected, semi-simple Lie groups. A Cartan subalgebra $\mathfrak h $, that is, a maximal Abelian subalgebra with the property that its adjoint representation on $\frak g$ is semi-simple, is chosen (cf. also Cartan subalgebra). A root system $\Delta \subset \mathfrak { h } ^ { * }$, corresponding to the resulting decomposition of $\frak g$, is obtained. A further choice of a positive root system $\Delta ^ { + } \subset \Delta$ determines subalgebras $\mathfrak n$ and $\mathfrak{n}^{-}$ corresponding to the positive and negative root spaces, respectively. The building blocks in the study of $G$ are the finite-dimensional irreducible $\frak g$-modules $L ( \lambda )$. They are indexed by the set $P ^ { + } \subset \mathfrak { h } ^ { * }$ of dominant integral weights $\lambda$ relative to $\Delta ^ { + }$.
For any ring $A$ with unity, a resolution of a left $A$-module $M$ is an exact chain complex of $A$-modules:
\begin{equation*} \ldots \rightarrow D _ { 2 } \stackrel { \delta _ { 2 } } { \rightarrow } D _ { 1 } \stackrel { \delta _ { 1 } } { \rightarrow } D _ { 0 } \stackrel { \delta _ { 0 } } { \rightarrow } M \rightarrow 0. \end{equation*}
For example, let $\frak a$ be a complex Lie algebra, and let $D _ { k } = U ( \mathfrak{a} ) \otimes _ { \mathbf{C} } \wedge ^ { k } ( \mathfrak{a} )$, where $\wedge ^ { k } (\mathfrak{a} )$ is the $k$th exterior power of $\frak a$, $k = 0 , \ldots , n = \operatorname { dim } \mathfrak{a}$. Let
\begin{equation*} \delta _ { k } ( X \bigotimes X _ { 1 } \bigwedge \ldots \bigwedge X _ { k } ) = \end{equation*}
\begin{equation*} = \sum _ { i = 1 } ^ { k } ( - 1 ) ^ { i + 1 } X X _ { i } \bigotimes X _ { 1 } \bigwedge \ldots \bigwedge \hat{X} _ { i } \bigwedge \ldots \bigwedge X _ { k } + \end{equation*}
\begin{equation*} + \sum _ { 1 \leq i < j \leq k } ( - 1 ) ^ { i + j } X \bigotimes [ X , X _ { j } ] \bigwedge \end{equation*}
where $X \in U ( \mathfrak{a} )$, $X _ { i } \in \mathfrak{a}$ and $\hat{X}_i$ means that $X_i$ has been omitted. Let $\delta _ { 0 } ( X )$ be the constant term of $X \in U ( \mathfrak{a} )$. Then
\begin{equation*} 0 \rightarrow D _ { n } \stackrel { \delta _ { n } } { \rightarrow } \ldots \stackrel { \delta _ { 1 } } { \rightarrow } D _ { 0 } \stackrel { \delta _ { 0 } } { \rightarrow } \mathbf{C} \rightarrow 0 \end{equation*}
is the standard resolution $V ( \mathfrak{a} )$ of the trivial $\frak a$-module $\mathbf{C}$. If $\mathfrak{p} \subset \mathfrak{a}$ is a subalgebra, one considers the relative version $V ( \mathfrak{a} , \mathfrak{p} )$ of $V ( \mathfrak{a} )$ by setting $\overline { D } _ { k } = U ( {\frak a} ) \otimes_{U ( {\frak p} )} \wedge ^ { k } ( {\frak a}/ \frak{p} )$. One observes that the obvious modification of the $\delta _ { k }$ produces mappings $\overline { \delta } _{k} : \overline { D } _ { k } \rightarrow \overline { D } _ { k - 1 }$, $k = 1 , \ldots , r = \operatorname { dim } \mathfrak{a} / \mathfrak{p}$, and that the resulting complex is similarly exact.
In [a3] two constructions of a resolution of $L = L ( \lambda )$, $\lambda \in P ^ { + }$, were obtained. They are described below.
Weak BGG resolution.
Let $\mathfrak { b } = \mathfrak { h } \oplus \mathfrak { n } \subset \mathfrak { g }$ and let $\mathcal{O}$ be the category of finitely-generated $\mathfrak h $-diagonalizable $U ( {\frak n} )$-finite $\frak g$-modules ([a2]). Let $Z ( {\frak g} )$ denote the centre of $U ( \mathfrak { g } )$. If M is a $\frak g$-module, let $\Theta( M ) \subset Z ( \mathfrak { g } ) ^ { * }$ denote the set of eigenvalues of $M$. For $\theta \in \Theta ( M )$, let $M _ { \theta }$ denote the eigenspace associated to $\theta$. The set $\Theta( L ( \lambda ) )$ consists of only one element, denoted by $\theta _ { \lambda }$. For $M \in \mathcal{O}$, $\mathfrak { F } _ { \lambda } ( M ) = ( M \otimes L ( \lambda ) ) _ { \theta _ { \lambda } }$ defines an exact functor in $\mathcal{O}$. If $r = \operatorname { dim } \mathfrak{n}^-$, let
\begin{equation*} ( B , \delta ) : 0 \rightarrow B _ { r } \stackrel { \delta _ { r } } { \rightarrow } \ldots \stackrel { \delta _ { 1 } } { \rightarrow } B _ { 1 } \stackrel { \delta _ { 0 } } { \rightarrow } L ( \lambda ) \rightarrow 0 \end{equation*}
be the image of $V ( \mathfrak { g } , \mathfrak { b } )$ under the functor $\mathfrak { F } _ { \lambda }$. $( B , \delta )$ is known as the weak BGG resolution. Its importance lies in the property of the $B _ { k }$ explained below. For $\mu \in \mathfrak { h } ^ { * }$, ${\bf C} ( \mu )$ denotes the trivially extended action of $\mu$ from $\mathfrak h $ to $\mathfrak{b}$. The $\frak g$-module $M ( \mu ) = U ( \mathfrak { g } ) \otimes_{ U ( \mathfrak { b } )} \mathbf{C} ( \mu )$ is the Verma module associated to $\mu$. Let $\Pi \subset \Delta ^ { + }$ denote the set of simple (i.e. indecomposable in $\Delta ^ { + }$, positive roots. Let $W$ be the group of automorphisms of $\mathfrak{h} ^ { * }$ generated by the reflections $\sigma _ { \alpha }$ relative to $\alpha \in \Pi$ (cf. also Weyl group). Let $W ^ { ( i ) }$ be the set of elements $w \in W$ that are minimally expressed as a product of $i$ reflections $\sigma _ { \alpha }$, $\alpha \in \Pi$. One writes $W ^ { ( i ) } = \{ w \in W : l ( w ) = i \}$. Each $B _ { k }$ has a filtration (cf. also Filtered algebra) $B _ { k } = M _ { 1 } \supset \ldots \supset M _ { s } = 0$ of $\frak g$-modules such that $M _ { i } / M _ { i - 1 } \simeq M ( \mu _ { i } )$ and $\{ \mu _ { i } \} _ { i = 1 } ^ { s - 1 } = \{ w \cdot \lambda \} _ { w \in W ^ { ( k ) } }$, where $w.\mu = w ( \mu + \rho ) - \rho$ and $\rho = ( 1 / 2 ) \sum _ { \alpha \in \Delta ^ { + } } \alpha$.
If $\frak a$ is a Lie algebra and
is a resolution of the $\frak a$-module $M$ by projective $\frak a$-modules, and $( \operatorname{Hom} _ {\frak a } ( D , N ) , \delta ^ { \prime } )$ is the image of $( D , \delta )$ under the functor $\operatorname { Hom } _ { a }( - , N ) : N ^ { \prime } \rightarrow \operatorname { Hom } _ { a } ( N ^ { \prime } , N )$, then $\operatorname { Ext } _ { a } ^ { i } ( M , N ) = \operatorname { Ker } \delta _ { i + 1 } ^ { \prime } / \operatorname { Im } \delta _ { i } ^ { \prime }$. The cohomology groups $H ^ { i } ( \mathfrak{a} , M )$ are defined as $\operatorname{Ext}_{\mathfrak{a}}^i( \mathbf{C} , M)$. If $L = L ( \lambda )$, and $\lambda \in P ^ { + }$, the weak BGG resolution implies that $\operatorname{dim} H ^ { i } ( \mathfrak { n } ^ { - } , L ) = \# W ^ { ( i ) }$.
Strong BGG resolution.
For $w _ { 1 } , w _ { 2 } \in W$ one writes $w _ { 1 } \leftarrow w _ { 2 }$ if there exists a $\gamma \in \Delta _ { + }$ such that $w _ { 1 } = \sigma _ { \gamma } w _ { 2 }$ and $l ( w _ { 1 } ) = l ( w _ { 2 } ) + 1$. This relation induces a partial ordering $\leq$ on $W$, by setting $w \leq w ^ { \prime }$ whenever there are $w _ { 1 } , \dots , w _ { k }$ in $W$ such that $w = w _ { 1 } \leftarrow \ldots \leftarrow w _ { k } = w ^ { \prime }$. It was shown in [a1] that
![](https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210130.png) |
if and only if $w _ { 1 } \leq w _ { 2 }$. Furthermore, every such homomorphism is zero or injective. One fixes, for each pair $( w _ { 1 } , w _ { 2 } )$, one such injection $i_{w _ { 1 } , w_ { 2 }}$. Let $C _ { k } = \oplus _ { w \in W ^ { ( i ) } } M ( w . \lambda )$. Therefore, a $\frak g$-homomorphism $d _ { k } : C _ { k } \rightarrow C _ { k - 1 }$ is determined by a complex matrix $( c _ { w _ { 1 } , w _ { 2 }} )$ with $w _ { 1 } \in W ^ { ( k ) }$ and $w _ { 2 } \in W ^ { ( k - 1 ) }$. It is shown in [a3] that there exist $c _{w_{ 1 } , w _ { 2 } } \in \{ \pm 1 \}$, $w _ { 1 } \in W ^ { ( k ) }$, $w _ { 1 } \leftarrow w _ { 2 }$, for $k = 1 , \dots , r = \operatorname { dim } \mathfrak{n} ^ { - }$, such that
![](https://www.encyclopediaofmath.org/legacyimages/b/b120/b120210/b120210144.png) |
where $d _ { 0 } : M ( \lambda ) \rightarrow L ( \lambda )$ is the canonical surjection, is exact. This strong BGG resolution refines the weak BGG resolution $( B , \delta )$ and, in particular, calculates the cohomology groups $H ^ { i } ( \mathfrak { h } ^ { - } , L )$. In [a4] it was proved that the weak and the strong BGG resolutions are isomorphic. The results of [a4] apply to the more general situation of parabolic subalgebras $\mathfrak { p } \supset \mathfrak{b}$. They imply the existence of a complex in terms of the degenerate principal series representations of $G$ that has the same cohomology as the de Rham complex [a4]. The BGG resolution has been extended to Kac–Moody algebras (see [a5] and also Kac–Moody algebra) and to the Lie algebra of vector fields on the circle [a6].
References
[a1] | I.N. Bernstein, I.M. Gelfand, S.I. Gelfand, "Structure of representations generated by vectors of highest weight" Funkts. Anal. Prilozh. , 5 : 1 (1971) pp. 1–9 |
[a2] | I.N. Bernstein, I.M. Gelfand, S.I. Gelfand, "A certain category of $\frak g$-modules" Funkts. Anal. Prilozh. , 10 : 2 (1976) pp. 1–8 |
[a3] | I.N. Bernstein, I.M. Gelfand, S.I. Gelfand, "Differential operators on the base affine space and a study of $\frak g$-modules" I.M. Gelfand (ed.) , Lie groups and their representations, Proc. Summer School on Group Representations , Janos Bolyai Math. Soc.&Wiley (1975) pp. 39–64 |
[a4] | A. Rocha-Caridi, "Splitting criteria for $\frak g$-modules induced from a parabolic and the Bernstein–Gelfand–Gelfand resolution of a finite dimensional, irreducible $\frak g$-module" Trans. Amer. Math. Soc. , 262 : 2 (1980) pp. 335–366 |
[a5] | A. Rocha-Caridi, N.R. Wallach, "Projective modules over graded Lie algebras" Math. Z. , 180 (1982) pp. 151–177 |
[a6] | A. Rocha-Caridi, N.R. Wallach, "Highest weight modules over graded Lie algebras: Resolutions, filtrations and character formulas" Trans. Amer. Math. Soc. , 277 : 1 (1983) pp. 133–162 |