BGG resolution

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 $\frak a$, $k = 0 , \ldots , n = \dim \mathfrak{a}$. Let

\begin{align*} \delta _ { k } ( X \bigotimes X _ { 1 } \bigwedge \ldots \bigwedge X _ { k } ) &= \sum _ { i = 1 } ^ { k } ( - 1 ) ^ { i + 1 } X X _ { i } \bigotimes X _ { 1 } \bigwedge \ldots \bigwedge \hat{X} _ { i } \bigwedge \ldots \bigwedge X _ { k } \\ &\qquad + \sum _ { 1 \leq i < j \leq k } ( - 1 ) ^ { i + j } X \bigotimes [ X , X _ { j } ] \bigwedge X_1 \bigwedge \dots \bigwedge \hat{X}_i \bigwedge \dots \bigwedge \hat{X}_j \bigwedge \dots \bigwedge X_k, \end{align*}

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 \to D_{n} \xrightarrow{\delta_{n}} \dots \xrightarrow{\delta_1} D_{0} \xrightarrow{\delta_0} \CC \to 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 $\overline { \delta } _{k} : \overline { D } _ { k } \rightarrow \overline { D } _ { k - 1 }$, $k = 1 , \ldots , r = \dim \mathfrak{a} / \mathfrak{p}$, 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 $r = \dim\mathfrak{n}^-$, let

\begin{equation*} ( B , \delta ) : 0 \to B_{r} \xrightarrow{\delta_{r}} \dots \xrightarrow{\delta_{1}} B_{1} \xrightarrow{\delta_{0}} L(\lambda) \to 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|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

\begin{equation*} (D, \delta) : \dots \to D_2 \xrightarrow{\delta_2} D_1 \xrightarrow{\delta_1} D_0 M \xrightarrow{\delta_0} 0 \end{equation*}

is a resolution of the $\frak a$-module $M$ by projective $\frak a$-modules, and $( \Hom_{\frak a}(D,N), \delta')$ is the image of $(D, \delta)$ under the functor $\Hom_{\frak a}(-, N): N' \rightarrow \Hom_{\frak a}(N', N)$, then $\Ext_{\frak a}^{i}(M, N) = \operatorname{Ker} \delta _ { i + 1 } ^ { \prime } / \Im \delta'_i$. The cohomology groups $H^{i}(\mathfrak{a}, M)$ are defined as $\Ext_{\mathfrak{a}}^i(\CC, M)$. If $L = L ( \lambda )$, and $\lambda \in P ^ { + }$, the weak BGG resolution implies that $\dim H^{i}(\mathfrak{n}^{-}, L ) = \# W^{(i)}$.

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

\[ \Hom_{\mathfrak{g}}(M(w_1 . \lambda), M(w_2 . \lambda)) = \CC \]

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

\[ (C, d) : 0 \to C_r \xrightarrow{d_r} \dots \xrightarrow{d_2} C_1 \xrightarrow{d_1} C_0 \xrightarrow{d_0} L(\lambda) \to 0, \]

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