Symmetry, Integrability and Geometry: Methods and Applications The BGG Complex on Projective Space ⋆

We give a complete construction of the Bernstein-Gelfand-Gelfand complex on real or complex projective space using minimal ingredients.


Introduction
This article is concerned with differential calculus on real or complex projective space, invariant under projective transformations. These transformations constitute a semisimple Lie group and projective space is hence a homogeneous space of the form G/P for G semisimple. The subgroup P is parabolic and, more generally, differential geometries modelled on homogeneous spaces of this form are known as 'parabolic' [5]. Projective space gives rise to projective differential geometry in this sense. Conformal and CR geometry are included amongst other examples of parabolic differential geometry. The interplay between the symmetries of projective space and its invariant differential operators is mediated by representation theory. In this article we present the 'BGG complex' on projective space as perhaps the simplest of these constructions in one of the simplest of settings. We anticipate that our approach will extend to G/P in general and perhaps to other homogeneous spaces. In [19], Olver introduced complexes of differential operators on Euclidean space acting between 'hyperforms'. These are BGG complexes constructed directly, employing only affine invariance and the associated Schur functors. He constructs some 'easy' examples and observes that "Other examples, of greater complexity, can of course be constructed at will, but the expressions rapidly get out of hand, even in low dimensional spaces". We maintain that the extra symmetry that the operators exhibit under projective transformations allows one to control these expressions more effectively. The fascinating combination of symmetries and differential operators is a defining feature in the work of Willard Miller to whom we dedicate this article.
The Bernstein-Gelfand-Gelfand (BGG) complexes on RP n or CP n are, by now, well-known complexes of vector bundles and differential operators between them generalising the de Rham complex on RP n and the holomorphic de Rham complex on CP n , respectively. An introduction to such complexes is given in [11] and the particular case of projective space is discussed in [12].
Usually, their construction involves choosing so-called 'splitting operators' [3,6] constructed from Kostant's Laplacian [17] or 'quabla operator' [3] or from the Jantzen-Zuckerman translation functor [24]. Here, we avoid the direct use of splitting operators, instead relying only on diagram chasing, as is already done in [11,12] in simple cases. In fact, the BGG complex of holomorphic differential operators on CP 2 was already constructed in this manner [10, p. 351] before it was realised by John Rice [21] that complexes like this were dual to Lepowsky's construction [18] on the level of Verma modules. (On CP 1 there is just one family of BGG operators, already singled out for their invariance in [13,Proposition 2.1].) In this article, for simplicity and cleanliness we employ a spectral sequence to effect the diagram chasing. We employ projective invariance to derive explicit formulae for the operators in the projective BGG complex on RP n in terms of the usual round metric on the sphere.
As is often done in differential geometry, when it is necessary to write out tensors and their natural operations and we shall adorn them with upper or lower indices corresponding to the tangent or cotangent bundle respectively. For example, a vector field can be written as X a , a one-form as ω a , and the natural pairing between them as X a ω a in accordance with the 'Einstein summation convention'. For any tensor φ abc we shall write its symmetric part as φ (abc) and its skew part as φ [abc] . For example, to say that ω ab is a two-form is to say that ω ab = −ω ba or, equivalently, ω ab = ω [ab] or, equivalently, ω (ab) = 0 and then for any torsion-free connection ∇ a , are the exterior derivative of ω ab and the Lie derivative of ω ab in the direction of the vector field X a , respectively. Such formulae are not meant to imply any choice of local coördinates. More precisely, this is Penrose's 'abstract index notation' [20] and it formalises the conventions used by many classical authors -see, for example, the discussion of projective differential geometry by Schouten [22]. We shall view RP n as a homogeneous space or as a quotient of the 'projective sphere' under the antipodal map. In either case, it is convenient to write the associated Lie algebra as where From now on we shall discuss only real projective space RP n or its double cover, the sphere S n . The complex case is completely parallel with real numbers being replaced by complex numbers everywhere and by working in the holomorphic category rather than the smooth.

An outline of the construction
If V is a finite-dimensional representation of P , we shall denote by V the induced homogeneous vector bundle on G/P constructed as Notice that if V is actually a G-module restricted to P , then V is canonically trivialised as a vector bundle (but not as a homogeneous vector bundle). Hence, in this case V is naturally equipped with a G-equivariant flat connection ∇ obtained by transporting to V the exterior derivative d : is exact on the level of germs and provides a resolution of V as a locally constant sheaf on G/P . This general reasoning holds on any homogeneous space G/P but the following discussion is specific to RP n or S n . Suppose V is irreducible as a G-module. In this case we shall see that as a P -module V is filtered meaning that these are the subquotients listed in a natural order, starting on the left with the smallest quotient of V. (In other words V N is the smallest P -submodule in the filtration, the quotient V/V N has a filtration V 0 + V 1 + V 2 + · · · + V N −1 , and the meaning is now clear by induction.) It follows that the bundle V is correspondingly filtered and now we claim that the connection ∇ : V → Λ 1 ⊗V , and consequently the whole complex (3), is compatible with this filtration (as detailed in Theorem 1 below). The spectral sequence of a filtered complex [9] now comes into play, having as its E 0 -level the following.
Here, the precise positioning of the coördinate axes is a matter of convention. The important property of the E 0 -level is that the differentials ∂ are simply homomorphisms of vector bundles and we shall show that they are induced by a complex of G 0 -modules Furthermore, we shall show that this complex defines the Lie algebra cohomology H r (g − , V), which in turn has been computed by Kostant [17]. It follows that the E 1 -level of the spectral sequence is rather sparse, typically where H 0 = V 0 and, in particular, there is precisely one irreducible bundle in each diagonal E p,d−p 1 for d fixed. Since (3) resolves V, we know that this spectral sequence is also converging to V and the only way that this can happen is if the differentials fit together as a resolution This is the required BGG resolution. The rest of the article is devoted to filling in the details of this argument.

The f iltering of V as a P -module
Recall the decomposition (1) of g = sl(n + 1, R) and consider the element . .
uniquely characterised as lying in the centre of g 0 with [H, X] = X, for X ∈ g + . It is called the grading element [5] of the |1|-graded Lie algebra (1). If the G-module V is restricted to G 0 , then it splits into eigenspaces under H. For the standard representation by matrix multiplication on column vectors for example, Rather than use the actual eigenvalues, which are rational in general, let us subtract the lowest eigenvalue and write for the eigenspace decomposition, noting that g + acts by V j → V j+1 for all j. It follows that are P -submodules of V for all j and we have our filtration (4).

The f iltered complex Λ • ⊗ V and its spectral sequence
The filtration (10) of V as a P -module certainly induces a filtration Proof . Because the assertion is local, without loss of generality it suffices to prove it on the standard affine coördinate patch, namely Recall (9) that V splits as a G 0 -module. Hence the same is true of V restricted to this patch: To proceed we need a formula for ∇. The appendix discusses various natural constructions on a general Lie group G, which we now specialise to be the Abelian Lie group and according to (23) we find that ∇ = d + θ, where • θ : V | R n → Λ 1 ⊗ V | R n is defined by the same trivialisation together with the Maurer-Cartan form θ on G − ; more specifically, Evidently, d preserves the splitting (12) and the conclusion is manifest.
According to this corollary, we may now consider the spectral sequence of the filtered complex where we have chosen to normalise (6) by placing V 0 at the origin. By construction, the bundle V k on G/P is the homogeneous bundle induced from V k = V k /V k+1 as a P -module (cf. (10)). We already know that, as an eigenspace for the grading element H in the centre of g 0 , the vector space V k is a G 0 -module. By regarding it as the quotient V k /V k+1 we are equivalently making V k into a P -module by decreeing that G + act trivially. By construction, the E 0 -differential is Gequivariant. Furthermore, its definition and the Leibniz rule ensure that it is linear over the functions. In other words ∂ is a G-equivariant homomorphism of homogeneous bundles and, as such, must be induced by a homomorphism of P -modules Λ p (g/p) * ⊗ V k → Λ p+1 (g/p) * ⊗ V k−1 , which we shall also denote by ∂. In fact, from the formula for ∇ displayed in the proof of Corollary 1, we see that ∂ is induced by where V is regarded as a P -module by restricting the G action to G 0 and decreeing that G + act trivially. Since g − is Abelian, this formula agrees with (26) for the Koszul differential (used in [8] to define Lie algebra cohomology). As observed in the appendix, this is a complex of G 0 -modules. Alternatively, we could come to the same conclusion by restricting attention to a standard affine coördinate chart G − ∼ = R n ֒→ RP n and noticing that the kernel of d : Λ p ⊗V → Λ p+1 ⊗V consists precisely of the left-invariant V-valued p-forms under the action of G − on itself. Since ∇ = d+∂, we may now invoke the second realisation of H r (g − , V) from Theorem 5 in the Appendix.
To proceed we need to be explicit concerning the irreducible representation V of SL(n + 1, R). In fact, one usually deals with complex representations of SL(n+1, C) (starting with sl(n+1, C)) but here there is no real difference and we shall adopt the notation from [2] in denoting such representations by attaching non-negative integers to its Dynkin diagram a 2 a 3 a 4 a 5 · · · a n−1 an (meaning that −[a 1 , a 2 , . . . , a n ] is the lowest weight of this representation with respect to the standard basis of fundamental weights). With this notation, here is the conclusion of Kostant's computation [17] of Lie algebra cohomology. We describe the result as an SL(n, R)-module (by attaching non-negative integers to an A-series Dynkin diagram with one fewer nodes) and fix the action of G 0 by specifying how the grading element H ∈ g 0 acts.
Theorem 2. The E 1 -level of the spectral sequence of the filtered complex ∇ : Λ • ⊗ V consists of irreducible homogeneous vector bundles on RP n under the action of G = SL(n + 1, R). Only the following terms are non-zero meaning that the bundle in question is induced by the G 0 -module as listed and extended trivially as a P -module.
Proof . According to (13), we already know that Λ r ⊗ V is spread along the r th diagonal of the E 0 -level of the spectral sequence and that the E 0 -differential is induced by the Koszul differential (14) defining the Lie algebra cohomology H r (g − , V). We see from (15) that each of these H r (g − , V) is irreducible and so it follows that the r th diagonal of the E 1 -level consists of a single irreducible homogeneous vector bundle. To complete the proof it suffices to locate the position of this bundle along this particular diagonal and, to do this, the action of the grading element H turns out to be sufficient. More precisely, Therefore H acts by −c + p on the G 0 -module inducing E p,q 1 and so, from the action of H in table (15), the bundle induced by H r (g − , V) is located at E a 1 +a 2 +···+ar+r,−a 1 −a 2 −···−ar 1 . Finally, we are claiming that N = a 1 + a 2 + · · · + a n . To see this we note that, since H acts on V 0 by −c, it acts on V N by N − c. On the other hand, in accordance with the action of the longest element of the Weyl group, an a n−1 · · · a 5 a 4 a 3 a 2 a and so H acts on V * N by −c ′ , where c ′ = na n + (n − 1)a n−1 + · · · + 3a 3 + 2a 2 + a 1 n + 1 .
As outlined in Section 2, the E 1 -level of this spectral sequence is rather sparse (7) and Theorem 2 says exactly how sparse. In particular, since there is only one non-zero bundle on each diagonal of the E 1 -level, the general theory of spectral sequences provides a complex of differential operators whose cohomology on the level of sheaves coincides with that of (3), namely V for ker : H 0 → H 1 and otherwise zero. The bundle H r is induced on RP n = G/P from the G 0 -module H r (g − , V) extended trivially as as P -module. This is the BGG resolution (constructed on a general G/P by Lepowsky [18] on the level of generalised Verma modules). More explicitly, in the discussion following Corollary 1, in any affine coördinate patch we identified ∇ : is the exterior derivative with values in V −q . Adding these operators to the diagram (6) gives a double complex as employed by Baston [1, p. 120]. We conclude that in any affine coördinate patch, our spectral sequence (of a filtered complex) coincides with Baston's spectral sequence (of a double complex). In particular, the operators H k → H k+1 are obtained as zigzag compositions of the first order differential operators d together with choices of algebraic splittings of the operators ∂ (Baston and others use the algebraic adjoint ∂ * introduced by Kostant [17]). This confirms that the resulting operators H k → H k+1 are differential. By using the spectral sequence of a filtered complex as we have done, it is manifest that the differential operators H k → H k+1 are independent of any choice of splittings and that the whole construction and resulting BGG complex is G-equivariant. It is also clear that the first steps in this approach can be taken on an arbitrary homogeneous space G/P .

Formulae for the BGG operators
Already, the affine invariance of the operators occurring in the BGG resolution on RP n fixes their formulae with respect to the flat connection on R n ⊂ RP n as follows (affine invariance being ensured by noting that Q as in (11) is acting on R n = Q/G 0 by affine transformations). Firstly, consider the symbol of the BGG differential operator H r → H r+1 . As a homomorphism of homogeneous bundles s Λ 1 ⊗ H r → H r+1 for some s, it is induced by a homomorphism of G 0 -modules for some s, which we can determine just from the action of the grading element. Specifically, we know that H acts by −c + a 1 + a 2 + · · · + a r + r on H r (g − , V) H acts by −1 on g − ∴ H acts by s on s g * − .
Theorem 3. Each bundle H r in the BGG complex on RP n is an irreducible tensor bundle and the differential operator ∇ (s) : H r → H r+1 is given by φ → π(∇ s φ) where ∇ is the flat affine connection and π : s Λ 1 ⊗ H r → H r+1 is the unique projection onto this irreducible summand.
Equivalently, the bundles H r and the operators between them are exhibited as Young tableau in [12]. It is often useful, however, to be able to write the BGG complex globally on RP n without recourse to affine coördinates and projective invariance. For this, we shall use the round metric on the sphere and the corresponding Levi Civita connection on S n or RP n . By way of normalisation, if g ab denotes the round metric and δ a b the identity endomorphism on the tangent bundle, let us choose the radius of the sphere so that defines the Riemann curvature tensor. For any irreducible covariant tensor bundle E on RP n , let us write ∇ 2 for the composition and g for the composition Theorem 4. If s is odd, the operator ∇ (s) : H r → H r+1 is given by and, if s is even, π (∇ 2 + (s − 1) 2 g) · · · (∇ 2 + 9g)(∇ 2 + g) .
Here, if it is the r th row to which the boxes on the right hand side are being added and there are a total of v boxes on the left hand side (so that v is the valence of the corresponding tensor), then w + r = s + v + b (see [12]). In particular, if we consider only first order operators · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ✛ ✲ b (w) ∇ −−−→ · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · More generally, taking conventions from [12], if ∇ and∇ are two torsion-free connections in the same projective clasŝ for some 1-form Υ a , then for φ bc···d having symmetries and projective weight specified by the left hand side of (17), where π is the Young projector corresponding to the right hand side of (17). We may iterate this formula, adding more boxes to the r th row (assuming that there is room to do so) and, each time, both b and v increase by one. Suppressing indices, after s iterations the result is that where we are writing k = w − (v + b + 1 − r). In particular, if s = w + r − v − b as it is in the case of a BGG operator, then this iteration reads where k = s−1. So far, this conclusion holds under any projective change of connection but now we specialise to the case of the round connection ∇ on the sphere, being projectively equivalent to the flat connection∇ (under gnomonic projection). The general formula [12, (3.4)] for the change in the Ricci tensor specialises to In this equation Υ is viewed as a tensor but if it is viewed as an operator φ Υ −→ Υφ, then we should write ∇Υ = Υ∇ + g + Υ 2 . This equation allows us to deal with the iterated formula above. For example, when k = 2, also bearing in mind that as operators ∇ and g commute, as advertised in the statement of the theorem. For higher k, direct calculations rapidly get out of hand. Instead, it suffices to prove the following lemma in which we have isolated the required algebra (and then we prove the lemma by indirect means). Remark 1. Our curvature normalisation (16) implies that the Ricci tensor on our round sphere is given by Thus, the metric g ab coincides with 1 n−1 R ab , a tensor generally known in projective differential geometry [12] as the Schouten tensor or Rho-tensor P ab . Replacing g by P in the formulae of Theorem 4 gives expressions that are valid on any space of constant curvature. More generally, there is a Rho-tensor that arises in similar contexts [4] within parabolic differential geometry [5]. Lemma 1. Define an associative algebra R = R ∇, Υ with generators subject to the 'Riccati relation' ∇Υ = Υ∇ + 1 + Υ 2 . Then the following identities hold in R. If k is even, then and, if k is odd, then Proof . The algebra R may be realised by the following differential operators on the circle To see this, note that these operators certainly satisfy the Riccati relation and we are required, therefore, to show that they satisfy no further relations. Within R we may normalise any element as follows. By induction, the Riccati relation extends to and it follows by induction on k that where the ellipsis · · · denotes terms of lower order in ∇ with coefficients that are real polynomial in Υ. It follows that every element of R can be written uniquely in the form k p=0 A p (Υ) ∇ p for suitable real polynomials A p (Υ). In our claimed realisation, such an element is represented by the differential operator k p=0 A p (tan θ) d p /dθ p and now it suffices to observe (by acting on 1, θ, θ 2 , . . . , θ k near 0) that such a differential operator vanishes if and only if all the polynomials A p (Υ) are zero. (More precisely, we should restrict the action of such operators to smooth functions on (−π/2, π/2) or some other suitable function space.) Having realised R by differential operators, we are reduced to proving identities amongst these operators. This is accomplished in the following lemma. Lemma 2. Let D denote the differential operator f (θ) → (cos 2 θ)(df (θ)/dθ) on the circle. Then the following identities hold. If k is even, then and, if k is odd, then Proof . Writing D k+1 for the operators on the left-hand-sides of the displays in this lemma, the following identity is easily verified It follows that, in our realisation of the algebra R, we obtain the expressions on the left-handsides of the displays in the Lemma 1.
this follows easily by induction.
Remark 2. More generally, the formula is used in [15] to derive expressions for BGG operators in conformal geometry and in [4] these expressions are extended to parabolic geometries in general. Lemma 2 concerns the case Φ = 1.

An example
The following is amongst the simplest of non-trivial examples. Let us take n = 2 so G = SL(3, R) and let V = R 3 , regarded as column vectors with SL(3, R) acting by left matrix multiplication. We saw in (8) how H splits R 3 and in Section 3 that this results in the filtering where SL(2, R) ⊂ P acts on R 2 as the standard representation and acts trivially on R. Dropping projective weights, the corresponding bundle V on RP 2 is filtered where T is the tangent bundle and Λ 0 is the trivial bundle. The E 0 -level (6) of our spectral sequence becomes and one checks that ∂ : Λ 0 → Λ 1 ⊗ T and ∂ : Λ 1 → Λ 2 ⊗ T are given by respectively. In this simple case, we do not need Kostant's Theorem to see that where (Λ 1 ⊗ T ) • denotes the trace-free part of Λ 1 ⊗ T . Therefore, the E 1 -level (7) is as the resulting BGG complex. Writing out the flat connection ∇ on V in terms of the round connection gives for the two operators in V → Λ 1 ⊗ V → Λ 2 ⊗ V and, noting that we obtain a formula for the projectively invariant operator (Λ 1 ⊗ T ) • ∇ (2) −→ Λ 2 in agreement with Theorem 4.
Appendix: Lie algebra cohomology as a geometrical construction Although in this article we shall need only the cohomology of an Abelian Lie algebra, we take the opportunity here to describe the cohomology of a general Lie algebra g in terms of differential geometry on G, a Lie group whose Lie algebra is g. We believe that for a general parabolic geometry, we shall need this geometric interpretation for a nilpotent Lie algebra. Suppose V is a G-module and use the same notation for the corresponding representation of g. Following but adapting [8], we are going to present the Lie algebra cohomology H r (g, V) as a geometrical construction on G. Beware that G is no longer the Lie group SL(n + 1, R) as it was until now. This section is written to be self-contained with the aim of being useful elsewhere. This material is well-known to experts and implicit in [8] but we believe it worthwhile laying out the details.
We shall view G as a homogeneous space under its own action on the left. Its tangent bundle T G is then regarded as a homogeneous bundle and can be identified as G × g, where g is the Lie algebra of G. It is convenient to write this isomorphism as where θ is a 1-form on G with values in g known as the Maurer-Cartan form [23]. To compute with θ it is convenient to write functions on G with values in g as X α and then (18) becomes where φ a α θ β a = δ α β and θ α a φ b α = δ a b . A vector field X a on G is left-invariant if and only if the corresponding function X α : G → g is constant. Choosing any torsion-free affine connection ∇ a on G and expanding the definition ∇ a (θ α b X b ) = 0, the left-invariant vector fields are those that satisfy from which it follows easily that the Lie bracket of two left-invariant vector fields is again leftinvariant. Since the left-invariant vector fields on G are of the form φ(X) for X ∈ g we may define the Lie bracket on g by transportation: For computational purposes, let us write [X, Y ] γ = Γ αβ γ X α Y β for the Lie bracket on g. Then we can write out (19) explicitly as or, in other words, as , we may rewrite this as and finally as This formula employs an arbitrary torsion-free connection. Without indices and without this connection it is more usually written as In other words, the definition (19) of the Lie bracket on g is equivalent to (20) or (21), usually known as the Maurer-Cartan equation [23]. Now suppose V is a G-module and use the same notation for the corresponding g-module. There are two canonically defined connections on the vector bundle V = G × V over G. One of them is the evident flat connection, ignoring the action of G on V. We shall denote it by d since it is the exterior derivative acting on functions with values in V. The other one takes the isomorphism from (2) and pulls back the evident flat connection on the right hand side as was done in Section 2 and we shall denote this one by ∇ as was done there. To relate these two connections more explicitly suppose f : G → V is a section of V that is constant after the twisting (22). It means that the function g → gf (g) is constant. If so, then for fixed g ∈ G and X ∈ g, the function is constant. Equivalently, the function t → e tX f (ge tX ) is constant and so where this last equality is due to the flow of the left-invariant vector field φ(X) being the oneparameter subgroup of right-translations g → ge tX (see, e.g. [25]). For computational purposes, let us write X α → X α ρ α where ρ α ∈ g * ⊗ End(V) for the action of g on V. Then g → gf (g) is constant if and only if 0 = φ(X)f + Xf = X α φ a α d a f + X α ρ α f = X a (d a f + θ α a ρ α f ) for all left-invariant vector fields X a . It follows that the connection ∇ a on V is given by where θ ∈ Λ 1 ⊗g is the Maurer-Cartan form. As a check, the differential in the coupled de Rham sequence (3) is and the Maurer-Cartan equation (21) shows that the composition V ∇ → Λ 1 ⊗ V ∇ → Λ 2 ⊗ V is given by ∇ 2 f = d(df + θf ) + θ ∧ (df + θf ) = (dθ + θ ∧ θ)f = 0 and the connection ∇ is flat, as expected. More generally, the whole sequence is a complex. Consider the linear mapping Λ p g * ⊗ V ∋ v αβ···γ θ −→ v ab···c ≡ θ α a θ β b · · · θ γ c v αβ···γ ∈ Γ(G, Λ p ⊗ V) in which V is just a passenger (i.e. plays no rôle). We shall refer to the resulting V-valued p-form as left-invariant just as we would if V were absent. Lemma 3. The connection ∇ : Λ p ⊗ V → Λ p+1 ⊗ V preserves left-invariance.
Proof . We use the Maurer-Cartan equation (20) to compute as required.
We have shown that there are two ways of defining Lie algebra cohomology as follows.
Theorem 5. The Lie algebra cohomology H r (g, V) may be defined as either • the cohomology of the Koszul complex (27), or • the cohomology of the complex (24) restricted to left-invariant forms.
Remark 3. Although we use this theorem in the main body of this article, it is easily avoided. In tackling a general parabolic geometry, however, we believe that Theorem 5 will be essential.
Remark 4. As a minor variation on this construction, suppose G is enlarged to Q, a semi-direct product and suppose that V extends to a representation of Q. We identify G with the Q-homogeneous space Q/G 0 , noting that when the action of Q on G = Q/G 0 is restricted to G it coincides with its usual action of G on itself by left translation. The Q-homogeneous bundle V ≡ Q × G 0 V on Q/G 0 is equipped with a flat connection ∇ by dint of the canonical trivialisation which clearly coincides with ∇ defined by (22). This connection is Q-equivariant. Consequently, not only does the twisted de Rham complex coincide with (24) and thereby compute the Lie algebra cohomology H r (g, V) when restricted to G-invariant forms, but also the complex (27) is automatically one of Q-modules where the Q-action on V is as supposed and the Q-action on g * is induced by the conjugation action of Q on G (bearing in mind that G is a normal subgroup of Q).