$k$-Dirac Complexes

This is the first paper in a series of two papers. In this paper we construct complexes of invariant differential operators which live on homogeneous spaces of $|2|$-graded parabolic geometries of some particular type. We call them $k$-Dirac complexes. More explicitly, we will show that each $k$-Dirac complex arises as the direct image of a relative BGG sequence and so this fits into the scheme of the Penrose transform. We will also prove that each $k$-Dirac complex is formally exact, i.e., it induces a long exact sequence of infinite (weighted) jets at any fixed point. In the second part of the series we use this information to show that each $k$-Dirac complex is exact at the level of formal power series at any point and that it descends to a resolution of the $k$-Dirac operator studied in Clifford analysis.


Introduction
Let h be a non-degenerate, symmetric and C-bilinear form on C 2m . The Grassmannian variety M of totally isotropic k-dimensional subspaces in C 2m is a homogeneous space of a |2|-graded parabolic geometry. We assume throughout the paper that n := m − k ≥ k ≥ 2. We will show that on M there is a complex of invariant differential operators which we call the k-Dirac complex. The main result of this article is (see Theorem 7.14) that the complex is formally exact (as explained above in the abstract) in the sense of [25].
This result is crucial for an application in [24] where it is shown that the complex is exact with formal power series at any fixed point and that it descends (as outlined in the recent series [5,6,7]) to a resolution of the k-Dirac operator studied in Clifford analysis (see [12,20]). As a potential application of the resolution, there is an open problem of characterizing the domains of monogenicity, i.e., an open set U is a domain of monogenicity if there is no open set U with U U such that each monogenic function 1 in U extends to a monogenic function in U . Recall from [16,Section 4] that the Dolbeault resolution together with some L 2 estimates are crucial in a proof of the statement that any pseudoconvex domain is a domain of holomorphy.
The k-Dirac complexes belong to the singular infinitesimal character and so the BGG machinery introduced in [9] is not available. However, we will show that each k-Dirac complex arises as the direct image of a relative BGG sequence (see [10,11] for a recent publication on this topic) and so, this paper fits into the scheme of the Penrose transform (see [2,26]). In particular, we will work here only in the setting of complex parabolic geometries.
The machinery of the Penrose transform is a main tool used in [1]. The main result of that article is a construction of families of locally exact complexes of invariant differential operators on quaternionic manifolds. One of these quaternionic complexes can (see [3,4,13]) be identified with a resolution of the k-Cauchy-Fueter operator which has been intensively studied in Clifford analysis (see again [12,20]). In order to prove the local exactness of this quaternionic complex, one uses that an almost quaternionic structure is a |1|-graded parabolic geometry and the theory of constant coefficient operators from [19].
Unfortunately, the parabolic geometry on M is |2|-graded and so there is canonical a 2-step filtration of the tangent bundle of M given by a bracket generating distribution. With such a structure, it is more natural to work with weighted jets (see [18]) rather than usual jets and we use this concept also here, i.e., we prove the formal exactness of the k-Dirac complexes with respect to the weighted jets. Nevertheless, we will prove in [24] that the formal exactness of the k-Dirac complex (or more precisely the exactness of (7.16) for each + j ≥ 0) is enough to conclude that it descends to a resolution of the k-Dirac operator.
We consider here only the even case C 2m as, due to the representation theory, the Penrose transform does not work in odd dimension C 2m+1 and it seems that this case has to be treated by completely different methods. The assumption n ≥ k is (see [12]) called the stable range. This assumption is needed only in Proposition 5.7 where we compute direct images of sheaves that appear in the relative BGG sequences. Hence, it is reasonable to expect that (see also [17]) the machinery of the Penrose transform provides formally exact complexes also in the unstable range n < k.
For the application in [24], we need to show that the k-Dirac complexes constructed in this paper give rise to complexes from [22] which live on the corresponding real parabolic geometries. This turns out to be rather easy since any linear G-invariant operator is determined by a certain P-equivariant homomorphism. As this correspondence works also in the smooth setting, passing from the holomorphic setting to the smooth setting is rather elementary.
The abstract approach of the Penrose transform is not very helpful when one is interested in local formulae of differential operators. Local formulae of the operators in the k-Dirac complexes can be found in [22]. Notice that in this article we construct only one half of each complex from [22]. This is due to the fact that the complex space of spinors decomposes into two irreducible so(2n, C)-sub-representations. The other half of each k-Dirac complex can (see Remark 5.8) be easily obtained by adapting results from this paper.
Finally, let us mention few more articles which deal with the k-Dirac complexes. The null solutions of the first operator in the k-Dirac complex were studied in [21,23]. The singular Hasse graphs and the corresponding homomorphisms of generalized Verma modules were computed in [14].

Preliminaries
In Section 2 we will review some well known material. Namely, in Section 2.1 we will summarize some theory of complex parabolic geometries. We will recall in Section 2.2 the concept of weighted jets on filtered manifolds and in Section 2.3 the definition of the normal bundle associated to analytic subvariety and the formal neighborhood. In Section 2.4 we will give a short summary of the Penrose transform.
See [8] for a thorough introduction into the theory of parabolic geometries. The concept of weighted jets was originally introduced in the smooth setting by Tohru Morimoto, see for example [18]. Sections 2.3 and 2.4 were taken mostly from [2,26].
Then p Σ is the parabolic subalgebra associated to the |k|-grading and p Σ = g 0 ⊕ p + is known as the Levi decomposition (see [2,Section 2.2]). This means that p + is the nilradical 2 and that g 0 is a maximal reductive subalgebra called the Levi factor. It is clear that each subspace g i is p Σ -invariant and that g − is a nilpotent subalgebra. Moreover, it can be shown that, as a Lie algebra, g − is generated by g −1 .
The algebra b := p 0 is called the standard Borel subalgebra. A subalgebra of g is called standard parabolic if it contains b and in particular, p Σ is such an algebra. More generally, a subalgebra of g is called a Borel subalgebra and a parabolic subalgebra if it is conjugated to the standard Borel subalgebra and to a standard parabolic subalgebra, respectively. We will for brevity sometimes write p instead of p Σ .
Let s i be the simple reflection associated to α i , i = 1, 2, . . . , m, W g be the Weyl group of g and W p be the subgroup of W g generated by {s i : α i ∈ Σ}. Then W p is isomorphic to the Weyl group of g ss 0 and the directed graph that encodes the Bruhat order on W g contains a subgraph called the Hasse diagram W p attached to p (see [2,Section 4.3]). The vertices of W p consist of those w ∈ W g such that w.λ is p-dominant for any g-dominant weight λ where the dot stands for the affine action, namely, w.λ = w(λ + ρ) − ρ where ρ is the lowest form. It turns out that each right coset of W p in W g contains a unique element from W p and it can be shown that this is the element of minimal length (see [2,Lemma 4.3.3]). This identifies W p with W p \W g .
We will need also the relative case. Assume that Σ ⊂ 0 and put r := p Σ . Then q := r ∩ p = p Σ∪Σ is a standard parabolic subalgebra and l := g ss 0 ∩ q is a parabolic subalgebra of g ss 0 (see [2,Section 2.4]). The definition of the Hasse diagram attached to p applies also to the pair (g ss 0 , l), namely an element w ∈ W p (as [2,Section 4.4]) belongs to the relative Hasse diagram W q p if it T. Salač is the element of minimal length in its right coset of W q in W p . Hence, W q p is a subset of W g which can be naturally identified with W q \W p .
There is (up to isomorphism) a unique connected and simply connected complex Lie group G with Lie algebra g. Assume that Σ = {α i 1 , . . . , α i j }. Let ω 1 , . . . , ω m be the fundamental weights associated to the simple roots and V be an irreducible g-module with highest weight λ := ω i 1 + · · · + ω i j . Since any g-representation integrates to G, V is also a G-module. The action descends to the projective space P(V) and the stabilizer of the line spanned by a non-zero highest weight vector v is the associated parabolic subgroup P. This is by definition a closed subgroup of G and its Lie algebra is p. The homogeneous space G/P is biholomorphic to the G-orbit of [v] ∈ P(V) and since it is completely determined by Σ, we denote it by crossing in the Dynkin diagram of g the simple roots from Σ. We will for brevity put M := G/P and denote by p : G → M the canonical projection.
On G lives a tautological g-valued 1-form ω which is known as the Maurer-Cartan form. This form is P-equivariant in the sense that for each p ∈ P : (r p ) * ω = Ad(p −1 ) • ω where Ad is the adjoint representation and r p is the principal action of p. If V is a subspace of g and g ∈ G, then ω −1 g (V) is a subspace of T g G and the disjoint union g∈G ω −1 g (V) determines a distribution on G which we for brevity denote by ω −1 (V). Since T p = T p • T r p , it follows that T p(ω −1 (V)) is a well-defined distribution on M provided that V is P-invariant. In particular, this applies to g i and we put F i := T p(ω −1 (g i )). Since ker(T p) = ω −1 (p), it follows that the filtration Since M is the homogeneous model, we have the following: • the filtration is compatible 3 with the Lie bracket of vector fields in the sense that the commutator of a section of F i and a section of F j is a section of F i+j , • the Lie bracket descends to a vector bundle map L : Λ 2 gr(T M ) → gr(T M ), called the Levi form, which is homogeneous of degree zero and Hence, (gr(T M ), L) is a locally trivial bundle of nilpotent Lie algebras with typical fiber g − and it follows that F −1 is a bracket generating distribution. We denote by T * M = Λ 1,0 T * M the vector bundle dual to T M , i.e., the fiber over x ∈ M is the space of C-linear maps T x M → C. The filtration of T M induces a filtration is the associated graded vector bundle and gr i (T * M ) is isomorphic to the dual of gr i (T M ).

Weighted dif ferential operators
Let M be the homogeneous space with the regular filtration {F −j : j = 0, . . . , k} as in Section 2.1. As M is a complex manifold, T M C := T M ⊗ C = T 1,0 M ⊕ T 0,1 M where the first and the second summand is the holomorphic and the anti-holomorphic part 4 , respectively. As each vector bundle F −j is a holomorhic sub-bundle of T M , we have F −j ⊗ C = F 1,0 −j ⊕ F 0,1 −j as above. Let U be an open subset of M and X be a holomorphic vector field on U. The weighted order wo(X) of X is the smallest integer j such that X ∈ Γ(F 1,0 −j | U ). A differential operator D acting on the space O(U) of holomorphic functions on U is called a differential operator of weighted order at most r if for each x ∈ U there is an open neighborhood U x of x with a local framing 5 {X 1 , . . . , X p } by holomorphic vector fields such that where N p 0 := {a = (a 1 , . . . , a p ) : a i ∈ Z, a i ≥ 0, i = 1, . . . , p}, f a ∈ O(U x ) and for all a in the sum with f a non-zero: Let O x be the space of germs of holomorphic functions at x. We denote by F i x the space of those germs f ∈ O x such that Df (x) = 0 for every differential operator D which is defined on an open neighborhood of x and wo(D) ≤ i. We put J i x the class of f and call it the i-th weighted jet of f . Then the disjoint union J i := ∪ x∈M J i x is naturally a holomorphic vector bundle over M , the canonical vector bundle map J i π i − → J i−1 has constant rank and thus, its kernel gr i is again a holomorphic vector bundle with fiber gr i x over x. Notice that for each integer i ≥ 0 there is a short exact sequence 0 → F i+1 Assume that V is a holomorphic vector bundle over M . We denote by V * the dual bundle, by −, − the canonical pairing between V and V * and finally, by x V the equivalence class of s and call it the i-th weighted jet of s. Then the disjoint union J i V := x∈M J i x M is naturally a holomorphic vector bundle over M , the canonical bundle map J i V π i − → J i−1 V has constant rank and thus, its kernel gr i V is again a holomorphic vector bundle and we denote by gr i x V its fiber over x. As above, there is for each integer i ≥ 0 a short exact sequence x V → 0 and just as in the smooth case, there is a canonical linear isomorphism gr i Remark 2.1. If the filtration is trivial, i.e., F −1 = T M , then the concept of weighted jets agrees with that of usual jets. In this case we will use calligraphic letters instead of Gothic letters, i.e., we write F i and J i and gr i and j i x f instead of F i and J i and gr i and j i x f , respectively. The vector bundle gr i is canonically isomorphic to the i-th symmetric power S i T * M .
Assume that there is a P-module V such that V is isomorphic to the G-homogeneous vector bundle G × P V. Let e be the identity element of G. Then we call the point x 0 := eP the origin of M and we put There are linear isomorphisms We will be interested in the sub-bundle S i gr 1 (T * M ) ⊗ V of gr i V . Notice that the fiber of this sub-bundle over , the vector space of all weighted i-th jets of germs of holomorphic sections at x whose usual (i − 1)-th jet vanishes. The fiber of this bundle over x 0 is isomorphic to S i g 1 ⊗ V and we denote it for brevity by gr i V.
Suppose that W is another P-module and W := G× P W be the associated homogeneous vector bundle. We say that the weighted order of a linear differential operator D : Γ(V ) → Γ(W ) is at most r if for each x ∈ M , s ∈ O(V ) x : j r x s = 0 ⇒ Ds(x) = 0. It is well known (see [18]) that D induces for each i ≥ 0 a vector bundle map gr i V → gr i−r W where we agree that gr W = 0 if < 0. The restriction of this map to the fibers over the origin is a linear map grD : gr i V → gr i−r W.

Ideal sheaf of an analytic subvariety
Let us first recall some basics from the theory of sheaves (see for example [26]). Suppose that F and G are sheaves on topological spaces X and Y , respectively, and that ι : X → Y is a continuous map. We denote by F x the stalk of F at x ∈ X and by F(U) or by Γ(U, F) the space of sections of F over an open set U. Then the pullback sheaf ι −1 G is a sheaf on X and the direct image ι * F is a sheaf on Y . The q-th direct image ι q * F is a sheaf on Y , it is defined as the sheafification of the pre-sheaf Suppose now that X and Y are complex manifolds with structure sheaves of holomorphic functions O X and O Y , respectively, that ι is holomorphic and that G is a sheaf of O Y -modules. Then ι −1 G is in general not a sheaf of O X -modules. To fix this problem, we use that Now we can continue with the definition of the ideal sheaf. Suppose that the holomorphic map ι is an embedding. The restriction T Y | X contains the tangent bundle T X of X. The normal bundle N X of X in Y is simply the quotient bundle, i.e., it fits into the short exact sequence 0 → T X → T Y | X → N X → 0 of holomorhic vector bundles. Dually, the co-normal bundle N * fits into the short exact sequence 0 is an ideal in the ring O Y (V) and hence, for each positive integer i there is the sheaf I i X whose space of sections over V is (I X (V)) i . Then there are short exact sequences of sheaves where S i N * is the i-th symmetric power of N * and we agree that I 0 X = O Y . We put F i X := ι −1 I i X . As ι −1 is an exact functor, we get short exact sequences of sheaves on X. Here we use that the adjunction morphism ι −1 ι * F → F is an isomorphism contains basically the same information as the sheaf O (i) X . These sheaves will be crucial in this article.
Since any sheaf over a point is completely determined by its stalk, there is no risk of confusion with the notation set in Remark 2.1.

The Penrose transform
Let us first set notation. Suppose that λ ∈ h * is a g-integral and p-dominant weight. Then there is (see [2, Remark 3.1.6]) an irreducible P-module V λ with lowest weight −λ. We denote by V λ := G × P V λ the induced vector bundle and by O p (λ) the associated sheaf of holomorphic sections.
Suppose that p, r are standard parabolic subalgebras. Then q := r ∩ p is also a standard parabolic subalgebra and we denote by P and R and Q the associated parabolic subgroups with Lie algebras p and r and q, respectively, as explained in Section 2.1. Then Q = R ∩ P and there is a double fibration diagram where η and τ are the canonical projections. The space G/R is called the twistor space T S and G/Q the correspondence space CS. Such a diagram is a starting point for the Penrose transform. Next we need to fix an r-dominant and integral weight λ ∈ h * . Then there is a relative BGG sequence * (λ) which is an exact sequence of holomorphic sections of associated vector bundles over CS and linear G-invariant differential operators such that η −1 O r (λ) is the kernel sheaf of the first operator in the sequence. In other words, there is a long exact sequence of sheaves The upshot of this is that although the pullback sheaf η −1 O r (λ) is not a sheaf of holomorphic sections of an associated vector bundle over CS, it is naturally a sub-sheaf of O q (λ) which is cut out by an invariant differential equation. Moreover, the graph of the relative BGG sequence is [2,Section 8.7] completely determined by the W q r -orbit of λ. Then we push down the relative BGG sequence by the direct image functor τ * . Computing higher direct images of sheaves in the relative BGG sequence is completely algorithmic and algebraic (see [2,Section 5.3]). On the other hand, there is no general algorithm which computes direct images of differential operators and it seems that this has to be treated in each case separately. Nevertheless, in this way one obtains a complex of operators on G/P.

Lie theory
In Section 3 we will provide an algebraic background which is needed in the construction of the k-Dirac complexes via the Penrose transform. We will work with complex parabolic geometries which are associated to gradings on the simple Lie algebra g = so(2m, C). Section 3 is organized as follows: in Section 3.1 we will set notation and study the gradings on g. In Section 3.2 we will compute the relative Hasse diagram W q r .

Lie algebra g and parabolic subalgebras
Let {e 1 , . . . , e m , e * k+1 , . . . , e * m , e * 1 , . . . , e * k } be the standard basis of C 2m , δ be the Kronecker delta and h be the complex bilinear form that satisfies h(e i , e * j ) = δ ij , h(e i , e j ) = h(e * i , e * j ) = 0 for all i, j = 1, . . . , m. A matrix belongs to the associated Lie algebra g := so(h) ∼ = so(2m, C) if and T. Salač The subspace of diagonal matrices h is a Cartan subalgebra of g. We denote by i the linear form on h defined by then we will also write λ = (λ 1 , . . . , λ m ). The simple reflection and We will be interested in the double fibration diagram where, going from left to right, the sets of simple roots are {α m }, {α k , α m } and {α k } and the associated gradings are respectively. With respect to the block decomposition from (3.1), we have 6 The associated standard parabolic subalgebras are respectively, and we have the following isomorphisms We for brevity put Notice that the bilinear form h induces dualities between C k and C k * and between C n and C n * which justifies the notation, that C m is a maximal, totally isotropic and r 0 -invariant subspace, that C k , C k * , C n and C n * are q 0 -invariant, that C k , C 2n and C k * are g 0 -invariant and finally, that h| C 2n is a non-degenerate, symmetric and g 0 -invariant bilinear form. We will for brevity write only h instead of h| C 2n as it will be always clear from the context what is meant.
Let us now consider the associated nilpotent subalgebras By the Jacobi identity, the Lie bracket is equivariant with respect to the adjoint action of the corresponding Levi factor and by the grading property following equation (2.1), it is homogeneous of degree zero. Hence, we can consider the Lie bracket in each homogeneity separately. The first algebra r − is abelian and so there is nothing to add. On the other hand, q − is 3-graded and, as q 0 -modules, we have q −1 ∼ = E⊕F, q −2 ∼ = C k * ⊗C n * , q −3 ∼ = Λ 2 C k * where we put E := C k * ⊗ C n and F := Λ 2 C n * . Using these isomorphisms, the Lie brackets in homogeneity −2 and −3 are the compositions of the canonical projections and respectively. Here we use the canonical pairing C n ⊗ C n * → C. Notice that Λ 2 E ⊕ Λ 2 F is contained in the kernel of (3.5).
In order to understand the Lie bracket on g − , first notice that there are isomorphisms g −1 ∼ = C k * ⊗ C 2n and g −2 ∼ = Λ 2 C k * ⊗ C of irreducible g 0 -modules where C is the trivial representation of so(2n, C). As g − is 2-graded, the Lie bracket is non-zero only in homogeneity −2. It is given by where in the last map we take the trace with respect to h.
Let us first set notation. By a partition we will mean an element of N k,n ++ := {(a 1 , . . . , a k ) : a i ∈ Z, n ≥ a 1 ≥ a 2 ≥ · · · ≥ a k ≥ 0}. For two partitions a = (a 1 , . . . , a k ) and a = (a 1 , . . . , a k ) we write a ≤ a if a i ≤ a i for all i = 1, . . . , k and a < a if a ≤ a and a = a . If a < a does not hold, then we write a ≮ a . We put To the partition a we associate the Young diagram (or the Ferrers diagram) Y consisting of k left-justified rows with a i -boxes in the i-th row. Let b i be the number of boxes in the i-th column of Y. Then we call b = (b 1 , . . . , b n ) ∈ N n,k ++ the partition conjugated to a and we say that a is symmetric if a i = b i , i = 1, . . . , k and b k+1 = · · · = b n = 0. As we assume n ≥ k, the set of symmetric partitions in N k,n ++ depends only on k, and thus, we denote it for simplicity by S k and put S k j := {a ∈ S k : r(a) = j}. Example 3.1.
(1) The empty partition is by definition always symmetric.
Notice that d(a) and q(a) are equal to the number of boxes in the associated Young diagram that are on and above the main diagonal, respectively and that a partition is symmetric if and only if its Young diagram is symmetric with respect to the reflection along the main diagonal.
We can now continue by investigating the relative Hasse graph W q r . The group W r is generated by s 1 , . . . , s m−1 while W q is generated by elements s 1 , . . . , s k−1 , s k+1 , . . . , s m−1 . By (3.2), it follows that W r is the permutation group S m on {1, . . . , m} and that W q ∼ = S k × S n is the stabilizer of {1, . . . , k}. Recall from Section 2.1 that in each left coset of W q in W r there is a unique element of minimal length and that we denote the set of all such distinguished representatives by W q r . Moreover, the Bruhat order on W g descends to a partial order on W r and on W r q . We will now show that there is an isomorphism N k,n ++ → W q r of partially ordered sets.
Let a = (a 1 , . . . , a k ) ∈ N k,n ++ and Y be the associated Young diagram. We will call the box in the i-th row and the j-th column of Y an (i, j)-box and we write into this box the number (i, j) := k − i + j. Notice that 1 ≤ (i, j) ≤ m. Then the set of boxes in Y is indexed by Ξ a := {(i, j) : i = 1, . . . , k, j = 1, . . . , a i } and we order this set lexicographically, i.e., (i, j) < (i , j ) if i < i or i = i and j < j . Then where Ψ : {1, 2, . . . , |a|} → Ξ a is the unique isomorphism of ordered sets. Let us now look at an example.
Example 3.2. The Young diagram from (3.7) is filled as We have the following preliminary observation. Then the permutation w a ∈ S m from (3.8) satisfies for each i = 1, . . . , k and j = 1, . . . , n.
Similarly, if j = 1, . . . , n and b j > 0, then there is c j := k + j − 1 in the (1, j)-box and Then it is easy to check that w a (r i + 1) = r i and w a (c j ) = c j + 1 which completes the proof.
Notice that the sets {k − i + 1 + a i : i = 1, . . . , k} and {k + j − b j : j = 1, . . . , n} are disjoint and that their union is {1, 2, . . . , m}. By (3.9), it follows that where ρ = (m − 1, . . . , 1, 0) is the lowest form of g and for clarity, we separate the first k and last n coefficients by |. Comparing this with Table 1, we see that w a ρ is q-dominant. As the same holds for any r-dominant weight, it follows that w a ∈ W q r .
Lemma 3.4. The map N k,n ++ → W q r , a → w a is an isomorphism of partially ordered sets.
Proof . The map a → w a is by (3.10) clearly injective. To show surjectivity, fix w ∈ W q r . Then the sequence . , k. This shows that w −1 a ω k = w −1 ω k and thus, w = w a . Now it remains to show that the map is compatible with the orders.
Assume that a = (a 1 , . . . , a k ), a = (a 1 , . . . , a k ) ∈ N k,n ++ satisfy |a | = |a| + 1 and a < a . Then there is a unique integer i ≤ k such that a i = a i + 1 and so w a = w a s k−i+a i . By (3.9), we have that w a α k−i+a i > 0 and thus by [8,Proposition 3.2.16], there is an arrow w a → w a in W q r . On the other hand, suppose that a = (a 1 , . . . , a k ) ∈ N k,n ++ satisfies a < a . In order to complete the proof, it is enough to show that there is no arrow w a → w a . By assumptions, there is j such that a 1 ≤ a 1 , . . . , a j−1 ≤ a j−1 and a j > a j . Without loss of generality we may assume that i = j.
On the other hand by (3.9), it follows that w −1 a (w a α k+a i −i ) > 0. We proved that Φ w a ⊂ Φ w a and thus by [8,Proposition 3.2.17], there cannot be any arrow w a → w a .
We will later need the following two observations. A permutation w ∈ S m is k-balanced, if the following is true: If a ∈ S k , then there is j such that Recall from [2] that given w ∈ S m , there exists a minimal integer (w), called the length of w, such that w can be expressed as a product of (w) simple reflections s 1 , . . . , s m . It is well known that (w) is equal to the number of pairs 1 ≤ i < j ≤ m such that w(i) > w(j).
Proof . By the definition of w a , it follows that (w a ) ≤ |a|. On the other hand, if a < a , then w a → w a and thus also (w a ) < (w a ). By induction on |a|, we have that (w a ) ≥ |a|.

Geometric structures attached to (3.3)
In Section 4 we will consider different geometric structures associated to (3.3). Namely, we will consider in Section 4.1 the associated homogeneous spaces, in Section 4.2 the filtrations of tangent bundles of these parabolic geometries and in Section 4.3 the projections η and τ .

Homogeneous spaces
A connected and simply connected Lie group G with Lie algebra g is isomorphic to Spin(2m, C). Let R, Q and P be the parabolic subgroups of G with Lie algebras r, q and p that are associated to {α m }, {α k , α m } and {α k }, respectively, as explained in Section 2.1. We for brevity put T S := G/R, CS := G/Q and M := G/P. Recall from Section 2.4 that we call T S the twistor space and CS the correspondence space.
The twistor space T S. Let us first recall (see [15,Section 6]) some well known facts about spinors. Recall from (3.4) that W := C m is a maximal totally isotropic subspace of C 2m . We can There is a canonical linear map C 2m → End(S) which is determined by w · ψ = i w ψ and w * · ψ = w * ∧ ψ where w ∈ W, w * ∈ W * , ψ ∈ S and i w stands for the contraction by w. If ψ ∈ S, then we put T ψ := {v ∈ C 2m : v · ψ = 0}. If ψ = 0, then T ψ is a totally isotropic subspace and we call ψ a pure spinor if dim T ψ = m (which is equivalent to saying that T ψ is a maximal totally isotropic subspace).
The standard linear isomorphism Λ 2 C 2m ∼ = g gives an injective linear map g → End(S). It is straightforward to verify that the map is a homomorphism of Lie algebras where the commutator in the associative algebra End(S) is the standard one. Hence, g is a Lie subalgebra of End(S) and it turns out that S is no longer irreducible under g but it decomposes as S + ⊕ S − where S + := m i=0 Λ 2i W * and S − := m i=0 Λ 2i+1 W * . Then S + and S − are irreducible non-isomorphic complex spinor representations of g with highest weights ω m and ω m−1 , respectively. It is well known that any pure spinor belongs to S + or to S − (which explains why the Grassmannian of maximal totally isotropic subspaces in C 2m has two connected components). Now we can easily describe the twistor space. The spinor 1 ∈ S + is annihilated by all positive roots in g and hence, it is a highest weight vector. Recall from Section 2.1 that the line spanned by 1 is invariant under R and since T 1 = W, we find that R is the stabilizer of W inside G. As G is connected, we conclude that T S is the connected component of W in the Grassmannian of maximal totally isotropic subspaces in C 2m .
The isotropic Grassmannian M . An irreducible g-module with highest weight ω k is isomorphic to Λ k C 2m . Then e 1 ∧ e 2 ∧ · · · ∧ e k is clearly a highest weight vector and the corresponding point in P(Λ k C 2m ) can be viewed as the totally isotropic subspace x 0 := C k . We see that M is the Grassmannian of totally isotropic k-dimensional subspaces in C 2m . We denote by p : G → M the canonical projection.
The correspondence space CS. The correspondence space CS is the generalized flag manifold of nested subspaces {(z, x) : z ∈ T S, x ∈ M, x ⊂ z} and Q is the stabilizer of (W, x 0 ). Let q : G → CS be the canonical projection.

Filtrations of the tangent bundles of M and CS
Recall from Section 2.1 that the |2|-grading g = g −2 ⊕g −1 ⊕· · ·⊕g 2 associated to {α k } determines a 2- There are linear isomorphisms Recall from Section 2.2 that gr r x 0 denotes the vector space of weighted r-jets of germs of holomorhic functions at x 0 whose weighted (r−1)-jet vanishes. Then the isomorphisms from (2.3) are gr 1 . .

for small r and in general
gr r is a locally trivial vector bundle of graded nilpotent Lie algebras with typical fiber q − . Dually, we get a filtration T * CS = F CS The Q-invariant subspaces E ⊕ q and F ⊕ q give a finer filtration of the tangent bundle, namely F CS −1 = E CS ⊕ F CS . Since the Lie bracket Λ 2 q −1 → q −2 vanishes on Λ 2 E ⊕ Λ 2 F, it follows that E CS and F CS are integrable distributions. This can be deduced also from the short exact sequences i.e., E CS = ker(T η) and

Projections τ and η
Recall from (3.4) that C 2n := [e k+1 , . . . , e m , e * k+1 , . . . , e * m ] and C n := [e k+1 , . . . , e m ], i.e., we view C 2n and C n as subspaces of C 2m . On C 2n we consider the non-degenerate bilinear form h| C 2n which we for brevity denote by h. Then C n is a maximal totally isotropic subspace of C 2n .
The fibers of τ and η are homogeneous spaces of parabolic geometries which (see [2]) can be recovered from the Dynkin diagrams given in (3.3). Proof . As the fibers over distinct points are biholomorphic, it suffices to look at the fibers of η and τ over W and x 0 , respectively.
(a) By definition, η −1 (W) is the set of k-dimensional totally isotropic subspaces in W. As W is already totally isotropic, the first claim follows. ( We will use the following notation. Assume that X ∈ M (2m, k, C) and Y ∈ M (2m, n, C) have maximal rank. Then we denote by [X] the k-dimensional subspace of C 2m that is spanned by the columns of the matrix and by [X|Y ] the flag of nested subspaces It is straightforward to verify that We see that X := p • exp(g − ) is an open, dense and affine subset of M and that any (z, x) ∈ τ −1 (X ) can be represented by where A, B ∈ M (n, C), C ∈ M (k, n, C) are such that A B ∈ Gr + h (n, n) and C = −(X T 1 B+X T 2 A). We immediately get the following observation.
Lemma 4.2. The set τ −1 (X ) is biholomorphic to X × τ −1 (x 0 ). The restriction of τ to this set is then the projection onto the first factor.
The set τ −1 (X ) is not affine as different choices of A and B might lead to the same element in τ −1 (X ). Let Y be the subset of τ −1 (X ) of those nested flags x ⊂ z which can be represented by a matrix as above with A regular. In that case we may assume A = 1 n which uniquely pins down B. It is straightforward to find that B = −B T and conversely, any skew-symmetric n × n matrix determines a totally isotropic n-dimensional subspace in C 2n . We see that Y is an open and affine set which is biholomorphic to g − × A(n, C). In order to write down also η as a canonical projection C ( m 2 )+nk → C ( m 2 ) , it will be convenient to choose a different coordinate system on Y. where pr 1 is the canonical projection and the horizontal arrows are biholomorphisms.
Proof . Let (z, x) be the nested flag corresponding to (4.5) where is clearly a biholomorphism. In order to have a geometric interpretation of the map, consider the following. Using Gaussian elimination on the columns of the matrix (4.5), we can eliminate the X 1 -block and get a new matrix The columns of the matrix span the same totally isotropic subspace z as the original matrix. Moreover, it is clear that z admits a unique basis of this form. From this we easily see that Z is indeed an open affine subset of T S which is biholomorphic to A(m, C). In these coordinate systems, the restriction of η is the projection onto the first factor.

The Penrose transform for the k-Dirac complexes
In Section 5 we will consider the relative BGG sequence associated to a particular r-dominant and integral weight as explained in Section 2.4. More explicitly, we will define in Section 5.1 for each p, q ≥ 0 a sheaf of relative (p, q)-forms and we get a Dolbeault-like double complex. Then we will show (see Section 5.2) that this double complex contains a relative holomorphic de Rham complex. Then in Section 5.3 we will twist each sheaf of relative (p, q)-forms as well as the holomorhic de Rham complex by a certain pullback sheaf. Using some elementary representation theory, we will turn (see Section 5.4) the twisted relative de Rham complex into the relative BGG sequence. In Section 5.5 we will compute direct images of sheaves in the relative BGG sequence. We will use the following notation. We denote by O q and E p,q q the structure sheaf and the sheaf of smooth (p, q)-forms, respectively, over CS. We denote the corresponding sheaves over T S by the subscript r. If W is a holomorphic vector bundle over CS, then we denote by O q (W ) the sheaf of holomorphic sections of W and by E p,q q (W ) the sheaf of smooth (p, q)-forms with values in W . We for brevity put E * := E 0,0 * and E p,q * (U) := Γ(U, E p,q * ) where U is an open set and * = q or r. Moreover, we put η * E p,q r := E q ⊗ η −1 Er E p,q r where we use that η −1 E r is naturally a sub-sheaf of E q .
We call the sheaf of relative (p + 1, q)-forms. By the G-action, it is clearly enough to understand the space of sections of this sheaf over the open set Y from Section 4.3. Given ω ∈ E p,q η (Y), it is easy to see that there is a unique (p, q)-form cohomologous to ω which can be written in the form where Σ denote that the summation is performed only over strictly increasing multi-indeces 7 and each ω I ∈ E 0,q q (Y). As η is holomorphic, ∂ and∂ commute with the pullback map η * . We see that ∂(η * E 1,0 q and thus, ∂ and∂ descend to differential operators respectively. From the definitions it easily follows that: where ω ∈ E p,q η (Y), ω ∈ E p ,q η (Y) and f ∈ E(Y). Recall from Section 4.2 that ∂ z αi ∈ Γ(E 1,0 | Y ) and thus, ∂ η f (x), x ∈ Y depends only on the first weighted jet j 1 x f of f at x (see Section 2.2). Recall from (4.3) that the distribution E CS is equal to ker(T η).  (ii) ∂ η is a linear G-invariant differential operator of weighted order one and the sequence of sheaves (E * ,q η , ∂ η ), q ≥ 0 is exact. (iii) The data (E p,q η , (−1) p∂ , ∂ η ) define a double complex of fine sheaves with exact rows and columns.
Proof . (i) By definition, the sequence of vector bundles 0 → E CS⊥ → T * CS → E CS * → 0 is short exact. Hence, also the sequence 0 → E CS⊥ ∧ Λ p T * CS → Λ p+1 T * CS → Λ p+1 E CS * → 0, p ≥ 1 is short exact. In view of the isomorphism E p+1,q r is a sub-sheaf of E 1,0 q = E q (T * CS) and since ker(T η) = E CS , it is contained in E q (E CS⊥ ). Using that ker(T η) = E CS again, it is easy to see that the map η * E 1,0 r → E q (E CS⊥ ) induces an isomorphism of stalks at any point. Hence, η * E 1,0 r ∼ = E q (E CS⊥ ) and the proof of the first claim is complete. The second claim is clear.
(ii) It is clear that ∂ η is C-linear. It is G-invariant as ∂ commutes with the pullback of any holomorphic map and since η is G-equivariant. As we already observed above that ∂ η f (x) depends only on j 1 x f when x ∈ Y, the G-invariance of ∂ η shows that the same holds on CS and thus, ∂ η is a differential operator of weighted order one. It remains to check the exactness of the complex and using the G-invariance, it is enough to do this at x ∈ Y. By Lemma 4.3, Y is biholomorphic to C where = m 2 + nk. Hence, we can view the standard coordinates w 1 , . . . , w on C as coordinates on Y. If J = (j 1 , . . . , j q ) where j 1 , . . . , j q ∈ {1, . . . , }, then we put dw J = dw j 1 ∧ · · · ∧ dw jq and |J| = q. Let ω =
(i) The relative de Rham complex is an exact sequence of sheaves which resolves the sheaf η −1 O r .
Proof . (i) Since [∂ η ,∂] = 0, the relative de Rham complex is a sub-complex of the zero-th row (E * ,0 q , ∂ η ) of the double complex from Proposition 5.3. By diagram chasing and using the exactness of columns and rows in the double complex, one easily proves the exactness of the relative de Rham complex. By (5.2), it easily follows that η −1 O r = E 0,0 η ∩ ker(∂ η ) ∩ ker(∂). (ii) The standard de Rham complex induces the Spencer complex (see [25]) which is known to be exact. As the complex (Ω * η , ∂ η ) is just a relative version of the (holomorphic) de Rham complex and ∂ η satisfies the usual properties of ∂, it is clear that the relative de Rham complex induces for each s 0 > 0, s 1 , s 2 and s 3 the long exact sequence (5.4). The sequence (5.3) is the direct sum of all such sequences as s 0 , s 1 , s 2 , and s 3 ranges over all quadruples of non-negative integers satisfying r = s 0 + s 1 + 2s 2 + 3s 3 .
(iii) This readily follows from the part (ii).

Twisted relative de Rham complex
The weight λ := (1 − 2n)ω m is g-integral and r-dominant. Hence, there is an irreducible Rmodule W λ with lowest weight −λ. Since r is associated to {α m }, it follows that dim W λ = 1 and so W λ is also an irreducible Q-module. We will denote by E q (λ) and O q (λ) the sheaves of smooth and holomorphic sections of W CS λ := G × Q W λ , respectively. If W is a vector bundle over CS, then we denote W (λ) := W ⊗ W CS λ , i.e., we twist W by tensoring with the line bundle W CS λ . It is not hard to see that η * E r (λ) ∼ = E q (λ) and η * O r (λ) = O q (λ) where we denote by the subscript r the corresponding sheaves over T S.
A section of E p,q η (λ) is by definition a finite sum of decomposable elements ω ⊗ v where ω and v are sections of E p,q η and η −1 E r (λ), respectively. As any section of η −1 E r as well as transition functions between sections of η −1 E r (λ) belong to ker(∂ η ), it follows that there is a unique linear differential operator We denote the operator also by ∂ η as there is no risk of confusion. It is clear that ∂ η is a linear G-invariant differential operator of weighted order one.   (iii) There is a double complex (E p,q η (λ), ∂ η , (−1) p∂ ) of fine sheaves with exact rows and columns.
Proof . (i) By construction, the sequence is a Dolbeault complex and the claim follows.
(iii) We need to verify that [∂, ∂ η ] = 0. To see this, notice that a section of E p,q η (λ) can be locally written as a finite sum of elements as above with v holomorphic. The claim then easily follows from Proposition 5.1(iii).

Relative BGG sequence
We know that Ω p η (λ) is isomorphic to the sheaf of holomorphic sections of Λ p E CS * (λ) = G × Q (Λ p E * ⊗ W λ ). The Q-module Λ p E * is not irreducible. Decomposing this module into irreducible Q-modules, we obtain from the relative twisted de Rham complex a relative BGG sequence and this will be crucial in the construction of the k-Dirac complexes. We will use notation from Section 3.2.
Proposition 5.5. Let a ∈ N k,n ++ and w a ∈ W q r be as in Section 3.2. Then where W λa is an irreducible Q-module with lowest weight −λ a := −w a .λ.
There is a linear G-invariant differential operator where the first map is the canonical inclusion and the last map is the canonical projection. If a ≮ a , then ∂ a a = 0. Proof . Recall from Section 3.1 that the semi-simple part r ss 0 of r 0 is isomorphic to sl(m, C) and that r ss 0 ∩q is a parabolic subalgebra of r ss 0 . The direct sum decomposition from (5.8) then follows at once from the Kostant's version of the Bott-Borel-Weyl theorem (see [8,Theorem 3.3.5]) applied to W λ and (r ss 0 , r ss 0 ∩ q) and the identity (w a ) = |a| from Lemma 3.6. Recall from [2, Section 8.7] that the graph of the relative BGG sequence coincides with the relative Hasse graph W q r . The last claim then follows from Lemma 3.4.
Proof . By definition, each w ∈ W q p fixes the first k coefficients of λ a and so it is enough to look at the last n coefficients. By Remark 5.6, it follows (3.2) and Table 1, if c i = c j for some i = j, then there cannot be a p-dominant weight in the W q p -orbit of λ a . If a ∈ S k , then by Lemma 3.5 there is s ∈ {0, . . . , k − 1} such that 8 w a (k − s) = k + i ≥ k and w a (k + s + 1) = k + j ≥ k for some distinct positive integers i and j. By (5.10), it follows that We may now suppose that a ∈ S k . By the definition of d such that c i > c j . Equivalently, there are (a) pairs i < j such that c i < c j . It follows that the length of the permutation that maps (c 1 , . . . , c n ) to 2n−1 2 , . . . , 3 2 , 1 2 is precisely (a). Now it is easy to see (recall (3.2)) that there is w ∈ W q p such that w.λ a is p-dominant and (w) = (a). As there are n − d(a) negative numbers in the sequence b 1 − 1 2 , . . . , b j − j + 1 2 , . . . , the last claim about the sign of the last coefficient of w.λ a also follows. This completes the proof.
Remark 5.8. In Proposition 5.7 we recovered the W p -orbit of the singular weight µ + if n is even and of µ − if n is odd which was computed in [14]. There is an automorphism of g which swaps α m and α m−1 and hence, it swaps also the associated parabolic subalgebras. If we cross in (3.3) the simple root α m−1 instead of α m , take (1−2n)ω m−1 as λ and follow the computations given above, we will get the W p -orbit of µ + if n is odd and of µ − if n is even. As also all other arguments presented in this paper work for the other case, we will obtain the other "half" of the k-Dirac complex from [22] as mentioned in Introduction.

Double complex of relative forms II
The direct sum decomposition from Proposition 5.5 together with the isomorphism in (5.5) gives a direct sum decomposition E p,q η (λ) = a∈N k,n ++ :|a|=p E 0,q q (λ a ). Let a, a ∈ N k,n ++ be such that p = |a| = |a | − 1. Then there is a linear differential operator where the first map is the canonical inclusion and the last map is the canonical projection as in (5.9). We denote the differential operator by ∂ a a as in (5.9) as there is no risk of confusion. Recall from Proposition 5.5 that ∂ a a = 0 if a ≮ a . Suppose that U is an open, contractible and Stein subset of M . We put E p,q η (τ −1 (U), λ) := Γ(τ −1 (U), E p,q η (λ)), i.e., this is the space of sections of the sheaf E p,q η (λ) over τ −1 (U). Then there is a double complex  Proof . The first claim follows from Proposition 5.7 and application of the Leray spectral sequence as explained in [2]. For the second claim, recall from [27,Theorem 3.20] that the sheaf cohomology is equal to the Dolbeault cohomology, i.e., there is an isomorphism 14) The cohomology group appears on the (|a| + (a)) = (d(a) + 2q(a) + n 2 − q(a)) = ( n 2 + r(a))-th diagonal of the double complex. Here, see Proposition 5.7, we use that (a) = n 2 − q(a), the notation from (3.6) and S k j = {a ∈ S k : r(a) = j}.

k-Dirac complexes
In Section 6 we will give the definition of differential operators in the k-Dirac complexes. It will be clear from the construction that the operators are linear, local and G-invariant. Later in Lemma 7.12 we will show that each operator is indeed a differential operator and we give an upper bound on its weighted order. The operators naturally form a sequence and we will prove in Theorem 6.2 that they form a complex which we call the k-Dirac complex.

T. Salač
Lemma 6.1. Let j ≥ 0, a ∈ S k j , a ∈ S k j+1 be such that a < a and U be the Stein set as above. Then there is a linear, local and G-invariant operator D a a : Γ(U, O p (µ a )) → Γ(U, O p (µ a )).
Proof . Let us for a moment put V := τ −1 (U). Using the isomorphisms from (5.13), it is enough to define a map H (a) (V, O q (λ a )) → H (a ) (V, O q (λ a )) which has the right properties. By assumption, we have |a | − |a| ∈ {1, 2}. Let us first consider |a | − |a| = 1. Then q := (a ) = (a) and by (5.11), we have the map ∂ a a : E 0,q q (V, λ a ) → E 0,q q (V, λ a ) in the double complex (5.12). The induced map on cohomology is D a a . If |a | − |a| = 2, then q := (a) = (a ) + 1 and we find that there are precisely two nonsymmetric partitions b, c ∈ N k,n ++ such that a < b < a and a < c < a . Then there is a diagram which lives in the double complex (5.12). Let α ∈ E 0,q q (V, λ a ) be∂-closed. Then ∂ a b α and ∂ a c α are also∂-closed and thus by Lemma 5.9 and the isomorphism (5.14), we can find β and γ such that ∂ a b α = (−1) p∂ β and ∂ a c α = (−1) p∂ γ where p = |a| + 1. Since the relative BGG sequence is a complex, we havē . We call the following complex (6.2) the k-Dirac complex. Theorem 6.2. With the notation set above, there is a complex Proof . Let a, a ∈ S k be such that a < a , r(a) = r(a ) − 2. We need to verify that a ∈S k : a<a <a D a a D a a = 0. Observe that |a |−|a| ∈ {3, 4}. Let us first assume that |a |−3 = |a|.
Then there are at most two symmetric partitions a such that a < a < a . If there is only one such symmetric partition a , then, since the relative BGG sequence is a complex, it follows easily that D a a D a a = 0. So we can assume that there are two symmetric partitions, say a 1 , a 2 . Consider for example Then we can find β and β so that ∂ a b α = (−1) p∂ β and ∂ a b α = (−1) p∂ β where p = |b| = |b |.
This completes the proof when |a | = |a| + 3 and now we may assume |a | = |a| + 4. We put A := {a ∈ S k | a < a < a }, B := {b ∈ N k,n ++ | ∃ a ∈ A : a < b < a }, B := {b ∈ N k,n ++ | ∃ a ∈ A : a < b < a } and finally C := {c ∈ N k,n : : where A = {a }, B = {b 1 , b 2 }, B = {b 1 , b 2 } and C = {c 1 , c 2 }. As above, the set A contains at most two elements but we will not need that. Now we can proceed as above. There are β i such that (−1) a j β i for every a j ∈ A . As the relative BGG sequence is a complex, we have for each c ∈ C:

T. Salač
As above, there is γ such that (−1) This implies that In the first equality we use the fact that given b s ∈ B , there is only one a j ∈ A such that a j < b s and in the third equality we use that the relative BGG sequence is a complex once more.

Formal exactness of k-Dirac complexes
We will proceed in Section 7 as follows. In Section 7.1 we will recall the definition of the normal bundle of the analytic subvariety X 0 := τ −1 (x 0 ) and give the definition of the weighted formal neighborhood of X 0 . In Section 7.2 we will consider the double complex of twisted relative forms from Section 5 and restrict it to the weighted formal neighborhood of X 0 . In Section 7.3 we will prove that the operators defined in Section 6 are differential operators and finally, in Theorem 7.14 we will prove that the k-Dirac complexes are formally exact.
7.1 Formal neighborhood of τ −1 (x 0 ) Let us first recall notation from Section 4.2. There is the 2-step filtration decomposes as E CS ⊕ F CS where E CS = ker(T η) and F CS = ker(T τ ). From this it follows that E CS and F CS are integrable distributions. Dually, there are filtrations Let us now briefly recall Section 2.3. If X is an analytic subvariety of a complex manifold Y , then the normal bundle N X of X in Y is the quotient (T Y | X )/T X and the co-normal bundle N * X is the annihilator of T X inside T * X. In particular, the origin x 0 can be viewed as an analytic subvariety of M with local defining equation X 1 = 0, X 2 = 0 and Y = 0 where the matrices are those as in (4.4). For each i ≥ 1 there is the associated (i-th power of the) ideal sheaf I i x 0 . This is a sheaf of O M -modules such that j i x f = 0} and the subscript x stands for the stalk at x ∈ M of the corresponding sheaf.
Also recall from Section 2.2 the definition of weighted jets. For each i ≥ 0, there is a short exact sequence of vector spaces We will view (7.1) also as a short exact sequence of sheaves over {x 0 }.
Put X 0 := τ −1 (x 0 ). Recall from Lemma 4.1 that X 0 is complex manifold which is biholomorphic to the connected component Gr + h (n, n) of C n in the Grassmannian of maximal totally isotropic subspaces in C 2n .
Remark 7.1. If V CS is a holomorphic vector bundle over CS, we will for brevity put V := V CS | X 0 . We also put τ 0 := τ | X 0 .

Lemma 7.2.
(i) X 0 is a closed analytic subvariety of CS and there is an isomorphism of sheaves (ii) There is an isomorphism of vector bundles 9 T X 0 ∼ = F .
(iii) The normal bundle N of X 0 in CS is isomorphic to τ * 0 T x 0 M . In particular, N is a trivial holomorphic vector bundle.
T. Salač over X 0 . Moreover, for each ≥ 0 there are isomorphisms of vector bundles Proof . There is a canonical injective vector bundle map τ * 0 T * x 0 M → T * CS and a moment of thought shows that its image is contained in N * . By comparing dimensions of both vector bundles, we have τ * 0 T * x 0 M ∼ = N * and thus the first claim. It is clear that N * 2 = τ * 0 g 2 is the annihilator of F −2 = (T τ ) −1 (g −1 ) and since G 3 = F ⊥ −2 , the second claim follows. The first sequence in (7.2) is the pullback of the short exact sequence 0 → g 2 → T * x 0 M → g 1 → 0 and thus, it is short exact. The exactness of the latter sequence follows from the The isomorphisms in (7.3) follow immediately from definitions and the isomorphism (4.2).
We know that S N * is a trivial holomorphic vector bundle over the compact base X 0 . It follows that any global holomorphic section of S N * is a constant gr -valued function on X 0 and that S N * is trivialized by such sections. The same is obviously true also for S N * 1 . Let us formulate this as lemma. Let us finish this section by recalling the concept of formal neighborhoods (see [1,26]). Let ι 0 : X 0 → CS be the inclusion. Then F X 0 := ι −1 0 I X 0 is a sheaf of O X 0 -modules whose stalk at x ∈ X 0 is the space of germs of holomorphic functions which are defined on some open neighborhood V of x in CS and which vanish on V ∩ X 0 . Let us now view the vector space The infinite-dimensional vector spaces from (7.1) form a decreasing filtration · · · ⊂ F i+1 Arguing as in Section 2.3, one can show that for each i ≥ 0 there is a short exact sequence of sheaves and thus, the graded sheaf associated to the filtration F X 0 is isomorphic to i≥1 O(S i N * ). Using the analogy with the classical formal neighborhood, we will call the pair (X, O

7.2
The double complex on the formal neighborhood of τ −1 (x 0 ) Recall from Section 5.4 that for each a ∈ N k,n ++ there is a Q-dominant and integral weight λ a , an irreducible Q-module W λa with lowest weight −λ a and an associated vector bundle W CS λa = G × Q W λa . We will denote by W λa the restriction of W CS λa to X 0 , by O(λ a ) the sheaf of holomorphic sections of W λa , by E p,q the sheaf of smooth (p, q)-forms over X 0 and by E p,q (λ a ) the sheaf of (p, q)-forms with values in W λa . If V is another vector bundle over X 0 , then we denote by V (λ a ) the tensor product of V with W λa . We will use the notation set in (2.2) and (2.3).
Lemma 7.6. There is for each r := + j ≥ 0 a long exact sequence a vector bundles over X 0 : This sequence contains a long exact subsequence Proof . In order to obtain the sequence (7.4), take the direct sum of all long exact sequences from (5.7) indexed by s 0 , s 1 , s 2 and s 3 where s 0 + s 2 + 2s 3 = + j, s 1 = 0 and restrict it to X 0 . The subsequence (7.5) is obtained similarly, we only add one more condition s 3 = 0.
Recall that each long exact sequence from (5.7) is induced by the relative twisted de Rham complex by restricting to weighted jets. Hence, also (7.4) and (7.5) are naturally induced by this complex.
Remark 7.7. Let E 0,q (Λ j E * ⊗ S N * (λ)) be the sheaf of smooth (0, q)-forms with values in the corresponding vector bundle over X 0 . The vector bundle map d j induces a map of sheaves which we also denote by d j as there is no risk of confusion. Recall from (5.8) that Λ j E * ⊗W λ = a∈N k,n ++ : |a|=j W λa which gives direct sum decomposition E 0,q (Λ j E * ⊗ S N * (λ)) = a∈N k,n ++ : |a|=j E 0,q (S N * (λ a )). We see that if a, a ∈ N k,n ++ are such that |a| = |a | − 1 = j, then d j induces d a a : E 0,q S N * (λ a ) → E 0,q S −1 N * (λ a ) (7.7) in the same way ∂ η induces in (5.9) the operator ∂ a a in the relative BGG sequence. By Proposition 5.5, d a a = 0 if a ≮ a . Remark 7.8. Replacing (7.4) by (7.5) in Remark 7.7, we get a map of sheaves If a, a are as above, then there is a map , which is induced in the same way d j induces d a a . Even though the proof of Lemma 7.9 is trivial, it will be crucial later on. Lemma 7.9. Let a ∈ N k,n ++ . Then and where 10 in (7.9) and (7.10) the first possibility holds if and only if a ∈ S k and q = (a). 10 As above, we identify a sheaf over {x0} with its stalk.

T. Salač
Proof . The first equality in (7.9) is just the definition of (τ 0 ) q * . The sheaf cohomology group in the middle is equal to the cohomology of the Dolbeault complex. In view of Lemma 7.5, Γ(E 0,q (S N * (λ a ))) ∼ = gr x 0 ⊗ Γ(E 0,q (λ a )) and thus, the sheaf cohomology group is isomorphic to gr x 0 ⊗ H q (X 0 , O(λ a )). By the Bott-Borel-Weil theorem, H q (X 0 , O(λ a )) ∼ = V µa if a ∈ S k , q = (a) and vanishes otherwise. The second equality in (7.9) then follows from the isomorphism The isomorphism in (7.10) is proved similarly. We only use the other isomorphism → gr x 0 from Lemma 7.5 and the isomorphism gr There is for each non-negative integer a certain double complex whose horizontal differential is (7.6) and the vertical differential is (up to sign) the Dolbeault differential. This is the double complex from Proposition 5.3 restricted to the weighted formal neighborhood of X 0 . Proposition 7.10. Let r ≥ 0 be an integer. Then there is a double complex (E p,q (r), d , d ) where: • E p,q (r) = Γ(E 0,q (Λ p E * ⊗ S r−p N * (λ))), • the vertical differential d is (−1) p∂ where∂ is the standard Dolbeault differential and • the horizontal differential d is d p from (7.6).
Moreover, we claim that: where T i (r) := p+q=i E p,q (r); (ii) the first page of the spectral sequence associated to the filtration by columns is (iii) the spectral sequence degenerates on the second page.
Proof . Recall from the proof of Proposition 5.2 that gr∂ η is induced from ∂ η by passing to weighted jets (as explained at the end of Section 2.2) and, see Lemma 7.6, that d = d p is the restriction of the map gr∂ η to the sub-complex (7.4). Since [∂ η ,∂] = 0, we have that [d,∂] = 0 and thus also d d = −d d . This shows the first claim.
(i) The rows of the double complex are exact as the sequence (7.4) is exact. Since dim X 0 = n 2 , it follows that E p,q 1 (r) = 0 whenever q > n 2 . This proves the claim. (ii) By definition, E p,q 1 (r) is the d -cohomology group in the p-th row and q-th column. The claim then follows from the direct sum decomposition from Remark 7.7 and Lemma 7.9.
(iii) The space gr r−p V µa lives on the |a|-th vertical line and (a) = ( n 2 − q(a))-th horizontal line of the first page of the spectral sequence and thus, on the (|a| + n 2 − q(a)) = (2q(a) + d(a) + n 2 − q(a)) = (r(a) + n 2 )-th diagonal. Choose a ∈ S k such that gr r−|a | V µ a lives on the next diagonal and a < a . This means that r(a ) = r(a) + 1 and so q(a) = q(a ) or q(a ) = q(a) + 1. In the first case, gr r−|a | V µ a lives on the (a)-th row. In the second case, it lives on the ( (a)−1)-th row. As d a a = 0 if a ≮ a , it follows from definition that the differential on the i-th page is zero if i > 2.
If we use the exactness of (7.5) instead of (7.4) and use the isomorphism (7.10) instead of (7.9), the proof of Proposition 7.10 gives the following.
Proposition 7.11. The double complex from Proposition 7.10 contains a double complex (F p,q (r), d , d ) where F p,q (r) := Γ(E 0,q (Λ p E * ⊗ S r−p N * 1 (λ))). Moreover we claim that: where T i (r) := p+q=i F p,q (r); (ii) the first page of the spectral sequence associated to the filtration by columns is (iii) the spectral sequence degenerates on the second page.

Long exact sequence of weighted jets
Let a ∈ S k and V µa be an irreducible P-module with lowest weight −µ a , see Proposition 5.7. Now we are ready to show that the linear operators defined in Lemma 6.1 are differential operators and we give an upper bound on their weighted order.
Lemma 7.12. Let a, a ∈ S k be such that a < a and r(a ) = r(a) + 1. Then the operator D a a from Lemma 6.1 is a differential operator of weighted order at most s := |a | − |a|.
Hence, D a a induces for each i ≥ 0 a linear map grD a a : gr i V µa → gr i−s V µ a , (7.11) which restricts to a linear map grD a a : gr i V µa → gr i−s V µ a . (7.12) Proof . Let us make a few preliminary observations. Let v ∈ O p (µ a ) x 0 . By the G-invariance of D a a , it is obviously enough to show that (D a a v)(x 0 ) depends only on j s x 0 v. We may assume that v is defined on the Stein set U from Section 6 and so we can view v as a cohomology class [α] = H (a) (τ −1 (U), O q (λ a )). A choice of Weyl structure (see [8]) and the isomorphisms (7.9) give for each integer i ≥ 0 isomorphisms O O (7.13) 11 We will at this point avoid discussion about the convergence of the sum as we will not need it.
where the lower vertical arrows are the isomorphisms from Lemma 7.9, the upper vertical arrows are the canonical projections and the map d a a is the one from (7.7). Let us now assume s = 2. In view of the diagram (6.1), we have to replace in (7.13) the map d a a by the diagram Γ(E 0,q (S i N * (λ a ))) ∩ Ker(∂) Γ(E 0,q−1 (S i−1 N * (λ b ⊕ λ c )))∂ where we for brevity put S • N * (λ b ⊕ λ c ) := S • N * (λ b ) ⊕ S • N * (λ c ). Following the same line of arguments as in the case s = 1, we easily find that D a a (v)(x 0 ) = 0 whenever j 2 x 0 v = 0. In order to prove the claim about grD a a , we need to replace everywhere d a a by its restriction δ a a and use (7.10) instead of (7.9).
In order to get rid of the factor s in (7.11) and (7.12), we shift the gradings by introducing gr i V µa [↑] := gr i−q(a) V µa and gr i V µa [↑] := gr i−q(a) V µa . We can now rewrite the maps from (7.11) and (7.12)  and where the first map is the canonical inclusion and the last map is the canonical projection. Recall from Section 3.2 that if a < a , a ∈ S k j , a ∈ S k j+1 , then q(a ) ≤ q(a) + 1. This implies that gr(D j ) i i = 0 if i = i or i = i + 1. Then grD j is  where the horizontal arrows and the diagonal arrows are gr(D j ) i i and gr(D j ) i i+1 , respectively. We similarly define linear maps grD j : gr V j [↑] → gr −1 V j+1 [↑] and gr(D j ) i i : Put p := i + j, q := n 2 − i and r := + j. Then we can view gr(D j ) i i and gr(D j ) i i+1 as maps E p,q 1 (r) → E p+1,q 1 (r) and E p,q 1 (r) → E p+2,q−1 1 (r), respectively. By the definition of gr(D j ) i i from Lemma 7.12, it follows that we can view it as the differential d 1 on the first page of the spectral sequence from Proposition 7.10.
Suppose that v ∈ E p,q 1 (s) satisfies d 1 (v) = 0. Then we can apply the differential d 2 living on the second page to v + im(d 1 ) and, comparing this with the definition of gr(D j ) i i+1 from Lemma 7.12, we find that d 2 (v + im(d 1 )) = gr(D j ) i i+1 (v) + im(d 1 ). (7.14) Similarly we find that gr V j,i = F p,q 1 (s) where p, q and s are as above. Moreover we can view gr(D j ) i i and gr(D j ) i i+1 as maps F p,q 1 (s) → F p+1,q 1 (s) and F p,q 1 (s) → F p+2,q−1 1 (s), respectively. As the double complex from Proposition 7.11 is a sub-complex of the double complex from Proposition 7.10 and gr(D j ) i i is the restriction of gr(D j ) i i to the corresponding subspace, we see that gr(D j ) i i coincides with the differential on the first page of the spectral sequence from Proposition 7.11 and that gr(D j ) i i+1 is related to the differential on the second page just as gr(D j ) i i+1 is related to d 2 .
The exactness of the complex (7.15) for each + j ≥ 0 implies (see [24]) the exactness of the k-Dirac complex at the level of infinite weighted jets at any fixed point. Following [25], we say that the k-Dirac complex is formally exact. Notice that for application in [24], the exactness of the sub-complex (7.16) for each + j ≥ 0 is a crucial point in the proof of the local exactness of the descended complex and thus, in constructing the resolution of the k-Dirac operator.
Theorem 7.14. The k-Dirac complex induces for each + j ≥ 0 a long exact sequence of finite-dimensional vector spaces. The complex contains a sub-complex which is also exact.
Proof . Let v ∈ gr V j [↑], j ≥ 1 be such that grD j (v) = 0. Write v = (v 0 , . . . , v j ) with respect to the decomposition given above, i.e., v i ∈ gr V j,i [↑]. Assume that v 0 = v 1 = · · · = v i−1 = 0 and that v i = 0. We have that gr(D i i )(v i ) = 0 and gr(D i i+1 )(v i ) + gr(D i+1 i+1 )(v i+1 ) = 0. If we view v i as an element of E p,q 1 (s) as in Remark 7.13, we see that d 1 (v i ) = 0 and by (7.14), we find that d 2 (v i ) = 0. By Proposition 7.10, the spectral sequence E p,q (r) collapses on the second page and by part (i), we have that ker(d 2 ) = im(d 2 ) beyond the n 2 -th diagonal. By Remark 7.13 again, T. Salač gr V j,i [↑] lives on the ( n 2 + j)-th diagonal. We see that there are t i−1 ∈ gr +1 V j−1,i−1 [↑] and t i ∈ gr +1 V j−1,i [↑] such that gr(D i−1 i−1 )(t i−1 ) = 0 and gr(D i−1 i )(t i−1 ) + gr(D i i )(t i ) = v i . Hence, we can kill the lowest non-zero component of v and repeating this argument finitely many times, we see that there is t ∈ gr +1 V j−1 [↑] such that v = grD j−1 (t).
The proof of the exactness of the second sequence (7.16) proceeds similarly. We only replace gr V j [↑] by gr V j [↑], gr V j,i [↑] by gr V j,i [↑], use that the second spectral sequence from Proposition 7.11 has the same key properties as the spectral sequence from Proposition 7.10 and the end of Remark 7.13.