k-Dirac Complexes
This is the first paper in a series of two papers. In this paper, we construct complexes of invariant differential operators that 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 com...
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| description | This is the first paper in a series of two papers. In this paper, we construct complexes of invariant differential operators that 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.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 14 (2018), 012, 33 pages
k-Dirac Complexes
Tomáš SALAČ
Mathematical Institute, Charles University, Sokolovská 49/83, Prague, Czech Republic
E-mail: salac@karlin.mff.cuni.cz
URL: http://www.karlin.mff.cuni.cz/~salac/
Received June 01, 2017, in final form February 06, 2018; Published online February 16, 2018
https://doi.org/10.3842/SIGMA.2018.012
Abstract. 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 ex-
plicitly, 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.
Key words: Penrose transform; complexes of invariant differential operators; relative BGG
complexes; formal exactness; weighted jets
2010 Mathematics Subject Classification: 58J10; 32N05; 32L25; 35A22; 53C28; 58A20
1 Introduction
Let h be a non-degenerate, symmetric and C-bilinear form on C2m. The Grassmannian varietyM
of totally isotropic k-dimensional subspaces in C2m 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 function1 in U extends to a monogenic function in U ′.
Recall from [16, Section 4] that the Dolbeault resolution together with some L2 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 ma-
chinery 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
1A monogenic function is a null solution of the k-Dirac operator.
mailto:salac@karlin.mff.cuni.cz
http://www.karlin.mff.cuni.cz/~salac/
https://doi.org/10.3842/SIGMA.2018.012
2 T. Salač
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 C2m as, due to the representation theory, the Penrose
transform does not work in odd dimension C2m+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 com-
plexes 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].
Notation
• M(n, k,C) matrices of size n× k with complex coefficients;
• M(n,C) := M(n, n,C);
• A(n,C) := {A ∈M(n,C) |AT = −A};
• 1n identity n× n-matrix;
• [v1, . . . , v`] linear span of vectors v1, . . . , v`.
2 Preliminaries
In Section 2 we will review some well known material. Namely, in Section 2.1 we will summa-
rize 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 asso-
ciated to analytic subvariety and the formal neighborhood. In Section 2.4 we will give a short
summary of the Penrose transform.
k-Dirac Complexes 3
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].
2.1 Review of parabolic geometries
Let g be a complex semi-simple Lie algebra, h be a Cartan subalgebra, 4 be the associated set
of roots, 4+ be a set of positive roots and 40 = {α1, . . . , αm} be the associated set of (pairwise
distinct) simple roots. We will denote by gα the root space associated to α ∈ 4 and we will write
α > 0 if α ∈ 4+ and α < 0 if −α ∈ 4+. Given α ∈ 4, there are unique integers λ1, . . . , λm
such that α = λ1α1 + · · ·+ λmαm. If Σ ⊂ 40, then the integer htΣ(α) :=
∑
i : αi∈Σ
λi is called the
Σ-height of α. We put gi :=
⊕
α : htΣ(α)=i
gα. Then there is an integer k ≥ 0 such that gk 6= {0},
g` = {0} whenever |`| > k and
g = g−k ⊕ g−k+1 ⊕ · · · ⊕ gk−1 ⊕ gk. (2.1)
The direct sum decomposition (2.1) is the |k|-grading on g associated to Σ.
Since [gα, gβ] ⊂ gα+β for every α, β ∈ 4, it follows that [gi, gj ] ⊂ gi+j for each i, j ∈ Z. In
particular, it follows that g0 is a subalgebra and it can be shown that it is always reductive,
i.e., g0 = gss0 ⊕ z(g0) where gss0 := [g0, g0] is semi-simple and z(g0) is the center (see [8, Corol-
lary 2.1.6]). Moreover, each subspace gi is g0-invariant and the Killing form of g induces an
isomorphism g−i ∼= g∗i of g0-modules where ∗ denotes the dual module. We put
gi :=
⊕
j≥i
gj , g− := g−k ⊕ · · · ⊕ g−1, pΣ := g0 and p+ := g1.
Then pΣ is the parabolic subalgebra associated to the |k|-grading and pΣ = g0 ⊕ p+ is known as
the Levi decomposition (see [2, Section 2.2]). This means that p+ is the nilradical2 and that g0
is a maximal reductive subalgebra called the Levi factor. It is clear that each subspace gi is
pΣ-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 := p40 is called the standard Borel subalgebra. A subalgebra of g is called
standard parabolic if it contains b and in particular, pΣ 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 pΣ.
Let si be the simple reflection associated to αi, i = 1, 2, . . . ,m, Wg be the Weyl group of g
and Wp be the subgroup of Wg generated by {si : αi 6∈ Σ}. Then Wp is isomorphic to the Weyl
group of gss0 and the directed graph that encodes the Bruhat order on Wg 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 ∈ Wg 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 Wp in Wg 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 Wp\Wg.
We will need also the relative case. Assume that Σ′ ⊂ 40 and put r := pΣ′ . Then q := r∩p =
pΣ∪Σ′ is a standard parabolic subalgebra and l := gss0 ∩ q is a parabolic subalgebra of gss0 (see [2,
Section 2.4]). The definition of the Hasse diagram attached to p applies also to the pair (gss0 , l),
namely an element w ∈ Wp (as [2, Section 4.4]) belongs to the relative Hasse diagram W q
p if it
2Recall that the nilradical is a maximal nilpotent ideal and that it is unique.
4 T. Salač
is the element of minimal length in its right coset of Wq in Wp. Hence, W q
p is a subset of Wg
which can be naturally identified with Wq\Wp.
There is (up to isomorphism) a unique connected and simply connected complex Lie group G
with Lie algebra g. Assume that Σ = {αi1 , . . . , αij}. Let ω1, . . . , ωm be the fundamental weights
associated to the simple roots and V be an irreducible g-module with highest weight λ :=
ωi1 + · · · + ωij . 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: (rp)∗ω = Ad(p−1) ◦ ω where Ad is the
adjoint representation and rp 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 TgG 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 Tp = Tp ◦ Trp, it follows that Tp(ω−1(V))
is a well-defined distribution on M provided that V is P-invariant. In particular, this applies
to gi and we put Fi := Tp(ω−1(gi)). Since ker(Tp) = ω−1(p), it follows that the filtration
g−k/p ⊃ · · · ⊃ g−1/p ⊃ g0/p gives a filtration TM = F−k ⊃ F−k+1 ⊃ · · · ⊃ F−1 ⊃ F0 = {0}
of the tangent bundle TM where {0} is the zero section. The graded tangent bundle associated
to the filtration {F−i : i = k, . . . , 0} is gr(TM) :=
⊕−1
i=−k gri(TM) where gri(TM) := Fi/Fi+1.
Since M is the homogeneous model, we have the following:
• the filtration is compatible3 with the Lie bracket of vector fields in the sense that the
commutator of a section of Fi and a section of Fj is a section of Fi+j ,
• the Lie bracket descends to a vector bundle map L : Λ2gr(TM) → gr(TM), called the
Levi form, which is homogeneous of degree zero and
• (gr(TxM),Lx), x ∈M is a nilpotent Lie algebra isomorphic to g−.
Hence, (gr(TM),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 = Λ1,0T ∗M the vector bundle dual to TM , i.e., the fiber over x ∈M is the
space of C-linear maps TxM → C. The filtration of TM induces a filtration T ∗M = F1 ⊃ F2 ⊃
· · · ⊃ Fk+1 = {0} where Fi := F⊥−i+1 is the annihilator of F−i+1. We put gri(T
∗M) := Fi/Fi+1,
i = 1, 2, . . . , k so that gr(T ∗M) =
⊕k
i=1 gri(T
∗M) is the associated graded vector bundle and
gri(T
∗M) is isomorphic to the dual of gri(TM).
2.2 Weighted differential 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, TMC := TM⊗C = T 1,0M⊕T 0,1M where the first and the second
summand is the holomorphic and the anti-holomorphic part4, respectively. As each vector
bundle F−j is a holomorhic sub-bundle of TM , 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
3Filtrations which satisfy this property are called regular.
4The holomorphic and anti-holomorphic part is the −i and the +i-eigenspace, respectively, for the canonical
almost complex structure on TMC.
k-Dirac Complexes 5
order at most r if for each x ∈ U there is an open neighborhood Ux of x with a local framing5
{X1, . . . , Xp} by holomorphic vector fields such that
D|Ux =
∑
a∈Np0
faX
a1
1 · · ·X
ap
p ,
where Np0 := {a = (a1, . . . , ap) : ai ∈ Z, ai ≥ 0, i = 1, . . . , p}, fa ∈ O(Ux) and for all a in the
sum with fa non-zero:
p∑
i=1
ai.wo(Xi) ≤ r. We write wo(D) ≤ r.
Let Ox be the space of germs of holomorphic functions at x. We denote by Fix the space of
those germs f ∈ Ox 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 Jix := Ox/Fi+1
x , denote by jixf ∈ Jix the class
of f and call it the i-th weighted jet of f . Then the disjoint union Ji := ∪x∈MJix is naturally
a holomorphic vector bundle over M , the canonical vector bundle map Ji
πi−→ Ji−1 has constant
rank and thus, its kernel gri is again a holomorphic vector bundle with fiber grix over x. Notice
that for each integer i ≥ 0 there is a short exact sequence 0→ Fi+1
x → Fix → gri+1
x → 0 of vector
spaces.
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 O(V )x the space of germs
of holomorphic sections of V at x. We define FixV as the space of germs s ∈ O(V )x such
that 〈λ, s〉 ∈ Fix for each λ ∈ O(V ∗)x. We put Jix := O(V )x/F
i+1
x V , denote by jixs ∈ JixV the
equivalence class of s and call it the i-th weighted jet of s. Then the disjoint union JiV :=⋃
x∈M JixM is naturally a holomorphic vector bundle over M , the canonical bundle map JiV
πi−→
Ji−1V has constant rank and thus, its kernel griV is again a holomorphic vector bundle and we
denote by grixV its fiber over x. As above, there is for each integer i ≥ 0 a short exact sequence
0 → Fi+1
x V → FixV → gri+1
x V → 0 and just as in the smooth case, there is a canonical linear
isomorphism grix ⊗ Vx → grixV .
Remark 2.1. If the filtration is trivial, i.e., F−1 = TM , 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 gri and jixf instead of Fi and Ji and gri and jixf , respectively. The
vector bundle gri is canonically isomorphic to the i-th symmetric power SiT ∗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 x0 := eP the origin
of M and we put
JiV := Jix0
V and griV := grix0
V. (2.2)
There are linear isomorphisms
grrV ∼= grrx0
⊗ V ∼=
⊕
i1+2i2+···+kik=r
Si1g1 ⊗ Si2g2 ⊗ · · · ⊗ Sikgk ⊗ V. (2.3)
We will be interested in the sub-bundle Sigr1(T ∗M)⊗V of griV . Notice that the fiber of this
sub-bundle over x ∈ M is {jixf : f ∈ O(V )x, j
i−1
x f = 0}, i.e., 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 x0 is isomorphic to Sig1 ⊗ V and we denote it for brevity by griV.
Suppose thatW is another P-module and W := G×PW 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 : jrxs = 0⇒ Ds(x) = 0. It is well known (see [18]) that D
5This means that the holomorphic vector fields X1, . . . , Xp trivialize T 1,0M over Ux.
6 T. Salač
induces for each i ≥ 0 a vector bundle map griV → gri−rW 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 : griV→ gri−rW.
2.3 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 Fx 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−1G is a sheaf on X and
the direct image ι∗F is a sheaf on Y . The q-th direct image ιq∗F is a sheaf on Y , it is defined as
the sheafification of the pre-sheaf V 7→ Hq(ι−1(V),F) where V is open in Y .
Suppose now that X and Y are complex manifolds with structure sheaves of holomorphic
functions OX and OY , respectively, that ι is holomorphic and that G is a sheaf of OY -modules.
Then ι−1G is in general not a sheaf of OX -modules. To fix this problem, we use that ι−1OY is
naturally a sub-sheaf of OX and define a new sheaf ι∗G := OX ⊗ι−1OY ι
−1G. Then ι∗G is by
construction a sheaf of OX -modules.
Now we can continue with the definition of the ideal sheaf. Suppose that the holomorphic
map ι is an embedding. The restriction TY |X contains the tangent bundle TX of X. The
normal bundle NX of X in Y is simply the quotient bundle, i.e., it fits into the short exact
sequence 0 → TX → TY |X → NX → 0 of holomorhic vector bundles. Dually, the co-normal
bundle N∗ fits into the short exact sequence 0→ N∗ → T ∗Y |X → T ∗X → 0.
The structure sheaf OY contains a sub-sheaf called the ideal sheaf IX . If V is an open subset
of Y , then IX(V) = {f ∈ OY (V) : f = 0 on V ∩ X}. Notice that IX(V) is an ideal in the
ring OY (V) and hence, for each positive integer i there is the sheaf IiX whose space of sections
over V is (IX(V))i. Then there are short exact sequences of sheaves
0→ IX → OY → ι∗OX → 0
and
0→ Ii+1
X → IiX → ι∗O(SiN∗)→ 0,
where SiN∗ is the i-th symmetric power of N∗ and we agree that I0
X = OY . We put F iX :=
ι−1IiX . As ι−1 is an exact functor, we get short exact sequences
0→ FX → ι−1OY → OX → 0
and
0→ F i+1
X → F iX → O(SiN∗)→ 0
of sheaves on X. Here we use that the adjunction morphism ι−1ι∗F → F is an isomorphism
when F = OX or O(SiN∗).
Put O(i)
X := OY /Ii+1
X . The pair (X,O(i)
X ) is called the i-th formal neighborhood of X in Y .
Then ι−1O(i)
X
∼= ι−1OY /F (i+1) and since the support of O(i)
X is contained in X, the sheaf ι−1O(i)
X
contains basically the same information as the sheaf O(i)
X . These sheaves will be crucial in this
article.
Remark 2.2. The stalk of F iX at x ∈ X is equal to {f ∈ Fx : jixf = 0}. Hence, if X = x is
a point, the stalk of F iX at x is {f ∈ (OY )x : jixf = 0}. 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.
k-Dirac Complexes 7
2.4 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 Vλ with lowest weight −λ. We denote
by Vλ := G×P Vλ the induced vector bundle and by Op(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
G/Q
η
||
τ
##
G/R G/P,
where η and τ are the canonical projections. The space G/R is called the twistor space TS 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 N∗(λ) which is an exact sequence of holomorphic sections of associated vector bundles
over CS and linear G-invariant differential operators such that η−1Or(λ) is the kernel sheaf of
the first operator in the sequence. In other words, there is a long exact sequence of sheaves
0→ η−1Or(λ)→ N∗(λ).
The upshot of this is that although the pullback sheaf η−1Or(λ) is not a sheaf of holomorphic
sections of an associated vector bundle over CS, it is naturally a sub-sheaf of Oq(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.
3 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 .
3.1 Lie algebra g and parabolic subalgebras
Let {e1, . . . , em, e
∗
k+1, . . . , e
∗
m, e
∗
1, . . . , e
∗
k} be the standard basis of C2m, δ be the Kronecker delta
and h be the complex bilinear form that satisfies h(ei, e
∗
j ) = δij , h(ei, ej) = 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
8 T. Salač
only if it is of the form
A Z1 Z2 W
X1 B D −ZT2
X2 C −BT −ZT1
Y −XT
2 −XT
1 −AT
, (3.1)
where A ∈ M(k,C), B ∈ M(n,C), C,D ∈ A(n,C), X1, X2, Z
T
1 , Z
T
2 ∈ M(n, k,C), W,Y ∈
A(k,C).
The subspace of diagonal matrices h is a Cartan subalgebra of g. We denote by εi the
linear form on h defined by εi : H = (hkl) 7→ hii. Then {ε1, . . . , εm} is a basis of h∗ and
4 = {±εi ± εj : i, j = 1, . . . ,m}. If we choose 4+ = {εi ± εj : 1 ≤ i < j ≤ m} as positive roots,
then the simple roots are αi := εi− εi+1, i = 1, . . . ,m− 1 and αm := εm−1 + εm. The associated
fundamental weights are ωi = ε1 + · · ·+ εi, i = 1, . . . ,m− 2, ωm−1 = 1
2(ε1 + · · ·+ εm−1− εm) and
ωm = 1
2(ε1 +· · ·+εm). The lowest form ρ is equal to 1
2
∑
α∈4+
α = ω1 +· · ·+ωm = (m−1, . . . , 1, 0).
If λ =
m∑
i=1
λiεi where λi ∈ C, then we will also write λ = (λ1, . . . , λm). The simple reflection
si ∈Wg associated to αi acts on h∗ by
si(λ) = (λ1, . . . , λi−1, λi+1, λi, λi+2, . . . , λm), i = 1, . . . ,m− 1 (3.2)
and
sm(λ) = (λ1, . . . , λm−2,−λm,−λm−1).
We will be interested in the double fibration diagram
• · · · × · · · •�
×
�•
↙ ↘
• · · · •�
×
�•
• · · · × · · · •�
•
�•
(3.3)
where, going from left to right, the sets of simple roots are {αm}, {αk, αm} and {αk} and the
associated gradings are
r−1 ⊕ r0 ⊕ r1, q−3 ⊕ q−2 ⊕ · · · ⊕ q3 and g−2 ⊕ · · · ⊕ g2,
respectively. With respect to the block decomposition from (3.1), we have6
q0 q1 q2 q3
q−1 q0 q1 q2
q−2 q−1 q0 q1
q−3 q−2 q−1 q0
↙ ↘
r0 r0 r1 r1
r0 r0 r1 r1
r−1 r−1 r0 r0
r−1 r−1 r0 r0
g0 g1 g1 g2
g−1 g0 g0 g1
g−1 g0 g0 g1
g−2 g−1 g−1 g0
.
6Here we mean q0 is the subspace of block diagonal matrices, q1 is the subspace of those block matrices where
only the matrices Z1, D are non-zero, etc.
k-Dirac Complexes 9
The associated standard parabolic subalgebras are
r = r0 ⊕ r1, q = q0 ⊕ q1 ⊕ q2 ⊕ q3 and p = g0 ⊕ g1 ⊕ g2,
respectively, and we have the following isomorphisms
r0 ∼= M(m,C), q0
∼= M(k,C)⊕M(n,C) and g0
∼= M(k,C)⊕ so(2n,C).
We for brevity put
Ck := [e1, . . . , ek], Ck∗ := [e∗1, . . . , e
∗
k],
Cn := [ek+1, . . . , em], Cn∗ := [e∗k+1, . . . , e
∗
m],
C2n := Cn ⊕ Cn∗ and Cm := Ck ⊕ Cn. (3.4)
Notice that the bilinear form h induces dualities between Ck and Ck∗ and between Cn and Cn∗
which justifies the notation, that Cm is a maximal, totally isotropic and r0-invariant subspace,
that Ck, Ck∗, Cn and Cn∗ are q0-invariant, that Ck, C2n and Ck∗ are g0-invariant and finally,
that h|C2n is a non-degenerate, symmetric and g0-invariant bilinear form. We will for brevity
write only h instead of h|C2n as it will be always clear from the context what is meant.
Let us now consider the associated nilpotent subalgebras
r− = r−1, q− = q−3 ⊕ q−2 ⊕ q−1 and g− = g−2 ⊕ g−1.
By the Jacobi identity, the Lie bracket is equivariant with respect to the adjoint action of the cor-
responding 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 q0-modules, we have q−1
∼= E⊕F, q−2
∼= Ck∗⊗Cn∗,
q−3
∼= Λ2Ck∗ where we put E := Ck∗ ⊗ Cn and F := Λ2Cn∗. Using these isomorphisms, the Lie
brackets in homogeneity −2 and −3 are the compositions of the canonical projections
Λ2q−1 → E⊗ F =
(
Ck∗ ⊗ Cn
)
⊗ Λ2Cn∗ → Ck∗ ⊗ Cn∗ = q−2 (3.5)
and
q−1 ⊗ q−2 → E⊗ q−2 =
(
Ck∗ ⊗ Cn
)
⊗
(
Ck∗ ⊗ Cn∗
)
→ Λ2Ck∗ = q−3,
respectively. Here we use the canonical pairing Cn ⊗ Cn∗ → C. Notice that Λ2E ⊕ Λ2F 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
∼=
Ck∗ ⊗C2n and g−2
∼= Λ2Ck∗ ⊗C of irreducible g0-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
Λ2g−1 = Λ2
(
Ck∗ ⊗ C2n
)
→ Λ2Ck∗ ⊗ S2C2n → Λ2Ck∗ ⊗ C = g−2,
where in the last map we take the trace with respect to h.
In the table below we specify when λ = (λ1, . . . , λm) ∈ h∗ is dominant for each parabolic
subalgebra p, q and r. We put N0 := N ∪ {0}.
10 T. Salač
Table 1. Dominant weights.
algebra dominant and integral weights
p λi − λi+1 ∈ N0, i 6= k, 2λm ∈ Z, λm−1 ≥ |λm|
r λi − λi+1 ∈ N0
q λi − λi+1 ∈ N0, i 6= k
3.2 Relative Hasse diagram W q
r
Let us first set notation. By a partition we will mean an element of Nk,n++ := {(a1, . . . , ak) : ai ∈ Z,
n ≥ a1 ≥ a2 ≥ · · · ≥ ak ≥ 0}. For two partitions a = (a1, . . . , ak) and a′ = (a′1, . . . , a
′
k) we write
a ≤ a′ if ai ≤ a′i for all i = 1, . . . , k and a < a′ if a ≤ a′ and a 6= a′. If a < a′ does not hold, then
we write a ≮ a′. We put
|a| = a1 + · · ·+ ak, d(a) := max{i : ai ≥ i}, (3.6)
q(a) =
k∑
i=1
max{ai − i, 0} and r(a) := d(a) + q(a).
To the partition a we associate the Young diagram (or the Ferrers diagram) Y consisting
of k left-justified rows with ai-boxes in the i-th row. Let bi be the number of boxes in the i-th
column of Y. Then we call b = (b1, . . . , bn) ∈ Nn,k++ the partition conjugated to a and we say that
a is symmetric if ai = bi, i = 1, . . . , k and bk+1 = · · · = bn = 0. As we assume n ≥ k, the set
of symmetric partitions in Nk,n++ depends only on k, and thus, we denote it for simplicity by Sk
and put Skj := {a ∈ Sk : r(a) = j}.
Example 3.1.
(1) The empty partition is by definition always symmetric.
(2) The Young diagram of a = (4, 3, 1, 0, 0) ∈ N5,6
++ is
(3.7)
and we find that d(a) = 2, q(a) = 4 and r(a) = 6. The conjugated partition is b =
(3, 2, 2, 1, 0, 0) with d(b) = 2, q(b) = 2 and r(b) = 4. The Young diagram of b is
.
We see that the partition is not 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 graphW q
r . The groupWr is generated
by s1, . . . , sm−1 while Wq is generated by elements s1, . . . , sk−1, sk+1, . . . , sm−1. By (3.2), it
follows that Wr is the permutation group Sm on {1, . . . ,m} and that Wq
∼= Sk × Sn is the
stabilizer of {1, . . . , k}. Recall from Section 2.1 that in each left coset of Wq in Wr 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 Wg descends to a partial order on Wr
and on W r
q . We will now show that there is an isomorphism Nk,n++ → W q
r of partially ordered
sets.
Let a = (a1, . . . , ak) ∈ Nk,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
k-Dirac Complexes 11
number ](i, j) := k − i + j. Notice that 1 ≤ ](i, j) ≤ m. Then the set of boxes in Y is
indexed by Ξa := {(i, j) : i = 1, . . . , k, j = 1, . . . , ai} and we order this set lexicographically, i.e.,
(i, j) < (i′, j′) if i < i′ or i = i′ and j < j′. Then
wa := s](Ψ(1))s](Ψ(2)) . . . s](Ψ(|a|)) ∈ Sm, (3.8)
where Ψ: {1, 2, . . . , |a|} → Ξa 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
k k + 1 k + 2 k + 3
k − 1 k k + 1
k − 2
and so wa := sksk+1sk+2sk+3sk−1sksk+1sk−2.
We have the following preliminary observation.
Lemma 3.3. Let a = (a1, . . . , ak) be as above and b = (b1, . . . , bn) be the conjugated partition.
Then the permutation wa ∈ Sm from (3.8) satisfies
wa(k − i+ 1 + ai) = k − i+ 1 and wa(k + j − bj) = k + j (3.9)
for each i = 1, . . . , k and j = 1, . . . , n.
Proof. Fix i = 1, . . . , k. If ai > 0, there is ri := k − i + 1 written in the (i, 1)-box and
ri := k − i + ai = ri + ai − 1 in the (i, ai)-box. We put ri − 1 = ri := k − i + 1 if ai = 0.
Similarly, if j = 1, . . . , n and bj > 0, then there is cj := k + j − 1 in the (1, j)-box and
cj := k + j − bj = cj − bj + 1 in the (bj , j)-box. We put cj − 1 = cj := k + j − 1 if bj = 0. Then
it is easy to check that wa(r
i + 1) = ri and wa(c
j) = cj + 1 which completes the proof. �
Notice that the sets {k − i+ 1 + ai : i = 1, . . . , k} and {k + j − bj : j = 1, . . . , n} are disjoint
and that their union is {1, 2, . . . ,m}. By (3.9), it follows that
waρ = ρ+ (−ak, . . . ,−a1 | b1, . . . , bn), (3.10)
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 waρ is q-dominant. As the
same holds for any r-dominant weight, it follows that wa ∈W q
r .
Lemma 3.4. The map Nk,n++ →W q
r , a 7→ wa is an isomorphism of partially ordered sets.
Proof. The map a 7→ wa is by (3.10) clearly injective. To show surjectivity, fix w ∈W q
r . Then
the sequence c1, . . . , ck where ci = w−1(i), i = 1, . . . , k is increasing. By [8, Proposition 3.2.16],
the map w ∈ W q
r 7→ w−1ωk is injective. It follows that w−1ωk is uniquely determined by the
sequence c1, . . . , ck. Then a := (a1, . . . , ak) ∈ Nk,n++ where ai := ck−i+1 + i − k − 1, i = 1, . . . , k
and from (3.9), it follows that w−1
a (k− i+ 1) = k− i+ 1 + ai = ck−i+1, i = 1, . . . , k. This shows
that w−1
a ωk = w−1ωk and thus, w = wa. Now it remains to show that the map is compatible
with the orders.
Assume that a = (a1, . . . , ak), 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 = ai + 1 and so wa′ = wask−i+a′i . By (3.9), we have
that waαk−i+a′i > 0 and thus by [8, Proposition 3.2.16], there is an arrow wa → wa′ in W q
r .
On the other hand, suppose that a′′ = (a′′1, . . . , a
′′
k) ∈ N
k,n
++ satisfies a′ 6< a′′. In order to
complete the proof, it is enough to show that there is no arrow wa′ → wa′′ . By assumptions,
12 T. Salač
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. Then w−1
a′ (waαk+ai−i) = sk+ai−iαk+ai−i < 0. On the other hand
by (3.9), it follows that w−1
a′′ (waαk+ai−i) > 0. We proved that Φw′a 6⊂ Φwa′′ and thus by [8,
Proposition 3.2.17], there cannot be any arrow wa′ → wa′′ . �
We will later need the following two observations. A permutation w ∈ Sm is k-balanced, if
the following is true: if w(k − i) > k for some i = 0, . . . , k − 1, then w(k + i+ 1) ≤ k.
Lemma 3.5. The permutation wa associated to a ∈ Nk,n++ is k-balanced if and only if a ∈ Sk.
Proof. Let b = (b1, . . . , bn) be the partition conjugated to a = (a1, . . . , ak). First notice that if
wa(k − i) = k + j > k, then by (3.9) we have i = bj − j.
If a ∈ Sk, then wa(k + i+ 1) = wa(k − j + bj + 1) = wa(k − j + aj + 1) = k − j + 1 ≤ k and
so a is k-balanced.
If a 6∈ Sk, then there is j such that a1 = b1, . . . , aj−1 = bj−1 and aj 6= bj . It follows that
bj ≥ j and so i := bj − j ≥ 0. Then wa(k − i) = wa(k − bj + j) = k + j > k. If aj > bj , then
wa(k + i+ 1) = k + bj + 1 > k. If aj < bj , then wa(k + i+ 1) = k + bj > k. �
Recall from [2] that given w ∈ Sm, there exists a minimal integer `(w), called the length of w,
such that w can be expressed as a product of `(w) simple reflections s1, . . . , sm. It is well known
that `(w) is equal to the number of pairs 1 ≤ i < j ≤ m such that w(i) > w(j).
Lemma 3.6. Let wa ∈ Sm. Then `(wa) = |a|.
Proof. By the definition of wa, it follows that `(wa) ≤ |a|. On the other hand, if a < a′, then
wa → wa′ and thus also `(wa) < `(wa′). By induction on |a|, we have that `(wa) ≥ |a|. �
4 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 .
4.1 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 {αm}, {αk, αm} and {αk}, respectively, as explained in Section 2.1. We for brevity put
TS := G/R, CS := G/Q and M := G/P. Recall from Section 2.4 that we call TS the twistor
space and CS the correspondence space.
The twistor space TS. Let us first recall (see [15, Section 6]) some well known facts about
spinors. Recall from (3.4) that W := Cm is a maximal totally isotropic subspace of C2m. We can
(via h) identify the dual space W∗ with the subspace [e∗1, . . . , e
∗
m]. Put S :=
⊕m
i=0 ΛiW∗. There
is a canonical linear map C2m → End(S) which is determined by w ·ψ = iwψ and w∗ ·ψ = w∗∧ψ
where w ∈ W, w∗ ∈ W∗, ψ ∈ S and iw stands for the contraction by w. If ψ ∈ S, then we put
Tψ := {v ∈ C2m : v · ψ = 0}. If ψ 6= 0, then Tψ is a totally isotropic subspace and we call ψ
a pure spinor if dimTψ = m (which is equivalent to saying that Tψ is a maximal totally isotropic
subspace).
The standard linear isomorphism Λ2C2m ∼= 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
k-Dirac Complexes 13
S+ :=
⊕m
i=0 Λ2iW∗ and S− :=
⊕m
i=0 Λ2i+1W∗. Then S+ and S− are irreducible non-isomorphic
complex spinor representations of g with highest weights ωm and ωm−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 C2m 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 T1 = W, we find that R is the stabilizer of W inside G.
As G is connected, we conclude that TS is the connected component of W in the Grassmannian
of maximal totally isotropic subspaces in C2m.
The isotropic Grassmannian M . An irreducible g-module with highest weight ωk is isomor-
phic to ΛkC2m. Then e1 ∧ e2 ∧ · · · ∧ ek is clearly a highest weight vector and the corresponding
point in P(ΛkC2m) can be viewed as the totally isotropic subspace x0 := Ck. We see that M is
the Grassmannian of totally isotropic k-dimensional subspaces in C2m. 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 ∈ TS, x ∈ M, x ⊂ z} and Q is the stabilizer of (W, x0). Let
q : G→ CS be the canonical projection.
4.2 Filtrations of the tangent bundles of M and CS
Recall from Section 2.1 that the |2|-grading g = g−2⊕g−1⊕· · ·⊕g2 associated to {αk} determines
a 2-step filtration {0} = FM0 ⊂ FM−1 ⊂ FM−2 = TM of the tangent bundle of M where {0} is
the zero section. We put GMi := FMi /FMi+1, i = −2,−1 so that the associated graded bundle
gr(TM) = GM−2 ⊕ GM−1 is a locally trivial bundle of graded nilpotent Lie algebras with typical
fiber g−. Dually, there is a filtration T ∗M = FM1 ⊃ FM2 ⊃ FM3 = {0} where FMi
∼= (FM−i+1)⊥.
We put GMi := FMi /FMi+1 so that GMi
∼= (GM−i)
∗. There are linear isomorphisms
gi ∼=
(
GMi
)
x0
, i = −2,−1, 1, 2. (4.1)
Recall from Section 2.2 that grrx0
denotes the vector space of weighted r-jets of germs of
holomorhic functions at x0 whose weighted (r−1)-jet vanishes. Then the isomorphisms from (2.3)
are
gr1x0
∼= g1, gr2x0
∼= S2g1 ⊕ g2, gr3x0
∼= S3g1 ⊕ g1 ⊗ g2, . . .
for small r and in general
grrx0
∼=
⊕
i+2j=r
Sig1 ⊗ Sjg2. (4.2)
The |3|-grading g =
⊕3
i=−3 qi determined by {αk, αm} induces a 3-step filtration TCS =
FCS−3 ⊃ FCS−2 ⊃ FCS−1 ⊃ FCS0 = {0}. We put GCSi := FCSi /FCSi+1 so that gr(TCS) =
⊕−1
i=−3G
CS
i
is a locally trivial vector bundle of graded nilpotent Lie algebras with typical fiber q−. Dually,
we get a filtration T ∗CS = FCS1 ⊃ FCS2 ⊃ FCS3 ⊃ FCS4 = {0} where FCSi := (FCS−i+1)⊥. The
associated graded vector bundle is gr(T ∗CS) =
⊕3
i=1G
CS
i where we put GCSi := FCSi /FCSi+1.
Then as above, GCSi
∼= (GCS−i )∗.
The Q-invariant subspaces E ⊕ q and F ⊕ q give a finer filtration of the tangent bundle,
namely FCS−1 = ECS ⊕ FCS . Since the Lie bracket Λ2q−1 → q−2 vanishes on Λ2E ⊕ Λ2F, it
follows that ECS and FCS are integrable distributions. This can be deduced also from the short
exact sequences
0→ ECS → TCS
Tη−−→ T (TS)→ 0 and 0→ FCS → TCS
Tτ−−→ TM → 0, (4.3)
i.e., ECS = ker(Tη) and FCS = ker(Tτ). Notice that (Tτ)−1(FM−1) = FCS−2 .
14 T. Salač
4.3 Projections τ and η
Recall from (3.4) that C2n := [ek+1, . . . , em, e
∗
k+1, . . . , e
∗
m] and Cn := [ek+1, . . . , em], i.e., we
view C2n and Cn as subspaces of C2m. On C2n we consider the non-degenerate bilinear form h|C2n
which we for brevity denote by h. Then Cn is a maximal totally isotropic subspace of C2n.
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).
Lemma 4.1.
(a) The fibers of τ are biholomorphic to the Grassmannian of k-dimensional subspaces in Cn+k.
(b) The fibers of η are biholomorphic to the connected component Gr+
h (n, n) of Cn in the
Grassmannian of maximal totally isotropic subspaces in C2n.
Proof. As the fibers over distinct points are biholomorphic, it suffices to look at the fibers of η
and τ over W and x0, 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.
(b) Notice that x⊥0 = x0 ⊕ C2n. Then it is easy to see that y ∈ Gr+
h (n, n) 7→ (x0 ⊕ y, x0) ∈
τ−1(x0) is a biholomorphism. �
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 C2m that is spanned
by the columns of the matrix and by [X|Y ] the flag of nested subspaces [X] ⊂ [X]⊕ [Y ].
It is straightforward to verify that
(p ◦ exp) : g− →M, (4.4)
(p ◦ exp)
0 0 0 0
X1 0 0 0
X2 0 0 0
Y −XT
2 −XT
1 0
=
1k
X1
X2
Y − 1
2(XT
1 X2 +XT
2 X1)
.
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
1k 0
X1 A
X2 B
Y − 1
2(XT
1 X2 +XT
2 X1) C
, (4.5)
where A,B ∈M(n,C), C ∈M(k, n,C) are such that
[
A
B
]
∈ Gr+
h (n, n) and C = −(XT
1 B+XT
2 A).
We immediately get the following observation.
Lemma 4.2. The set τ−1(X ) is biholomorphic to X × τ−1(x0). 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 = 1n which uniquely
pins down B. It is straightforward to find that B = −BT and conversely, any skew-symmetric
n× n matrix determines a totally isotropic n-dimensional subspace in C2n. 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(m2 )+nk → C(m2 ), it will be convenient to choose a different coordinate
system on Y.
k-Dirac Complexes 15
Lemma 4.3. Let Y be as above and put Z := η(Y). Then Y and Z are open affine sets and
there is a commutative diagram of holomorphic maps
Y
η|Y
��
// A(m,C)×M(n, k,C)
pr1
��
Z // A(m,C),
(4.6)
where pr1 is the canonical projection and the horizontal arrows are biholomorphisms.
Proof. Let (z, x) be the nested flag corresponding to (4.5) where A = 1n so that B = −BT .
Put for brevity Y ′ := Y − 1
2(XT
1 X2 +XT
2 X1) and C := −XT
2 −XT
1 B. The map in the first row
in (4.6) is (z, x) 7→ (W,Z) where
W =
(
W1 W2
W0 −W T
1
)
and Z := X1, W1 := X2 − BX1, W0 := Y ′ − CX1, W2 := B. Then W = −W T and the
map Y → A(m,C) × M(n, k,C) 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 X1-block and get a new matrix
1k 0
0 1n
X2 −BX1 B
Y ′ − CX1 C
=
(
1m
W
)
.
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 TS which is biholomorphic to A(m,C). In these coordinate
systems, the restriction of η is the projection onto the first factor. �
5 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 byOq and Ep,qq the structure sheaf and the sheaf
of smooth (p, q)-forms, respectively, over CS. We denote the corresponding sheaves over TS by
the subscript r. If W is a holomorphic vector bundle over CS, then we denote by Oq(W ) the
sheaf of holomorphic sections of W and by Ep,qq (W ) the sheaf of smooth (p, q)-forms with values
in W . We for brevity put E∗ := E0,0
∗ and Ep,q∗ (U) := Γ(U , Ep,q∗ ) where U is an open set and ∗ = q
or r. Moreover, we put η∗Ep,qr := Eq⊗η−1Er E
p,q
r where we use that η−1Er is naturally a sub-sheaf
of Eq.
16 T. Salač
5.1 Double complex of relative forms
Recall from Lemma 4.3 that Y is biholomorphic to A(m,C) ×M(n, k,C), that η(Y) = Z is
biholomorphic to A(m,C) and that the canonical map Y → Z is the projection onto the first
factor. In this way we can use matrix coefficients on M(n, k,C) as coordinates on the fibers
of η. We will write Z = (zαi) ∈ M(n, k,C) and if I = ((α1, i1), . . . , (αp, ip)) is a multi-index
where (αj , ij) ∈ I(n, k) := {(α, i) : α = 1, . . . , n, i = 1, . . . , k}, j = 1, . . . , p, then we put
dzI := dz(α1,i1) ∧ · · · ∧ dz(αp,ip) and |I| = p.
We call
Ep+1,q
η := Ep+1,q
q /(η∗E1,0
r ∧ Ep,qq ), p, q ≥ 0
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 ∈ Ep,qη (Y), it is easy
to see that there is a unique (p, q)-form cohomologous to ω which can be written in the form∑′
|I|=p
dzI ∧ ωI , (5.1)
where Σ′ denote that the summation is performed only over strictly increasing multi-indeces7
and each ωI ∈ E0,q
q (Y).
As η is holomorphic, ∂ and ∂̄ commute with the pullback map η∗. We see that ∂(η∗E1,0
r ∧
Ep,qq ) ⊂ η∗E1,0
r ∧ Ep+1,q
q and ∂̄(η∗E1,0
r ∧ Ep,qq ) ⊂ η∗E1,0
r ∧ Ep,q+1
q and thus, ∂ and ∂̄ descend to
differential operators
∂η : Ep,qη → Ep+1,q
η and ∂̄ : Ep,qη → Ep,q+1
η ,
respectively. From the definitions it easily follows that:
∂η(ω ∧ ω′) = (∂ηω) ∧ ω′ + (−1)p+qω ∧ ∂ηω′, (5.2)
∂ηf =
∑
(α,i)∈I(n,k)
∂f
∂zαi
dzαi,
where ω ∈ Ep,qη (Y), ω′ ∈ Ep
′,q′
η (Y) and f ∈ E(Y). Recall from Section 4.2 that ∂zαi ∈ Γ(E1,0|Y)
and thus, ∂ηf(x), x ∈ Y depends only on the first weighted jet j1xf of f at x (see Section 2.2).
Recall from (4.3) that the distribution ECS is equal to ker(Tη).
Proposition 5.1.
(i) The sheaf Ep,qη is naturally isomorphic to the sheaf E0,q
q (ΛpECS∗) of smooth (0, q)-forms
with values in the vector bundle ΛpECS∗ and (Ep,∗η , ∂̄) is a resolution of Oq(Λ
pECS∗) by
fine sheaves.
(ii) ∂η is a linear G-invariant differential operator of weighted order one and the sequence of
sheaves (E∗,qη , ∂η), q ≥ 0 is exact.
(iii) The data (Ep,qη , (−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 → ECS⊥ → T ∗CS → ECS∗ → 0 is
short exact. Hence, also the sequence 0 → ECS⊥ ∧ ΛpT ∗CS → Λp+1T ∗CS → Λp+1ECS∗ → 0,
p ≥ 1 is short exact. In view of the isomorphism Ep+1,q
q
∼= E0,q
q (Λp+1T ∗CS), it is enough to show
7We order the set I(n, k) lexicographically, i.e., (α, i) < (α′, i′) if α < α′ or α = α′ and i < i′.
k-Dirac Complexes 17
that η∗E1,0
r ∧ Ep,qq is isomorphic to E0,q
q (ECS⊥ ∧ ΛpT ∗CS) ∼= Eq(ECS⊥) ∧ E0,q
q (ΛpT ∗CS). Now
η∗E1,0
r is a sub-sheaf of E1,0
q = Eq(T ∗CS) and since ker(Tη) = ECS , it is contained in Eq(ECS⊥).
Using that ker(Tη) = ECS again, it is easy to see that the map η∗E1,0
r → Eq(ECS⊥) induces an
isomorphism of stalks at any point. Hence, η∗E1,0
r
∼= Eq(ECS⊥) 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 ∂ηf(x)
depends only on j1xf 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
w1, . . . , w` on C` as coordinates on Y. If J = (j1, . . . , jq) where j1, . . . , jq ∈ {1, . . . , `}, then
we put dw̄J = dw̄j1 ∧ · · · ∧ dw̄jq and |J | = q. Let ω =
∑′
|I|=p
dzI ∧ ωI ∈ Ep,qη (Y) be the relative
form as in (5.1). Then there are unique functions fI,J ∈ Eq(Y) so that ωI =
∑′
|J |=q
fI,Jdw̄J .
Assume that ∂ηω = 0 on some open neighborhood Ux of x. This is equivalent to saying that
for each increasing multi-index J : ∂ησJ = 0 on Ux where σJ :=
∑′
I
fI,JdzI . Now using the
same arguments as in the proof of the Dolbeault lemma, see [16, Theorem 2.3.3], we can for
each J find a (p − 1, 0)-form φJ such that ∂ηφJ = σJ on some open neighborhood of x. Then
∂η
(∑′
J
φJ ∧ dw̄J
)
=
∑′
J
σJ ∧ dw̄J = ω on some open neighborhood of x and the proof is
complete.
(iii) This follows from [∂̄, ∂η] = 0 and the observations made above. �
5.2 Relative de Rham complex
By definition, Ω∗η := E∗,0η ∩ ker ∂̄ is a sheaf of holomorphic sections. Since [∂̄, ∂η] = 0, there is
a complex of sheaves (Ω∗η, ∂η) and we call it the relative de Rham complex.
Proposition 5.2.
(i) The relative de Rham complex is an exact sequence of sheaves which resolves the sheaf
η−1Or.
(ii) The relative de Rham complex induces for each r := ` + j ≥ 0 a long exact sequence of
vector bundles
gr`+j
gr∂η−−→ ECS∗ ⊗ gr`+j−1 → · · ·
→ ΛjECS∗ ⊗ gr`
gr∂η−−→ Λj+1ECS∗ ⊗ gr`+j−1 → · · · . (5.3)
Let s0 > 0, s1, s2, s3 ≥ 0 be integers such that s0 + s1 + 2s2 + 3s3 = r. Then the se-
quence (5.3) contains a long exact subsequence
0→ Ss0ECS∗ ⊗ Ss1,s2,s3 → ECS∗ ⊗ Ss0−1ECS∗ ⊗ Ss1,s2,s3 → · · · (5.4)
→ ΛjECS∗ ⊗ Ss0−jECS∗ ⊗ Ss1,s2,s3→ Λj+1ECS∗ ⊗ Ss0−j−1ECS∗ ⊗ Ss1,s2,s3→ · · · ,
where Ss1,s2,s3 := Ss1FCS∗ ⊗ Ss2GCS2 ⊗ Ss3GCS3 .
(iii) The kernel of the first map in (5.3) is
⊕
s1+2s2+3s3=r S
s1,s2,s3.
Proof. (i) Since [∂η, ∂̄] = 0, the relative de Rham complex is a sub-complex of the zero-th
row (E∗,0q , ∂η) of the double complex from Proposition 5.3. By diagram chasing and using the
18 T. Salač
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−1Or = E0,0
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 s0 > 0, s1, s2 and s3 the long exact sequence (5.4). The sequence (5.3) is the
direct sum of all such sequences as s0, s1, s2, and s3 ranges over all quadruples of non-negative
integers satisfying r = s0 + s1 + 2s2 + 3s3.
(iii) This readily follows from the part (ii). �
5.3 Twisted relative de Rham complex
The weight λ := (1 − 2n)ωm is g-integral and r-dominant. Hence, there is an irreducible R-
module Wλ with lowest weight −λ. Since r is associated to {αm}, it follows that dimWλ = 1
and so Wλ is also an irreducible Q-module. We will denote by Eq(λ) and Oq(λ) the sheaves of
smooth and holomorphic sections of WCS
λ := G ×Q Wλ, respectively. If W is a vector bundle
over CS, then we denote W (λ) := W ⊗ WCS
λ , i.e., we twist W by tensoring with the line
bundle WCS
λ . It is not hard to see that η∗Er(λ) ∼= Eq(λ) and η∗Or(λ) = Oq(λ) where we denote
by the subscript r the corresponding sheaves over TS.
We call Ep,qη (λ) := Ep,qη ⊗η−1Er η
−1Er(λ) the sheaf of twisted relative (p, q)-forms. Consider the
following sequence of canonical isomorphisms:
Ep,qη (λ)→ Ep,qη ⊗Eq Eq ⊗η−1Er η
−1Er(λ)→ Ep,qη ⊗Eq η∗Er(λ)
→ Ep,qη ⊗Eq Eq(λ)→ E0,q
q ⊗Eq
(
Ep,0η ⊗Eq Eq(λ)
)
→ E0,q
q
(
ΛpECS∗(λ)
)
. (5.5)
We see that Ep,qη (λ) is isomorphic to the sheaf of smooth (0, q)-forms with values in ΛpECS∗(λ).
Hence, the Dolbeault differential induces a differential ∂̄ : Ep,qη (λ) → Ep,q+1
η (λ) and a complex
(Ep,∗η (λ), ∂̄).
A section of Ep,qη (λ) is by definition a finite sum of decomposable elements ω⊗v where ω and
v are sections of Ep,qη and η−1Er(λ), respectively. As any section of η−1Er as well as transition
functions between sections of η−1Er(λ) belong to ker(∂η), it follows that there is a unique linear
differential operator
Ep,qη (λ)→ Ep+1,q
η (λ),
which satisfies ω ⊗ v 7→ ∂ηω ⊗ v. 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.
Proposition 5.3. Let p, q ≥ 0 be integers.
(i) The sequence of sheaves (E(p,∗)
η (λ), ∂̄) is exact.
(ii) The sequence of sheaves (E(∗,q)
η (λ), ∂η) is exact.
(iii) There is a double complex (Ep,qη (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.
(ii) The exactness follows immediately from Proposition 5.1(ii).
(iii) We need to verify that [∂̄, ∂η] = 0. To see this, notice that a section of Ep,qη (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). �
k-Dirac Complexes 19
Put Ω∗η(λ) := E∗,0η (λ)∩ ker(∂̄). The complex (E∗,0η (λ), ∂η) contains a sub-complex (Ω∗η(λ), ∂η)
which we call the twisted relative de Rham complex. As in Proposition 5.2, one can easily see
that (Ω∗η(λ), ∂η) is an exact sequence of sheaves of holomorhic sections. Following the proof of
Proposition 5.2, we obtain the following:
Proposition 5.4. The relative de Rham complex (Ω∗η(λ), ∂η) induces for each r ≥ 0 a long exact
sequence of vector bundles(
Λ•ECS∗ ⊗ grr−•(λ), gr∂η
)
. (5.6)
Let s0 > 0, s1, s2, s3 ≥ 0 be integers such that s0 + s1 + 2s2 + 3s3 = r. Then the sequence (5.6)
contains a long exact subsequence(
Λ•ECS∗ ⊗ Ss0−•ECS∗ ⊗ Ss1,s2,s3(λ), gr∂η
)
, (5.7)
where Ss1,s2,s3 is defined in Proposition 5.2. The kernel of the first map in (5.6) is the bundle⊕
s1+2s2+3s3=r S
s1,s2,s3(λ).
5.4 Relative BGG sequence
We know that Ωp
η(λ) is isomorphic to the sheaf of holomorphic sections of ΛpECS∗(λ) = G ×Q
(ΛpE∗⊗Wλ). The Q-module ΛpE∗ 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 ∈ Nk,n++ and wa ∈W q
r be as in Section 3.2. Then
ΛpE∗ ⊗Wλ =
⊕
a∈Nk,n++:|a|=p
Wλa and thus Ωp
η(λ) =
⊕
a∈Nn,k++:|a|=p
Oq(λa), (5.8)
where Wλa is an irreducible Q-module with lowest weight −λa := −wa.λ.
There is a linear G-invariant differential operator
∂aa′ : Oq(λa)→ Ωp
η(λ)
∂η−→ Ωp
η(λ)→ Oq(λa′) (5.9)
where the first map is the canonical inclusion and the last map is the canonical projection. If
a ≮ a′, then ∂aa′ = 0.
Proof. Recall from Section 3.1 that the semi-simple part rss0 of r0 is isomorphic to sl(m,C) and
that rss0 ∩q is a parabolic subalgebra of rss0 . 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 Wλ and (rss0 , r
ss
0 ∩ q) and the identity `(wa) = |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. �
Remark 5.6. Let a = (a1, . . . , ak) ∈ Nk,n++ and b = (b1, . . . , bn) ∈ Nn,k++ be the conjugated
partition. In order to compute λa from Proposition 5.5, notice that
λ+ ρ =
(
2k − 1
2
, . . . ,
3
2
,
1
2
∣∣∣∣ − 1
2
,−3
2
, . . . ,
−2n+ 1
2
)
. (5.10)
Since wa(ωm) = ωm, we have wa(λ) = λ and thus λa = wa(λ+ ρ)− ρ = λ+waρ− ρ. By (3.10),
it follows that
λa = λ+ (−ak, . . . ,−a1 | b1, b2, . . . , bn).
20 T. Salač
5.5 Direct image of the relative BGG sequence
Recall from [2, Section 5.3] that given a g-integral and q-dominant weight ν, there is at most
one p-dominant weight in the W q
p -orbit of ν. If there is no p-dominant weight, then all direct
images of Oq(ν) vanish. If there is a p-dominant weight, say µ = w.ν where w ∈ W q
p , then
τ
`(w)
∗ Oq(ν) ∼= Op(µ) is the unique non-zero direct image of Oq(ν).
Proposition 5.7. Let n ≥ k ≥ 2 and a = (a1, . . . , ak) ∈ Nk,n++. Put µ± := 1
2(−2n+ 1, . . . ,−2n+
1 | 1, . . . , 1,±1) and µa := µ∗ − (ak, . . . , a1 | 0, . . . , 0) where ∗ = + if d(a) ≡ n mod 2 and ∗ = −
otherwise. Then
τ q∗ (Oq(λa)) =
{
Op(µa), if a ∈ Sk, q = `(a) :=
(
n
2
)
− q(a),
{0}, otherwise.
Proof. By definition, each w ∈ W q
p fixes the first k coefficients of λa and so it is enough to
look at the last n coefficients. By Remark 5.6, it follows
wa(λ+ ρ) = λa + ρ =
(
. . . , i− ai −
1
2
, . . . ,
1
2
− a1
∣∣∣∣ b1 − 1
2
, . . . , bj − j +
1
2
, . . .
)
,
where b = (b1, . . . , bn) is the conjugated partition. Put c := (c1, . . . , cn) where cj :=
∣∣bj − j + 1
2
∣∣.
By (3.2) and Table 1, if ci = cj for some i 6= j, then there cannot be a p-dominant weight
in the W q
p -orbit of λa. If a 6∈ Sk, then by Lemma 3.5 there is s ∈ {0, . . . , k − 1} such that8
wa(k− s) = k+ i ≥ k and wa(k+ s+ 1) = k+ j ≥ k for some distinct positive integers i and j.
By (5.10), it follows that bi − i + 1
2 = 1
2 + s, bj − j − 1
2 = −1
2 − s and thus ci = cj . Hence, all
direct images of Oq(λa) are zero.
We may now suppose that a ∈ Sk. By the definition of d(a), we have bd(a) − d(a) + 1
2 > 0 >
bd(a)+1 − d(a) − 1
2 . By Lemma 3.5 and (5.10), it follows that (c1, . . . , cn) is a permutation of(
2n−1
2 , . . . , 3
2 ,
1
2
)
. As λa is q-dominant, we know that the sequence
(
b1 − 1
2 , · · · , bj − j + 1
2 , . . .
)
is decreasing. Thus for each integer i = 1, . . . , d(a), the set {j : i < j ≤ n, ci > cj} contains
precisely bi − i distinct elements. Altogether, there are precisely
d(a)∑̀
=1
(b` − `) = q(a) pairs i < j
such that ci > cj . Equivalently, there are `(a) pairs i < j such that ci < cj . It follows that the
length of the permutation that maps (c1, . . . , cn) 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.λa is p-dominant and `(w) = `(a).
As there are n − d(a) negative numbers in the sequence
(
b1 − 1
2 , . . . , bj − j + 1
2 , . . .
)
, the last
claim about the sign of the last coefficient of w.λa 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 αm and αm−1 and hence, it swaps also the associated parabolic subalgebras. If we cross
in (3.3) the simple root αm−1 instead of αm, take (1−2n)ωm−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.
5.6 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 Ep,qη (λ) =
⊕
a∈Nk,n++:|a|=p E
0,q
q (λa). Let a, a′ ∈ Nk,n++ be such that
8Notice that at this point we need that n ≥ k.
k-Dirac Complexes 21
p = |a| = |a′| − 1. Then there is a linear differential operator
∂aa′ : E0,q
q (λa)→ Ep,qη (λ)
∂η−→ Ep+1,q
η (λ)→ E0,q
q (λa′) (5.11)
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 ∂aa′ as in (5.9) as there is no risk of confusion.
Recall from Proposition 5.5 that ∂aa′ = 0 if a ≮ a′.
Suppose that U is an open, contractible and Stein subset of M . We put Ep,qη (τ−1(U), λ) :=
Γ(τ−1(U), Ep,qη (λ)), i.e., this is the space of sections of the sheaf Ep,qη (λ) over τ−1(U). Then there
is a double complex
· · · // Ep,q+1
η (τ−1(U), λ)
D′′ //
OO
Ep+1,q+1
η (τ−1(U), λ)
OO
// · · ·
· · · // Ep,qη (τ−1(U), λ)
D′′ //
D′
OO
Ep+1,q
η (τ−1(U), λ)
D′
OO
// · · · ,
OO OO
(5.12)
where D′ = (−1)p∂̄ and D′′ = ∂η. Put T i(U) :=
⊕
p+q=i E
p,q
η (τ−1(U), λ). We obtain three
complexes (T ∗(U), D′), (T ∗(U), D′′) and (T ∗(U), D′ +D′′).
Lemma 5.9. Let a ∈ Nk,n++ and U be the open, contractible and Stein subset of M as above.
Then
Hq
(
τ−1(U),Oq(λa)
)
=
{
Γ(U ,Op(µa)), a ∈ Sk, q = `(a),
{0}, otherwise,
(5.13)
and thus also
H(n2)+j(T ∗(U), D′) =
⊕
a∈Skj
H`(a)
(
τ−1(U),Oq(λa)
)
.
Proof. The first claim follows from Proposition 5.7 and application of the Leray spectral se-
quence 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
Hq
(
τ−1(U),Oq(λa)
) ∼= Hq
(
E0,∗
q
(
τ−1(U), λa
)
, ∂̄
)
. (5.14)
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 Skj = {a ∈ Sk : r(a) = j}. �
6 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.
Recall from Section 5.6 that E0,q
q (τ−1(U), λa) is the space of sections of the sheaf E0,q
q (λa)
over τ−1(U). If α ∈ E0,q
q (τ−1(U), λa) is ∂̄-closed, then we will denote by [α] ∈ Hq(τ−1(U),Oq(λa))
the corresponding cohomology class.
22 T. Salač
Lemma 6.1. Let j ≥ 0, a ∈ Skj , a′ ∈ Skj+1 be such that a < a′ and U be the Stein set as above.
Then there is a linear, local and G-invariant operator
Da
a′ : Γ(U ,Op(µa))→ Γ(U ,Op(µa′)).
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,Oq(λa)) → H`(a′)(V,Oq(λa′)) 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 ∂aa′ : E
0,q
q (V, λa) → E0,q
q (V, λa′) in the double complex (5.12).
The induced map on cohomology is Da
a′ .
If |a′| − |a| = 2, then q := `(a) = `(a′) + 1 and we find that there are precisely two non-
symmetric partitions b, c ∈ Nk,n++ such that a < b < a′ and a < c < a′. Then there is a diagram
E0,q
q (V, λa)
(∂ab ,∂
a
c )
// E0,q
q (V, λb)⊕ E0,q
q (V, λc)
E0,q−1
q (V, λb)⊕ E0,q−1
q (V, λc)
(−1)p∂̄
OO
∂b
a′+∂
c
a′// E0,q−1
q (V, λa′),
(6.1)
which lives in the double complex (5.12). Let α ∈ E0,q
q (V, λa) be ∂̄-closed. Then ∂abα and ∂acα
are also ∂̄-closed and thus by Lemma 5.9 and the isomorphism (5.14), we can find β and γ such
that ∂abα = (−1)p∂̄β and ∂acα = (−1)p∂̄γ where p = |a|+ 1. Since the relative BGG sequence is
a complex, we have
∂̄
(
∂ba′β + ∂ca′γ
)
= (−1)p(∂ba′∂
a
b + ∂ca′∂
a
c )α = 0,
which shows that ∂ba′β + ∂ca′γ is a cocycle. Of course this elements depends on choices but we
claim that [∂ba′β+∂ca′γ] depends only on [α]. It is easy to see that [∂ba′β+∂ca′γ] does not depend on
the choices of β and γ. If [α] = 0, say α = (−1)p−1∂̄%, then we may put β = −∂ab % and γ = −∂ac %
and thus ∂ba′β + ∂ca′γ = −(∂ba′∂
a
b + ∂ca′∂
a
c )% = 0. Hence, we can put Da
a′ [α] := [∂ba′β + ∂ca′γ].
From the construction is clear that Da
a′ is linear. The locality follows from the fact that Da
a′
is compatible with restrictions to smaller Stein subsets of U . As the operators in the double
complex (5.12) are G-invariant, it is easy to verify that each operator Da
a′ is G-invariant. �
Put Oj :=
⊕
a∈Skj
Op(µa) and Oj(U) := Γ(U ,Oj). If a ∈ Skj and s ∈ Oj(U), then we denote
by sa the a-th component of s so that we may write s = (sa)a∈Skj
. We call the following
complex (6.2) the k-Dirac complex.
Theorem 6.2. With the notation set above, there is a complex
O0(U)
D0−−→ O1(U)→ · · · → Oj(U)
Dj−−→ Oj+1(U)→ · · · (6.2)
of linear G-invariant operators where
(Djs)a′ =
∑
a<a′
Da
a′sa.
Proof. Let a, a′ ∈ Sk be such that a < a′, r(a) = r(a′) − 2. We need to verify that∑
a′′∈Sk : a<a′′<a′
Da′′
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
k-Dirac Complexes 23
easily that Da′′
a′ D
a
a′′ = 0. So we can assume that there are two symmetric partitions, say a′′1, a′′2.
Consider for example
b =
%%
// c =
%%
a =
::
//
$$
a′′1 =
99
%%
a′′2 = // a′ = .
b′ = //
99
c′ =
99
Then we can find β and β′ so that ∂abα = (−1)p∂̄β and ∂ab′α = (−1)p∂̄β′ where p = |b| = |b′|.
Then [Da
a′′2
α] = [∂ba′′2
β + ∂b
′
a′′2
β′] which implies
(−1)p+1∂̄
(
∂bcβ
)
= (−1)p+1∂bc ∂̄β = −∂bc∂abα = ∂
a′′1
c ∂aa′′1
α
and similarly (−1)p+1∂̄(∂b
′
c′β
′) = ∂
a′′1
c′ ∂
a
a′′1
α. Hence, we conclude that
D
a′′1
a′ D
a
a′′1
[α] =
[
∂ca′∂
b
cβ + ∂c
′
a′∂
b′
c′β
′]
and thus
D
a′′2
a′ D
a
a′′2
[α] =
[
∂
a′′2
a′
(
∂ba′′2
β + ∂b
′
a′′2
β′
)]
= −
[
∂ca′∂
b
cβ + ∂c
′
a′∂
b′
c′β
′] = −Da′′1
a′ D
a
a′′1
[α].
This completes the proof when |a′| = |a|+ 3 and now we may assume |a′| = |a|+ 4.
We put A′′ := {a′′ ∈ Sk| a < a′′ < a′}, B := {b ∈ Nk,n++ | ∃ a′′ ∈ A′′ : a < b < a′′}, B′ := {b′ ∈
Nk,n++ | ∃ a′′ ∈ A′′ : a′′ < b′ < a′} and finally C := {c ∈ Nk,n++ \ Sk | ∃ b ∈ B, ∃ b′ ∈ B′ : b′ < c < b′′}.
Consider for example the diagram
b1 = //
$$
c1 = // b′1 =
%%
a =
##
;;
a′′ =
%%
::
a′ = ,
b2 = //
::
c2 = // b′2 =
99
where A′′ = {a′′}, B = {b1, b2}, B′ = {b′1, b′2} and C = {c1, c2}. As above, the set A′′ contains
at most two elements but we will not need that.
Now we can proceed as above. There are βi such that (−1)p∂̄βi = ∂abiα where p = |bi| = |a|+1
and so [Da
a′′j
α] =
[ ∑
bi∈B
∂bi
a′′j
βi
]
for every a′′j ∈ A′′. As the relative BGG sequence is a complex,
we have for each c` ∈ C:
∂̄
(∑
bi∈B
∂bic`βi
)
=
∑
bi∈B
∂bic` ∂̄βi = (−1)p
∑
bi∈B
∂bic`∂
a
bi
α = 0.
24 T. Salač
As above, there is γ` such that (−1)p+1∂̄γ` =
∑
bi∈B
∂bic`βi. Then for b′s ∈ B′:
(−1)p+2∂̄
( ∑
c`∈C
∂c`b′s
γ`
)
= −
∑
c`∈C
∂c`b′s
((−1)p+1∂̄γ`) = −
∑
c`∈C, bi∈B
∂c`b′s
∂bic`βi
=
∑
bi∈B, a′′j ∈A′′j : bi<a′′j<b
′
s
∂
a′′j
b′s
∂bi
a′′j
βi =
∑
a′′j ∈A : a′′j<b
′
s
∂
a′′j
b′s
(∑
bi∈B
∂bi
a′′j
βi
)
=
∑
a′′j ∈A : a′′j<b
′
s
∂
a′′j
b′s
(Da
a′′j
α).
This implies that
∑
a′′j ∈A′′
Da′′
a′ D
a
a′′([α]) is the cohomology class of
∑
a′′j ∈A′′
( ∑
b′s∈B′ : a′′j<b′s
∂
b′s
a′
( ∑
c`∈C
∂c`b′s
γ`
))
=
∑
b′s∈B′
∑
c`∈C
∂
b′s
a′ ∂
c`
b′s
γ`
=
∑
c`∈C
∑
b′s∈B′
∂
b′s
a′ ∂
c`
b′s
γ` =
∑
c`∈C
0 = 0.
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. �
7 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 X0 := τ−1(x0) and give the definition of the weighted formal
neighborhood of X0. 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 X0. 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(x0)
Let us first recall notation from Section 4.2. There is the 2-step filtration {0} = FM0 ⊂ FM−1 ⊂
FM−2 = TM and the 3-step filtration {0} = FCS0 ⊂ FCS−1 ⊂ FCS−2 ⊂ FCS−3 = TM . Moreover, FCS−1
decomposes as ECS ⊕ FCS where ECS = ker(Tη) and FCS = ker(Tτ). From this it follows
that ECS and FCS are integrable distributions. Dually, there are filtrations T ∗M = FM1 ⊃
FM2 ⊃ FM3 = {0} and T ∗CS = FCS1 ⊃ FCS2 ⊃ FCS3 ⊃ FCS4 = {0} where FMi is the annihilator
of FM−i+1 and similarly for FCSi . We put GMi := FMi /FMi+1 and GCSi := FCSi /FCSi+1 so that
gr(TM) = GM−2 ⊕GM−1, gr(T ∗M) = GM1 ⊕GM2 ,
gr(TCS) = GCS−3 ⊕GCS−2 ⊕GCS−1 , gr(T ∗CS) = GCS1 ⊕GCS2 ⊕GCS3 ,
GMi
∼=
(
GM−i
)∗
, i = 1, 2 and GCSi
∼=
(
GCS−i
)∗
, i = 1, 2, 3.
Let us now briefly recall Section 2.3. If X is an analytic subvariety of a complex manifold Y ,
then the normal bundle NX of X in Y is the quotient (TY |X)/TX and the co-normal bundle N∗X
is the annihilator of TX inside T ∗X. In particular, the origin x0 can be viewed as an analytic
subvariety of M with local defining equation X1 = 0, X2 = 0 and Y = 0 where the matrices are
k-Dirac Complexes 25
those as in (4.4). For each i ≥ 1 there is the associated (i-th power of the) ideal sheaf Iix0
. This
is a sheaf of OM -modules such that
(Ix0)ix =
{
(OM )x, x 6= x0,
F ix0
, x = x0,
where OM is the structure sheaf on M , F ix = {f ∈ (OM )x : jixf = 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
0→ Fi+1
x0
→ Fix0
→ gri+1
x0
→ 0, (7.1)
where Fix0
:= {f ∈ Ox0 : jix0
f = 0}. We will view (7.1) also as a short exact sequence of sheaves
over {x0}.
Put X0 := τ−1(x0). Recall from Lemma 4.1 that X0 is complex manifold which is biholo-
morphic to the connected component Gr+
h (n, n) of Cn in the Grassmannian of maximal totally
isotropic subspaces in C2n.
Remark 7.1. If V CS is a holomorphic vector bundle over CS, we will for brevity put V :=
V CS |X0 . We also put τ0 := τ |X0 .
Lemma 7.2.
(i) X0 is a closed analytic subvariety of CS and there is an isomorphism of sheaves IX0
∼=
τ∗Ix0.
(ii) There is an isomorphism of vector bundles9 TX0
∼= F .
(iii) The normal bundle N of X0 in CS is isomorphic to τ∗0Tx0M . In particular, N is a trivial
holomorphic vector bundle.
Proof. (i) By Lemma 4.2, τ−1(X ) = X × X0 where X = (p ◦ exp)(g−). From this the claim
easily follows.
(ii) As X0 = τ−1(x0), it is clear that TX0 = ker(Tτ)|X0 . But we know that ker(Tτ) = F and
the claim follows.
(iii) By definition, τ∗0Tx0M = {(x, v) |x ∈ X0, v ∈ Tx0M}. Hence, there is an obvious
projection TCS|X0 → τ∗0Tx0M , (x, v) 7→ (x, Txτ(v)) which descends to an isomorphism N →
τ∗0Tx0M . �
Recall now the linear isomorphisms gi ∼= (GMi )x0 , i = −2,−1, 1, 2, from (4.1). In particular,
we can view gi as the fiber of GMi over {x0} and thus also as a vector bundle over {x0}. We use
this point of view in the following definition.
Definition 7.3. Put N∗i := τ∗0 gi, i = 1, 2 and S`N∗ := τ∗0 gr
`, ` = 0, 1, 2, . . . .
Notice that N∗i , i = 1, 2 and S`N∗, ` ≥ 0 are by definition trivial holomorphic vector bundles
over X0. Recall from the end of Section 2.2 that gr`x0
is the subspace of gr`x0
that is isomorphic
to S`g1.
Lemma 7.4. The co-normal bundle N∗ of X0 in CS is isomorphic to τ∗0T
∗
x0
M and the bundle N∗2
is isomorphic to G3. There are short exact sequences of vector bundles
0→ N∗2 → N∗ → N∗1 → 0 and 0→ G2 → N∗1 → E∗ → 0 (7.2)
9Here we use notation set in Remark 7.1.
26 T. Salač
over X0. Moreover, for each ` ≥ 0 there are isomorphisms of vector bundles
S`N∗ =
⊕
`1+2`2=`
S`1N∗1 ⊗ S`2N∗2 and S`N∗1 = τ∗gr`x0
. (7.3)
Proof. There is a canonical injective vector bundle map τ∗0T
∗
x0
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∗0T
∗
x0
M ∼= N∗ and thus the first claim. It is clear that N∗2 = τ∗0 g2 is the
annihilator of F−2 = (Tτ)−1(g−1) and since G3 = F⊥−2, the second claim follows.
The first sequence in (7.2) is the pullback of the short exact sequence 0 → g2 → T ∗x0
M →
g1 → 0 and thus, it is short exact. The exactness of the latter sequence follows from the
exactness of 0→ GCS2 → F⊥/GCS3 → ECS∗ → 0 and the isomorphisms N∗ ∼= F⊥, G3
∼= N∗2 and
N∗1
∼= N∗/N∗2 .
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 X0. It
follows that any global holomorphic section of S`N∗ is a constant gr`-valued function on X0
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.
Lemma 7.5. The holomorphic vector bundles S`N∗ and S`N∗1 are trivial and there are canonical
isomorphisms gr`x0
→ Γ(O(S`N∗)) and gr`x0
→ Γ(O(S`N∗1 )) of finite-dimensional vector spaces.
Let us finish this section by recalling the concept of formal neighborhoods (see [1, 26]). Let
ι0 : X0 ↪→ CS be the inclusion. Then FX0 := ι−1
0 IX0 is a sheaf of OX0-modules whose stalk
at x ∈ X0 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 ∩ X0. Let us now view the vector space
Fx0 = {f ∈ (OM )x0 : f(x0) = 0} also as a sheaf over {x0}. Then (recall from Lemma 7.2)
it is easy to see that FX0 = τ∗0Fx0 . Observe that Γ(FX0) is the space of equivalence classes
of holomorphic functions which are defined on an open neighborhood of X0 in CS where two
such functions belong to the same equivalence class if they agree on some possibly smaller open
neighborhood of X0.
The infinite-dimensional vector spaces from (7.1) form a decreasing filtration · · · ⊂ Fi+1
x0
⊂
Fix0
⊂ · · · of Fx0 = F0
x0
. Then FiX0
:= τ∗Fix0
is a sheaf of OX0-modules which is naturally
a sub-sheaf of FX0 . This induces a filtration · · · ⊂ Fi+1
X0
⊂ FiX0
⊂ · · · of FX0 = F0
X0
. Arguing as
in Section 2.3, one can show that for each i ≥ 0 there is a short exact sequence of sheaves
0→ Fi+1
X0
→ FiX0
→ O
(
Si+1N∗
)
→ 0
and thus, the graded sheaf associated to the filtration FX0 is isomorphic to
⊕
i≥1O(SiN∗).
Using the analogy with the classical formal neighborhood, we will call the pair (X,O(i)
X ) where
O(i)
X := ι−1
0 OCS/F
i+1
X0
the i-th weighted formal neighborhood of X0. Notice that the filtration
{FiX0
: i = 0, 1, 2, . . . } descends to a filtration of O(i)
X and that the associated graded sheaf is
isomorphic to
⊕i+1
j≥0O(SjN∗).
7.2 The double complex on the formal neighborhood of τ−1(x0)
Recall from Section 5.4 that for each a ∈ Nk,n++ there is a Q-dominant and integral weight λa,
an irreducible Q-module Wλa with lowest weight −λa and an associated vector bundle WCS
λa
=
G ×Q Wλa . We will denote by Wλa the restriction of WCS
λa
to X0, by O(λa) the sheaf of
holomorphic sections of Wλa , by Ep,q the sheaf of smooth (p, q)-forms over X0 and by Ep,q(λa)
k-Dirac Complexes 27
the sheaf of (p, q)-forms with values in Wλa . If V is another vector bundle over X0, then we
denote by V (λa) the tensor product of V with Wλa . 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 X0:
SrN∗(λ)
d0−→ E∗ ⊗Sr−1N∗(λ)
d1−→ Λ2E∗ ⊗Sr−2N∗(λ)
d2−→ · · · . (7.4)
This sequence contains a long exact subsequence
SrN∗1 (λ)
δ0−→ E∗ ⊗ Sr−1N∗1 (λ)
δ1−→ Λ2E∗ ⊗ Sr−2N∗1 (λ)
δ2−→ · · · . (7.5)
Proof. In order to obtain the sequence (7.4), take the direct sum of all long exact sequences
from (5.7) indexed by s0, s1, s2 and s3 where s0 + s2 + 2s3 = `+ j, s1 = 0 and restrict it to X0.
The subsequence (7.5) is obtained similarly, we only add one more condition s3 = 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 E0,q(ΛjE∗⊗S`N∗(λ)) be the sheaf of smooth (0, q)-forms with values in the
corresponding vector bundle over X0. The vector bundle map dj induces a map of sheaves
E0,q
(
ΛjE∗ ⊗S`N∗(λ)
)
→ E0,q
(
Λj+1E∗ ⊗S`−1N∗(λ)
)
, (7.6)
which we also denote by dj as there is no risk of confusion.
Recall from (5.8) that ΛjE∗⊗Wλ =
⊕
a∈Nk,n++ : |a|=jWλa which gives direct sum decomposition
E0,q(ΛjE∗ ⊗ S`N∗(λ)) =
⊕
a∈Nk,n++ : |a|=j E
0,q(S`N∗(λa)). We see that if a, a′ ∈ Nk,n++ are such
that |a| = |a′| − 1 = j, then dj induces
daa′ : E0,q
(
S`N∗(λa)
)
→ E0,q
(
S`−1N∗(λa′)
)
(7.7)
in the same way ∂η induces in (5.9) the operator ∂aa′ in the relative BGG sequence. By Propo-
sition 5.5, daa′ = 0 if a ≮ a′.
Remark 7.8. Replacing (7.4) by (7.5) in Remark 7.7, we get a map of sheaves
δj : E0,q
(
ΛjE∗ ⊗ S`N∗1 (λ)
)
→ E0,q
(
Λj+1E∗ ⊗ S`−1N∗1 (λ)
)
. (7.8)
If a, a′ are as above, then there is a map
δaa′ : E0,q
(
S`N∗1 (λa)
)
→ E0,q
(
S`−1N∗1 (λa′)
)
,
which is induced in the same way dj induces daa′ .
Even though the proof of Lemma 7.9 is trivial, it will be crucial later on.
Lemma 7.9. Let a ∈ Nk,n++. Then
(τ0)q∗
(
O
(
S`N∗(λa)
))
= Hq
(
X0,O
(
S`N∗(λa)
))
=
{
gr`Vµa ,
{0}
(7.9)
and
(τ0)q∗
(
O(S`N∗1 (λa))
)
= Hq
(
X0,O(S`N∗1 (λa))
)
=
{
gr`Vµa ,
{0},
(7.10)
where10 in (7.9) and (7.10) the first possibility holds if and only if a ∈ Sk and q = `(a).
10As above, we identify a sheaf over {x0} with its stalk.
28 T. Salač
Proof. The first equality in (7.9) is just the definition of (τ0)q∗. The sheaf cohomology group
in the middle is equal to the cohomology of the Dolbeault complex. In view of Lemma 7.5,
Γ(E0,q(S`N∗(λa))) ∼= gr`x0
⊗ Γ(E0,q(λa)) and thus, the sheaf cohomology group is isomorphic
to gr`x0
⊗ Hq(X0,O(λa)). By the Bott–Borel–Weil theorem, Hq(X0,O(λa)) ∼= Vµa if a ∈ Sk,
q = `(a) and vanishes otherwise. The second equality in (7.9) then follows from the isomorphism
gr`x0
⊗ Vµa → gr`Vµa from (2.3).
The isomorphism in (7.10) is proved similarly. We only use the other isomorphism
Γ
(
O
(
S`N∗1
))
→ gr`x0
from Lemma 7.5 and the isomorphism gr`x0
⊗ Vµa → gr`Vµa . �
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 X0.
Proposition 7.10. Let r ≥ 0 be an integer. Then there is a double complex (Ep,q(r), d′, d′′)
where:
• Ep,q(r) = Γ(E0,q(ΛpE∗ ⊗Sr−pN∗(λ))),
• the vertical differential d′ is (−1)p∂̄ where ∂̄ is the standard Dolbeault differential and
• the horizontal differential d′′ is dp from (7.6).
Moreover, we claim that:
(i) Hj(T ∗(r), d′ + d′′) = 0 if j >
(
n
2
)
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
Ep,q1 (r) =
⊕
a∈Sk : |a|=p, `(a)=q
grr−pVµa ;
(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 = dp 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 dimX0 =(
n
2
)
, it follows that Ep,q1 (r) = 0 whenever q >
(
n
2
)
. This proves the claim.
(ii) By definition, Ep,q1 (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 grr−pVµa 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′ ∈ Sk such that grr−|a
′|Vµa′ 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, grr−|a
′|Vµa′ lives on the `(a)-th row. In the second case, it lives on the (`(a)−1)-th
row. As daa′ = 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.
k-Dirac Complexes 29
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(E0,q(ΛpE∗ ⊗ Sr−pN∗1 (λ))). Moreover we claim that:
(i) Hj(T ∗(r), d′ + d′′) = 0 if j >
(
n
2
)
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
F p,q1 (r) :=
⊕
a∈Sk : |a|=p, `(a)=q
grr−pVµa ;
(iii) the spectral sequence degenerates on the second page.
7.3 Long exact sequence of weighted jets
Let a ∈ Sk and Vµa be an irreducible P-module with lowest weight −µa, 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′ ∈ Sk be such that a < a′ and r(a′) = r(a) + 1. Then the operator Da
a′
from Lemma 6.1 is a differential operator of weighted order at most s := |a′| − |a|.
Hence, Da
a′ induces for each i ≥ 0 a linear map
grDa
a′ : griVµa → gri−sVµa′ , (7.11)
which restricts to a linear map
grDa
a′ : griVµa → gri−sVµa′ . (7.12)
Proof. Let us make a few preliminary observations. Let v ∈ Op(µa)x0 . By the G-invariance
of Da
a′ , it is obviously enough to show that (Da′
a v)(x0) depends only on jsx0
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),Oq(λa)). A choice of Weyl structure (see [8]) and the isomorphisms (7.9)
give for each integer i ≥ 0 isomorphisms
JiVµa →
i⊕
j=0
grjVµa →
i⊕
j=0
H`(a)
(
X0,S
jN∗(λa)
)
.
Hence, the Taylor series of v at x0 determines an infinite11 sum
∞∑
j=0
[vj ] where each [vj ] belong
to H`(a)(X0,S
j(λa)).
Now we can proceed with the proof. By assumption, s ∈ {1, 2}. If s = 1, then `(a) = `(a′).
By definition, Da
a′v corresponds to [∂aa′α] ∈ H`(a′)(τ−1(U),Oq(λa′)) and jix0
(Da
a′v) can be viewed
as
i∑
j=0
[(daa′)vj+1]. But since [daa′(vj)] ∈ H`(a)(X0,S
j−1(λa)), it is clear that Da
a′(v)(x0) = 0 if
j1x0
v = 0. This completes the proof when s = 1.
Notice that the linear map grDa
a′ fits into a commutative diagram
Γ(E0,`(a)
(
SiN∗(λa))
)
∩ ker ∂̄
��
da
a′ // Γ
(
E0,`(a′)
(
Si−1N∗(λa′)
))
∩ ker ∂̄
��
H`(a)
(
X0,S
iN∗(λa)
)
// H`(a′)
(
X0,S
i−1N∗(λa′)
)
griVµa
OO
grDa
a′ // gri−1Vµa′ ,
OO
(7.13)
11We will at this point avoid discussion about the convergence of the sum as we will not need it.
30 T. Salač
where the lower vertical arrows are the isomorphisms from Lemma 7.9, the upper vertical arrows
are the canonical projections and the map daa′ 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 daa′ by the diagram
Γ(E0,q(SiN∗(λa))) ∩Ker(∂̄)
(dab )⊕(dac )
// Γ(E0,q(Si−1N∗(λb ⊕ λc)))
Γ(E0,q−1(Si−1N∗(λb ⊕ λc)))
∂̄
33
(db
a′ )+(dc
a′ )// Γ(E0,q−1(Si−2N∗(λa′))) ∩Ker(∂̄),
where we for brevity put S•N∗(λb ⊕ λc) := S•N∗(λb)⊕S•N∗(λc). Following the same line of
arguments as in the case s = 1, we easily find that Da
a′(v)(x0) = 0 whenever j2x0
v = 0.
In order to prove the claim about grDa
a′ , we need to replace everywhere daa′ by its restric-
tion δaa′ 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
griVµa [↑] := gri−q(a)Vµa and griVµa [↑] := gri−q(a)Vµa . We can now rewrite the maps from (7.11)
and (7.12) as
grDa
a′ : gr`Vµa [↑]→ gr`−1Vµa′ [↑] and grDa
a′ : gr`Vµa [↑]→ gr`−1Vµa′ [↑],
respectively, where ` ≥ 0 is the corresponding integer. We also put
gr`Vj,i[↑] =
⊕
a∈Skj : q(a)=i
gr`Vµa [↑], gr`Vj [↑] =
j⊕
i=0
gr`Vj,i[↑]
and
gr`Vj,i[↑] =
⊕
a∈Skj : q(a)=i
gr`Vµa [↑], gr`Vj [↑] =
j⊕
i=0
gr`Vj,i[↑].
We view grDa
a′ also as a map gr`Vj [↑]→ gr`−1Vj+1[↑] by extending it from gr`Vµa [↑] by zero to
all the other summands. We put
grDj :=
∑
a∈Skj , a′∈Skj+1 : a<a′
grDa
a′ : gr`Vj [↑]→ gr`−1Vj+1[↑]
and
gr(Dj)
i
i′ : gr`Vj,i[↑]→ gr`Vj [↑]
grDj−−−→ gr`−1Vj+1[↑]→ gr`−1Vj+1,i′ [↑],
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 ∈ Skj , a′ ∈ Skj+1, then q(a′) ≤ q(a) + 1. This implies
that gr(Dj)
i
i′ = 0 if i 6= i′ or i′ 6= i+ 1. Then grDj is
gr`Vj,0[↑]
⊕
gr`Vj,1[↑]
⊕
. . .
−→
↘
−→
↘
. . .
gr`−1Vj+1,0[↑]
⊕
gr`−1Vj+1,1[↑]
⊕
. . .
,
where the horizontal arrows and the diagonal arrows are gr(Dj)
i
i and gr(Dj)
i
i+1, respectively.
We similarly define linear maps grDj : gr`Vj [↑] → gr`−1Vj+1[↑] and gr(Dj)
i
i′ : gr
`Vj,i[↑] →
gr`−1Vj+1,i′ [↑].
k-Dirac Complexes 31
Remark 7.13. Notice that
gr`Vj,i[↑] =
⊕
a∈Skj : q(a)=i
gr`Vµa [↑] =
⊕
a∈Sk : r(a)=j, q(a)=i
gr`−q(a)Vµa
=
⊕
a∈Sk : |a|=i+j, `(a)=(n2)−i
gr`−iVµa = E
i+j,(n2)−i
1 (`+ j).
Put p := i+ j, q :=
(
n
2
)
− i and r := `+ j. Then we can view gr(Dj)
i
i and gr(Dj)
i
i+1 as maps
Ep,q1 (r)→ Ep+1,q
1 (r) and Ep,q1 (r)→ Ep+2,q−1
1 (r),
respectively. By the definition of gr(Dj)
i
i from Lemma 7.12, it follows that we can view it as
the differential d1 on the first page of the spectral sequence from Proposition 7.10.
Suppose that v ∈ Ep,q1 (s) satisfies d1(v) = 0. Then we can apply the differential d2 living
on the second page to v + im(d1) and, comparing this with the definition of gr(Dj)
i
i+1 from
Lemma 7.12, we find that
d2(v + im(d1)) = gr(Dj)
i
i+1(v) + im(d1). (7.14)
Similarly we find that gr`Vj,i = F p,q1 (s) where p, q and s are as above. Moreover we can view
gr(Dj)
i
i and gr(Dj)
i
i+1 as maps
F p,q1 (s)→ F p+1,q
1 (s) and F p,q1 (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(Dj)
i
i′ is the restriction of gr(Dj)
i
i′ to the corresponding
subspace, we see that gr(Dj)
i
i coincides with the differential on the first page of the spectral
sequence from Proposition 7.11 and that gr(Dj)
i
i+1 is related to the differential on the second
page just as gr(Dj)
i
i+1 is related to d2.
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
gr`+jV0[↑] grD0−−−→ gr`+j−1V1[↑]→ · · · → gr`Vj [↑]
grDj−−−→ gr`−1Vj+1[↑]→ · · · (7.15)
of finite-dimensional vector spaces. The complex contains a sub-complex
gr`+jV0[↑] grD0−−−→ gr`+j−1V1[↑]→ · · · → gr`Vj [↑]
grDj−−−→ gr`−1Vj+1[↑]→ · · · , (7.16)
which is also exact.
Proof. Let v ∈ gr`Vj [↑], j ≥ 1 be such that grDj(v) = 0. Write v = (v0, . . . , vj) with respect to
the decomposition given above, i.e., vi ∈ gr`Vj,i[↑]. Assume that v0 = v1 = · · · = vi−1 = 0 and
that vi 6= 0. We have that gr(Di
i)(vi) = 0 and gr(Di
i+1)(vi) + gr(Di+1
i+1)(vi+1) = 0. If we view vi
as an element of Ep,q1 (s) as in Remark 7.13, we see that d1(vi) = 0 and by (7.14), we find that
d2(vi) = 0. By Proposition 7.10, the spectral sequence Ep,q(r) collapses on the second page and
by part (i), we have that ker(d2) = im(d2) beyond the
(
n
2
)
-th diagonal. By Remark 7.13 again,
32 T. Salač
gr`Vj,i[↑] lives on the (
(
n
2
)
+ j)-th diagonal. We see that there are ti−1 ∈ gr`+1Vj−1,i−1[↑] and
ti ∈ gr`+1Vj−1,i[↑] such that gr(Di−1
i−1)(ti−1) = 0 and gr(Di−1
i )(ti−1) +gr(Di
i)(ti) = vi. 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`+1Vj−1[↑] such that v = grDj−1(t).
The proof of the exactness of the second sequence (7.16) proceeds similarly. We only replace
gr`Vj [↑] by gr`Vj [↑], gr`Vj,i[↑] by gr`Vj,i[↑], use that the second spectral sequence from Propo-
sition 7.11 has the same key properties as the spectral sequence from Proposition 7.10 and the
end of Remark 7.13. �
Acknowledgements
The author is grateful to Vladimı́r Souček for his support and many useful conversations. The
author would also like to thank to Lukáš Krump for the possibility of using his package for
the Young diagrams. The author wishes to thank to the unknown referees for many helpful
suggestions which considerably improved this article. The research was partially supported by
the grant 17-01171S of the Grant Agency of the Czech Republic.
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1 Introduction
2 Preliminaries
2.1 Review of parabolic geometries
2.2 Weighted differential operators
2.3 Ideal sheaf of an analytic subvariety
2.4 The Penrose transform
3 Lie theory
3.1 Lie algebra g and parabolic subalgebras
3.2 Relative Hasse diagram Wrq
4 Geometric structures attached to (3.3)
4.1 Homogeneous spaces
4.2 Filtrations of the tangent bundles of M and CS
4.3 Projections and
5 The Penrose transform for the k-Dirac complexes
5.1 Double complex of relative forms
5.2 Relative de Rham complex
5.3 Twisted relative de Rham complex
5.4 Relative BGG sequence
5.5 Direct image of the relative BGG sequence
5.6 Double complex of relative forms II
6 k-Dirac complexes
7 Formal exactness of k-Dirac complexes
7.1 Formal neighborhood of -1(x0)
7.2 The double complex on the formal neighborhood of -1(x0)
7.3 Long exact sequence of weighted jets
References
|
| id | nasplib_isofts_kiev_ua-123456789-209452 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T18:40:32Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Salač, T. 2025-11-21T19:04:49Z 2018 k-Dirac Complexes / T. Salač // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 27 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 58J10; 32N05; 32L25; 35A22; 53C28; 58A20 arXiv: 1705.09469 https://nasplib.isofts.kiev.ua/handle/123456789/209452 https://doi.org/10.3842/SIGMA.2018.012 This is the first paper in a series of two papers. In this paper, we construct complexes of invariant differential operators that 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. The author is grateful to Vladimír Souček for his support and many useful conversations. The author would also like to thank Lukáš Krump for the possibility of using his package for the Young diagrams. The author wishes to thank the unknown referees for many helpful suggestions, which considerably improved this article. The research was partially supported by the grant 17-01171S of the Grant Agency of the Czech Republic. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications k-Dirac Complexes Article published earlier |
| spellingShingle | k-Dirac Complexes Salač, T. |
| title | k-Dirac Complexes |
| title_full | k-Dirac Complexes |
| title_fullStr | k-Dirac Complexes |
| title_full_unstemmed | k-Dirac Complexes |
| title_short | k-Dirac Complexes |
| title_sort | k-dirac complexes |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209452 |
| work_keys_str_mv | AT salact kdiraccomplexes |