$k$-Dirac Complexes

$\boldsymbol{k}$-Dirac Complexes

Tomáš SALAČ

T. Salač

Mathematical Institute, Charles University, Sokolovská 49/83, Prague, Czech
Republic<EMAIL_ADDRESS>http://www.karlin.mff.cuni.cz/~salac/

Received June 01, 2017, in final form February 06, 2018; Published online
February 16, 2018

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

Penrose transform; complexes of invariant differential operators; relative BGG
complexes; formal exactness; weighted jets

58J10; 32N05; 32L25; 35A22; 53C28; 58A20

##  1 Introduction

Let $h$ be a non-degenerate, symmetric and $\mathbb{C}$-bilinear form on
$\mathbb{C}^{2m}$. The Grassmannian variety $M$ of totally isotropic
$k$-dimensional subspaces in $\mathbb{C}^{2m}$ is a homogeneous space of a
$|2|$-graded parabolic geometry. We assume throughout the paper that
$n:=m-k\geq k\geq 2$. 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 $\mathcal{U}$ is a domain of
monogenicity if there is no open set $\mathcal{U}^{\prime}$ with
$\mathcal{U}\subsetneq\mathcal{U}^{\prime}$ such that each monogenic
function111A monogenic function is a null solution of the $k$-Dirac operator.
in $\mathcal{U}$ extends to a monogenic function in $\mathcal{U}^{\prime}$.
Recall from [16, Section 4] that the Dolbeault resolution together with some
$L^{2}$ estimates are crucial in a proof of the statement that any
pseudoconvex domain is a domain of holomorphy.

The $k$-Dirac complexes belong to the singular infinitesimal character and so
the BGG machinery introduced in [9] is not available. However, we will show
that each $k$-Dirac complex arises as the direct image of a relative BGG
sequence (see [10, 11] for a recent publication on this topic) and so, this
paper fits into the scheme of the Penrose transform (see [2, 26]). In
particular, we will work here only in the setting of complex parabolic
geometries.

The machinery of the Penrose transform is a main tool used in [1]. The main
result of that article is a construction of families of locally exact
complexes of invariant differential operators on quaternionic manifolds. One
of these quaternionic complexes can (see [3, 4, 13]) be identified with a
resolution of the $k$-Cauchy–Fueter operator which has been intensively
studied in Clifford analysis (see again [12, 20]). In order to prove the local
exactness of this quaternionic complex, one uses that an almost quaternionic
structure is a $|1|$-graded parabolic geometry and the theory of constant
coefficient operators from [19].

Unfortunately, the parabolic geometry on $M$ is $|2|$-graded and so there is
canonical a 2-step filtration of the tangent bundle of $M$ given by a bracket
generating distribution. With such a structure, it is more natural to work
with weighted jets (see [18]) rather than usual jets and we use this concept
also here, i.e., we prove the formal exactness of the $k$-Dirac complexes with
respect to the weighted jets. Nevertheless, we will prove in [24] that the
formal exactness of the $k$-Dirac complex (or more precisely the exactness of
(7.16) for each $\ell+j\geq 0$) is enough to conclude that it descends to a
resolution of the $k$-Dirac operator.

We consider here only the even case $\mathbb{C}^{2m}$ as, due to the
representation theory, the Penrose transform does not work in odd dimension
$\mathbb{C}^{2m+1}$ and it seems that this case has to be treated by
completely different methods. The assumption $n\geq k$ 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 $\mathrm{G}$-invariant operator is determined by a certain
$\mathrm{P}$-equivariant homomorphism. As this correspondence works also in
the smooth setting, passing from the holomorphic setting to the smooth setting
is rather elementary.

The abstract approach of the Penrose transform is not very helpful when one is
interested in local formulae of differential operators. Local formulae of the
operators in the $k$-Dirac complexes can be found in [22]. Notice that in this
article we construct only one half of each complex from [22]. This is due to
the fact that the complex space of spinors decomposes into two irreducible
$\mathfrak{so}(2n,\mathbb{C})$-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

  * •

$M(n,k,\mathbb{C})$ matrices of size $n\times k$ with complex coefficients;

  * •

$M(n,\mathbb{C}):=M(n,n,\mathbb{C})$;

  * •

$A(n,\mathbb{C}):=\\{A\in M(n,\mathbb{C})\,|\,A^{T}=-A\\}$;

  * •

$1_{n}$ identity $n\times n$-matrix;

  * •

$[v_{1},\dots,v_{\ell}]$ linear span of vectors $v_{1},\dots,v_{\ell}$.

##  2 Preliminaries

In Section 2 we will review some well known material. Namely, in Section 2.1
we will summarize some theory of complex parabolic geometries. We will recall
in Section 2.2 the concept of weighted jets on filtered manifolds and in
Section 2.3 the definition of the normal bundle associated to analytic
subvariety and the formal neighborhood. In Section 2.4 we will give a short
summary of the Penrose transform.

See [8] for a thorough introduction into the theory of parabolic geometries.
The concept of weighted jets was originally introduced in the smooth setting
by Tohru Morimoto, see for example [18]. Sections 2.3 and 2.4 were taken
mostly from [2, 26].

###  2.1 Review of parabolic geometries

Let $\mathfrak{g}$ be a complex semi-simple Lie algebra, $\mathfrak{h}$ be a
Cartan subalgebra, $\triangle$ be the associated set of roots, $\triangle^{+}$
be a set of positive roots and
$\triangle^{0}=\\{\alpha_{1},\dots,\alpha_{m}\\}$ be the associated set of
(pairwise distinct) simple roots. We will denote by $\mathfrak{g}_{\alpha}$
the root space associated to $\alpha\in\triangle$ and we will write $\alpha>0$
if $\alpha\in\triangle^{+}$ and $\alpha<0$ if $-\alpha\in\triangle^{+}$. Given
$\alpha\in\triangle$, there are unique integers
$\lambda_{1},\dots,\lambda_{m}$ such that
$\alpha=\lambda_{1}\alpha_{1}+\dots+\lambda_{m}\alpha_{m}$. If
$\Sigma\subset\triangle^{0}$, then the integer
$ht_{\Sigma}(\alpha):=\sum\limits_{i\colon\alpha_{i}\in\Sigma}\lambda_{i}$ is
called the $\Sigma$-height of $\alpha$. We put
$\mathfrak{g}_{i}:=\bigoplus\limits_{\alpha\colon
ht_{\Sigma}(\alpha)=i}\mathfrak{g}_{\alpha}$. Then there is an integer $k\geq
0$ such that $\mathfrak{g}_{k}\neq\\{0\\}$, $\mathfrak{g}_{\ell}=\\{0\\}$
whenever $|\ell|>k$ and

$\displaystyle\mathfrak{g}=\mathfrak{g}_{-k}\oplus\mathfrak{g}_{-k+1}\oplus\dots\oplus\mathfrak{g}_{k-1}\oplus\mathfrak{g}_{k}.$
(2.1)

The direct sum decomposition (2.1) is the $|k|$-grading on $\mathfrak{g}$
associated to $\Sigma$.

Since
$[\mathfrak{g}_{\alpha},\mathfrak{g}_{\beta}]\subset\mathfrak{g}_{\alpha+\beta}$
for every $\alpha,\beta\in\triangle$, it follows that
$[\mathfrak{g}_{i},\mathfrak{g}_{j}]\subset\mathfrak{g}_{i+j}$ for each
$i,j\in\mathbb{Z}$. In particular, it follows that $\mathfrak{g}_{0}$ is a
subalgebra and it can be shown that it is always reductive, i.e.,
$\mathfrak{g}_{0}=\mathfrak{g}_{0}^{ss}\oplus\mathfrak{z}(\mathfrak{g}_{0})$
where $\mathfrak{g}_{0}^{ss}:=[\mathfrak{g}_{0},\mathfrak{g}_{0}]$ is semi-
simple and $\mathfrak{z}(\mathfrak{g}_{0})$ is the center (see [8, Corollary
2.1.6]). Moreover, each subspace $\mathfrak{g}_{i}$ is
$\mathfrak{g}_{0}$-invariant and the Killing form of $\mathfrak{g}$ induces an
isomorphism $\mathfrak{g}_{-i}\cong\mathfrak{g}_{i}^{\ast}$ of
$\mathfrak{g}_{0}$-modules where ∗ denotes the dual module. We put

$\displaystyle\mathfrak{g}^{i}:=\bigoplus_{j\geq
i}\mathfrak{g}_{j},\qquad\mathfrak{g}_{-}:=\mathfrak{g}_{-k}\oplus\dots\oplus\mathfrak{g}_{-1},\qquad\mathfrak{p}_{\Sigma}:=\mathfrak{g}^{0}\qquad\mathrm{and}\qquad\mathfrak{p}_{+}:=\mathfrak{g}^{1}.$

Then $\mathfrak{p}_{\Sigma}$ is the parabolic subalgebra associated to the
$|k|$-grading and
$\mathfrak{p}_{\Sigma}=\mathfrak{g}_{0}\oplus\mathfrak{p}_{+}$ is known as the
Levi decomposition (see [2, Section 2.2]). This means that $\mathfrak{p}_{+}$
is the nilradical222Recall that the nilradical is a maximal nilpotent ideal
and that it is unique. and that $\mathfrak{g}_{0}$ is a maximal reductive
subalgebra called the Levi factor. It is clear that each subspace
$\mathfrak{g}^{i}$ is $\mathfrak{p}_{\Sigma}$-invariant and that
$\mathfrak{g}_{-}$ is a nilpotent subalgebra. Moreover, it can be shown that,
as a Lie algebra, $\mathfrak{g}_{-}$ is generated by $\mathfrak{g}_{-1}$.

The algebra $\mathfrak{b}:=\mathfrak{p}_{\triangle^{0}}$ is called the
standard Borel subalgebra. A subalgebra of $\mathfrak{g}$ is called standard
parabolic if it contains $\mathfrak{b}$ and in particular,
$\mathfrak{p}_{\Sigma}$ is such an algebra. More generally, a subalgebra of
$\mathfrak{g}$ 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 $\mathfrak{p}$
instead of $\mathfrak{p}_{\Sigma}$.

Let $s_{i}$ be the simple reflection associated to $\alpha_{i}$,
$i=1,2,\dots,m$, $W_{\mathfrak{g}}$ be the Weyl group of $\mathfrak{g}$ and
$W_{\mathfrak{p}}$ be the subgroup of $W_{\mathfrak{g}}$ generated by
$\\{s_{i}\colon\alpha_{i}\not\in\Sigma\\}$. Then $W_{\mathfrak{p}}$ is
isomorphic to the Weyl group of $\mathfrak{g}_{0}^{ss}$ and the directed graph
that encodes the Bruhat order on $W_{\mathfrak{g}}$ contains a subgraph called
the Hasse diagram $W^{\mathfrak{p}}$ attached to $\mathfrak{p}$ (see [2,
Section 4.3]). The vertices of $W^{\mathfrak{p}}$ consist of those $w\in
W_{\mathfrak{g}}$ such that $w.\lambda$ is $\mathfrak{p}$-dominant for any
$\mathfrak{g}$-dominant weight $\lambda$ where the dot stands for the affine
action, namely, $w.\lambda=w(\lambda+\rho)-\rho$ where $\rho$ is the lowest
form. It turns out that each right coset of $W_{\mathfrak{p}}$ in
$W_{\mathfrak{g}}$ contains a unique element from $W^{\mathfrak{p}}$ and it
can be shown that this is the element of minimal length (see [2, Lemma
4.3.3]). This identifies $W^{\mathfrak{p}}$ with $W_{\mathfrak{p}}\backslash
W_{\mathfrak{g}}$.

We will need also the relative case. Assume that
$\Sigma^{\prime}\subset\triangle^{0}$ and put
$\mathfrak{r}:=\mathfrak{p}_{\Sigma^{\prime}}$. Then
$\mathfrak{q}:=\mathfrak{r}\cap\mathfrak{p}=\mathfrak{p}_{\Sigma\cup\Sigma^{\prime}}$
is a standard parabolic subalgebra and
$\mathfrak{l}:=\mathfrak{g}_{0}^{ss}\cap\mathfrak{q}$ is a parabolic
subalgebra of $\mathfrak{g}_{0}^{ss}$ (see [2, Section 2.4]). The definition
of the Hasse diagram attached to $\mathfrak{p}$ applies also to the pair
$(\mathfrak{g}_{0}^{ss},\mathfrak{l})$, namely an element $w\in
W_{\mathfrak{p}}$ (as [2, Section 4.4]) belongs to the relative Hasse diagram
$W^{\mathfrak{q}}_{\mathfrak{p}}$ if it is the element of minimal length in
its right coset of $W_{\mathfrak{q}}$ in $W_{\mathfrak{p}}$. Hence,
$W^{\mathfrak{q}}_{\mathfrak{p}}$ is a subset of $W_{\mathfrak{g}}$ which can
be naturally identified with $W_{\mathfrak{q}}\backslash W_{\mathfrak{p}}$.

There is (up to isomorphism) a unique connected and simply connected complex
Lie group $\mathrm{G}$ with Lie algebra $\mathfrak{g}$. Assume that
$\Sigma=\\{\alpha_{i_{1}},\dots,\alpha_{i_{j}}\\}$. Let
$\omega_{1},\dots,\omega_{m}$ be the fundamental weights associated to the
simple roots and $\mathbb{V}$ be an irreducible $\mathfrak{g}$-module with
highest weight $\lambda:=\omega_{i_{1}}+\dots+\omega_{i_{j}}$. Since any
$\mathfrak{g}$-representation integrates to $\mathrm{G}$, $\mathbb{V}$ is also
a $\mathrm{G}$-module. The action descends to the projective space
$\mathbb{P}(\mathbb{V})$ and the stabilizer of the line spanned by a non-zero
highest weight vector $v$ is the associated parabolic subgroup $\mathrm{P}$.
This is by definition a closed subgroup of $\mathrm{G}$ and its Lie algebra is
$\mathfrak{p}$. The homogeneous space $\mathrm{G}/\mathrm{P}$ is biholomorphic
to the $\mathrm{G}$-orbit of $[v]\in\mathbb{P}(\mathbb{V})$ and since it is
completely determined by $\Sigma$, we denote it by crossing in the Dynkin
diagram of $\mathfrak{g}$ the simple roots from $\Sigma$. We will for brevity
put $M:=\mathrm{G}/\mathrm{P}$ and denote by
$\textbf{p}\colon\mathrm{G}\rightarrow M$ the canonical projection.

On $\mathrm{G}$ lives a tautological $\mathfrak{g}$-valued 1-form $\omega$
which is known as the Maurer-Cartan form. This form is
$\mathrm{P}$-equivariant in the sense that for each
$p\in\mathrm{P}\colon(r^{p})^{\ast}\omega=\operatorname{Ad}(p^{-1})\circ\omega$
where $\operatorname{Ad}$ is the adjoint representation and $r^{p}$ is the
principal action of $p$. If $\mathbb{V}$ is a subspace of $\mathfrak{g}$ and
$g\in\mathrm{G}$, then $\omega^{-1}_{g}(\mathbb{V})$ is a subspace of
$T_{g}\mathrm{G}$ and the disjoint union
$\bigcup_{g\in\mathrm{G}}\omega_{g}^{-1}(\mathbb{V})$ determines a
distribution on $\mathrm{G}$ which we for brevity denote by
$\omega^{-1}(\mathbb{V})$. Since $T\textbf{p}=T\textbf{p}\circ Tr^{p}$, it
follows that $T\textbf{p}(\omega^{-1}(\mathbb{V}))$ is a well-defined
distribution on $M$ provided that $\mathbb{V}$ is $\mathrm{P}$-invariant. In
particular, this applies to $\mathfrak{g}^{i}$ and we put
$F_{i}:=T\textbf{p}(\omega^{-1}(\mathfrak{g}^{i}))$. Since
$\ker(T\textbf{p})=\omega^{-1}(\mathfrak{p})$, it follows that the filtration
$\mathfrak{g}^{-k}/\mathfrak{p}\supset\dots\supset\mathfrak{g}^{-1}/\mathfrak{p}\supset\mathfrak{g}^{0}/\mathfrak{p}$
gives a filtration $TM=F_{-k}\supset F_{-k+1}\supset\dots\supset F_{-1}\supset
F_{0}=\\{0\\}$ of the tangent bundle $TM$ where $\\{0\\}$ is the zero section.
The graded tangent bundle associated to the filtration $\\{F_{-i}\colon
i=k,\dots,0\\}$ is $gr(TM):=\bigoplus_{i=-k}^{-1}gr_{i}(TM)$ where
$gr_{i}(TM):=F_{i}/F_{i+1}$. Since $M$ is the homogeneous model, we have the
following:

  * •

the filtration is compatible333Filtrations which satisfy this property are
called regular. with the Lie bracket of vector fields in the sense that the
commutator of a section of $F_{i}$ and a section of $F_{j}$ is a section of
$F_{i+j}$,

  * •

the Lie bracket descends to a vector bundle map
$\mathcal{L}\colon\Lambda^{2}gr(TM)\rightarrow gr(TM)$, called the Levi form,
which is homogeneous of degree zero and

  * •

$(gr(T_{x}M),\mathcal{L}_{x})$, $x\in M$ is a nilpotent Lie algebra isomorphic
to $\mathfrak{g}_{-}$.

Hence, $(gr(TM),\mathcal{L})$ is a locally trivial bundle of nilpotent Lie
algebras with typical fiber $\mathfrak{g}_{-}$ and it follows that $F_{-1}$ is
a bracket generating distribution.

We denote by $T^{\ast}M=\Lambda^{1,0}T^{\ast}M$ the vector bundle dual to
$TM$, i.e., the fiber over $x\in M$ is the space of $\mathbb{C}$-linear maps
$T_{x}M\rightarrow\mathbb{C}$. The filtration of $TM$ induces a filtration
$T^{\ast}M=F_{1}\supset F_{2}\supset\dots\supset F_{k+1}=\\{0\\}$ where
$F_{i}:=F_{-i+1}^{\perp}$ is the annihilator of $F_{-i+1}$. We put
$gr_{i}(T^{\ast}M):=F_{i}/F_{i+1}$, $i=1,2,\dots,k$ so that
$gr(T^{\ast}M)=\bigoplus_{i=1}^{k}gr_{i}(T^{\ast}M)$ is the associated graded
vector bundle and $gr_{i}(T^{\ast}M)$ is isomorphic to the dual of
$gr_{i}(TM)$.

###  2.2 Weighted differential operators

Let $M$ be the homogeneous space with the regular filtration $\\{F_{-j}\colon
j=0,\dots,k\\}$ as in Section 2.1. As $M$ is a complex manifold,
$TM_{\mathbb{C}}:=TM\otimes\mathbb{C}=T^{1,0}M\oplus T^{0,1}M$ where the first
and the second summand is the holomorphic and the anti-holomorphic part444The
holomorphic and anti-holomorphic part is the $-i$ and the $+i$-eigenspace,
respectively, for the canonical almost complex structure on
$TM_{\mathbb{C}}$., respectively. As each vector bundle $F_{-j}$ is a
holomorhic sub-bundle of $TM$, we have
$F_{-j}\otimes\mathbb{C}=F^{1,0}_{-j}\oplus F^{0,1}_{-j}$ as above.

Let $\mathcal{U}$ be an open subset of $M$ and $X$ be a holomorphic vector
field on $\mathcal{U}$. The weighted order $\mathfrak{wo}(X)$ of $X$ is the
smallest integer $j$ such that $X\in\Gamma(F_{-j}^{1,0}|_{\mathcal{U}})$. A
differential operator $D$ acting on the space $\mathcal{O}(\mathcal{U})$ of
holomorphic functions on $\mathcal{U}$ is called a differential operator of
weighted order at most $r$ if for each $x\in\mathcal{U}$ there is an open
neighborhood $\mathcal{U}_{x}$ of $x$ with a local framing555This means that
the holomorphic vector fields $X_{1},\dots,X_{p}$ trivialize $T^{1,0}M$ over
$\mathcal{U}_{x}$. $\\{X_{1},\dots,X_{p}\\}$ by holomorphic vector fields such
that

$\displaystyle
D|_{\mathcal{U}_{x}}=\sum_{a\in\mathbb{N}_{0}^{p}}f_{a}X_{1}^{a_{1}}\cdots
X_{p}^{a_{p}},$

where $\mathbb{N}_{0}^{p}:=\\{a=(a_{1},\dots,a_{p})\colon a_{i}\in\mathbb{Z},\
a_{i}\geq 0,\ i=1,\dots,p\\}$, $f_{a}\in\mathcal{O}(\mathcal{U}_{x})$ and for
all $a$ in the sum with $f_{a}$ non-zero:
$\sum\limits_{i=1}^{p}a_{i}.\mathfrak{wo}(X_{i})\leq r$. We write
$\mathfrak{wo}(D)\leq r$.

Let $\mathcal{O}_{x}$ be the space of germs of holomorphic functions at $x$.
We denote by $\mathfrak{F}_{x}^{i}$ the space of those germs
$f\in\mathcal{O}_{x}$ such that $Df(x)=0$ for every differential operator $D$
which is defined on an open neighborhood of $x$ and $\mathfrak{wo}(D)\leq i$.
We put $\mathfrak{J}^{i}_{x}:=\mathcal{O}_{x}/\mathfrak{F}_{x}^{i+1}$, denote
by $\mathfrak{j}^{i}_{x}f\in\mathfrak{J}^{i}_{x}$ the class of $f$ and call it
the $i$-th weighted jet of $f$. Then the disjoint union
$\mathfrak{J}^{i}:=\cup_{x\in M}\mathfrak{J}_{x}^{i}$ is naturally a
holomorphic vector bundle over $M$, the canonical vector bundle map
$\mathfrak{J}^{i}\xrightarrow{\pi_{i}}\mathfrak{J}^{i-1}$ has constant rank
and thus, its kernel $\mathfrak{gr}^{i}$ is again a holomorphic vector bundle
with fiber $\mathfrak{gr}^{i}_{x}$ over $x$. Notice that for each integer
$i\geq 0$ there is a short exact sequence
$0\rightarrow\mathfrak{F}^{i+1}_{x}\rightarrow\mathfrak{F}^{i}_{x}\rightarrow\mathfrak{gr}^{i+1}_{x}\rightarrow
0$ of vector spaces.

Assume that $V$ is a holomorphic vector bundle over $M$. We denote by
$V^{\ast}$ the dual bundle, by $\langle-,-\rangle$ the canonical pairing
between $V$ and $V^{\ast}$ and finally, by $\mathcal{O}(V)_{x}$ the space of
germs of holomorphic sections of $V$ at $x$. We define $\mathfrak{F}_{x}^{i}V$
as the space of germs $s\in\mathcal{O}(V)_{x}$ such that
$\langle\lambda,s\rangle\in\mathfrak{F}_{x}^{i}$ for each
$\lambda\in\mathcal{O}(V^{\ast})_{x}$. We put
$\mathfrak{J}^{i}_{x}:=\mathcal{O}(V)_{x}/\mathfrak{F}_{x}^{i+1}V$, denote by
$\mathfrak{j}^{i}_{x}s\in\mathfrak{J}^{i}_{x}V$ the equivalence class of $s$
and call it the $i$-th weighted jet of $s$. Then the disjoint union
$\mathfrak{J}^{i}V:=\bigcup_{x\in M}\mathfrak{J}^{i}_{x}M$ is naturally a
holomorphic vector bundle over $M$, the canonical bundle map
$\mathfrak{J}^{i}V\xrightarrow{\pi_{i}}\mathfrak{J}^{i-1}V$ has constant rank
and thus, its kernel $\mathfrak{gr}^{i}V$ is again a holomorphic vector bundle
and we denote by $\mathfrak{gr}^{i}_{x}V$ its fiber over $x$. As above, there
is for each integer $i\geq 0$ a short exact sequence
$0\rightarrow\mathfrak{F}^{i+1}_{x}V\rightarrow\mathfrak{F}^{i}_{x}V\rightarrow\mathfrak{gr}^{i+1}_{x}V\rightarrow
0$ and just as in the smooth case, there is a canonical linear isomorphism
$\mathfrak{gr}^{i}_{x}\otimes V_{x}\rightarrow\mathfrak{gr}^{i}_{x}V$.

######  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 $\mathcal{F}^{i}$ and
$\mathcal{J}^{i}$ and $gr^{i}$ and $j^{i}_{x}f$ instead of $\mathfrak{F}^{i}$
and $\mathfrak{J}^{i}$ and $\mathfrak{gr}^{i}$ and $\mathfrak{j}^{i}_{x}f$,
respectively. The vector bundle $gr^{i}$ is canonically isomorphic to the
$i$-th symmetric power $S^{i}T^{\ast}M$.

Assume that there is a $\mathrm{P}$-module $\mathbb{V}$ such that $V$ is
isomorphic to the $\mathrm{G}$-homogeneous vector bundle
$\mathrm{G}\times_{\mathrm{P}}\mathbb{V}$. Let $e$ be the identity element of
$\mathrm{G}$. Then we call the point $x_{0}:=e\mathrm{P}$ the origin of $M$
and we put

$\displaystyle\mathfrak{J}^{i}\mathbb{V}:=\mathfrak{J}^{i}_{x_{0}}V\qquad\mathrm{and}\qquad\mathfrak{gr}^{i}\mathbb{V}:=\mathfrak{gr}^{i}_{x_{0}}V.$
(2.2)

There are linear isomorphisms

$\displaystyle\mathfrak{gr}^{r}\mathbb{V}\cong\mathfrak{gr}^{r}_{x_{0}}\otimes\mathbb{V}\cong\bigoplus_{i_{1}+2i_{2}+\dots+ki_{k}=r}S^{i_{1}}\mathfrak{g}_{1}\otimes
S^{i_{2}}\mathfrak{g}_{2}\otimes\dots\otimes
S^{i_{k}}\mathfrak{g}_{k}\otimes\mathbb{V}.$ (2.3)

We will be interested in the sub-bundle $S^{i}gr_{1}(T^{\ast}M)\otimes V$ of
$\mathfrak{gr}^{i}V$. Notice that the fiber of this sub-bundle over $x\in M$
is $\\{\mathfrak{j}^{i}_{x}f\colon f\in\mathcal{O}(V)_{x},\
j^{i-1}_{x}f=0\\}$, 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 $x_{0}$ is isomorphic to
$S^{i}\mathfrak{g}_{1}\otimes\mathbb{V}$ and we denote it for brevity by
$gr^{i}\mathbb{V}$.

Suppose that $\mathbb{W}$ is another $\mathrm{P}$-module and
$W:=\mathrm{G}\times_{\mathrm{P}}\mathbb{W}$ be the associated homogeneous
vector bundle. We say that the weighted order of a linear differential
operator $D\colon\Gamma(V)\rightarrow\Gamma(W)$ is at most $r$ if for each
$x\in M$, $s\in\mathcal{O}(V)_{x}\colon\mathfrak{j}^{r}_{x}s=0\Rightarrow
Ds(x)=0$. It is well known (see [18]) that $D$ induces for each $i\geq 0$ a
vector bundle map $\mathfrak{gr}^{i}V\rightarrow\mathfrak{gr}^{i-r}W$ where we
agree that $\mathfrak{gr}^{\ell}W=0$ if $\ell<0$. The restriction of this map
to the fibers over the origin is a linear map

$\displaystyle\mathfrak{gr}D\colon\
\mathfrak{gr}^{i}\mathbb{V}\rightarrow\mathfrak{gr}^{i-r}\mathbb{W}.$

###  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 $\mathcal{F}$ and $\mathcal{G}$ are sheaves on topological
spaces $X$ and $Y$, respectively, and that $\iota\colon X\rightarrow Y$ is a
continuous map. We denote by $\mathcal{F}_{x}$ the stalk of $\mathcal{F}$ at
$x\in X$ and by $\mathcal{F}(\mathcal{U})$ or by
$\Gamma(\mathcal{U},\mathcal{F})$ the space of sections of $\mathcal{F}$ over
an open set $\mathcal{U}$. Then the pullback sheaf $\iota^{-1}\mathcal{G}$ is
a sheaf on $X$ and the direct image $\iota_{\ast}\mathcal{F}$ is a sheaf on
$Y$. The $q$-th direct image $\iota^{q}_{\ast}\mathcal{F}$ is a sheaf on $Y$,
it is defined as the sheafification of the pre-sheaf $\mathcal{V}\mapsto
H^{q}(\iota^{-1}(\mathcal{V}),\mathcal{F})$ where $\mathcal{V}$ is open in
$Y$.

Suppose now that $X$ and $Y$ are complex manifolds with structure sheaves of
holomorphic functions $\mathcal{O}_{X}$ and $\mathcal{O}_{Y}$, respectively,
that $\iota$ is holomorphic and that $\mathcal{G}$ is a sheaf of
$\mathcal{O}_{Y}$-modules. Then $\iota^{-1}\mathcal{G}$ is in general not a
sheaf of $\mathcal{O}_{X}$-modules. To fix this problem, we use that
$\iota^{-1}\mathcal{O}_{Y}$ is naturally a sub-sheaf of $\mathcal{O}_{X}$ and
define a new sheaf
$\iota^{\ast}\mathcal{G}:=\mathcal{O}_{X}\otimes_{\iota^{-1}\mathcal{O}_{Y}}\iota^{-1}\mathcal{G}$.
Then $\iota^{\ast}\mathcal{G}$ is by construction a sheaf of
$\mathcal{O}_{X}$-modules.

Now we can continue with the definition of the ideal sheaf. Suppose that the
holomorphic map $\iota$ is an embedding. The restriction $TY|_{X}$ contains
the tangent bundle $TX$ of $X$. The normal bundle $N_{X}$ of $X$ in $Y$ is
simply the quotient bundle, i.e., it fits into the short exact sequence
$0\rightarrow TX\rightarrow TY|_{X}\rightarrow NX\rightarrow 0$ of holomorhic
vector bundles. Dually, the co-normal bundle $N^{\ast}$ fits into the short
exact sequence $0\rightarrow N^{\ast}\rightarrow T^{\ast}Y|_{X}\rightarrow
T^{\ast}X\rightarrow 0$.

The structure sheaf $\mathcal{O}_{Y}$ contains a sub-sheaf called the ideal
sheaf $\mathcal{I}_{X}$. If $\mathcal{V}$ is an open subset of $Y$, then
$\mathcal{I}_{X}(\mathcal{V})=\\{f\in\mathcal{O}_{Y}(\mathcal{V})\colon f=0$
on $\mathcal{V}\cap X\\}$. Notice that $\mathcal{I}_{X}(\mathcal{V})$ is an
ideal in the ring $\mathcal{O}_{Y}(\mathcal{V})$ and hence, for each positive
integer $i$ there is the sheaf $\mathcal{I}^{i}_{X}$ whose space of sections
over $\mathcal{V}$ is $(\mathcal{I}_{X}(\mathcal{V}))^{i}$. Then there are
short exact sequences of sheaves

$\displaystyle
0\rightarrow\mathcal{I}_{X}\rightarrow\mathcal{O}_{Y}\rightarrow\iota_{\ast}\mathcal{O}_{X}\rightarrow
0$

and

$\displaystyle
0\rightarrow\mathcal{I}^{i+1}_{X}\rightarrow\mathcal{I}^{i}_{X}\rightarrow\iota_{\ast}\mathcal{O}(S^{i}N^{\ast})\rightarrow
0,$

where $S^{i}N^{\ast}$ is the $i$-th symmetric power of $N^{\ast}$ and we agree
that $\mathcal{I}^{0}_{X}=\mathcal{O}_{Y}$. We put
$\mathcal{F}^{i}_{X}:=\iota^{-1}\mathcal{I}_{X}^{i}$. As $\iota^{-1}$ is an
exact functor, we get short exact sequences

$\displaystyle
0\rightarrow\mathcal{F}_{X}\rightarrow\iota^{-1}\mathcal{O}_{Y}\rightarrow\mathcal{O}_{X}\rightarrow
0$

and

$\displaystyle
0\rightarrow\mathcal{F}^{i+1}_{X}\rightarrow\mathcal{F}^{i}_{X}\rightarrow\mathcal{O}(S^{i}N^{\ast})\rightarrow
0$

of sheaves on $X$. Here we use that the adjunction morphism
$\iota^{-1}\iota_{\ast}\mathcal{F}\rightarrow\mathcal{F}$ is an isomorphism
when $\mathcal{F}=\mathcal{O}_{X}$ or $\mathcal{O}(S^{i}N^{\ast})$.

Put $\mathcal{O}_{X}^{(i)}:=\mathcal{O}_{Y}/\mathcal{I}^{i+1}_{X}$. The pair
$(X,\mathcal{O}^{(i)}_{X})$ is called the $i$-th formal neighborhood of $X$ in
$Y$. Then
$\iota^{-1}\mathcal{O}_{X}^{(i)}\cong\iota^{-1}\mathcal{O}_{Y}/\mathcal{F}^{(i+1)}$
and since the support of $\mathcal{O}_{X}^{(i)}$ is contained in $X$, the
sheaf $\iota^{-1}\mathcal{O}_{X}^{(i)}$ contains basically the same
information as the sheaf $\mathcal{O}_{X}^{(i)}$. These sheaves will be
crucial in this article.

######  Remark 2.2.

The stalk of $\mathcal{F}^{i}_{X}$ at $x\in X$ is equal to
$\\{f\in\mathcal{F}_{x}\colon j^{i}_{x}f=0\\}$. Hence, if $X=x$ is a point,
the stalk of $\mathcal{F}^{i}_{X}$ at $x$ is
$\\{f\in(\mathcal{O}_{Y})_{x}\colon j^{i}_{x}f=0\\}$. 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.

###  2.4 The Penrose transform

Let us first set notation. Suppose that $\lambda\in\mathfrak{h}^{\ast}$ is a
$\mathfrak{g}$-integral and $\mathfrak{p}$-dominant weight. Then there is (see
[2, Remark 3.1.6]) an irreducible $\mathrm{P}$-module $\mathbb{V}_{\lambda}$
with lowest weight $-\lambda$. We denote by
$V_{\lambda}:=\mathrm{G}\times_{\mathrm{P}}\mathbb{V}_{\lambda}$ the induced
vector bundle and by $\mathcal{O}_{\mathfrak{p}}(\lambda)$ the associated
sheaf of holomorphic sections.

Suppose that $\mathfrak{p}$, $\mathfrak{r}$ are standard parabolic
subalgebras. Then $\mathfrak{q}:=\mathfrak{r}\cap\mathfrak{p}$ is also a
standard parabolic subalgebra and we denote by $\mathrm{P}$ and $\mathrm{R}$
and $\mathrm{Q}$ the associated parabolic subgroups with Lie algebras
$\mathfrak{p}$ and $\mathfrak{r}$ and $\mathfrak{q}$, respectively, as
explained in Section 2.1. Then $\mathrm{Q}=\mathrm{R}\cap\mathrm{P}$ and there
is a double fibration diagram

$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathrm{G}/\mathrm{Q}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\scriptstyle{\tau}$$\textstyle{\mathrm{G}/\mathrm{R}}$$\textstyle{\mathrm{G}/\mathrm{P},}$

where $\eta$ and $\tau$ are the canonical projections. The space
$\mathrm{G}/\mathrm{R}$ is called the twistor space $TS$ and
$\mathrm{G}/\mathrm{Q}$ the correspondence space $CS$. Such a diagram is a
starting point for the Penrose transform.

Next we need to fix an $\mathfrak{r}$-dominant and integral weight
$\lambda\in\mathfrak{h}^{\ast}$. Then there is a relative BGG sequence
$\blacktriangle^{\ast}(\lambda)$ which is an exact sequence of holomorphic
sections of associated vector bundles over $CS$ and linear
$\mathrm{G}$-invariant differential operators such that
$\eta^{-1}\mathcal{O}_{\mathfrak{r}}(\lambda)$ is the kernel sheaf of the
first operator in the sequence. In other words, there is a long exact sequence
of sheaves

$\displaystyle
0\rightarrow\eta^{-1}\mathcal{O}_{\mathfrak{r}}(\lambda)\rightarrow\blacktriangle^{\ast}(\lambda).$

The upshot of this is that although the pullback sheaf
$\eta^{-1}\mathcal{O}_{\mathfrak{r}}(\lambda)$ is not a sheaf of holomorphic
sections of an associated vector bundle over $CS$, it is naturally a sub-sheaf
of $\mathcal{O}_{\mathfrak{q}}(\lambda)$ 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
$W^{\mathfrak{q}}_{\mathfrak{r}}$-orbit of $\lambda$.

Then we push down the relative BGG sequence by the direct image functor
$\tau_{\ast}$. 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
$\mathrm{G}/\mathrm{P}$.

##  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 $\mathfrak{g}=\mathfrak{so}(2m,\mathbb{C})$. Section 3 is
organized as follows: in Section 3.1 we will set notation and study the
gradings on $\mathfrak{g}$. In Section 3.2 we will compute the relative Hasse
diagram $W^{\mathfrak{q}}_{\mathfrak{r}}$.

###  3.1 Lie algebra $\mathfrak{g}$ and parabolic subalgebras

Let
$\\{e_{1},\dots,e_{m},e^{\ast}_{k+1},\dots,e^{\ast}_{m},e_{1}^{\ast},\dots,e_{k}^{\ast}\\}$
be the standard basis of $\mathbb{C}^{2m}$, $\delta$ be the Kronecker delta
and $h$ be the complex bilinear form that satisfies
$h(e_{i},e^{\ast}_{j})=\delta_{ij}$,
$h(e_{i},e_{j})=h(e^{\ast}_{i},e^{\ast}_{j})=0$ for all $i,j=1,\dots,m$. A
matrix belongs to the associated Lie algebra
$\mathfrak{g}:=\mathfrak{so}(h)\cong\mathfrak{so}(2m,\mathbb{C})$ if and only
if it is of the form

$\displaystyle\left(\begin{matrix}A&Z_{1}&Z_{2}&W\\\ X_{1}&B&D&-Z_{2}^{T}\\\
X_{2}&C&-B^{T}&-Z_{1}^{T}\\\
Y&-X_{2}^{T}&-X_{1}^{T}&-A^{T}\end{matrix}\right),$ (3.1)

where $A\in M(k,\mathbb{C})$, $B\in M(n,\mathbb{C})$, $C,D\in
A(n,\mathbb{C})$, $X_{1},X_{2},Z_{1}^{T},Z_{2}^{T}\in M(n,k,\mathbb{C})$,
$W,Y\in A(k,\mathbb{C})$.

The subspace of diagonal matrices $\mathfrak{h}$ is a Cartan subalgebra of
$\mathfrak{g}$. We denote by $\epsilon_{i}$ the linear form on $\mathfrak{h}$
defined by $\varepsilon_{i}\colon H=(h_{kl})\mapsto h_{ii}$. Then
$\\{\epsilon_{1},\dots,\epsilon_{m}\\}$ is a basis of $\mathfrak{h}^{\ast}$
and $\triangle=\\{\pm\epsilon_{i}\pm\epsilon_{j}\colon i,j=1,\dots,m\\}$. If
we choose $\triangle^{+}=\\{\epsilon_{i}\pm\epsilon_{j}\colon 1\leq i<j\leq
m\\}$ as positive roots, then the simple roots are
$\alpha_{i}:=\epsilon_{i}-\epsilon_{i+1}$, $i=1,\dots,m-1$ and
$\alpha_{m}:=\epsilon_{m-1}+\epsilon_{m}$. The associated fundamental weights
are $\omega_{i}=\epsilon_{1}+\dots+\epsilon_{i}$, $i=1,\dots,m-2$,
$\omega_{m-1}=\frac{1}{2}(\epsilon_{1}+\dots+\epsilon_{m-1}-\epsilon_{m})$ and
$\omega_{m}=\frac{1}{2}(\epsilon_{1}+\dots+\epsilon_{m})$. The lowest form
$\rho$ is equal to
$\frac{1}{2}\sum\limits_{\alpha\in\triangle^{+}}\alpha=\omega_{1}+\dots+\omega_{m}=(m-1,\dots,1,0)$.
If $\lambda=\sum\limits_{i=1}^{m}\lambda_{i}\epsilon_{i}$ where
$\lambda_{i}\in\mathbb{C}$, then we will also write
$\lambda=(\lambda_{1},\dots,\lambda_{m})$. The simple reflection $s_{i}\in
W_{\mathfrak{g}}$ associated to $\alpha_{i}$ acts on $\mathfrak{h}^{\ast}$ by

$\displaystyle
s_{i}(\lambda)=(\lambda_{1},\dots,\lambda_{i-1},\lambda_{i+1},\lambda_{i},\lambda_{i+2},\dots,\lambda_{m}),\qquad
i=1,\dots,m-1$ (3.2)

and

$\displaystyle
s_{m}(\lambda)=(\lambda_{1},\dots,\lambda_{m-2},-\lambda_{m},-\lambda_{m-1}).$

We will be interested in the double fibration diagram

$\displaystyle\begin{matrix}{{\hskip 5.0pt\hskip-2.5pt\raise 7.50002pt\hbox
to0.0pt{\hss\hskip
4.49997pt$\scriptstyle$\hss}\hbox{$\bullet$}\hskip-2.5pt\raise 2.20001pt\hbox
to17.7778pt{\hrulefill}\hskip 5.0pt\cdots\hskip 5.0pt\raise 2.20001pt\hbox
to17.7778pt{\hrulefill}\hskip-3.6pt\raise 7.50002pt\hbox to0.0pt{\hss\hskip
6.00006pt$\scriptstyle$\hss}\hbox{$\times$}\hskip-3.80005pt\raise
2.20001pt\hbox to17.7778pt{\hrulefill}\hskip 5.0pt\cdots\hskip 5.0pt\raise
2.20001pt\hbox to17.7778pt{\hrulefill}\hskip-2.5pt\raise 7.50002pt\hbox
to0.0pt{\hss\hskip
4.49997pt$\scriptstyle$\hss}\hbox{$\bullet$}\hskip-2.5pt\hbox to0.0pt{\raise
4.49997pt\hbox{$\diagup$}\hskip-3.6pt\raise
8.95557pt\hbox{\hbox{$\times$}\hskip 3.00003pt$\scriptstyle$}\hss}\hbox
to0.0pt{\raise-5.0pt\hbox{$\diagdown$}\hskip-2.5pt\raise-9.30003pt\hbox{\hbox{$\bullet$}\hskip
3.00003pt$\scriptstyle$}\hss}\hskip 11.88893pt}}\\\ \swarrow\ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \searrow\\\ \\\ {{\hskip 5.0pt\hskip-2.5pt\raise
7.50002pt\hbox to0.0pt{\hss\hskip
4.49997pt$\scriptstyle$\hss}\hbox{$\bullet$}\hskip-2.5pt\raise 2.20001pt\hbox
to17.7778pt{\hrulefill}\hskip 5.0pt\cdots\hskip 5.0pt\raise 2.20001pt\hbox
to17.7778pt{\hrulefill}\hskip-2.5pt\raise 7.50002pt\hbox to0.0pt{\hss\hskip
4.49997pt$\scriptstyle$\hss}\hbox{$\bullet$}\hskip-2.5pt\hbox to0.0pt{\raise
4.49997pt\hbox{$\diagup$}\hskip-3.6pt\raise
8.95557pt\hbox{\hbox{$\times$}\hskip 3.00003pt$\scriptstyle$}\hss}\hbox
to0.0pt{\raise-5.0pt\hbox{$\diagdown$}\hskip-2.5pt\raise-9.30003pt\hbox{\hbox{$\bullet$}\hskip
3.00003pt$\scriptstyle$}\hss}\hskip 11.88893pt}}\ \ \ \ \ \ \ \ \ \ {{\hskip
5.0pt\hskip-2.5pt\raise 7.50002pt\hbox to0.0pt{\hss\hskip
4.49997pt$\scriptstyle$\hss}\hbox{$\bullet$}\hskip-2.5pt\raise 2.20001pt\hbox
to17.7778pt{\hrulefill}\hskip 5.0pt\cdots\hskip 5.0pt\raise 2.20001pt\hbox
to17.7778pt{\hrulefill}\hskip-3.6pt\raise 7.50002pt\hbox to0.0pt{\hss\hskip
6.00006pt$\scriptstyle$\hss}\hbox{$\times$}\hskip-3.80005pt\raise
2.20001pt\hbox to17.7778pt{\hrulefill}\hskip 5.0pt\cdots\hskip 5.0pt\raise
2.20001pt\hbox to17.7778pt{\hrulefill}\hskip-2.5pt\raise 7.50002pt\hbox
to0.0pt{\hss\hskip
4.49997pt$\scriptstyle$\hss}\hbox{$\bullet$}\hskip-2.5pt\hbox to0.0pt{\raise
4.49997pt\hbox{$\diagup$}\hskip-2.5pt\raise
8.95557pt\hbox{\hbox{$\bullet$}\hskip 3.00003pt$\scriptstyle$}\hss}\hbox
to0.0pt{\raise-5.0pt\hbox{$\diagdown$}\hskip-2.5pt\raise-9.30003pt\hbox{\hbox{$\bullet$}\hskip
3.00003pt$\scriptstyle$}\hss}\hskip 11.88893pt}}\end{matrix}$ (3.3)

where, going from left to right, the sets of simple roots are
$\\{\alpha_{m}\\}$, $\\{\alpha_{k},\alpha_{m}\\}$ and $\\{\alpha_{k}\\}$ and
the associated gradings are

$\displaystyle\mathfrak{r}_{-1}\oplus\mathfrak{r}_{0}\oplus\mathfrak{r}_{1},\qquad\mathfrak{q}_{-3}\oplus\mathfrak{q}_{-2}\oplus\dots\oplus\mathfrak{q}_{3}\qquad\mathrm{and}\qquad\mathfrak{g}_{-2}\oplus\dots\oplus\mathfrak{g}_{2},$

respectively. With respect to the block decomposition from (3.1), we
have666Here we mean $\mathfrak{q}_{0}$ is the subspace of block diagonal
matrices, $\mathfrak{q}_{1}$ is the subspace of those block matrices where
only the matrices $Z_{1}$, $D$ are non-zero, etc.

$\displaystyle\hskip
48.36967pt\left(\begin{matrix}\mathfrak{q}_{0}&\mathfrak{q}_{1}&\mathfrak{q}_{2}&\mathfrak{q}_{3}\\\
\mathfrak{q}_{-1}&\mathfrak{q}_{0}&\mathfrak{q}_{1}&\mathfrak{q}_{2}\\\
\mathfrak{q}_{-2}&\mathfrak{q}_{-1}&\mathfrak{q}_{0}&\mathfrak{q}_{1}\\\
\mathfrak{q}_{-3}&\mathfrak{q}_{-2}&\mathfrak{q}_{-1}&\mathfrak{q}_{0}\end{matrix}\right)$
$\displaystyle\hskip 71.13188pt\swarrow\ \ \ \ \ \ \ \ \ \ \ \ \ \searrow$
$\displaystyle\left(\begin{matrix}\mathfrak{r}_{0}&\mathfrak{r}_{0}&\mathfrak{r}_{1}&\mathfrak{r}_{1}\\\
\mathfrak{r}_{0}&\mathfrak{r}_{0}&\mathfrak{r}_{1}&\mathfrak{r}_{1}\\\
\mathfrak{r}_{-1}&\mathfrak{r}_{-1}&\mathfrak{r}_{0}&\mathfrak{r}_{0}\\\
\mathfrak{r}_{-1}&\mathfrak{r}_{-1}&\mathfrak{r}_{0}&\mathfrak{r}_{0}\\\
\end{matrix}\right)\ \ \
\left(\begin{matrix}\mathfrak{g}_{0}&\mathfrak{g}_{1}&\mathfrak{g}_{1}&\mathfrak{g}_{2}\\\
\mathfrak{g}_{-1}&\mathfrak{g}_{0}&\mathfrak{g}_{0}&\mathfrak{g}_{1}\\\
\mathfrak{g}_{-1}&\mathfrak{g}_{0}&\mathfrak{g}_{0}&\mathfrak{g}_{1}\\\
\mathfrak{g}_{-2}&\mathfrak{g}_{-1}&\mathfrak{g}_{-1}&\mathfrak{g}_{0}\\\
\end{matrix}\right).$

The associated standard parabolic subalgebras are

$\displaystyle\mathfrak{r}=\mathfrak{r}_{0}\oplus\mathfrak{r}_{1},\qquad\mathfrak{q}=\mathfrak{q}_{0}\oplus\mathfrak{q}_{1}\oplus\mathfrak{q}_{2}\oplus\mathfrak{q}_{3}\qquad\mathrm{and}\qquad\mathfrak{p}=\mathfrak{g}_{0}\oplus\mathfrak{g}_{1}\oplus\mathfrak{g}_{2},$

respectively, and we have the following isomorphisms

$\displaystyle\mathfrak{r}_{0}\cong
M(m,\mathbb{C}),\qquad\mathfrak{q}_{0}\cong M(k,\mathbb{C})\oplus
M(n,\mathbb{C})\qquad\mathrm{and}\qquad\mathfrak{g}_{0}\cong
M(k,\mathbb{C})\oplus\mathfrak{so}(2n,\mathbb{C}).$

We for brevity put

$\displaystyle\mathbb{C}^{k}:=[e_{1},\dots,e_{k}],\qquad\mathbb{C}^{k\ast}:=[e_{1}^{\ast},\dots,e_{k}^{\ast}],$
$\displaystyle\mathbb{C}^{n}:=[e_{k+1},\dots,e_{m}],\qquad\mathbb{C}^{n\ast}:=[e_{k+1}^{\ast},\dots,e_{m}^{\ast}],$
$\displaystyle\mathbb{C}^{2n}:=\mathbb{C}^{n}\oplus\mathbb{C}^{n\ast}\qquad\mathrm{and}\qquad\mathbb{C}^{m}:=\mathbb{C}^{k}\oplus\mathbb{C}^{n}.$
(3.4)

Notice that the bilinear form $h$ induces dualities between $\mathbb{C}^{k}$
and $\mathbb{C}^{k\ast}$ and between $\mathbb{C}^{n}$ and $\mathbb{C}^{n\ast}$
which justifies the notation, that $\mathbb{C}^{m}$ is a maximal, totally
isotropic and $\mathfrak{r}_{0}$-invariant subspace, that $\mathbb{C}^{k}$,
$\mathbb{C}^{k\ast}$, $\mathbb{C}^{n}$ and $\mathbb{C}^{n\ast}$ are
$\mathfrak{q}_{0}$-invariant, that $\mathbb{C}^{k}$, $\mathbb{C}^{2n}$ and
$\mathbb{C}^{k\ast}$ are $\mathfrak{g}_{0}$-invariant and finally, that
$h|_{\mathbb{C}^{2n}}$ is a non-degenerate, symmetric and
$\mathfrak{g}_{0}$-invariant bilinear form. We will for brevity write only $h$
instead of $h|_{\mathbb{C}^{2n}}$ as it will be always clear from the context
what is meant.

Let us now consider the associated nilpotent subalgebras

$\displaystyle\mathfrak{r}_{-}=\mathfrak{r}_{-1},\qquad\mathfrak{q}_{-}=\mathfrak{q}_{-3}\oplus\mathfrak{q}_{-2}\oplus\mathfrak{q}_{-1}\qquad\mathrm{and}\qquad\mathfrak{g}_{-}=\mathfrak{g}_{-2}\oplus\mathfrak{g}_{-1}.$

By the Jacobi identity, the Lie bracket is equivariant with respect to the
adjoint action of the corresponding Levi factor and by the grading property
following equation (2.1), it is homogeneous of degree zero. Hence, we can
consider the Lie bracket in each homogeneity separately.

The first algebra $\mathfrak{r}_{-}$ is abelian and so there is nothing to
add.

On the other hand, $\mathfrak{q}_{-}$ is 3-graded and, as
$\mathfrak{q}_{0}$-modules, we have
$\mathfrak{q}_{-1}\cong\mathbb{E}\oplus\mathbb{F}$,
$\mathfrak{q}_{-2}\cong\mathbb{C}^{k\ast}\otimes\mathbb{C}^{n\ast}$,
$\mathfrak{q}_{-3}\cong\Lambda^{2}\mathbb{C}^{k\ast}$ where we put
$\mathbb{E}:=\mathbb{C}^{k\ast}\otimes\mathbb{C}^{n}$ and
$\mathbb{F}:=\Lambda^{2}\mathbb{C}^{n\ast}$. Using these isomorphisms, the Lie
brackets in homogeneity $-2$ and $-3$ are the compositions of the canonical
projections

$\displaystyle\Lambda^{2}\mathfrak{q}_{-1}\rightarrow\mathbb{E}\otimes\mathbb{F}=\big{(}\mathbb{C}^{k\ast}\otimes\mathbb{C}^{n}\big{)}\otimes\Lambda^{2}\mathbb{C}^{n\ast}\rightarrow\mathbb{C}^{k\ast}\otimes\mathbb{C}^{n\ast}=\mathfrak{q}_{-2}$
(3.5)

and

$\displaystyle\mathfrak{q}_{-1}\otimes\mathfrak{q}_{-2}\rightarrow\mathbb{E}\otimes\mathfrak{q}_{-2}=\big{(}\mathbb{C}^{k\ast}\otimes\mathbb{C}^{n}\big{)}\otimes\big{(}\mathbb{C}^{k\ast}\otimes\mathbb{C}^{n\ast}\big{)}\rightarrow\Lambda^{2}\mathbb{C}^{k\ast}=\mathfrak{q}_{-3},$

respectively. Here we use the canonical pairing
$\mathbb{C}^{n}\otimes\mathbb{C}^{n\ast}\rightarrow\mathbb{C}$. Notice that
$\Lambda^{2}\mathbb{E}\oplus\Lambda^{2}\mathbb{F}$ is contained in the kernel
of (3.5).

In order to understand the Lie bracket on $\mathfrak{g}_{-}$, first notice
that there are isomorphisms
$\mathfrak{g}_{-1}\cong\mathbb{C}^{k\ast}\otimes\mathbb{C}^{2n}$ and
$\mathfrak{g}_{-2}\cong\Lambda^{2}\mathbb{C}^{k\ast}\otimes\mathbb{C}$ of
irreducible $\mathfrak{g}_{0}$-modules where $\mathbb{C}$ is the trivial
representation of $\mathfrak{so}(2n,\mathbb{C})$. As $\mathfrak{g}_{-}$ is
2-graded, the Lie bracket is non-zero only in homogeneity $-2$. It is given by

$\displaystyle\Lambda^{2}\mathfrak{g}_{-1}=\Lambda^{2}\big{(}\mathbb{C}^{k\ast}\otimes\mathbb{C}^{2n}\big{)}\rightarrow\Lambda^{2}\mathbb{C}^{k\ast}\otimes
S^{2}\mathbb{C}^{2n}\rightarrow\Lambda^{2}\mathbb{C}^{k\ast}\otimes\mathbb{C}=\mathfrak{g}_{-2},$

where in the last map we take the trace with respect to $h$.

In the table below we specify when
$\lambda=(\lambda_{1},\dots,\lambda_{m})\in\mathfrak{h}^{\ast}$ is dominant
for each parabolic subalgebra $\mathfrak{p},\mathfrak{q}$ and $\mathfrak{r}$.
We put $\mathbb{N}_{0}:=\mathbb{N}\cup\\{0\\}$.

Table 1: Dominant weights. algebra | dominant and integral weights
---|---
$\mathfrak{p}$ | $\lambda_{i}-\lambda_{i+1}\in\mathbb{N}_{0},\ i\neq k,\ 2\lambda_{m}\in\mathbb{Z},\ \lambda_{m-1}\geq|\lambda_{m}|$
$\mathfrak{r}$ | $\lambda_{i}-\lambda_{i+1}\in\mathbb{N}_{0}$
$\mathfrak{q}$ | $\lambda_{i}-\lambda_{i+1}\in\mathbb{N}_{0},i\neq k$

###  3.2 Relative Hasse diagram $\boldsymbol{W_{\mathfrak{r}}^{\mathfrak{q}}}$

Let us first set notation. By a partition we will mean an element of
$\mathbb{N}^{k,n}_{++}:=\\{(a_{1},\dots,a_{k})\colon a_{i}\in\mathbb{Z}$,
$n\geq a_{1}\geq a_{2}\geq\dots\geq a_{k}\geq 0\\}$. For two partitions
$a=(a_{1},\dots,a_{k})$ and $a^{\prime}=(a_{1}^{\prime},\dots,a^{\prime}_{k})$
we write $a\leq a^{\prime}$ if $a_{i}\leq a^{\prime}_{i}$ for all
$i=1,\dots,k$ and $a<a^{\prime}$ if $a\leq a^{\prime}$ and $a\neq a^{\prime}$.
If $a<a^{\prime}$ does not hold, then we write $a\nless a^{\prime}$. We put

$\displaystyle|a|=a_{1}+\dots+a_{k},\qquad d(a):=\max\\{i\colon a_{i}\geq
i\\},$ (3.6) $\displaystyle
q(a)=\sum_{i=1}^{k}\max\\{a_{i}-i,0\\}\qquad\mathrm{and}\qquad
r(a):=d(a)+q(a).$

To the partition $a$ we associate the Young diagram (or the Ferrers diagram)
$\mathrm{Y}$ consisting of $k$ left-justified rows with $a_{i}$-boxes in the
$i$-th row. Let $b_{i}$ be the number of boxes in the $i$-th column of
$\mathrm{Y}$. Then we call $b=(b_{1},\dots,b_{n})\in\mathbb{N}^{n,k}_{++}$ the
partition conjugated to $a$ and we say that $a$ is symmetric if $a_{i}=b_{i}$,
$i=1,\dots,k$ and $b_{k+1}=\dots=b_{n}=0$. As we assume $n\geq k$, the set of
symmetric partitions in $\mathbb{N}^{k,n}_{++}$ depends only on $k$, and thus,
we denote it for simplicity by $S^{k}$ and put $S^{k}_{j}:=\\{a\in S^{k}\colon
r(a)=j\\}$.

######  Example 3.1.

  1. (1)

The empty partition is by definition always symmetric.

  2. (2)

The Young diagram of $a=(4,3,1,0,0)\in\mathbb{N}^{5,6}_{++}$ 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

$\displaystyle\begin{picture}(12.0,16.0)\put(0.0,16.0){\line(1,0){12.0}}
\put(0.0,12.0){\line(1,0){12.0}} \put(0.0,8.0){\line(1,0){8.0}}
\put(0.0,4.0){\line(1,0){8.0}} \put(0.0,0.0){\line(1,0){4.0}}
\put(0.0,16.0){\line(0,-1){16.0}} \put(4.0,16.0){\line(0,-1){16.0}}
\put(8.0,16.0){\line(0,-1){12.0}} \put(12.0,16.0){\line(0,-1){4.0}}
\end{picture}.$

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 graph
$W_{\mathfrak{r}}^{\mathfrak{q}}$. The group $W_{\mathfrak{r}}$ is generated
by $s_{1},\dots,s_{m-1}$ while $W_{\mathfrak{q}}$ is generated by elements
$s_{1},\dots,s_{k-1},s_{k+1},\dots,s_{m-1}$. By (3.2), it follows that
$W_{\mathfrak{r}}$ is the permutation group $S_{m}$ on $\\{1,\dots,m\\}$ and
that $W_{\mathfrak{q}}\cong S_{k}\times S_{n}$ is the stabilizer of
$\\{1,\dots,k\\}$. Recall from Section 2.1 that in each left coset of
$W_{\mathfrak{q}}$ in $W_{\mathfrak{r}}$ there is a unique element of minimal
length and that we denote the set of all such distinguished representatives by
$W_{\mathfrak{r}}^{\mathfrak{q}}$. Moreover, the Bruhat order on
$W_{\mathfrak{g}}$ descends to a partial order on $W_{\mathfrak{r}}$ and on
$W_{\mathfrak{q}}^{\mathfrak{r}}$. We will now show that there is an
isomorphism $\mathbb{N}^{k,n}_{++}\rightarrow W^{\mathfrak{q}}_{\mathfrak{r}}$
of partially ordered sets.

Let $a=(a_{1},\dots,a_{k})\in\mathbb{N}^{k,n}_{++}$ and $\mathrm{Y}$ be the
associated Young diagram. We will call the box in the $i$-th row and the
$j$-th column of $Y$ an $(i,j)$-box and we write into this box the number
$\sharp(i,j):=k-i+j$. Notice that $1\leq\sharp(i,j)\leq m$. Then the set of
boxes in $Y$ is indexed by $\Xi_{a}:=\\{(i,j)\colon i=1,\dots,k,\
j=1,\dots,a_{i}\\}$ and we order this set lexicographically, i.e.,
$(i,j)<(i^{\prime},j^{\prime})$ if $i<i^{\prime}$ or $i=i^{\prime}$ and
$j<j^{\prime}$. Then

$\displaystyle w_{a}:=s_{\sharp(\Psi(1))}s_{\sharp(\Psi(2))}\dots
s_{\sharp(\Psi(|a|))}\in S_{m},$ (3.8)

where $\Psi\colon\\{1,2,\dots,|a|\\}\rightarrow\Xi_{a}$ is the unique
isomorphism of ordered sets. Let us now look at an example.

######  Example 3.2.

The Young diagram from (3.7) is filled as

$\displaystyle\begin{matrix}k&k+1&k+2&k+3\\\ k-1&k&k+1&\\\ k-2&&&\\\
\end{matrix}$

and so $w_{a}:=s_{k}s_{k+1}s_{k+2}s_{k+3}s_{k-1}s_{k}s_{k+1}s_{k-2}$.

We have the following preliminary observation.

######  Lemma 3.3.

Let $a=(a_{1},\dots,a_{k})$ be as above and $b=(b_{1},\dots,b_{n})$ be the
conjugated partition. Then the permutation $w_{a}\in S_{m}$ from (3.8)
satisfies

$\displaystyle w_{a}(k-i+1+a_{i})=k-i+1\qquad\mathrm{and}\qquad
w_{a}(k+j-b_{j})=k+j$ (3.9)

for each $i=1,\dots,k$ and $j=1,\dots,n$.

###### Proof.

Fix $i=1,\dots,k$. If $a_{i}>0$, there is $r_{i}:=k-i+1$ written in the
$(i,1)$-box and $r^{i}:=k-i+a_{i}=r_{i}+a_{i}-1$ in the $(i,a_{i})$-box. We
put $r^{i}-1=r_{i}:=k-i+1$ if $a_{i}=0$. Similarly, if $j=1,\dots,n$ and
$b_{j}>0$, then there is $c_{j}:=k+j-1$ in the $(1,j)$-box and
$c^{j}:=k+j-b_{j}=c_{j}-b_{j}+1$ in the $(b_{j},j)$-box. We put
$c^{j}-1=c_{j}:=k+j-1$ if $b_{j}=0$. Then it is easy to check that
$w_{a}(r^{i}+1)=r_{i}$ and $w_{a}(c^{j})=c_{j}+1$ which completes the proof. ∎

Notice that the sets $\\{k-i+1+a_{i}\colon i=1,\dots,k\\}$ and
$\\{k+j-b_{j}\colon j=1,\dots,n\\}$ are disjoint and that their union is
$\\{1,2,\dots,m\\}$. By (3.9), it follows that

$\displaystyle w_{a}\rho=\rho+(-a_{k},\dots,-a_{1}\,|\,b_{1},\dots,b_{n}),$
(3.10)

where $\rho=(m-1,\dots,1,0)$ is the lowest form of $\mathfrak{g}$ and for
clarity, we separate the first $k$ and last $n$ coefficients by $|$. Comparing
this with Table 1, we see that $w_{a}\rho$ is $\mathfrak{q}$-dominant. As the
same holds for any $\mathfrak{r}$-dominant weight, it follows that $w_{a}\in
W^{\mathfrak{q}}_{\mathfrak{r}}$.

######  Lemma 3.4.

The map $\mathbb{N}^{k,n}_{++}\rightarrow W^{\mathfrak{q}}_{\mathfrak{r}}$,
$a\mapsto w_{a}$ is an isomorphism of partially ordered sets.

###### Proof.

The map $a\mapsto w_{a}$ is by (3.10) clearly injective. To show surjectivity,
fix $w\in W^{\mathfrak{q}}_{\mathfrak{r}}$. Then the sequence
$c_{1},\dots,c_{k}$ where $c_{i}=w^{-1}(i)$, $i=1,\dots,k$ is increasing. By
[8, Proposition 3.2.16], the map $w\in W^{\mathfrak{q}}_{\mathfrak{r}}\mapsto
w^{-1}\omega_{k}$ is injective. It follows that $w^{-1}\omega_{k}$ is uniquely
determined by the sequence $c_{1},\dots,c_{k}$. Then
$a:=(a_{1},\dots,a_{k})\in\mathbb{N}^{k,n}_{++}$ where
$a_{i}:=c_{k-i+1}+i-k-1$, $i=1,\dots,k$ and from (3.9), it follows that
$w_{a}^{-1}(k-i+1)=k-i+1+a_{i}=c_{k-i+1}$, $i=1,\dots,k$. This shows that
$w_{a}^{-1}\omega_{k}=w^{-1}\omega_{k}$ and thus, $w=w_{a}$. Now it remains to
show that the map is compatible with the orders.

Assume that $a=(a_{1},\dots,a_{k})$,
$a^{\prime}=(a^{\prime}_{1},\dots,a^{\prime}_{k})\in\mathbb{N}^{k,n}_{++}$
satisfy $|a^{\prime}|=|a|+1$ and $a<a^{\prime}$. Then there is a unique
integer $i\leq k$ such that $a^{\prime}_{i}=a_{i}+1$ and so
$w_{a^{\prime}}=w_{a}s_{k-i+a^{\prime}_{i}}$. By (3.9), we have that
$w_{a}\alpha_{k-i+a^{\prime}_{i}}>0$ and thus by [8, Proposition 3.2.16],
there is an arrow $w_{a}\rightarrow w_{a^{\prime}}$ in
$W^{\mathfrak{q}}_{\mathfrak{r}}$.

On the other hand, suppose that
$a^{\prime\prime}=(a_{1}^{\prime\prime},\dots,a_{k}^{\prime\prime})\in\mathbb{N}^{k,n}_{++}$
satisfies $a^{\prime}\not<a^{\prime\prime}$. In order to complete the proof,
it is enough to show that there is no arrow $w_{a^{\prime}}\rightarrow
w_{a^{\prime\prime}}$. By assumptions, there is $j$ such that
$a^{\prime}_{1}\leq a_{1}^{\prime\prime},\dots,a_{j-1}^{\prime}\leq
a^{\prime\prime}_{j-1}$ and $a^{\prime}_{j}>a_{j}^{\prime\prime}$. Without
loss of generality we may assume that $i=j$. Then
$w_{a^{\prime}}^{-1}(w_{a}\alpha_{k+a_{i}-i})=s_{k+a_{i}-i}\alpha_{k+a_{i}-i}<0$.
On the other hand by (3.9), it follows that
$w_{a^{\prime\prime}}^{-1}(w_{a}\alpha_{k+a_{i}-i})>0$. We proved that
$\Phi_{w_{a}^{\prime}}\not\subset\Phi_{w_{a^{\prime\prime}}}$ and thus by [8,
Proposition 3.2.17], there cannot be any arrow $w_{a^{\prime}}\rightarrow
w_{a^{\prime\prime}}$. ∎

We will later need the following two observations. A permutation $w\in S_{m}$
is $k$-balanced, if the following is true: if $w(k-i)>k$ for some
$i=0,\dots,k-1$, then $w(k+i+1)\leq k$.

######  Lemma 3.5.

The permutation $w_{a}$ associated to $a\in\mathbb{N}^{k,n}_{++}$ is
$k$-balanced if and only if $a\in S^{k}$.

###### Proof.

Let $b=(b_{1},\dots,b_{n})$ be the partition conjugated to
$a=(a_{1},\dots,a_{k})$. First notice that if $w_{a}(k-i)=k+j>k$, then by
(3.9) we have $i=b_{j}-j$.

If $a\in S^{k}$, then
$w_{a}(k+i+1)=w_{a}(k-j+b_{j}+1)=w_{a}(k-j+a_{j}+1)=k-j+1\leq k$ and so $a$ is
$k$-balanced.

If $a\not\in S^{k}$, then there is $j$ such that
$a_{1}=b_{1},\dots,a_{j-1}=b_{j-1}$ and $a_{j}\neq b_{j}$. It follows that
$b_{j}\geq j$ and so $i:=b_{j}-j\geq 0$. Then
$w_{a}(k-i)=w_{a}(k-b_{j}+j)=k+j>k$. If $a_{j}>b_{j}$, then
$w_{a}(k+i+1)=k+b_{j}+1>k$. If $a_{j}<b_{j}$, then $w_{a}(k+i+1)=k+b_{j}>k$. ∎

Recall from [2] that given $w\in S_{m}$, there exists a minimal integer
$\ell(w)$, called the length of $w$, such that $w$ can be expressed as a
product of $\ell(w)$ simple reflections $s_{1},\dots,s_{m}$. It is well known
that $\ell(w)$ is equal to the number of pairs $1\leq i<j\leq m$ such that
$w(i)>w(j)$.

######  Lemma 3.6.

Let $w_{a}\in S_{m}$. Then $\ell(w_{a})=|a|$.

###### Proof.

By the definition of $w_{a}$, it follows that $\ell(w_{a})\leq|a|$. On the
other hand, if $a<a^{\prime}$, then $w_{a}\rightarrow w_{a^{\prime}}$ and thus
also $\ell(w_{a})<\ell(w_{a^{\prime}})$. By induction on $|a|$, we have that
$\ell(w_{a})\geq|a|$. ∎

##  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 $\eta$ and $\tau$.

###  4.1 Homogeneous spaces

A connected and simply connected Lie group $\mathrm{G}$ with Lie algebra
$\mathfrak{g}$ is isomorphic to $\mathrm{Spin}(2m,\mathbb{C})$. Let
$\mathrm{R}$, $\mathrm{Q}$ and $\mathrm{P}$ be the parabolic subgroups of
$\mathrm{G}$ with Lie algebras $\mathfrak{r}$, $\mathfrak{q}$ and
$\mathfrak{p}$ that are associated to $\\{\alpha_{m}\\}$,
$\\{\alpha_{k},\alpha_{m}\\}$ and $\\{\alpha_{k}\\}$, respectively, as
explained in Section 2.1. We for brevity put $TS:=\mathrm{G}/\mathrm{R}$,
$CS:=\mathrm{G}/\mathrm{Q}$ and $M:=\mathrm{G}/\mathrm{P}$. 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 $\mathbb{W}:=\mathbb{C}^{m}$
is a maximal totally isotropic subspace of $\mathbb{C}^{2m}$. We can (via $h$)
identify the dual space $\mathbb{W}^{\ast}$ with the subspace
$[e_{1}^{\ast},\dots,e_{m}^{\ast}]$. Put
$\mathbb{S}:=\bigoplus_{i=0}^{m}\Lambda^{i}\mathbb{W}^{\ast}$. There is a
canonical linear map
$\mathbb{C}^{2m}\rightarrow\operatorname{End}(\mathbb{S})$ which is determined
by $w\cdot\psi=i_{w}\psi$ and $w^{\ast}\cdot\psi=w^{\ast}\wedge\psi$ where
$w\in\mathbb{W}$, $w^{\ast}\in\mathbb{W}^{\ast}$, $\psi\in\mathbb{S}$ and
$i_{w}$ stands for the contraction by $w$. If $\psi\in\mathbb{S}$, then we put
$T_{\psi}:=\\{v\in\mathbb{C}^{2m}\colon v\cdot\psi=0\\}$. If $\psi\neq 0$,
then $T_{\psi}$ is a totally isotropic subspace and we call $\psi$ a pure
spinor if $\dim T_{\psi}=m$ (which is equivalent to saying that $T_{\psi}$ is
a maximal totally isotropic subspace).

The standard linear isomorphism $\Lambda^{2}\mathbb{C}^{2m}\cong\mathfrak{g}$
gives an injective linear map
$\mathfrak{g}\rightarrow\operatorname{End}(\mathbb{S})$. It is straightforward
to verify that the map is a homomorphism of Lie algebras where the commutator
in the associative algebra $\operatorname{End}(\mathbb{S})$ is the standard
one. Hence, $\mathfrak{g}$ is a Lie subalgebra of
$\operatorname{End}(\mathbb{S})$ and it turns out that $\mathbb{S}$ is no
longer irreducible under $\mathfrak{g}$ but it decomposes as
$\mathbb{S}_{+}\oplus\mathbb{S}_{-}$ where
$\mathbb{S}_{+}:=\bigoplus_{i=0}^{m}\Lambda^{2i}\mathbb{W}^{\ast}$ and
$\mathbb{S}_{-}:=\bigoplus_{i=0}^{m}\Lambda^{2i+1}\mathbb{W}^{\ast}$. Then
$\mathbb{S}_{+}$ and $\mathbb{S}_{-}$ are irreducible non-isomorphic complex
spinor representations of $\mathfrak{g}$ with highest weights $\omega_{m}$ and
$\omega_{m-1}$, respectively. It is well known that any pure spinor belongs to
$\mathbb{S}_{+}$ or to $\mathbb{S}_{-}$ (which explains why the Grassmannian
of maximal totally isotropic subspaces in $\mathbb{C}^{2m}$ has two connected
components).

Now we can easily describe the twistor space. The spinor $1\in\mathbb{S}_{+}$
is annihilated by all positive roots in $\mathfrak{g}$ and hence, it is a
highest weight vector. Recall from Section 2.1 that the line spanned by 1 is
invariant under $\mathrm{R}$ and since $T_{1}=\mathbb{W}$, we find that
$\mathrm{R}$ is the stabilizer of $\mathbb{W}$ inside $\mathrm{G}$. As
$\mathrm{G}$ is connected, we conclude that $TS$ is the connected component of
$\mathbb{W}$ in the Grassmannian of maximal totally isotropic subspaces in
$\mathbb{C}^{2m}$.

The isotropic Grassmannian $M$. An irreducible $\mathfrak{g}$-module with
highest weight $\omega_{k}$ is isomorphic to $\Lambda^{k}\mathbb{C}^{2m}$.
Then $e_{1}\wedge e_{2}\wedge\dots\wedge e_{k}$ is clearly a highest weight
vector and the corresponding point in $\mathbb{P}(\Lambda^{k}\mathbb{C}^{2m})$
can be viewed as the totally isotropic subspace $x_{0}:=\mathbb{C}^{k}$. We
see that $M$ is the Grassmannian of totally isotropic $k$-dimensional
subspaces in $\mathbb{C}^{2m}$. We denote by
$\textbf{p}\colon\mathrm{G}\rightarrow M$ the canonical projection.

The correspondence space $CS$. The correspondence space $CS$ is the
generalized flag manifold of nested subspaces $\\{(z,x)\colon z\in TS,\ x\in
M,\ x\subset z\\}$ and $\mathrm{Q}$ is the stabilizer of $(\mathbb{W},x_{0})$.
Let $\textbf{q}\colon\mathrm{G}\rightarrow CS$ be the canonical projection.

###  4.2 Filtrations of the tangent bundles of $\boldsymbol{M}$ and
$\boldsymbol{CS}$

Recall from Section 2.1 that the $|2|$-grading
$\mathfrak{g}=\mathfrak{g}_{-2}\oplus\mathfrak{g}_{-1}\oplus\dots\oplus\mathfrak{g}_{2}$
associated to $\\{\alpha_{k}\\}$ determines a 2-step filtration
$\\{0\\}=F_{0}^{M}\subset F_{-1}^{M}\subset F_{-2}^{M}=TM$ of the tangent
bundle of $M$ where $\\{0\\}$ is the zero section. We put
$G_{i}^{M}:=F_{i}^{M}/F_{i+1}^{M}$, $i=-2,-1$ so that the associated graded
bundle $gr(TM)=G_{-2}^{M}\oplus G_{-1}^{M}$ is a locally trivial bundle of
graded nilpotent Lie algebras with typical fiber $\mathfrak{g}_{-}$. Dually,
there is a filtration $T^{\ast}M=F_{1}^{M}\supset F_{2}^{M}\supset
F_{3}^{M}=\\{0\\}$ where $F_{i}^{M}\cong(F^{M}_{-i+1})^{\perp}$. We put
$G_{i}^{M}:=F_{i}^{M}/F_{i+1}^{M}$ so that
$G_{i}^{M}\cong(G_{-i}^{M})^{\ast}$. There are linear isomorphisms

$\displaystyle\mathfrak{g}_{i}\cong\big{(}G^{M}_{i}\big{)}_{x_{0}},\qquad
i=-2,-1,1,2.$ (4.1)

Recall from Section 2.2 that $\mathfrak{gr}^{r}_{x_{0}}$ denotes the vector
space of weighted $r$-jets of germs of holomorhic functions at $x_{0}$ whose
weighted $(r-1)$-jet vanishes. Then the isomorphisms from (2.3) are

$\displaystyle\mathfrak{gr}^{1}_{x_{0}}\cong\mathfrak{g}_{1},\qquad\mathfrak{gr}^{2}_{x_{0}}\cong
S^{2}\mathfrak{g}_{1}\oplus\mathfrak{g}_{2},\qquad\mathfrak{gr}^{3}_{x_{0}}\cong
S^{3}\mathfrak{g}_{1}\oplus\mathfrak{g}_{1}\otimes\mathfrak{g}_{2},\qquad\dots$

for small $r$ and in general

$\displaystyle\mathfrak{gr}^{r}_{x_{0}}\cong\bigoplus_{i+2j=r}S^{i}\mathfrak{g}_{1}\otimes
S^{j}\mathfrak{g}_{2}.$ (4.2)

The $|3|$-grading $\mathfrak{g}=\bigoplus_{i=-3}^{3}\mathfrak{q}_{i}$
determined by $\\{\alpha_{k},\alpha_{m}\\}$ induces a 3-step filtration
$TCS=F_{-3}^{CS}\supset F_{-2}^{CS}\supset F_{-1}^{CS}\supset
F_{0}^{CS}=\\{0\\}$. We put $G_{i}^{CS}:=F_{i}^{CS}/F^{CS}_{i+1}$ so that
$gr(TCS)=\bigoplus_{i=-3}^{-1}G_{i}^{CS}$ is a locally trivial vector bundle
of graded nilpotent Lie algebras with typical fiber $\mathfrak{q}_{-}$.
Dually, we get a filtration $T^{\ast}CS=F_{1}^{CS}\supset F_{2}^{CS}\supset
F_{3}^{CS}\supset F_{4}^{CS}=\\{0\\}$ where
$F_{i}^{CS}:=(F^{CS}_{-i+1})^{\perp}$. The associated graded vector bundle is
$gr(T^{\ast}CS)=\bigoplus_{i=1}^{3}G_{i}^{CS}$ where we put
$G_{i}^{CS}:=F_{i}^{CS}/F^{CS}_{i+1}$. Then as above,
$G_{i}^{CS}\cong(G_{-i}^{CS})^{\ast}$.

The $\mathrm{Q}$-invariant subspaces $\mathbb{E}\oplus\mathfrak{q}$ and
$\mathbb{F}\oplus\mathfrak{q}$ give a finer filtration of the tangent bundle,
namely $F_{-1}^{CS}=E^{CS}\oplus F^{CS}$. Since the Lie bracket
$\Lambda^{2}\mathfrak{q}_{-1}\rightarrow\mathfrak{q}_{-2}$ vanishes on
$\Lambda^{2}\mathbb{E}\oplus\Lambda^{2}\mathbb{F}$, it follows that $E^{CS}$
and $F^{CS}$ are integrable distributions. This can be deduced also from the
short exact sequences

$\displaystyle 0\rightarrow E^{CS}\rightarrow
TCS\xrightarrow{T\eta}T(TS)\rightarrow 0\qquad\mathrm{and}\qquad 0\rightarrow
F^{CS}\rightarrow TCS\xrightarrow{T\tau}TM\rightarrow 0,$ (4.3)

i.e., $E^{CS}=\ker(T\eta)$ and $F^{CS}=\ker(T\tau)$. Notice that
$(T\tau)^{-1}(F_{-1}^{M})=F^{CS}_{-2}$.

###  4.3 Projections $\boldsymbol{\tau}$ and $\boldsymbol{\eta}$

Recall from (3.4) that
$\mathbb{C}^{2n}:=[e_{k+1},\dots,e_{m},e_{k+1}^{\ast},\dots,e_{m}^{\ast}]$ and
$\mathbb{C}^{n}:=[e_{k+1},\dots,e_{m}]$, i.e., we view $\mathbb{C}^{2n}$ and
$\mathbb{C}^{n}$ as subspaces of $\mathbb{C}^{2m}$. On $\mathbb{C}^{2n}$ we
consider the non-degenerate bilinear form $h|_{\mathbb{C}^{2n}}$ which we for
brevity denote by $h$. Then $\mathbb{C}^{n}$ is a maximal totally isotropic
subspace of $\mathbb{C}^{2n}$.

The fibers of $\tau$ and $\eta$ are homogeneous spaces of parabolic geometries
which (see [2]) can be recovered from the Dynkin diagrams given in (3.3).

######  Lemma 4.1.

  1. $(a)$

The fibers of $\tau$ are biholomorphic to the Grassmannian of $k$-dimensional
subspaces in $\mathbb{C}^{n+k}$.

  2. $(b)$

The fibers of $\eta$ are biholomorphic to the connected component
$\mathrm{Gr}^{+}_{h}(n,n)$ of $\mathbb{C}^{n}$ in the Grassmannian of maximal
totally isotropic subspaces in $\mathbb{C}^{2n}$.

###### Proof.

As the fibers over distinct points are biholomorphic, it suffices to look at
the fibers of $\eta$ and $\tau$ over $\mathbb{W}$ and $x_{0}$, respectively.

(a) By definition, $\eta^{-1}(\mathbb{W})$ is the set of $k$-dimensional
totally isotropic subspaces in $\mathbb{W}$. As $\mathbb{W}$ is already
totally isotropic, the first claim follows.

(b) Notice that $x_{0}^{\bot}=x_{0}\oplus\mathbb{C}^{2n}$. Then it is easy to
see that $y\in\mathrm{Gr}^{+}_{h}(n,n)\mapsto(x_{0}\oplus
y,x_{0})\in\tau^{-1}(x_{0})$ is a biholomorphism. ∎

We will use the following notation. Assume that $X\in M(2m,k,\mathbb{C})$ and
$Y\in M(2m,n,\mathbb{C})$ have maximal rank. Then we denote by $[X]$ the
$k$-dimensional subspace of $\mathbb{C}^{2m}$ that is spanned by the columns
of the matrix and by $[X|Y]$ the flag of nested subspaces
$[X]\subset[X]\oplus[Y]$.

It is straightforward to verify that

$\displaystyle(\textbf{p}\circ\exp)\colon\ \mathfrak{g}_{-}\rightarrow M,$
(4.4) $\displaystyle(\textbf{p}\circ\exp)\left(\begin{matrix}0&0&0&0\\\
X_{1}&0&0&0\\\ X_{2}&0&0&0\\\ Y&-X_{2}^{T}&-X_{1}^{T}&0\\\
\end{matrix}\right)=\left[\begin{matrix}1_{k}\\\ X_{1}\\\ X_{2}\\\
Y-\frac{1}{2}(X_{1}^{T}X_{2}+X^{T}_{2}X_{1})\\\ \end{matrix}\right].$

We see that $\mathcal{X}:=\textbf{p}\circ\exp(\mathfrak{g}_{-})$ is an open,
dense and affine subset of $M$ and that any $(z,x)\in\tau^{-1}(\mathcal{X})$
can be represented by

$\displaystyle\left[\begin{array}[]{c|c}1_{k}&0\\\ X_{1}&A\\\ X_{2}&B\\\
Y-\frac{1}{2}(X_{1}^{T}X_{2}+X_{2}^{T}X_{1})&C\\\ \end{array}\right],$ (4.9)

where $A,B\in M(n,\mathbb{C})$, $C\in M(k,n,\mathbb{C})$ are such that
$\left[\begin{matrix}A\\\ B\\\ \end{matrix}\right]\in\mathrm{Gr}^{+}_{h}(n,n)$
and $C=-(X_{1}^{T}B+X_{2}^{T}A)$. We immediately get the following
observation.

######  Lemma 4.2.

The set $\tau^{-1}(\mathcal{X})$ is biholomorphic to
$\mathcal{X}\times\tau^{-1}(x_{0})$. The restriction of $\tau$ to this set is
then the projection onto the first factor.

The set $\tau^{-1}(\mathcal{X})$ is not affine as different choices of $A$ and
$B$ might lead to the same element in $\tau^{-1}(\mathcal{X})$. Let
$\mathcal{Y}$ be the subset of $\tau^{-1}(\mathcal{X})$ of those nested flags
$x\subset z$ which can be represented by a matrix as above with $A$ regular.
In that case we may assume $A=1_{n}$ which uniquely pins down $B$. It is
straightforward to find that $B=-B^{T}$ and conversely, any skew-symmetric
$n\times n$ matrix determines a totally isotropic $n$-dimensional subspace in
$\mathbb{C}^{2n}$. We see that $\mathcal{Y}$ is an open and affine set which
is biholomorphic to $\mathfrak{g}_{-}\times A(n,\mathbb{C})$. In order to
write down also $\eta$ as a canonical projection $\mathbb{C}^{{m\choose
2}+nk}\rightarrow\mathbb{C}^{m\choose 2}$, it will be convenient to choose a
different coordinate system on $\mathcal{Y}$.

######  Lemma 4.3.

Let $\mathcal{Y}$ be as above and put $\mathcal{Z}:=\eta(\mathcal{Y})$. Then
$\mathcal{Y}$ and $\mathcal{Z}$ are open affine sets and there is a
commutative diagram of holomorphic maps

$\displaystyle\begin{split}&\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
6.75pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
0.0pt\raise-20.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\eta|_{\mathcal{Y}}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-30.16669pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
30.75pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
30.75pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{A(m,\mathbb{C})\times
M(n,k,\mathbb{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
79.15858pt\raise-20.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8264pt\hbox{$\scriptstyle{pr_{1}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 79.15858pt\raise-29.5pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
57.7407pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
57.7407pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{A(m,\mathbb{C}),}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{split}$
(4.10)

where $pr_{1}$ is the canonical projection and the horizontal arrows are
biholomorphisms.

###### Proof.

Let $(z,x)$ be the nested flag corresponding to (4.9) where $A=1_{n}$ so that
$B=-B^{T}$. Put for brevity
$Y^{\prime}:=Y-\frac{1}{2}(X_{1}^{T}X_{2}+X_{2}^{T}X_{1})$ and
$C:=-X_{2}^{T}-X_{1}^{T}B$. The map in the first row in (4.10) is
$(z,x)\mapsto(W,Z)$ where

$\displaystyle W=\left(\begin{matrix}W_{1}&W_{2}\\\
W_{0}&-W_{1}^{T}\end{matrix}\right)$

and $Z:=X_{1}$, $W_{1}:=X_{2}-BX_{1}$, $W_{0}:=Y^{\prime}-CX_{1}$, $W_{2}:=B$.
Then $W=-W^{T}$ and the map $\mathcal{Y}\rightarrow A(m,\mathbb{C})\times
M(n,k,\mathbb{C})$ 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 (4.9), we can eliminate the $X_{1}$-block and get
a new matrix

$\displaystyle\left(\begin{matrix}1_{k}&0\\\ 0&1_{n}\\\ X_{2}-BX_{1}&B\\\
Y^{\prime}-CX_{1}&C\end{matrix}\right)=\left(\begin{matrix}1_{m}\\\
W\end{matrix}\right).$

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 $\mathcal{Z}$ is indeed an open affine
subset of $TS$ which is biholomorphic to $A(m,\mathbb{C})$. In these
coordinate systems, the restriction of $\eta$ is the projection onto the first
factor. ∎

##  5 The Penrose transform for the $\boldsymbol{k}$-Dirac complexes

In Section 5 we will consider the relative BGG sequence associated to a
particular $\mathfrak{r}$-dominant and integral weight as explained in Section
2.4. More explicitly, we will define in Section 5.1 for each $p,q\geq 0$ a
sheaf of relative $(p,q)$-forms and we get a Dolbeault-like double complex.
Then we will show (see Section 5.2) that this double complex contains a
relative holomorphic de Rham complex. Then in Section 5.3 we will twist each
sheaf of relative $(p,q)$-forms as well as the holomorhic de Rham complex by a
certain pullback sheaf. Using some elementary representation theory, we will
turn (see Section 5.4) the twisted relative de Rham complex into the relative
BGG sequence. In Section 5.5 we will compute direct images of sheaves in the
relative BGG sequence.

We will use the following notation. We denote by $\mathcal{O}_{\mathfrak{q}}$
and $\mathcal{E}^{p,q}_{\mathfrak{q}}$ the structure sheaf and the sheaf of
smooth $(p,q)$-forms, respectively, over $CS$. We denote the corresponding
sheaves over $TS$ by the subscript $\mathfrak{r}$. If $W$ is a holomorphic
vector bundle over $CS$, then we denote by $\mathcal{O}_{\mathfrak{q}}(W)$ the
sheaf of holomorphic sections of $W$ and by
$\mathcal{E}^{p,q}_{\mathfrak{q}}(W)$ the sheaf of smooth $(p,q)$-forms with
values in $W$. We for brevity put
$\mathcal{E}_{\ast}:=\mathcal{E}^{0,0}_{\ast}$ and
$\mathcal{E}^{p,q}_{\ast}(\mathcal{U}):=\Gamma(\mathcal{U},\mathcal{E}_{\ast}^{p,q})$
where $\mathcal{U}$ is an open set and $\ast=\mathfrak{q}$ or $\mathfrak{r}$.
Moreover, we put
$\eta^{\ast}\mathcal{E}^{p,q}_{\mathfrak{r}}:=\mathcal{E}_{\mathfrak{q}}\otimes_{\eta^{-1}\mathcal{E}_{\mathfrak{r}}}\mathcal{E}^{p,q}_{\mathfrak{r}}$
where we use that $\eta^{-1}\mathcal{E}_{\mathfrak{r}}$ is naturally a sub-
sheaf of $\mathcal{E}_{\mathfrak{q}}$.

###  5.1 Double complex of relative forms

Recall from Lemma 4.3 that $\mathcal{Y}$ is biholomorphic to
$A(m,\mathbb{C})\times M(n,k,\mathbb{C})$, that
$\eta(\mathcal{Y})=\mathcal{Z}$ is biholomorphic to $A(m,\mathbb{C})$ and that
the canonical map $\mathcal{Y}\rightarrow\mathcal{Z}$ is the projection onto
the first factor. In this way we can use matrix coefficients on
$M(n,k,\mathbb{C})$ as coordinates on the fibers of $\eta$. We will write
$Z=(z_{\alpha i})\in M(n,k,\mathbb{C})$ and if
$I=((\alpha_{1},i_{1}),\dots,(\alpha_{p},i_{p}))$ is a multi-index where
$(\alpha_{j},i_{j})\in I(n,k):=\\{(\alpha,i)\colon\alpha=1,\dots,n,\
i=1,\dots,k\\}$, $j=1,\dots,p$, then we put
$dz_{I}:=dz_{(\alpha_{1},i_{1})}\wedge\dots\wedge dz_{(\alpha_{p},i_{p})}$ and
$|I|=p$.

We call

$\displaystyle\mathcal{E}^{p+1,q}_{\eta}:=\mathcal{E}^{p+1,q}_{\mathfrak{q}}/(\eta^{\ast}\mathcal{E}^{1,0}_{\mathfrak{r}}\wedge\mathcal{E}_{\mathfrak{q}}^{p,q}),\qquad
p,q\geq 0$

the sheaf of relative $(p+1,q)$-forms. By the $\mathrm{G}$-action, it is
clearly enough to understand the space of sections of this sheaf over the open
set $\mathcal{Y}$ from Section 4.3. Given
$\omega\in\mathcal{E}^{p,q}_{\eta}(\mathcal{Y})$, it is easy to see that there
is a unique $(p,q)$-form cohomologous to $\omega$ which can be written in the
form

$\displaystyle\sideset{}{{}^{\prime}}{\sum}_{|I|=p}dz_{I}\wedge\omega_{I},$
(5.1)

where $\Sigma^{\prime}$ denote that the summation is performed only over
strictly increasing multi-indeces777We order the set $I(n,k)$
lexicographically, i.e., $(\alpha,i)<(\alpha^{\prime},i^{\prime})$ if
$\alpha<\alpha^{\prime}$ or $\alpha=\alpha^{\prime}$ and $i<i^{\prime}$. and
each $\omega_{I}\in\mathcal{E}^{0,q}_{\mathfrak{q}}(\mathcal{Y})$.

As $\eta$ is holomorphic, $\partial$ and $\bar{\partial}$ commute with the
pullback map $\eta^{\ast}$. We see that
$\partial(\eta^{\ast}\mathcal{E}^{1,0}_{\mathfrak{r}}\wedge\mathcal{E}_{\mathfrak{q}}^{p,q})\subset\eta^{\ast}\mathcal{E}^{1,0}_{\mathfrak{r}}\wedge\mathcal{E}_{\mathfrak{q}}^{p+1,q}$
and
$\bar{\partial}(\eta^{\ast}\mathcal{E}^{1,0}_{\mathfrak{r}}\wedge\mathcal{E}_{\mathfrak{q}}^{p,q})\subset\eta^{\ast}\mathcal{E}^{1,0}_{\mathfrak{r}}\wedge\mathcal{E}_{\mathfrak{q}}^{p,q+1}$
and thus, $\partial$ and $\bar{\partial}$ descend to differential operators

$\displaystyle\partial_{\eta}\colon\
\mathcal{E}^{p,q}_{\eta}\rightarrow\mathcal{E}^{p+1,q}_{\eta}\qquad\mathrm{and}\qquad\bar{\partial}\colon\
\mathcal{E}^{p,q}_{\eta}\rightarrow\mathcal{E}^{p,q+1}_{\eta},$

respectively. From the definitions it easily follows that:

$\displaystyle\partial_{\eta}(\omega\wedge\omega^{\prime})=(\partial_{\eta}\omega)\wedge\omega^{\prime}+(-1)^{p+q}\omega\wedge\partial_{\eta}\omega^{\prime},$
(5.2) $\displaystyle\partial_{\eta}f=\sum_{(\alpha,i)\in I(n,k)}\frac{\partial
f}{\partial z_{\alpha i}}dz_{\alpha i},$

where $\omega\in\mathcal{E}^{p,q}_{\eta}(\mathcal{Y})$,
$\omega^{\prime}\in\mathcal{E}^{p^{\prime},q^{\prime}}_{\eta}(\mathcal{Y})$
and $f\in\mathcal{E}(\mathcal{Y})$. Recall from Section 4.2 that
$\partial_{z_{\alpha i}}\in\Gamma(E^{1,0}|_{\mathcal{Y}})$ and thus,
$\partial_{\eta}f(x)$, $x\in\mathcal{Y}$ depends only on the first weighted
jet $\mathfrak{j}^{1}_{x}f$ of $f$ at $x$ (see Section 2.2).

Recall from (4.3) that the distribution $E^{CS}$ is equal to $\ker(T\eta)$.

######  Proposition 5.1.

  1. $(i)$

The sheaf $\mathcal{E}^{p,q}_{\eta}$ is naturally isomorphic to the sheaf
$\mathcal{E}^{0,q}_{\mathfrak{q}}(\Lambda^{p}E^{CS\ast})$ of smooth
$(0,q)$-forms with values in the vector bundle $\Lambda^{p}E^{CS\ast}$ and
$(\mathcal{E}^{p,\ast}_{\eta},\bar{\partial})$ is a resolution of
$\mathcal{O}_{\mathfrak{q}}(\Lambda^{p}E^{CS\ast})$ by fine sheaves.

  2. $(ii)$

$\partial_{\eta}$ is a linear $\mathrm{G}$-invariant differential operator of
weighted order one and the sequence of sheaves
$(\mathcal{E}^{\ast,q}_{\eta},\partial_{\eta})$, $q\geq 0$ is exact.

  3. $(iii)$

The data $(\mathcal{E}^{p,q}_{\eta},(-1)^{p}\bar{\partial},\partial_{\eta})$
define a double complex of fine sheaves with exact rows and columns.

###### Proof.

$(i)$ By definition, the sequence of vector bundles $0\rightarrow
E^{CS\bot}\rightarrow T^{\ast}CS\rightarrow E^{CS\ast}\rightarrow 0$ is short
exact. Hence, also the sequence $0\rightarrow
E^{CS\bot}\wedge\Lambda^{p}T^{\ast}CS\rightarrow\Lambda^{p+1}T^{\ast}CS\rightarrow\Lambda^{p+1}E^{CS\ast}\rightarrow
0$, $p\geq 1$ is short exact. In view of the isomorphism
$\mathcal{E}^{p+1,q}_{\mathfrak{q}}\cong\mathcal{E}^{0,q}_{\mathfrak{q}}(\Lambda^{p+1}T^{\ast}CS)$,
it is enough to show that
$\eta^{\ast}\mathcal{E}^{1,0}_{\mathfrak{r}}\wedge\mathcal{E}_{\mathfrak{q}}^{p,q}$
is isomorphic to
$\mathcal{E}^{0,q}_{\mathfrak{q}}(E^{CS\bot}\wedge\Lambda^{p}T^{\ast}CS)\cong\mathcal{E}_{\mathfrak{q}}(E^{CS\bot})\wedge\mathcal{E}^{0,q}_{\mathfrak{q}}(\Lambda^{p}T^{\ast}CS)$.
Now $\eta^{\ast}\mathcal{E}^{1,0}_{\mathfrak{r}}$ is a sub-sheaf of
$\mathcal{E}_{\mathfrak{q}}^{1,0}=\mathcal{E}_{\mathfrak{q}}(T^{\ast}CS)$ and
since $\ker(T\eta)=E^{CS}$, it is contained in
$\mathcal{E}_{\mathfrak{q}}(E^{CS\bot})$. Using that $\ker(T\eta)=E^{CS}$
again, it is easy to see that the map
$\eta^{\ast}\mathcal{E}^{1,0}_{\mathfrak{r}}\rightarrow\mathcal{E}_{\mathfrak{q}}(E^{CS\bot})$
induces an isomorphism of stalks at any point. Hence,
$\eta^{\ast}\mathcal{E}^{1,0}_{\mathfrak{r}}\cong\mathcal{E}_{\mathfrak{q}}(E^{CS\bot})$
and the proof of the first claim is complete. The second claim is clear.

(ii) It is clear that $\partial_{\eta}$ is $\mathbb{C}$-linear. It is
$\mathrm{G}$-invariant as $\partial$ commutes with the pullback of any
holomorphic map and since $\eta$ is $\mathrm{G}$-equivariant. As we already
observed above that $\partial_{\eta}f(x)$ depends only on
$\mathfrak{j}^{1}_{x}f$ when $x\in\mathcal{Y}$, the $\mathrm{G}$-invariance of
$\partial_{\eta}$ shows that the same holds on $CS$ and thus,
$\partial_{\eta}$ is a differential operator of weighted order one. It remains
to check the exactness of the complex and using the $\mathrm{G}$-invariance,
it is enough to do this at $x\in\mathcal{Y}$. By Lemma 4.3, $\mathcal{Y}$ is
biholomorphic to $\mathbb{C}^{\ell}$ where $\ell={m\choose 2}+nk$. Hence, we
can view the standard coordinates $w_{1},\dots,w_{\ell}$ on
$\mathbb{C}^{\ell}$ as coordinates on $\mathcal{Y}$. If
$J=(j_{1},\dots,j_{q})$ where $j_{1},\dots,j_{q}\in\\{1,\dots,\ell\\}$, then
we put $d\bar{w}_{J}=d\bar{w}_{j_{1}}\wedge\cdots\wedge d\bar{w}_{j_{q}}$ and
$|J|=q$. Let
$\omega=\sideset{}{{}^{\prime}}{\sum}\limits_{|I|=p}dz_{I}\wedge\omega_{I}\in\mathcal{E}^{p,q}_{\eta}(\mathcal{Y})$
be the relative form as in (5.1). Then there are unique functions
$f_{I,J}\in\mathcal{E}_{\mathfrak{q}}(\mathcal{Y})$ so that
$\omega_{I}=\sideset{}{{}^{\prime}}{\sum}\limits_{|J|=q}f_{I,J}d\bar{w}_{J}$.
Assume that $\partial_{\eta}\omega=0$ on some open neighborhood
$\mathcal{U}_{x}$ of $x$. This is equivalent to saying that for each
increasing multi-index $J\colon\partial_{\eta}\sigma_{J}=0$ on
$\mathcal{U}_{x}$ where
$\sigma_{J}:=\sideset{}{{}^{\prime}}{\sum}_{I}f_{I,J}dz_{I}$. 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 $\phi_{J}$ such that
$\partial_{\eta}\phi_{J}=\sigma_{J}$ on some open neighborhood of $x$. Then
$\partial_{\eta}\big{(}\sideset{}{{}^{\prime}}{\sum}_{J}\phi_{J}\wedge
d\bar{w}_{J}\big{)}=\sideset{}{{}^{\prime}}{\sum}_{J}\sigma_{J}\wedge
d\bar{w}_{J}=\omega$ on some open neighborhood of $x$ and the proof is
complete.

(iii) This follows from $[\bar{\partial},\partial_{\eta}]=0$ and the
observations made above. ∎

###  5.2 Relative de Rham complex

By definition,
$\Omega^{\ast}_{\eta}:=\mathcal{E}^{\ast,0}_{\eta}\cap\ker\bar{\partial}$ is a
sheaf of holomorphic sections. Since $[\bar{\partial},\partial_{\eta}]=0$,
there is a complex of sheaves $(\Omega^{\ast}_{\eta},\partial_{\eta})$ and we
call it the relative de Rham complex.

######  Proposition 5.2.

  1. $(i)$

The relative de Rham complex is an exact sequence of sheaves which resolves
the sheaf $\eta^{-1}\mathcal{O}_{\mathfrak{r}}$.

  2. $(ii)$

The relative de Rham complex induces for each $r:=\ell+j\geq 0$ a long exact
sequence of vector bundles

$\displaystyle\mathfrak{gr}^{\ell+j}\xrightarrow{\mathfrak{gr}\partial_{\eta}}E^{CS\ast}\otimes\mathfrak{gr}^{\ell+j-1}\rightarrow\cdots$
$\displaystyle\hphantom{\mathfrak{gr}^{\ell+j}}{}\rightarrow\Lambda^{j}E^{CS\ast}\otimes\mathfrak{gr}^{\ell}\xrightarrow{\mathfrak{gr}\partial_{\eta}}\Lambda^{j+1}E^{CS\ast}\otimes\mathfrak{gr}^{\ell+j-1}\rightarrow\cdots.$
(5.3)

Let $s_{0}>0$, $s_{1},s_{2},s_{3}\geq 0$ be integers such that
$s_{0}+s_{1}+2s_{2}+3s_{3}=r$. Then the sequence (5.3) contains a long exact
subsequence

$\displaystyle 0\rightarrow{S}^{s_{0}}E^{CS\ast}\otimes
S^{s_{1},s_{2},s_{3}}\rightarrow
E^{CS\ast}\otimes{S}^{s_{0}-1}E^{CS\ast}\otimes
S^{s_{1},s_{2},s_{3}}\rightarrow\cdots$ (5.4)
$\displaystyle\hphantom{0}\rightarrow\Lambda^{j}E^{CS\ast}\otimes{S}^{s_{0}-j}E^{CS\ast}\otimes
S^{s_{1},s_{2},s_{3}}\\!\rightarrow\Lambda^{j+1}E^{CS\ast}\otimes{S}^{s_{0}-j-1}E^{CS\ast}\otimes
S^{s_{1},s_{2},s_{3}}\\!\rightarrow\cdots,$

where
${S}^{s_{1},s_{2},s_{3}}:={S}^{s_{1}}F^{CS\ast}\otimes{S}^{s_{2}}G^{CS}_{2}\otimes{S}^{s_{3}}G^{CS}_{3}$.

  3. $(iii)$

The kernel of the first map in (5.3) is
$\bigoplus_{s_{1}+2s_{2}+3s_{3}=r}{S}^{s_{1},s_{2},s_{3}}$.

###### Proof.

(i) Since $[\partial_{\eta},\bar{\partial}]=0$, the relative de Rham complex
is a sub-complex of the zero-th row
$(\mathcal{E}_{\mathfrak{q}}^{\ast,0},\partial_{\eta})$ of the double complex
from Proposition 5.3. By diagram chasing and using the exactness of columns
and rows in the double complex, one easily proves the exactness of the
relative de Rham complex. By (5.2), it easily follows that
$\eta^{-1}\mathcal{O}_{\mathfrak{r}}=\mathcal{E}^{0,0}_{\eta}\cap\ker(\partial_{\eta})\cap\ker(\bar{\partial})$.

(ii) The standard de Rham complex induces the Spencer complex (see [25]) which
is known to be exact. As the complex $(\Omega^{\ast}_{\eta},\partial_{\eta})$
is just a relative version of the (holomorphic) de Rham complex and
$\partial_{\eta}$ satisfies the usual properties of $\partial$, it is clear
that the relative de Rham complex induces for each ${s_{0}}>0$, $s_{1}$,
$s_{2}$ and $s_{3}$ the long exact sequence (5.4). The sequence (5.3) is the
direct sum of all such sequences as ${s_{0}}$, $s_{1}$, $s_{2},$ and $s_{3}$
ranges over all quadruples of non-negative integers satisfying
$r={s_{0}}+s_{1}+2s_{2}+3s_{3}$.

(iii) This readily follows from the part (ii). ∎

###  5.3 Twisted relative de Rham complex

The weight $\lambda:=(1-2n)\omega_{m}$ is $\mathfrak{g}$-integral and
$\mathfrak{r}$-dominant. Hence, there is an irreducible $\mathrm{R}$-module
$\mathbb{W}_{\lambda}$ with lowest weight $-\lambda$. Since $\mathfrak{r}$ is
associated to $\\{\alpha_{m}\\}$, it follows that $\dim\mathbb{W}_{\lambda}=1$
and so $\mathbb{W}_{\lambda}$ is also an irreducible $\mathrm{Q}$-module. We
will denote by $\mathcal{E}_{\mathfrak{q}}(\lambda)$ and
$\mathcal{O}_{\mathfrak{q}}(\lambda)$ the sheaves of smooth and holomorphic
sections of
$W_{\lambda}^{CS}:=\mathrm{G}\times_{\mathrm{Q}}\mathbb{W}_{\lambda}$,
respectively. If $W$ is a vector bundle over $CS$, then we denote
$W(\lambda):=W\otimes W_{\lambda}^{CS}$, i.e., we twist $W$ by tensoring with
the line bundle $W_{\lambda}^{CS}$. It is not hard to see that
$\eta^{\ast}\mathcal{E}_{\mathfrak{r}}(\lambda)\cong\mathcal{E}_{\mathfrak{q}}(\lambda)$
and
$\eta^{\ast}\mathcal{O}_{\mathfrak{r}}(\lambda)=\mathcal{O}_{\mathfrak{q}}(\lambda)$
where we denote by the subscript $\mathfrak{r}$ the corresponding sheaves over
$TS$.

We call
$\mathcal{E}^{p,q}_{\eta}(\lambda):=\mathcal{E}^{p,q}_{\eta}\otimes_{\eta^{-1}\mathcal{E}_{\mathfrak{r}}}\eta^{-1}\mathcal{E}_{\mathfrak{r}}(\lambda)$
the sheaf of twisted relative $(p,q)$-forms. Consider the following sequence
of canonical isomorphisms:

$\displaystyle\mathcal{E}^{p,q}_{\eta}(\lambda)\rightarrow\mathcal{E}^{p,q}_{\eta}\otimes_{\mathcal{E}_{\mathfrak{q}}}\mathcal{E}_{\mathfrak{q}}\otimes_{\eta^{-1}\mathcal{E}_{\mathfrak{r}}}\eta^{-1}\mathcal{E}_{\mathfrak{r}}(\lambda)\rightarrow\mathcal{E}^{p,q}_{\eta}\otimes_{\mathcal{E}_{\mathfrak{q}}}\eta^{\ast}\mathcal{E}_{\mathfrak{r}}(\lambda)$
$\displaystyle\hphantom{\mathcal{E}^{p,q}_{\eta}(\lambda)}{}\rightarrow\mathcal{E}^{p,q}_{\eta}\otimes_{\mathcal{E}_{\mathfrak{q}}}\mathcal{E}_{\mathfrak{q}}(\lambda)\rightarrow\mathcal{E}^{0,q}_{\mathfrak{q}}\otimes_{\mathcal{E}_{\mathfrak{q}}}\big{(}\mathcal{E}^{p,0}_{\eta}\otimes_{\mathcal{E}_{\mathfrak{q}}}\mathcal{E}_{\mathfrak{q}}(\lambda)\big{)}\rightarrow\mathcal{E}^{0,q}_{\mathfrak{q}}\big{(}\Lambda^{p}E^{CS\ast}(\lambda)\big{)}.$
(5.5)

We see that $\mathcal{E}^{p,q}_{\eta}(\lambda)$ is isomorphic to the sheaf of
smooth $(0,q)$-forms with values in $\Lambda^{p}E^{CS\ast}(\lambda)$. Hence,
the Dolbeault differential induces a differential
$\bar{\partial}\colon\mathcal{E}^{p,q}_{\eta}(\lambda)\rightarrow\mathcal{E}^{p,q+1}_{\eta}(\lambda)$
and a complex $(\mathcal{E}^{p,\ast}_{\eta}(\lambda),\bar{\partial})$.

A section of $\mathcal{E}^{p,q}_{\eta}(\lambda)$ is by definition a finite sum
of decomposable elements $\omega\otimes v$ where $\omega$ and $v$ are sections
of $\mathcal{E}^{p,q}_{\eta}$ and
$\eta^{-1}\mathcal{E}_{\mathfrak{r}}(\lambda)$, respectively. As any section
of $\eta^{-1}\mathcal{E}_{\mathfrak{r}}$ as well as transition functions
between sections of $\eta^{-1}\mathcal{E}_{\mathfrak{r}}(\lambda)$ belong to
$\ker(\partial_{\eta})$, it follows that there is a unique linear differential
operator

$\displaystyle\mathcal{E}^{p,q}_{\eta}(\lambda)\rightarrow\mathcal{E}^{p+1,q}_{\eta}(\lambda),$

which satisfies $\omega\otimes v\mapsto\partial_{\eta}\omega\otimes v$. We
denote the operator also by $\partial_{\eta}$ as there is no risk of
confusion. It is clear that $\partial_{\eta}$ is a linear
$\mathrm{G}$-invariant differential operator of weighted order one.

######  Proposition 5.3.

Let $p,q\geq 0$ be integers.

  1. $(i)$

The sequence of sheaves
$(\mathcal{E}^{(p,\ast)}_{\eta}(\lambda),\bar{\partial})$ is exact.

  2. $(ii)$

The sequence of sheaves
$(\mathcal{E}^{(\ast,q)}_{\eta}(\lambda),\partial_{\eta})$ is exact.

  3. $(iii)$

There is a double complex
$(\mathcal{E}_{\eta}^{p,q}(\lambda),\partial_{\eta},(-1)^{p}\bar{\partial})$
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 $[\bar{\partial},\partial_{\eta}]=0$. To see
this, notice that a section of $\mathcal{E}^{p,q}_{\eta}(\lambda)$ 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). ∎

Put
$\Omega^{\ast}_{\eta}(\lambda):=\mathcal{E}^{\ast,0}_{\eta}(\lambda)\cap\ker(\bar{\partial})$.
The complex $(\mathcal{E}^{\ast,0}_{\eta}(\lambda),\partial_{\eta})$ contains
a sub-complex $(\Omega^{\ast}_{\eta}(\lambda),\partial_{\eta})$ which we call
the twisted relative de Rham complex. As in Proposition 5.2, one can easily
see that $(\Omega_{\eta}^{\ast}(\lambda),\partial_{\eta})$ 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 $(\Omega_{\eta}^{\ast}(\lambda),\partial_{\eta})$
induces for each $r\geq 0$ a long exact sequence of vector bundles

$\displaystyle\big{(}\Lambda^{\bullet}E^{CS\ast}\otimes\mathfrak{gr}^{r-\bullet}(\lambda),\mathfrak{gr}\partial_{\eta}\big{)}.$
(5.6)

Let $s_{0}>0$, $s_{1},s_{2},s_{3}\geq 0$ be integers such that
$s_{0}+s_{1}+2s_{2}+3s_{3}=r$. Then the sequence (5.6) contains a long exact
subsequence

$\displaystyle\big{(}\Lambda^{\bullet}E^{CS\ast}\otimes{S}^{s_{0}-\bullet}E^{CS\ast}\otimes{S}^{s_{1},s_{2},s_{3}}(\lambda),\mathfrak{gr}\partial_{\eta}\big{)},$
(5.7)

where ${S}^{s_{1},s_{2},s_{3}}$ is defined in Proposition 5.2. The kernel of
the first map in (5.6) is the bundle
$\bigoplus_{s_{1}+2s_{2}+3s_{3}=r}{S}^{s_{1},s_{2},s_{3}}(\lambda)$.

###  5.4 Relative BGG sequence

We know that $\Omega^{p}_{\eta}(\lambda)$ is isomorphic to the sheaf of
holomorphic sections of
$\Lambda^{p}E^{CS\ast}(\lambda)=\mathrm{G}\times_{\mathrm{Q}}(\Lambda^{p}\mathbb{E}^{\ast}\otimes\mathbb{W}_{\lambda})$.
The $\mathrm{Q}$-module $\Lambda^{p}\mathbb{E}^{\ast}$ is not irreducible.
Decomposing this module into irreducible $\mathrm{Q}$-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 $a\in\mathbb{N}^{k,n}_{++}$ and $w_{a}\in W^{\mathfrak{q}}_{\mathfrak{r}}$
be as in Section 3.2. Then

$\displaystyle\Lambda^{p}\mathbb{E}^{\ast}\otimes\mathbb{W}_{\lambda}=\\!\\!\bigoplus_{a\in\mathbb{N}^{k,n}_{++}:|a|=p}\\!\\!\mathbb{W}_{\lambda_{a}}\qquad\text{and
thus}\qquad\Omega^{p}_{\eta}(\lambda)=\\!\\!\bigoplus_{a\in\mathbb{N}^{n,k}_{++}:|a|=p}\\!\\!\mathcal{O}_{\mathfrak{q}}(\lambda_{a}),$
(5.8)

where $\mathbb{W}_{\lambda_{a}}$ is an irreducible $\mathrm{Q}$-module with
lowest weight $-\lambda_{a}:=-w_{a}.\lambda$.

There is a linear $\mathrm{G}$-invariant differential operator

$\displaystyle\partial_{a^{\prime}}^{a}\colon\
\mathcal{O}_{\mathfrak{q}}(\lambda_{a})\rightarrow\Omega^{p}_{\eta}(\lambda)\xrightarrow{\partial_{\eta}}\Omega^{p}_{\eta}(\lambda)\rightarrow\mathcal{O}_{\mathfrak{q}}(\lambda_{a^{\prime}})$
(5.9)

where the first map is the canonical inclusion and the last map is the
canonical projection. If $a\nless a^{\prime}$, then
$\partial_{a^{\prime}}^{a}=0$.

###### Proof.

Recall from Section 3.1 that the semi-simple part $\mathfrak{r}_{0}^{ss}$ of
$\mathfrak{r}_{0}$ is isomorphic to $\mathfrak{sl}(m,\mathbb{C})$ and that
$\mathfrak{r}_{0}^{ss}\cap\mathfrak{q}$ is a parabolic subalgebra of
$\mathfrak{r}_{0}^{ss}$. 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 $\mathbb{W}_{\lambda}$ and
$(\mathfrak{r}_{0}^{ss},\mathfrak{r}_{0}^{ss}\cap\mathfrak{q})$ and the
identity $\ell(w_{a})=|a|$ from Lemma 3.6. Recall from [2, Section 8.7] that
the graph of the relative BGG sequence coincides with the relative Hasse graph
$W^{\mathfrak{q}}_{\mathfrak{r}}$. The last claim then follows from Lemma 3.4.
∎

######  Remark 5.6.

Let $a=(a_{1},\dots,a_{k})\in\mathbb{N}^{k,n}_{++}$ and
$b=(b_{1},\dots,b_{n})\in\mathbb{N}^{n,k}_{++}$ be the conjugated partition.
In order to compute $\lambda_{a}$ from Proposition 5.5, notice that

$\displaystyle\lambda+\rho=\bigg{(}\frac{2k-1}{2},\dots,\frac{3}{2},\frac{1}{2}\,\bigg{|}\,-\frac{1}{2},-\frac{3}{2},\dots,\frac{-2n+1}{2}\bigg{)}.$
(5.10)

Since $w_{a}(\omega_{m})=\omega_{m}$, we have $w_{a}(\lambda)=\lambda$ and
thus $\lambda_{a}=w_{a}(\lambda+\rho)-\rho=\lambda+w_{a}\rho-\rho$. By (3.10),
it follows that

$\displaystyle\lambda_{a}=\lambda+(-a_{k},\dots,-a_{1}\,|\,b_{1},b_{2},\dots,b_{n}).$

###  5.5 Direct image of the relative BGG sequence

Recall from [2, Section 5.3] that given a $\mathfrak{g}$-integral and
$\mathfrak{q}$-dominant weight $\nu$, there is at most one
$\mathfrak{p}$-dominant weight in the $W_{\mathfrak{p}}^{\mathfrak{q}}$-orbit
of $\nu$. If there is no $\mathfrak{p}$-dominant weight, then all direct
images of $\mathcal{O}_{\mathfrak{q}}(\nu)$ vanish. If there is a
$\mathfrak{p}$-dominant weight, say $\mu=w.\nu$ where $w\in
W^{\mathfrak{q}}_{\mathfrak{p}}$, then
$\tau^{\ell(w)}_{\ast}\mathcal{O}_{\mathfrak{q}}(\nu)\cong\mathcal{O}_{\mathfrak{p}}(\mu)$
is the unique non-zero direct image of $\mathcal{O}_{\mathfrak{q}}(\nu)$.

######  Proposition 5.7.

Let $n\geq k\geq 2$ and $a=(a_{1},\dots,a_{k})\in\mathbb{N}^{k,n}_{++}$. Put
$\mu^{\pm}:=\frac{1}{2}(-2n+1,\dots,-2n+1\,|\,1,\dots,1,\pm 1)$ and
$\mu_{a}:=\mu^{\ast}-(a_{k},\dots,a_{1}\,|\,0,\dots,0)$ where $\ast=+$ if
$d(a)\equiv n\mod 2$ and $\ast=-$ otherwise. Then

$\displaystyle\tau_{\ast}^{q}(\mathcal{O}_{\mathfrak{q}}(\lambda_{a}))=\begin{cases}\mathcal{O}_{\mathfrak{p}}(\mu_{a}),&\mathrm{if}\
a\in S^{k},\ q=\ell(a):={n\choose 2}-q(a),\\\
\\{0\\},&\mathrm{otherwise}.\end{cases}$

###### Proof.

By definition, each $w\in W_{\mathfrak{p}}^{\mathfrak{q}}$ fixes the first $k$
coefficients of $\lambda_{a}$ and so it is enough to look at the last $n$
coefficients. By Remark 5.6, it follows

$\displaystyle
w_{a}(\lambda+\rho)=\lambda_{a}+\rho=\bigg{(}\dots,i-a_{i}-\frac{1}{2},\dots,\frac{1}{2}-a_{1}\,\bigg{|}\,b_{1}-\frac{1}{2},\dots,b_{j}-j+\frac{1}{2},\dots\bigg{)},$

where $b=(b_{1},\dots,b_{n})$ is the conjugated partition. Put
$c:=(c_{1},\dots,c_{n})$ where $c_{j}:=\big{|}b_{j}-j+\frac{1}{2}\big{|}$. By
(3.2) and Table 1, if $c_{i}=c_{j}$ for some $i\neq j$, then there cannot be a
$\mathfrak{p}$-dominant weight in the $W^{\mathfrak{q}}_{\mathfrak{p}}$-orbit
of $\lambda_{a}$. If $a\not\in S^{k}$, then by Lemma 3.5 there is
$s\in\\{0,\dots,k-1\\}$ such that888Notice that at this point we need that
$n\geq k$. $w_{a}(k-s)=k+i\geq k$ and $w_{a}(k+s+1)=k+j\geq k$ for some
distinct positive integers $i$ and $j$. By (5.10), it follows that
$b_{i}-i+\frac{1}{2}=\frac{1}{2}+s$, $b_{j}-j-\frac{1}{2}=-\frac{1}{2}-s$ and
thus $c_{i}=c_{j}$. Hence, all direct images of
$\mathcal{O}_{\mathfrak{q}}(\lambda_{a})$ are zero.

We may now suppose that $a\in S^{k}$. By the definition of $d(a)$, we have
$b_{d(a)}-d(a)+\frac{1}{2}>0>b_{d(a)+1}-d(a)-\frac{1}{2}$. By Lemma 3.5 and
(5.10), it follows that $(c_{1},\dots,c_{n})$ is a permutation of
$\big{(}\frac{2n-1}{2},\dots,\frac{3}{2},\frac{1}{2}\big{)}$. As $\lambda_{a}$
is $\mathfrak{q}$-dominant, we know that the sequence
$\big{(}b_{1}-\frac{1}{2},\cdots,b_{j}-j+\frac{1}{2},\dots\big{)}$ is
decreasing. Thus for each integer $i=1,\dots,d(a)$, the set $\\{j\colon
i<j\leq n,\ c_{i}>c_{j}\\}$ contains precisely $b_{i}-i$ distinct elements.
Altogether, there are precisely
$\sum\limits_{\ell=1}^{d(a)}(b_{\ell}-\ell)=q(a)$ pairs $i<j$ such that
$c_{i}>c_{j}$. Equivalently, there are $\ell(a)$ pairs $i<j$ such that
$c_{i}<c_{j}$. It follows that the length of the permutation that maps
$(c_{1},\dots,c_{n})$ to
$\big{(}\frac{2n-1}{2},\dots,\frac{3}{2},\frac{1}{2}\big{)}$ is precisely
$\ell(a)$. Now it is easy to see (recall (3.2)) that there is $w\in
W^{\mathfrak{q}}_{\mathfrak{p}}$ such that $w.\lambda_{a}$ is
$\mathfrak{p}$-dominant and $\ell(w)=\ell(a)$. As there are $n-d(a)$ negative
numbers in the sequence
$\big{(}b_{1}-\frac{1}{2},\dots,b_{j}-j+\frac{1}{2},\dots\big{)}$, the last
claim about the sign of the last coefficient of $w.\lambda_{a}$ also follows.
This completes the proof.∎

######  Remark 5.8.

In Proposition 5.7 we recovered the $W^{\mathfrak{p}}$-orbit of the singular
weight $\mu^{+}$ if $n$ is even and of $\mu^{-}$ if $n$ is odd which was
computed in [14]. There is an automorphism of $\mathfrak{g}$ which swaps
$\alpha_{m}$ and $\alpha_{m-1}$ and hence, it swaps also the associated
parabolic subalgebras. If we cross in (3.3) the simple root $\alpha_{m-1}$
instead of $\alpha_{m}$, take $(1-2n)\omega_{m-1}$ as $\lambda$ and follow the
computations given above, we will get the $W^{\mathfrak{p}}$-orbit of
$\mu^{+}$ if $n$ is odd and of $\mu^{-}$ 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
$\mathcal{E}^{p,q}_{\eta}(\lambda)=\bigoplus_{a\in\mathbb{N}^{k,n}_{++}:|a|=p}\mathcal{E}^{0,q}_{\mathfrak{q}}(\lambda_{a})$.
Let $a,a^{\prime}\in\mathbb{N}^{k,n}_{++}$ be such that
$p=|a|=|a^{\prime}|-1$. Then there is a linear differential operator

$\displaystyle\partial_{a^{\prime}}^{a}\colon\
\mathcal{E}^{0,q}_{\mathfrak{q}}(\lambda_{a})\rightarrow\mathcal{E}^{p,q}_{\eta}(\lambda)\xrightarrow{\partial_{\eta}}\mathcal{E}^{p+1,q}_{\eta}(\lambda)\rightarrow\mathcal{E}^{0,q}_{\mathfrak{q}}(\lambda_{a^{\prime}})$
(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
$\partial_{a^{\prime}}^{a}$ as in (5.9) as there is no risk of confusion.
Recall from Proposition 5.5 that $\partial^{a}_{a^{\prime}}=0$ if $a\nless
a^{\prime}$.

Suppose that $\mathcal{U}$ is an open, contractible and Stein subset of $M$.
We put
$\mathcal{E}^{p,q}_{\eta}(\tau^{-1}(\mathcal{U}),\lambda):=\Gamma(\tau^{-1}(\mathcal{U}),\mathcal{E}^{p,q}_{\eta}(\lambda))$,
i.e., this is the space of sections of the sheaf
$\mathcal{E}^{p,q}_{\eta}(\lambda)$ over $\tau^{-1}(\mathcal{U})$. Then there
is a double complex

$\displaystyle\begin{split}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
6.75pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 62.45331pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 159.4377pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 229.85767pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-6.75pt\raise-35.56888pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
30.75pt\raise-35.56888pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
30.75pt\raise-35.56888pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{E}^{p,q+1}_{\eta}(\tau^{-1}(\mathcal{U}),\lambda)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
106.85059pt\raise-29.62166pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.94722pt\hbox{$\scriptstyle{D^{\prime\prime}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
124.15662pt\raise-35.56888pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
65.45331pt\raise-3.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
124.15662pt\raise-35.56888pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{E}^{p+1,q+1}_{\eta}(\tau^{-1}(\mathcal{U}),\lambda)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
162.4377pt\raise-3.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
226.10767pt\raise-35.56888pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
226.10767pt\raise-35.56888pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\cdots}$}}}}}}}{\hbox{\kern-6.75pt\raise-76.70663pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
34.32779pt\raise-76.70663pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
34.32779pt\raise-76.70663pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{E}^{p,q}_{\eta}(\tau^{-1}(\mathcal{U}),\lambda)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
106.85059pt\raise-70.75941pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.94722pt\hbox{$\scriptstyle{D^{\prime\prime}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
127.73439pt\raise-76.70663pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
52.36346pt\raise-56.13776pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.94722pt\hbox{$\scriptstyle{D^{\prime}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
65.45331pt\raise-41.06888pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
127.73439pt\raise-76.70663pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{E}^{p+1,q}_{\eta}(\tau^{-1}(\mathcal{U}),\lambda)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
149.34785pt\raise-56.13776pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.94722pt\hbox{$\scriptstyle{D^{\prime}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
162.4377pt\raise-41.06888pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
224.71878pt\raise-76.70663pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
224.71878pt\raise-76.70663pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\cdots,}$}}}}}}}{\hbox{\kern-3.0pt\raise-112.27551pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
62.45331pt\raise-112.27551pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
65.45331pt\raise-82.20663pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
159.4377pt\raise-112.27551pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
162.4377pt\raise-82.20663pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
229.85767pt\raise-112.27551pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{split}$
(5.12)

where $D^{\prime}=(-1)^{p}\bar{\partial}$ and
$D^{\prime\prime}=\partial_{\eta}$. Put
$T^{i}(\mathcal{U}):=\bigoplus_{p+q=i}\mathcal{E}^{p,q}_{\eta}(\tau^{-1}(\mathcal{U}),\lambda)$.
We obtain three complexes $(T^{\ast}(\mathcal{U}),D^{\prime})$,
$(T^{\ast}(\mathcal{U}),D^{\prime\prime})$ and
$(T^{\ast}(\mathcal{U}),D^{\prime}+D^{\prime\prime})$.

######  Lemma 5.9.

Let $a\in\mathbb{N}^{k,n}_{++}$ and $\mathcal{U}$ be the open, contractible
and Stein subset of $M$ as above. Then

$\displaystyle
H^{q}\big{(}\tau^{-1}(\mathcal{U}),\mathcal{O}_{\mathfrak{q}}(\lambda_{a})\big{)}=\begin{cases}\Gamma(\mathcal{U},\mathcal{O}_{\mathfrak{p}}(\mu_{a})),&a\in
S^{k},\ q=\ell(a),\\\ \\{0\\},&\mathrm{otherwise},\end{cases}$ (5.13)

and thus also

$\displaystyle H^{{n\choose
2}+j}(T^{\ast}(\mathcal{U}),D^{\prime})=\bigoplus_{a\in
S^{k}_{j}}H^{\ell(a)}\big{(}\tau^{-1}(\mathcal{U}),\mathcal{O}_{\mathfrak{q}}(\lambda_{a})\big{)}.$

###### Proof.

The first claim follows from Proposition 5.7 and application of the Leray
spectral sequence as explained in [2]. For the second claim, recall from [27,
Theorem 3.20] that the sheaf cohomology is equal to the Dolbeault cohomology,
i.e., there is an isomorphism

$\displaystyle
H^{q}\big{(}\tau^{-1}(\mathcal{U}),\mathcal{O}_{\mathfrak{q}}(\lambda_{a})\big{)}\cong
H^{q}\big{(}\mathcal{E}^{0,\ast}_{\mathfrak{q}}\big{(}\tau^{-1}(\mathcal{U}),\lambda_{a}\big{)},\bar{\partial}\big{)}.$
(5.14)

The cohomology group appears on the $(|a|+\ell(a))=(d(a)+2q(a)+{n\choose
2}-q(a))=({n\choose 2}+r(a))$-th diagonal of the double complex. Here, see
Proposition 5.7, we use that $\ell(a)={n\choose 2}-q(a)$, the notation from
(3.6) and $S^{k}_{j}=\\{a\in S^{k}\colon r(a)=j\\}$. ∎

##  6 $\boldsymbol{k}$-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 $\mathrm{G}$-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
$\mathcal{E}^{0,q}_{\mathfrak{q}}(\tau^{-1}(\mathcal{U}),\lambda_{a})$ is the
space of sections of the sheaf $\mathcal{E}^{0,q}_{\mathfrak{q}}(\lambda_{a})$
over $\tau^{-1}(\mathcal{U})$. If
$\alpha\in\mathcal{E}^{0,q}_{\mathfrak{q}}(\tau^{-1}(\mathcal{U}),\lambda_{a})$
is $\bar{\partial}$-closed, then we will denote by $[\alpha]\in
H^{q}(\tau^{-1}(\mathcal{U}),\mathcal{O}_{\mathfrak{q}}(\lambda_{a}))$ the
corresponding cohomology class.

######  Lemma 6.1.

Let $j\geq 0$, $a\in S^{k}_{j}$, $a^{\prime}\in S^{k}_{j+1}$ be such that
$a<a^{\prime}$ and $\mathcal{U}$ be the Stein set as above. Then there is a
linear, local and $\mathrm{G}$-invariant operator

$\displaystyle D_{a^{\prime}}^{a}\colon\
\Gamma(\mathcal{U},\mathcal{O}_{\mathfrak{p}}(\mu_{a}))\rightarrow\Gamma(\mathcal{U},\mathcal{O}_{\mathfrak{p}}(\mu_{a^{\prime}})).$

###### Proof.

Let us for a moment put $\mathcal{V}:=\tau^{-1}(\mathcal{U})$. Using the
isomorphisms from (5.13), it is enough to define a map
$H^{\ell(a)}(\mathcal{V},\mathcal{O}_{\mathfrak{q}}(\lambda_{a}))\rightarrow
H^{\ell(a^{\prime})}(\mathcal{V},\mathcal{O}_{\mathfrak{q}}(\lambda_{a^{\prime}}))$
which has the right properties. By assumption, we have
$|a^{\prime}|-|a|\in\\{1,2\\}$. Let us first consider $|a^{\prime}|-|a|=1$.
Then $q:=\ell(a^{\prime})=\ell(a)$ and by (5.11), we have the map
$\partial_{a^{\prime}}^{a}\colon\mathcal{E}^{0,q}_{\mathfrak{q}}(\mathcal{V},\lambda_{a})\rightarrow\mathcal{E}^{0,q}_{\mathfrak{q}}(\mathcal{V},\lambda_{a^{\prime}})$
in the double complex (5.12). The induced map on cohomology is
$D_{a^{\prime}}^{a}$.

If $|a^{\prime}|-|a|=2$, then $q:=\ell(a)=\ell(a^{\prime})+1$ and we find that
there are precisely two non-symmetric partitions $b,c\in\mathbb{N}^{k,n}_{++}$
such that $a<b<a^{\prime}$ and $a<c<a^{\prime}$. Then there is a diagram

$\displaystyle\begin{split}&\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
24.18881pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{E}^{0,q}_{\mathfrak{q}}(\mathcal{V},\lambda_{a})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
51.50941pt\raise 6.73611pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.98611pt\hbox{$\scriptstyle{\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!(\partial^{a}_{b},\partial_{c}^{a})}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 52.85548pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
52.85548pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{E}^{0,q}_{\mathfrak{q}}(\mathcal{V},\lambda_{b})\oplus\mathcal{E}^{0,q}_{\mathfrak{q}}(\mathcal{V},\lambda_{c})}$}}}}}}}{\hbox{\kern
206.535pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern-3.0pt\raise-41.13776pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
48.18881pt\raise-41.13776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{E}^{0,q-1}_{\mathfrak{q}}(\mathcal{V},\lambda_{b})\oplus\mathcal{E}^{0,q-1}_{\mathfrak{q}}(\mathcal{V},\lambda_{c})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
78.39609pt\raise-20.56888pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.18056pt\hbox{$\scriptstyle{(-1)^{p}\bar{\partial}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 102.68639pt\raise-5.5pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
127.09447pt\raise-34.35999pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.47221pt\hbox{$\scriptstyle{\ \ \ \ \ \ \ \ \
\partial_{a^{\prime}}^{b}+\partial_{a^{\prime}}^{c}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
181.18396pt\raise-41.13776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
181.18396pt\raise-41.13776pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{E}^{0,q-1}_{\mathfrak{q}}(\mathcal{V},\lambda_{a^{\prime}}),}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{split}$
(6.1)

which lives in the double complex (5.12). Let
$\alpha\in\mathcal{E}^{0,q}_{\mathfrak{q}}(\mathcal{V},\lambda_{a})$ be
$\bar{\partial}$-closed. Then $\partial_{b}^{a}\alpha$ and
$\partial_{c}^{a}\alpha$ are also $\bar{\partial}$-closed and thus by Lemma
5.9 and the isomorphism (5.14), we can find $\beta$ and $\gamma$ such that
$\partial_{b}^{a}\alpha=(-1)^{p}\bar{\partial}\beta$ and
$\partial_{c}^{a}\alpha=(-1)^{p}\bar{\partial}\gamma$ where $p=|a|+1$. Since
the relative BGG sequence is a complex, we have

$\displaystyle\bar{\partial}\big{(}\partial_{a^{\prime}}^{b}\beta+\partial_{a^{\prime}}^{c}\gamma\big{)}=(-1)^{p}(\partial_{a^{\prime}}^{b}\partial_{b}^{a}+\partial_{a^{\prime}}^{c}\partial_{c}^{a})\alpha=0,$

which shows that
$\partial_{a^{\prime}}^{b}\beta+\partial_{a^{\prime}}^{c}\gamma$ is a cocycle.
Of course this elements depends on choices but we claim that
$[\partial_{a^{\prime}}^{b}\beta+\partial_{a^{\prime}}^{c}\gamma]$ depends
only on $[\alpha]$. It is easy to see that
$[\partial_{a^{\prime}}^{b}\beta+\partial_{a^{\prime}}^{c}\gamma]$ does not
depend on the choices of $\beta$ and $\gamma$. If $[\alpha]=0$, say
$\alpha=(-1)^{p-1}\bar{\partial}\varrho$, then we may put
$\beta=-\partial_{b}^{a}\varrho$ and $\gamma=-\partial_{c}^{a}\varrho$ and
thus
$\partial_{a^{\prime}}^{b}\beta+\partial_{a^{\prime}}^{c}\gamma=-(\partial_{a^{\prime}}^{b}\partial^{a}_{b}+\partial^{c}_{a^{\prime}}\partial_{c}^{a})\varrho=0$.
Hence, we can put
$D_{a^{\prime}}^{a}[\alpha]:=[\partial_{a^{\prime}}^{b}\beta+\partial_{a^{\prime}}^{c}\gamma]$.

From the construction is clear that $D_{a^{\prime}}^{a}$ is linear. The
locality follows from the fact that $D_{a^{\prime}}^{a}$ is compatible with
restrictions to smaller Stein subsets of $\mathcal{U}$. As the operators in
the double complex (5.12) are $\mathrm{G}$-invariant, it is easy to verify
that each operator $D_{a^{\prime}}^{a}$ is $\mathrm{G}$-invariant. ∎

Put $\mathcal{O}_{j}:=\bigoplus_{a\in
S^{k}_{j}}\mathcal{O}_{\mathfrak{p}}(\mu_{a})$ and
$\mathcal{O}_{j}(\mathcal{U}):=\Gamma(\mathcal{U},\mathcal{O}_{j})$. If $a\in
S^{k}_{j}$ and $s\in\mathcal{O}_{j}(\mathcal{U})$, then we denote by $s_{a}$
the $a$-th component of $s$ so that we may write $s=(s_{a})_{a\in S^{k}_{j}}$.
We call the following complex (6.2) the $k$-Dirac complex.

######  Theorem 6.2.

With the notation set above, there is a complex

$\displaystyle\mathcal{O}_{0}(\mathcal{U})\xrightarrow{D_{0}}\mathcal{O}_{1}(\mathcal{U})\rightarrow\cdots\rightarrow\mathcal{O}_{j}(\mathcal{U})\xrightarrow{D_{j}}\mathcal{O}_{j+1}(\mathcal{U})\rightarrow\cdots$
(6.2)

of linear $\mathrm{G}$-invariant operators where

$\displaystyle(D_{j}s)_{a^{\prime}}=\sum_{a<a^{\prime}}D_{a^{\prime}}^{a}s_{a}.$

###### Proof.

Let $a,a^{\prime}\in S^{k}$ be such that $a<a^{\prime}$,
$r(a)=r(a^{\prime})-2$. We need to verify that
$\sum\limits_{a^{\prime\prime}\in S^{k}\colon
a<a^{\prime\prime}<a^{\prime}}D_{a^{\prime}}^{a^{\prime\prime}}D_{a^{\prime\prime}}^{a}=0$.
Observe that $|a^{\prime}|-|a|\in\\{3,4\\}$. Let us first assume that
$|a^{\prime}|-3=|a|$. Then there are at most two symmetric partitions
$a^{\prime\prime}$ such that $a<a^{\prime\prime}<a^{\prime}$. If there is only
one such symmetric partition $a^{\prime\prime}$, then, since the relative BGG
sequence is a complex, it follows easily that
$D_{a^{\prime}}^{a^{\prime\prime}}D_{a^{\prime\prime}}^{a}=0$. So we can
assume that there are two symmetric partitions, say $a^{\prime\prime}_{1}$,
$a^{\prime\prime}_{2}$. Consider for example

$\textstyle{b=\begin{picture}(9.0,6.0)\put(0.0,6.0){\line(1,0){9.0}}
\put(0.0,3.0){\line(1,0){9.0}} \put(0.0,0.0){\line(1,0){3.0}}
\put(0.0,6.0){\line(0,-1){6.0}} \put(3.0,6.0){\line(0,-1){6.0}}
\put(6.0,6.0){\line(0,-1){3.0}} \put(9.0,6.0){\line(0,-1){3.0}}
\end{picture}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{c=\begin{picture}(9.0,6.0)\put(0.0,6.0){\line(1,0){9.0}}
\put(0.0,3.0){\line(1,0){9.0}} \put(0.0,0.0){\line(1,0){6.0}}
\put(0.0,6.0){\line(0,-1){6.0}} \put(3.0,6.0){\line(0,-1){6.0}}
\put(6.0,6.0){\line(0,-1){6.0}} \put(9.0,6.0){\line(0,-1){3.0}}
\end{picture}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{a=\begin{picture}(6.0,6.0)\put(0.0,6.0){\line(1,0){6.0}}
\put(0.0,3.0){\line(1,0){6.0}} \put(0.0,0.0){\line(1,0){3.0}}
\put(0.0,6.0){\line(0,-1){6.0}} \put(3.0,6.0){\line(0,-1){6.0}}
\put(6.0,6.0){\line(0,-1){3.0}}
\end{picture}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{a^{\prime\prime}_{1}=\begin{picture}(6.0,6.0)\put(0.0,6.0){\line(1,0){6.0}}
\put(0.0,3.0){\line(1,0){6.0}} \put(0.0,0.0){\line(1,0){6.0}}
\put(0.0,6.0){\line(0,-1){6.0}} \put(3.0,6.0){\line(0,-1){6.0}}
\put(6.0,6.0){\line(0,-1){6.0}}
\end{picture}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{a^{\prime\prime}_{2}=\begin{picture}(9.0,9.0)\put(0.0,9.0){\line(1,0){9.0}}
\put(0.0,6.0){\line(1,0){9.0}} \put(0.0,3.0){\line(1,0){3.0}}
\put(0.0,0.0){\line(1,0){3.0}} \put(0.0,9.0){\line(0,-1){9.0}}
\put(3.0,9.0){\line(0,-1){9.0}} \put(6.0,9.0){\line(0,-1){3.0}}
\put(9.0,9.0){\line(0,-1){3.0}}
\end{picture}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{a^{\prime}=\begin{picture}(9.0,9.0)\put(0.0,9.0){\line(1,0){9.0}}
\put(0.0,6.0){\line(1,0){9.0}} \put(0.0,3.0){\line(1,0){6.0}}
\put(0.0,0.0){\line(1,0){3.0}} \put(0.0,9.0){\line(0,-1){9.0}}
\put(3.0,9.0){\line(0,-1){9.0}} \put(6.0,9.0){\line(0,-1){6.0}}
\put(9.0,9.0){\line(0,-1){3.0}}
\end{picture}.}$$\textstyle{b^{\prime}=\begin{picture}(6.0,9.0)\put(0.0,9.0){\line(1,0){6.0}}
\put(0.0,6.0){\line(1,0){6.0}} \put(0.0,3.0){\line(1,0){3.0}}
\put(0.0,0.0){\line(1,0){3.0}} \put(0.0,9.0){\line(0,-1){9.0}}
\put(3.0,9.0){\line(0,-1){9.0}} \put(6.0,9.0){\line(0,-1){3.0}}
\end{picture}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{c^{\prime}=\begin{picture}(6.0,9.0)\put(0.0,9.0){\line(1,0){6.0}}
\put(0.0,6.0){\line(1,0){6.0}} \put(0.0,3.0){\line(1,0){6.0}}
\put(0.0,0.0){\line(1,0){3.0}} \put(0.0,9.0){\line(0,-1){9.0}}
\put(3.0,9.0){\line(0,-1){9.0}} \put(6.0,9.0){\line(0,-1){6.0}}
\end{picture}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Then we can find $\beta$ and $\beta^{\prime}$ so that
$\partial_{b}^{a}\alpha=(-1)^{p}\bar{\partial}\beta$ and
$\partial_{b^{\prime}}^{a}\alpha=(-1)^{p}\bar{\partial}\beta^{\prime}$ where
$p=|b|=|b^{\prime}|$. Then
$[D_{a_{2}^{\prime\prime}}^{a}\alpha]=[\partial_{a^{\prime\prime}_{2}}^{b}\beta+\partial_{a^{\prime\prime}_{2}}^{b^{\prime}}\beta^{\prime}]$
which implies

$\displaystyle(-1)^{p+1}\bar{\partial}\big{(}\partial_{c}^{b}\beta\big{)}=(-1)^{p+1}\partial_{c}^{b}\bar{\partial}\beta=-\partial_{c}^{b}\partial_{b}^{a}\alpha=\partial_{c}^{a_{1}^{\prime\prime}}\partial_{a_{1}^{\prime\prime}}^{a}\alpha$

and similarly
$(-1)^{p+1}\bar{\partial}(\partial_{c^{\prime}}^{b^{\prime}}\beta^{\prime})=\partial_{c^{\prime}}^{a_{1}^{\prime\prime}}\partial_{a^{\prime\prime}_{1}}^{a}\alpha$.
Hence, we conclude that

$\displaystyle
D^{a_{1}^{\prime\prime}}_{a^{\prime}}D^{a}_{a_{1}^{\prime\prime}}[\alpha]=\big{[}\partial_{a^{\prime}}^{c}\partial_{c}^{b}\beta+\partial_{a^{\prime}}^{c^{\prime}}\partial_{c^{\prime}}^{b^{\prime}}\beta^{\prime}\big{]}$

and thus

$\displaystyle
D_{a^{\prime}}^{a_{2}^{\prime\prime}}D_{a_{2}^{\prime\prime}}^{a}[\alpha]=\big{[}\partial_{a^{\prime}}^{a_{2}^{\prime\prime}}\big{(}\partial_{a_{2}^{\prime\prime}}^{b}\beta+\partial_{a_{2}^{\prime\prime}}^{b^{\prime}}\beta^{\prime}\big{)}\big{]}=-\big{[}\partial_{a^{\prime}}^{c}\partial_{c}^{b}\beta+\partial_{a^{\prime}}^{c^{\prime}}\partial_{c^{\prime}}^{b^{\prime}}\beta^{\prime}\big{]}=-D^{a_{1}^{\prime\prime}}_{a^{\prime}}D^{a}_{a_{1}^{\prime\prime}}[\alpha].$

This completes the proof when $|a^{\prime}|=|a|+3$ and now we may assume
$|a^{\prime}|=|a|+4$.

We put $A^{\prime\prime}:=\\{a^{\prime\prime}\in S^{k}|\
a<a^{\prime\prime}<a^{\prime}\\}$,
$B:=\\{b\in\mathbb{N}^{k,n}_{++}\,|\,\exists\,a^{\prime\prime}\in
A^{\prime\prime}\colon a<b<a^{\prime\prime}\\}$,
$B^{\prime}:=\\{b^{\prime}\in\mathbb{N}^{k,n}_{++}\,|\,\exists\,a^{\prime\prime}\in
A^{\prime\prime}\colon a^{\prime\prime}<b^{\prime}<a^{\prime}\\}$ and finally
$C:=\\{c\in\mathbb{N}^{k,n}_{++}\setminus S^{k}\,|\,\exists\,b\in B,\
\exists\,b^{\prime}\in B^{\prime}\colon b^{\prime}<c<b^{\prime\prime}\\}$.
Consider for example the diagram

$\textstyle{b_{1}=\begin{picture}(3.0,6.0)\put(0.0,6.0){\line(1,0){3.0}}
\put(0.0,3.0){\line(1,0){3.0}} \put(0.0,0.0){\line(1,0){3.0}}
\put(0.0,6.0){\line(0,-1){6.0}} \put(3.0,6.0){\line(0,-1){6.0}}
\end{picture}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{c_{1}=\begin{picture}(3.0,9.0)\put(0.0,9.0){\line(1,0){3.0}}
\put(0.0,6.0){\line(1,0){3.0}} \put(0.0,3.0){\line(1,0){3.0}}
\put(0.0,0.0){\line(1,0){3.0}} \put(0.0,9.0){\line(0,-1){9.0}}
\put(3.0,9.0){\line(0,-1){9.0}}
\end{picture}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{b^{\prime}_{1}=\begin{picture}(6.0,9.0)\put(0.0,9.0){\line(1,0){6.0}}
\put(0.0,6.0){\line(1,0){6.0}} \put(0.0,3.0){\line(1,0){3.0}}
\put(0.0,0.0){\line(1,0){3.0}} \put(0.0,9.0){\line(0,-1){9.0}}
\put(3.0,9.0){\line(0,-1){9.0}} \put(6.0,9.0){\line(0,-1){3.0}}
\end{picture}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{a=\begin{picture}(3.0,3.0)\put(0.0,3.0){\line(1,0){3.0}}
\put(0.0,0.0){\line(1,0){3.0}} \put(0.0,3.0){\line(0,-1){3.0}}
\put(3.0,3.0){\line(0,-1){3.0}}
\end{picture}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{a^{\prime\prime}=\begin{picture}(6.0,6.0)\put(0.0,6.0){\line(1,0){6.0}}
\put(0.0,3.0){\line(1,0){6.0}} \put(0.0,0.0){\line(1,0){3.0}}
\put(0.0,6.0){\line(0,-1){6.0}} \put(3.0,6.0){\line(0,-1){6.0}}
\put(6.0,6.0){\line(0,-1){3.0}}
\end{picture}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{a^{\prime}=\begin{picture}(9.0,9.0)\put(0.0,9.0){\line(1,0){9.0}}
\put(0.0,6.0){\line(1,0){9.0}} \put(0.0,3.0){\line(1,0){3.0}}
\put(0.0,0.0){\line(1,0){3.0}} \put(0.0,9.0){\line(0,-1){9.0}}
\put(3.0,9.0){\line(0,-1){9.0}} \put(6.0,9.0){\line(0,-1){3.0}}
\put(9.0,9.0){\line(0,-1){3.0}}
\end{picture},}$$\textstyle{b_{2}=\begin{picture}(6.0,3.0)\put(0.0,3.0){\line(1,0){6.0}}
\put(0.0,0.0){\line(1,0){6.0}} \put(0.0,3.0){\line(0,-1){3.0}}
\put(3.0,3.0){\line(0,-1){3.0}} \put(6.0,3.0){\line(0,-1){3.0}}
\end{picture}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{c_{2}=\begin{picture}(9.0,3.0)\put(0.0,3.0){\line(1,0){9.0}}
\put(0.0,0.0){\line(1,0){9.0}} \put(0.0,3.0){\line(0,-1){3.0}}
\put(3.0,3.0){\line(0,-1){3.0}} \put(6.0,3.0){\line(0,-1){3.0}}
\put(9.0,3.0){\line(0,-1){3.0}}
\end{picture}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{b_{2}^{\prime}=\begin{picture}(9.0,6.0)\put(0.0,6.0){\line(1,0){9.0}}
\put(0.0,3.0){\line(1,0){9.0}} \put(0.0,0.0){\line(1,0){3.0}}
\put(0.0,6.0){\line(0,-1){6.0}} \put(3.0,6.0){\line(0,-1){6.0}}
\put(6.0,6.0){\line(0,-1){3.0}} \put(9.0,6.0){\line(0,-1){3.0}}
\end{picture}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where $A^{\prime\prime}=\\{a^{\prime\prime}\\}$, $B=\\{b_{1},b_{2}\\}$,
$B^{\prime}=\\{b_{1}^{\prime},b_{2}^{\prime}\\}$ and $C=\\{c_{1},c_{2}\\}$. As
above, the set $A^{\prime\prime}$ contains at most two elements but we will
not need that.

Now we can proceed as above. There are $\beta_{i}$ such that
$(-1)^{p}\bar{\partial}\beta_{i}=\partial_{b_{i}}^{a}\alpha$ where
$p=|b_{i}|=|a|+1$ and so
$[D_{a^{\prime\prime}_{j}}^{a}\alpha]=\big{[}\sum\limits_{b_{i}\in
B}\partial_{a^{\prime\prime}_{j}}^{b_{i}}\beta_{i}\big{]}$ for every
$a^{\prime\prime}_{j}\in A^{\prime\prime}$. As the relative BGG sequence is a
complex, we have for each $c_{\ell}\in C$:

$\displaystyle\bar{\partial}\bigg{(}\sum_{b_{i}\in
B}\partial_{c_{\ell}}^{b_{i}}\beta_{i}\bigg{)}=\sum_{b_{i}\in
B}\partial_{c_{\ell}}^{b_{i}}\bar{\partial}\beta_{i}=(-1)^{p}\sum_{b_{i}\in
B}\partial_{c_{\ell}}^{b_{i}}\partial_{b_{i}}^{a}\alpha=0.$

As above, there is $\gamma_{\ell}$ such that
$(-1)^{p+1}\bar{\partial}\gamma_{\ell}=\sum\limits_{b_{i}\in
B}\partial_{c_{\ell}}^{b_{i}}\beta_{i}$. Then for $b^{\prime}_{s}\in
B^{\prime}$:

$\displaystyle(-1)^{p+2}\bar{\partial}\bigg{(}\sum_{c_{\ell}\in
C}\partial_{b^{\prime}_{s}}^{c_{\ell}}\gamma_{\ell}\bigg{)}=-\sum_{c_{\ell}\in
C}\partial_{b^{\prime}_{s}}^{c_{\ell}}((-1)^{p+1}\bar{\partial}\gamma_{\ell})=-\sum_{c_{\ell}\in
C,\ b_{i}\in
B}\partial_{b^{\prime}_{s}}^{c_{\ell}}\partial_{c_{\ell}}^{b_{i}}\beta_{i}$
$\displaystyle\hphantom{(-1)^{p+2}\bar{\partial}\bigg{(}\sum_{c_{\ell}\in
C}\partial_{b^{\prime}_{s}}^{c_{\ell}}\gamma_{\ell}\bigg{)}}{}=\sum_{b_{i}\in
B,\ a^{\prime\prime}_{j}\in A^{\prime\prime}_{j}\colon
b_{i}<a^{\prime\prime}_{j}<b_{s}^{\prime}}\partial_{b^{\prime}_{s}}^{a^{\prime\prime}_{j}}\partial_{a^{\prime\prime}_{j}}^{b_{i}}\beta_{i}=\sum_{a^{\prime\prime}_{j}\in
A\colon
a^{\prime\prime}_{j}<b^{\prime}_{s}}\partial_{b^{\prime}_{s}}^{a^{\prime\prime}_{j}}\bigg{(}\sum_{b_{i}\in
B}\partial_{a^{\prime\prime}_{j}}^{b_{i}}\beta_{i}\bigg{)}$
$\displaystyle\hphantom{(-1)^{p+2}\bar{\partial}\bigg{(}\sum_{c_{\ell}\in
C}\partial_{b^{\prime}_{s}}^{c_{\ell}}\gamma_{\ell}\bigg{)}}{}=\sum_{a^{\prime\prime}_{j}\in
A\colon
a^{\prime\prime}_{j}<b^{\prime}_{s}}\partial_{b^{\prime}_{s}}^{a^{\prime\prime}_{j}}(D_{a^{\prime\prime}_{j}}^{a}\alpha).$

This implies that $\sum\limits_{a^{\prime\prime}_{j}\in
A^{\prime\prime}}D_{a^{\prime}}^{a^{\prime\prime}}D_{a^{\prime\prime}}^{a}([\alpha])$
is the cohomology class of

$\displaystyle\sum_{a^{\prime\prime}_{j}\in
A^{\prime\prime}}\bigg{(}\sum_{b^{\prime}_{s}\in B^{\prime}\colon
a^{\prime\prime}_{j}<b_{s}^{\prime}}\partial_{a^{\prime}}^{b^{\prime}_{s}}\bigg{(}\sum_{c_{\ell}\in
C}\partial_{b^{\prime}_{s}}^{c_{\ell}}\gamma_{\ell}\bigg{)}\bigg{)}=\sum_{b^{\prime}_{s}\in
B^{\prime}}\sum_{c_{\ell}\in
C}\partial_{a^{\prime}}^{b^{\prime}_{s}}\partial_{b^{\prime}_{s}}^{c_{\ell}}\gamma_{\ell}$
$\displaystyle\hphantom{\sum_{a^{\prime\prime}_{j}\in
A^{\prime\prime}}\bigg{(}\sum_{b^{\prime}_{s}\in B^{\prime}\colon
a^{\prime\prime}_{j}<b_{s}^{\prime}}\partial_{a^{\prime}}^{b^{\prime}_{s}}\bigg{(}\sum_{c_{\ell}\in
C}\partial_{b^{\prime}_{s}}^{c_{\ell}}\gamma_{\ell}\bigg{)}\bigg{)}}{}=\sum_{c_{\ell}\in
C}\sum_{b^{\prime}_{s}\in
B^{\prime}}\partial_{a^{\prime}}^{b^{\prime}_{s}}\partial_{b^{\prime}_{s}}^{c_{\ell}}\gamma_{\ell}=\sum_{c_{\ell}\in
C}0=0.$

In the first equality we use the fact that given $b^{\prime}_{s}\in
B^{\prime}$, there is only one $a^{\prime\prime}_{j}\in A^{\prime\prime}$ such
that $a^{\prime\prime}_{j}<b^{\prime}_{s}$ and in the third equality we use
that the relative BGG sequence is a complex once more. ∎

##  7 Formal exactness of $\boldsymbol{k}$-Dirac complexes

We will proceed in Section 7 as follows. In Section 7.1 we will recall the
definition of the normal bundle of the analytic subvariety
$X_{0}:=\tau^{-1}(x_{0})$ and give the definition of the weighted formal
neighborhood of $X_{0}$. In Section 7.2 we will consider the double complex of
twisted relative forms from Section 5 and restrict it to the weighted formal
neighborhood of $X_{0}$. In Section 7.3 we will prove that the operators
defined in Section 6 are differential operators and finally, in Theorem 7.14
we will prove that the $k$-Dirac complexes are formally exact.

###  7.1 Formal neighborhood of $\boldsymbol{\tau^{-1}(x_{0})}$

Let us first recall notation from Section 4.2. There is the 2-step filtration
$\\{0\\}=F_{0}^{M}\subset F_{-1}^{M}\subset F_{-2}^{M}=TM$ and the 3-step
filtration $\\{0\\}=F_{0}^{CS}\subset F^{CS}_{-1}\subset F^{CS}_{-2}\subset
F^{CS}_{-3}=TM$. Moreover, $F_{-1}^{CS}$ decomposes as $E^{CS}\oplus F^{CS}$
where $E^{CS}=\ker(T\eta)$ and $F^{CS}=\ker(T\tau)$. From this it follows that
$E^{CS}$ and $F^{CS}$ are integrable distributions. Dually, there are
filtrations $T^{\ast}M=F_{1}^{M}\supset F_{2}^{M}\supset F_{3}^{M}=\\{0\\}$
and $T^{\ast}CS=F_{1}^{CS}\supset F_{2}^{CS}\supset F_{3}^{CS}\supset
F_{4}^{CS}=\\{0\\}$ where $F_{i}^{M}$ is the annihilator of $F_{-i+1}^{M}$ and
similarly for $F_{i}^{CS}$. We put $G_{i}^{M}:=F_{i}^{M}/F_{i+1}^{M}$ and
$G_{i}^{CS}:=F_{i}^{CS}/F_{i+1}^{CS}$ so that

$\displaystyle gr(TM)=G_{-2}^{M}\oplus G_{-1}^{M},\qquad
gr(T^{\ast}M)=G_{1}^{M}\oplus G_{2}^{M},$ $\displaystyle
gr(TCS)=G_{-3}^{CS}\oplus G_{-2}^{CS}\oplus G_{-1}^{CS},\qquad
gr(T^{\ast}CS)=G_{1}^{CS}\oplus G_{2}^{CS}\oplus G_{3}^{CS},$ $\displaystyle
G_{i}^{M}\cong\big{(}G_{-i}^{M}\big{)}^{\ast},\quad
i=1,2\qquad\mathrm{and}\qquad
G_{i}^{CS}\cong\big{(}G_{-i}^{CS}\big{)}^{\ast},\quad i=1,2,3.$

Let us now briefly recall Section 2.3. If $X$ is an analytic subvariety of a
complex manifold $Y$, then the normal bundle $N_{X}$ of $X$ in $Y$ is the
quotient $(TY|_{X})/TX$ and the co-normal bundle $N^{\ast}_{X}$ is the
annihilator of $TX$ inside $T^{\ast}X$. In particular, the origin $x_{0}$ can
be viewed as an analytic subvariety of $M$ with local defining equation
$X_{1}=0$, $X_{2}=0$ and $Y=0$ where the matrices are those as in (4.4). For