# Gelfand-Fuchs cohomology for affine superspaces $\mathbb{A}^{m,n}$

Slava Pimenov

###### Contents

  1. 0 Introduction
  2. 1 Notations and recollections
  3. 2 Cohomology of $\mathfrak{gl}(m,n)$
  4. 3 Cohomology of $\mathcal{V}_{m,n}$

##  0 Introduction

Let $\mathbb{A}^{m,n}$ be the the affine super space of even dimension $m$ and
odd dimension $n$ over an algebraically closed field $\mathbf{k}$ of
characteristic $0$. Consider Lie superalgebras $\mathcal{V}_{m,n}$ of vector
fields in the formal neighborhood of $0\in\mathbb{A}^{m,n}$. This is a
topological Lie superalgebra with $\mathbf{k}$-linear topology induced by the
defining ideal of point $0\in\mathbb{A}^{m,n}$. We are interested in the
continuous cohomology groups $H^{\bullet}(\mathcal{V}_{m,n},\mathbf{k})$.

Previously established results cover cases $0\leqslant m\leqslant n$, as well
as $n=0$ and $n=1$ and arbitrary $m\geqslant 0$ ([GF], [Fuk], [Ko], [AF],
[Pi1]). These results are collected below in theorem 1.2. The main result of
this paper is the following theorem that has been stated as a conjecture in
the previous paper ([Pi1]).

For any $m\geqslant n\geqslant 0$ we have an isomorphism

$H^{\bullet}(\mathcal{V}_{m,n},\mathbf{k})\ \simeq\
H^{\bullet}(\SS^{2n}X_{2(m-n)},\mathbf{k}).$

Here $\SS$ denotes the topological suspension functor, and $X_{2(m-n)}$ is the
pullback of the tautological $GL(m,\mathbb{C})$-torsor over
$BGL(m,\mathbb{C})$ to the $2(m-n)$-dimensional skeleton of
$BGL(m,\mathbb{C})$ consisting of cells of dimensions up to $2(m-n)$.

Combined with the previously established results this completely settles the
question of local Gelfand-Fuchs cohomology for super-manifolds.

The main tool in the calculation is the following theorem regarding cohomology
of Lie superalgebras $\mathfrak{gl}(m,n)$. Let $V$ be the standard
representation of $\mathfrak{gl}(m,n)$, and denote $\Sigma^{\lambda}(V)$ the
Schur functor corresponding to a diagram $\lambda$. We will write
$\mathcal{H}_{m,n}$ for the set of diagrams contained in a thick hook with $m$
rows and $n$ columns.

Let $\mathfrak{g}=\mathfrak{gl}(m,n)$ with $m\geqslant n\geqslant 0$, and

$\lambda\in\mathcal{H}_{m-n+k,k}-\mathcal{H}_{m-n+k-1,k-1}$

for some $0\leqslant k\leqslant n$. Then

$H^{\bullet}(\mathfrak{g},\Sigma^{\lambda}(V)\mathop{\otimes}\limits\Sigma^{\lambda}(V^{*}))\
\simeq\
\mathbf{k}[e_{1},\ldots,e_{2m-1}]\mathop{\otimes}\limits\mathbf{k}[e^{\prime}_{2(n-k)+1},\ldots,e^{\prime}_{2n-1}].$

This theorem appears to be a new result and may be of interest beyond the
Gelfand-Fuchs cohomology theory.

[Outline of the paper.] In section 1 we recall the relevant notations and
results that are used in this paper.

The section 2 is dedicated to the proof of theorem  ‣ 0 Introduction. We
proceed by induction on the number of odd variables $n$ and use the spectral
sequence for the Lie subalgebra

$\mathfrak{gl}(m,n-1)\oplus\mathfrak{gl}(1)\hookrightarrow\mathfrak{gl}(m,n)$

to reduce the question to $\mathfrak{gl}(m,n-1)$.

First we observe that the first layer of this spectral sequence has a
universal structure, which allows us to compare spectral sequences for a fixed
diagram $\lambda$ but different values of $m$ and $n$. We combine this with
the special case of theorem  ‣ 0 Introduction for $n=1$ that was established
in ([Pi1]) to identify all the diagrams contributing to the second layer of
the spectral sequence. Then by direct examination of the second and third
layers we establish the required isomorphism.

We would like to point out that this does not provide an independent proof of
theorem  ‣ 0 Introduction for $n=1$ as we use this result to greatly simplify
analysis of the first layer of the spectral sequence. Combining the proof in
this paper and the proof of the special case for $n=1$, the overall process
looks as follows. We start with $\mathfrak{gl}(1,1)$ as the base of induction,
then we grow number of even variables to get to $\mathfrak{gl}(m,1)$ then we
grow number of odd variables and arrive to $\mathfrak{gl}(m,n)$.

In the case of $\mathfrak{gl}(1,1)$ we have a description of the
indecomposable components of the coefficient module
$\Sigma^{\lambda}(V)\mathop{\otimes}\limits\Sigma^{\lambda}(V^{*})$ that give
rise to the cohomology groups in theorem  ‣ 0 Introduction. Presently, we do
not have a similar description for other $\mathfrak{gl}(n,n)$.

The section 3 deals with the proof of theorem  ‣ 0 Introduction. It follows
the general process developed in ([Pi1]) which in turn is a refinement of the
proof of Gelfand and Fuchs in the classical case. We consider the spectral
sequence for the Lie subalgebra
$\mathfrak{gl}(m,n)\hookrightarrow\mathcal{V}_{m,n}$ and identify all the
diagrams contributing to its first layer. We observe that up to transposition
these diagrams are the same as in the spectral sequence for
$\mathfrak{gl}(n-1,m+1)\hookrightarrow\mathcal{V}_{n-1,m+1}$, which allows us
to compare these two spectral sequences.

In the latter case we have $n-1<m+1$, so this is covered by the previous
result of Astashkevich and Fuchs ([AF]) which says that the cohomology
$H^{\bullet}(\mathcal{V}_{n-1,m+1},\mathbf{k})$ is isomorphic to the
cohomology of $(2m+1)$-dimensional sphere. This makes the analysis of the
original spectral sequence for $\mathcal{V}_{m,n}$ much simpler and allows us
to establish the required isomorphism.

The author would like to thank BIMSA (Beijing Institute of Mathematical
Sciences and Applications) for providing excellent working conditions during
preparation of this paper.

##  1 Notations and recollections

We will retain conventions and notations from [Pi1]. Here we will briefly
recall them and state the relevant results that will be used in this paper.

[Young diagrams and Schur functors.] Let $\lambda$ be a Young diagram of size
$d$, in other words it is an unordered partition
$\lambda=(\lambda_{1},\ldots,\lambda_{k})$ of $d$, with
$\lambda_{1}\geqslant\lambda_{2}\geqslant\ldots\geqslant\lambda_{k}>0$ and
$\sum_{i=1}^{k}\lambda_{i}=d$. We will refer to $k$ as the height of $\lambda$
and write $k=\mathrm{ht}(\lambda)$, and $d=|\lambda|$. We will denote
$\lambda^{\prime}$ the transposed Young diagram, specifically we put
$\lambda^{\prime}_{j}=\max\\{i\mid\lambda_{i}\geqslant j\\}$.

For any diagram $\lambda$ we construct a truncated diagram
$\overline{\lambda}$ obtained from $\lambda$ by removing the first column. In
other words we put $\overline{\lambda}_{i}=\max\\{\lambda_{i}-1,0\\}$.
Furthermore, we construct an extended diagram $\widetilde{\lambda}$ by adding
to $\lambda$ the first column of height $d=|\lambda|$. Formally, we put
$\widetilde{\lambda}_{i}=\lambda_{i}+1$ for $1\leqslant i\leqslant d$.

For any $m,n\geqslant 0$ we consider a subset $\mathcal{H}_{m,n}$ of Young
diagrams of arbitrary size, consisting of diagrams contained in a thick hook
with $m$ rows and $n$ columns. More precisely $\lambda\in\mathcal{H}_{m,n}$ if
and only if $\lambda_{i}\leqslant n$ whenever $i>m$. By convention, if either
$m$ or $n$ is negative we put $\mathcal{H}_{m,n}$ to be an empty set.

For any partition $\lambda$ we will denote $\Sigma^{\lambda}$ the
corresponding Schur functor acting on the symmetric monoidal category of super
vector spaces. For a super vector space $V=(V_{0},V_{1})$ of dimension
$(m,n)$, with $m=\mathop{\mathrm{dim}}\nolimits V_{0}$ and
$n=\mathop{\mathrm{dim}}\nolimits V_{1}$, the Schur functor
$\Sigma^{\lambda}(V)$ is non-zero if and only if
$\lambda\in\mathcal{H}_{m,n}$.

We will denote by $S^{n}$ and $\Lambda^{n}$ the functors of symmetric and
exterior powers respectively and recall that for any two super vector spaces
$V$ and $W$ we have isomorphisms of
$(\mathfrak{gl}(V)\times\mathfrak{gl}(W))$-modules

(1.0.1) $S^{n}(V\mathop{\otimes}\limits
W)=\bigoplus_{|\lambda|=n}\Sigma^{\lambda}(V)\mathop{\otimes}\limits\Sigma^{\lambda}(W),\quad\quad\Lambda^{n}(V\mathop{\otimes}\limits
W)=\bigoplus_{|\lambda|=n}\Sigma^{\lambda}(V)\mathop{\otimes}\limits\Sigma^{\lambda^{\prime}}(W).$

[Lie superalgebra $\mathfrak{gl}(m,n)$.] Let $V$ be a super vector space of
dimension $(m,n)$, then we write $\mathfrak{gl}(m,n)$ for the Lie superalgebra
of endomorphisms $\mathrm{End}(V)\simeq V\mathop{\otimes}\limits V^{*}$. We
will refer to $V$ as the standard representation of $\mathfrak{gl}(m,n)$. We
would like to point out here that even though Lie superalgebras
$\mathfrak{gl}(m,n)$ and $\mathfrak{gl}(n,m)$ are isomorphic, their standard
representations are different. Specifically, the standard representation $W$
of $\mathfrak{gl}(n,m)$ is obtained from $V$ by the change of parity
$W\simeq\Pi(V)$. Therefore, up to change of parity we have isomorphisms
$\Sigma^{\lambda}(W)=\Sigma^{\lambda}(\Pi
V)\simeq\Sigma^{\lambda^{\prime}}(V)$.

Let $\mathfrak{g}=\mathfrak{gl}(m,n)$, and $V$ its standard representation, we
are interested in the cohomology spaces
$H^{\bullet}(\mathfrak{g},\Sigma^{\lambda}(V)\mathop{\otimes}\limits\Sigma^{\lambda}(V^{*}))$.
In [Pi1] we have established the following result.

######  Theorem 1.1.

Let $\mathfrak{g}=\mathfrak{gl}(m,1)$, $V$ the standard representation of
$\mathfrak{g}$ and $\lambda\in\mathcal{H}_{m,1}$, then

$H^{\bullet}(\mathfrak{g},\Sigma^{\lambda}(V)\mathop{\otimes}\limits\Sigma^{\lambda}(V^{*}))=\begin{cases}\mathbf{k}[e_{1},\ldots,e_{2m-1}],&\text{if
$\mathrm{ht}(\lambda)\leqslant m-1$,}\\\
\mathbf{k}[e_{1},\ldots,e_{2m-1},e^{\prime}_{1}],&\text{otherwise},\end{cases}$

where generators $e_{i}$ are of cohomological degree $i$ and $e^{\prime}_{1}$
is of degree $1$.

Notice that the condition $\mathrm{ht}(\lambda)\leqslant m-1$ here can be
rewritten as $\lambda\in\mathcal{H}_{m-1,0}$.

[Lie superalgebra $\mathcal{V}_{m,n}$.] Consider a (super)commutative
superalgebra

$\mathcal{O}_{\mathbb{A}^{m,n}}=\mathbf{k}[x_{1},\ldots
x_{m},\xi_{1},\ldots\xi_{n}]$

of algebraic functions on the affine superspace $\mathbb{A}^{m,n}$. We assume
that variables $x_{i}$ are even and $\xi_{j}$ are odd. We denote
$\widehat{\mathcal{O}}_{\mathbb{A}^{m,n}}=\mathbf{k}[[x_{1},\ldots,x_{m},\xi_{1},\ldots,\xi_{n}]]$
its completion at zero, equipped with the inverse limit topology. We are
interested in the Lie superalgebra of continuous derivations

$\mathcal{V}_{m,n}=\mathrm{Der}_{\mathrm{cont}}(\widehat{\mathcal{O}}_{\mathbb{A}^{m,n}}).$

Explicitly, $\mathcal{V}_{m,n}$ is formed by elements $\sum
f_{i}{\partial_{x_{i}}}+\sum g_{j}{\partial_{\xi_{j}}}$, with
$f_{i},g_{j}\in\widehat{\mathcal{O}}_{\mathbb{A}^{m,n}}$. The bracket is given
by the action of derivations $\partial_{x_{i}}$ and $\partial_{\xi_{j}}$ on
functions. It contains $\mathfrak{gl}(m,n)$ as a subalgebra spanned by
elements with linear coefficients:
$\\{x_{i}\partial_{x_{j}},x_{i}\partial_{\xi_{j}},\xi_{i}\partial_{x_{j}},\xi_{i}\partial_{\xi_{j}}\\}$.

We will write $H^{\bullet}(\mathcal{V}_{m,n},\mathbf{k})$ for the continuous
cohomology spaces with respect to topology on $\mathcal{V}_{m,n}$ induced by
the inverse limit topology on $\widehat{\mathcal{O}}_{\mathbb{A}^{m,n}}$. We
recall the previously established results regarding this cohomology, for
details we refer to [Fuk], [AF], [Pi1].

The cohomology of $\mathcal{V}_{m,n}$ will be related to the cohomology of
various topological spaces, that can be constructed using the following
procedure. Consider the topological group $GL(m,\mathbb{C})$, and let $BGL(m)$
be its classifying space. Denote by $p\colon EGL(m)\to BGL(m)$ the
tautological principal $GL(m)$-bundle over the classifying space. Let us write
$\mathop{\mathrm{sk}}\nolimits_{d}BGL(m)$ for the $d$-dimensional skeleton of
$BGL(m)$, i.e. the subspace formed by all cells of dimension up to $d$. The
spaces that will be of interest to us are
$X_{d}=p^{-1}(\mathop{\mathrm{sk}}\nolimits_{d}BGL(m))$ for various $d$, i.e.
the pullbacks of the tautological bundle to the $d$-dimensional skeleta.
Furthermore, denote by $\SS$ the topological suspension functor.

######  Theorem 1.2.

We have the following isomorphisms.

  1. a)

[Fuk] If $n=0$, then

$H^{\bullet}(\mathcal{V}_{m,0},\mathbf{k})\ \simeq\
H^{\bullet}(X_{2m},\mathbf{k}).$

  2. b)

[AF] If $m<n$, then

$H^{\bullet}(\mathcal{V}_{m,n},\mathbf{k})\simeq
H^{\bullet}(S^{2n-1},\mathbf{k}).$

  3. c)

[AF] If $m=n$, then

$H^{\bullet}(\mathcal{V}_{n,n},\mathbf{k})\ \simeq\
H^{\bullet}(\SS^{2n}GL(n,\mathbb{C}),\mathbf{k}).$

  4. d)

[Pi1] If $n=1$, then

$H^{\bullet}(\mathcal{V}_{m,1},\mathbf{k})\ \simeq\
H^{\bullet}(\SS^{2}X_{2(m-1)},\mathbf{k}).$

[Spectral sequence.] The main tool used in calculations in this paper is the
spectral sequence relating the cohomology of a Lie superalgebra $\mathfrak{g}$
and its subalgebra $\mathfrak{h}$. For any $\mathfrak{g}$-module $M$ we have
an increasing filtration of the chain complex
$\Lambda^{\bullet}\mathfrak{g}\mathop{\otimes}\limits M$ by the number of
elements from $\mathfrak{h}$. It induces a decreasing filtration on the
cochain complex $C^{\bullet}(\mathfrak{g},M)$, giving rise to a spectral
sequence

$E_{1}^{pq}=H^{q}(\mathfrak{h},\
\mathrm{Hom}(\Lambda^{p}(\mathfrak{g}/\mathfrak{h}),M)\ )\Rightarrow
H^{p+q}(\mathfrak{g},M).$

We will use cohomological indexing convention for the spectral sequence: on
the layer $E_{r}$ we have differentials

$d_{r}\colon E_{r}^{p,q}\to E_{r}^{p+r,q-r+1}.$

##  2 Cohomology of $\mathfrak{gl}(m,n)$

Let $\mathfrak{g}=\mathfrak{gl}(m,n)$, and $V$ its standard representation.
Dimension of the super vector space $V$ is $(m,n)$. Consider a direct sum
decomposition of $V$ into two subspaces $V=W\oplus E$, such that
$\mathop{\mathrm{dim}}\nolimits W=(m,n-1)$ and $\mathop{\mathrm{dim}}\nolimits
E=(0,1)$. Denote by $\mathfrak{h}$ the subalgebra of $\mathfrak{g}$ that
preserves this decomposition, in other words

$\mathfrak{h}=\mathrm{End}(W)\oplus\mathrm{End}(E)\simeq\mathfrak{gl}(m,n-1)\oplus\mathfrak{gl}(1)\hookrightarrow\mathfrak{g}.$

We identify $W$ with the standard representation of $\mathfrak{gl}(m,n-1)$ and
$E$ with the standard representation of
$\mathfrak{gl}(0,1)\simeq\mathfrak{gl}(1)$. The quotient space
$\mathfrak{g}/\mathfrak{h}$ is isomorphic to $W\mathop{\otimes}\limits
E^{*}\oplus W^{*}\mathop{\otimes}\limits E$. We have

$\Lambda^{p}(\mathfrak{g}/\mathfrak{h})\ \simeq\
\bigoplus_{i+j=p}\Lambda^{i}(W\mathop{\otimes}\limits
E^{*})\mathop{\otimes}\limits\Lambda^{j}(W^{*}\mathop{\otimes}\limits E).$

Since $E$ is of odd dimension $1$, we can rewrite this as follows

(2.0.1) $\Lambda^{p}(\mathfrak{g}/\mathfrak{h})\ \simeq\
\bigoplus_{i+j=p}S^{i}(W)\mathop{\otimes}\limits
S^{j}(W^{*})\mathop{\otimes}\limits\Lambda^{i}(E^{*})\mathop{\otimes}\limits\Lambda^{j}(E).$

We are interested in the cohomology of $\mathfrak{g}$ with coefficients in
$\Sigma^{\lambda}(V)\mathop{\otimes}\limits\Sigma^{\lambda}(V^{*})$. Using
decomposition $V=W\oplus E$ we can write

(2.0.2) $\Sigma^{\lambda}(V)\ \simeq\
\bigoplus_{\mu}\Sigma^{\mu}(W)\mathop{\otimes}\limits\Lambda^{p}(E),$

where the sum is taken over all diagrams $\mu$ obtained from $\lambda$ by
removing at most one box from each row, and $p=|\lambda|-|\mu|$ is the total
number of removed boxes. We also have a similar expansion for
$\Sigma^{\lambda}(V^{*})$.

Let us consider the spectral sequence $E$ for the Lie subalgebra
$\mathfrak{h}\hookrightarrow\mathfrak{g}$ and coefficients
$\Sigma^{\lambda}(V)\mathop{\otimes}\limits\Sigma^{\lambda}(V^{*})$:

$E_{1}^{pq}=H^{\bullet}(\mathfrak{gl}(m,n-1)\oplus\mathfrak{gl}(1),\Lambda^{p}(\mathfrak{g}/\mathfrak{h})^{*}\mathop{\otimes}\limits\Sigma^{\lambda}(V)\mathop{\otimes}\limits\Sigma^{\lambda}(V^{*})).$

Combining 2.0.1 and 2.0.2 we find that diagrams $\mu$ contributing to the
first layer of this spectral sequence are obtained from $\lambda$ by first
removing say $i$ boxes in such a way that from each row we remove at most one
box, and then adding say $j$ boxes in such a way that in each column we add at
most one box. In particular, we immediately see that the diagrams $\mu$
appearing in $E_{1}$ are of height at most $\mathrm{ht}(\lambda)+1$.

Furthermore, the weight with respect to the action of the subalgebra
$\mathfrak{gl}(1)\hookrightarrow\mathfrak{h}$ of a component corresponding to
a diagram $\mu$ is $|\lambda|-|\mu|$, in other words it depends only on the
diagram $\mu$ itself, and not on the specific way it was obtained from
$\lambda$ by the procedure described above. We have a similar picture on the
dual side with components containing $\Sigma^{\nu}(W^{*})$, however since the
cohomology
$H^{\bullet}(\mathfrak{gl}(m,n-1),\Sigma^{\mu}(W)\mathop{\otimes}\limits\Sigma^{\nu}(W^{*}))$
vanishes unless $\mu=\nu$ it is sufficient to keep track only of the diagrams
$\mu$. We will refer to such components as components of type $\mu$.

The differential on the first layer of the spectral sequence is induced by
maps between the coefficients of $H^{\bullet}(\mathfrak{h},-)$, which in turn
corresponds to the action of $\mathfrak{g}/\mathfrak{h}$ on the coefficients
$\Sigma^{\lambda}(V)\mathop{\otimes}\limits\Sigma^{\lambda}(V^{*})$. In terms
of the above decomposition a differential between two components of type $\mu$
corresponds to two different ways of obtaining diagram $\mu$ from $\lambda$.
More precisely, let $\nu$ be the intermediate diagram in the process of
constructing $\mu$ from $\lambda$, i.e. $\nu$ is a common subdiagram of $\mu$
and $\lambda$ such that $\lambda-\nu$ has at most one box in each row, and
$\mu-\nu$ has at most one box in each column.

We say that a box of the diagram $\nu$ is flippable if the diagram $\nu_{1}$
obtained from $\nu$ by removing this box provides another valid way of
obtaining $\mu$ from $\lambda$. The differential in $E_{1}$ is a linear
combination of maps into components obtained by flipping a box (either for
$\Sigma^{\mu}(W)$ or for $\Sigma^{\mu}(W^{*})$). It is clear from the
construction that for each $\mu$ the subcomplex of components of type $\mu$ is
bounded.

[Universal complex.] The previous discussion of the first layer of the
spectral sequence can by summarized by saying that it is “universal” in a
certain sense. We will make this statement more precise. Let $S_{d}$ be a
symmetric group on $d$ elements and $S_{\bullet}$ denote the collection of all
$S_{d}$ for $d\geqslant 0$. An $S_{\bullet}$-module $M$ is a direct sum
$M=\bigoplus_{d\geqslant 0}M_{d}$, where each $M_{d}$ is an $S_{d}$-module.
Since the category of $S_{\bullet}$-modules is semisimple, and simple modules
correspond to partitions $\lambda$ of arbitrary size, we can further decompose

$M\ =\
\bigoplus_{\lambda}L_{\lambda}\mathop{\otimes}\limits\mathrm{Hom}_{S_{\bullet}}(L_{\lambda},M),$

where $L_{\lambda}$ is the simple module corresponding to a partition
$\lambda$. To simplify notation we will write
$M_{\lambda}=\mathrm{Hom}_{S_{\bullet}}(L_{\lambda},M)$.

For a super vector space $V$ the Schur functor $\Sigma(M,V)$ is defined by

$\Sigma(M,V)\ =\ \bigoplus_{d\geqslant
0}M_{d}\mathop{\otimes}\limits_{\mathbf{k}[S_{d}]}V^{\mathop{\otimes}\limits
d}\ \simeq\ \bigoplus_{\lambda}\Sigma^{\lambda}(V)\mathop{\otimes}\limits
M_{\lambda},$

where $S_{d}$ acts on the tensor power $V^{\mathop{\otimes}\limits d}$ by
permuting factors. Of course, as was already mentioned in the previous
section, the dimension of $V$ imposes restriction on what terms contribute to
the direct sum. If $\mathop{\mathrm{dim}}\nolimits V=(m,n)$, then the
contribution comes only from diagrams $\lambda\in\mathcal{H}_{m,n}$.

The above discussion of the first layer of the spectral sequence can be stated
as the following lemma.

######  Lemma 2.1.

There exists a complex of $S_{\bullet}$-modules
$\mathbb{E}=\mathbb{E}(\lambda)$, such that for any $(m,n)$ the first layer of
the spectral sequence associated to the Lie subalgebra
$\mathfrak{gl}(m,n-1)\oplus\mathfrak{gl}(1)\hookrightarrow\mathfrak{gl}(m,n)$
and coefficients
$\Sigma^{\lambda}(V)\mathop{\otimes}\limits\Sigma^{\lambda}(V^{*})$ has the
form

$E_{1}^{pq}\ \simeq\ \bigoplus_{\mu\atop
i+j=q}H^{i}(\mathfrak{gl}(m,n-1),\Sigma^{\mu}(W)\mathop{\otimes}\limits\Sigma^{\mu}(W^{*}))\mathop{\otimes}\limits
H^{j}(\mathfrak{gl}(1),\mathbf{k})\mathop{\otimes}\limits\mathbb{E}^{p}_{\mu}.$

where $W$ is the standard representation of $\mathfrak{gl}(m,n-1)$, and the
sum is over $\mu\in\mathcal{H}_{m,n-1}$.

Moreover, the differentials in $E_{1}$ are induced by the differentials in
$\mathbb{E}$, so the second layer $E_{2}$ (without differentials) has the form

$E_{2}^{pq}\ \simeq\ \bigoplus_{\mu\atop
i+j=q}H^{i}(\mathfrak{gl}(m,n-1),\Sigma^{\mu}(W)\mathop{\otimes}\limits\Sigma^{\mu}(W^{*}))\mathop{\otimes}\limits
H^{j}(\mathfrak{gl}(1),\mathbf{k})\mathop{\otimes}\limits
H^{p}(\mathbb{E}_{\mu}).$

$\square$

Because of the universal nature of the complex $\mathbb{E}$ we will be able to
obtain information about $H^{\bullet}(\mathbb{E})$ by comparing spectral
sequences for different values of $(m,n)$. First, we will need the following
simple lemma.

######  Lemma 2.2.

Consider a spectral sequence $E_{\bullet}$ concentrated in the quadrant with
$p,q\geqslant 0$, such that $E_{2}^{0,q}$ is an algebra isomorphic to
$A\mathop{\otimes}\limits\mathbf{k}[e_{1}]$ for some algebra $A$, and
$\mathop{\mathrm{deg}}\nolimits e_{1}=1$. Assume that the second layer is a
free $A\mathop{\otimes}\limits\mathbf{k}[e_{1}]$-module, with generators in
degrees $(p,0)$, and the differential on $E_{2}$ is compatible with the module
structure.

  1. a)

If $E_{\infty}^{0,q}\simeq E_{2}^{0,q}$ and $E_{\infty}^{pq}=0$ for $p>0$,
then $E_{2}^{pq}=0$ for $p>0$.

  2. b)

If $E_{\infty}^{0,q}\simeq A\hookrightarrow E_{2}^{0,q}$ and
$E_{\infty}^{pq}=0$ for $p>0$, then

$E_{2}^{pq}\simeq
A\mathop{\otimes}\limits\mathbf{k}[e_{1}]\mathop{\otimes}\limits\mathbf{k}[c_{1}]$

as a $A\mathop{\otimes}\limits\mathbf{k}[e_{1}]$-module. Here
$\mathop{\mathrm{deg}}\nolimits c_{1}=(2,0)$ and the differential in $E_{2}$
sends $e_{1}$ to $c_{1}$.

Proof: For part (a) let us assume that $E_{2}^{pq}\neq 0$ for some $p>0$, and
let us denote $p_{0}$ the minimal such $p$. By our assumption $E_{2}$ is
generated by elements in degrees $(p,0)$, therefore we must have
$E_{2}^{p_{0},0}\neq 0$. Since these elements do not survive to $E_{\infty}$
and $p_{0}$ is minimal, we must have non-zero differentials starting from the
first column. But this is impossible since $E_{\infty}^{0,q}\simeq
E_{2}^{0,q}$.

For part (b) observe that since $e_{1}$ doesn’t survive until $E_{\infty}$ it
must by killed by some differential, however, since our spectral sequence has
non-zero terms only for $p,q\geqslant 0$, we see that the differential of
$E_{2}$ doesn’t vanish on $e_{1}$. Denote by $c_{1}\in E_{2}^{2,0}$ its image.
Since by assumption the differential is compatible with the module structure
and $A$ is contained in the kernel of the differential, this completely
determines restriction of $d_{2}$ to the first column. Furthermore, since
$E_{2}$ is a free $A\mathop{\otimes}\limits\mathbf{k}[e_{1}]$-module, we see
that the image $\mathop{\mathrm{Im}}\nolimits d_{2}\colon E_{2}^{0\bullet}\to
E_{2}^{2\bullet}$ is identified with
$Ac_{1}\subset(A\mathop{\otimes}\limits\mathbf{k}[e_{1}])c_{1}$, and it is
surjective on $E_{2}^{2,0}$.

The element $e_{1}c_{1}$ does not survive to $E_{\infty}$, therefore as
before, $d_{2}$ doesn’t vanish on it and we identify $d_{2}(e_{1}c_{1})$ with
$c_{1}^{2}$. Repeating this argument we obtain the required isomorphism.

$\square$

These two lemmas allow us to establish the following result concerning the
universal complex $\mathbb{E}(\lambda)$.

######  Lemma 2.3.

Let $\mathbb{E}=\mathbb{E}(\lambda)$, then

  1. a)

for all $p>0$ we have $H^{2p}(\mathbb{E})\simeq L_{\mu}$, for some $\mu$ with
$\mathrm{ht}(\mu)=\mathrm{ht}(\lambda)+1$,

  2. b)

$H^{0}(\mathbb{E})\simeq L_{\overline{\lambda}}$, where $\overline{\lambda}$
is the truncation of $\lambda$,

  3. c)

for all $p\geqslant 0$ we have $H^{2p}(\mathbb{E})\simeq L_{(\lambda/p+1)}$,
where $(\lambda/k)$ denotes the diagram obtained from $\lambda$ by adding one
box in the first $k-1$ columns and removing $k$’th column. In other words

$(\lambda/k)^{\prime}=(\lambda^{\prime}_{1}+1,\ldots,\lambda^{\prime}_{k-1}+1,\lambda^{\prime}_{k+1},\ldots).$

Proof: Consider Lie superalgebra $\mathfrak{g}=\mathfrak{gl}(k,1)$ and its
subalgebra $\mathfrak{h}=\mathfrak{gl}(k)\oplus\mathfrak{gl}(1)$. Now, as
usual, let $W$ be the standard representation of $\mathfrak{gl}(k)$ then for
any diagram $\mu$ with $\mathrm{ht}(\mu)\leqslant k$ we have

$H^{\bullet}(\mathfrak{gl}(k),\Sigma^{\mu}(W)\mathop{\otimes}\limits\Sigma^{\mu}(W^{*}))\simeq\mathbf{k}[e_{1},\ldots,e_{2k-1}].$

We also have $H^{\bullet}(\mathfrak{gl}(1))\simeq\mathbf{k}[e_{1}]$. Using
lemma 2.1 we see that the spectral sequence for the Lie subalgebra
$\mathfrak{h}\hookrightarrow\mathfrak{g}$ satisfies conditions of lemma 2.2.

First, let $k\geqslant\mathrm{ht}(\lambda)+1$. As we already saw, all the
diagrams $\mu$ appearing in the universal complex $\mathbb{E}$ have
$\mathrm{ht}(\mu)\leqslant\mathrm{ht}(\lambda)+1$. Using theorem 1.1 we find
that the spectral sequence converges to $\mathbf{k}[e_{1},\ldots,e_{2k-1}]$.
Therefore, we are in the situation of lemma 2.2(b).

On the other hand, let $k=\mathrm{ht}(\lambda)$. Then again using theorem 1.1
we find that the spectral sequence converges to
$\mathbf{k}[e_{1},\ldots,e_{2k-1},e^{\prime}_{1}]$, so we are in the situation
of lemma 2.2(a). The difference between these two spectral sequences is that
in the latter case we have a restriction imposed on the diagrams $\mu$ by the
dimension of $W$. Namely, we lose diagrams $\mu$ with $\mathrm{ht}(\mu)>k$.
Therefore, diagrams $\mu$ contributing to $H^{p}(\mathbb{E})$ for $p>0$ must
be of height $\mathrm{ht}(\lambda)+1$. Combining this with isomorphism from
lemma 2.2(b) we prove part (a).

For part (b), notice that since
$H^{0}(\mathfrak{gl}(k,1),\Sigma^{\lambda}(V)\mathop{\otimes}\limits\Sigma^{\lambda}(V^{*}))=\mathbf{k}$
for any $k$, the cohomology $H^{0}(\mathbb{E})$ is isomorphic to a simple
module. Furthermore, from the construction of the first layer $E_{1}$ we see
that it must by isomorphic to $L_{\mu}$ for the smallest possible diagram
$\mu$ in the decomposition (2.0.2). It is straightforward to see that this
diagram is precisely the truncated diagram $\overline{\lambda}$.

Clearly, $(\lambda/1)=\overline{\lambda}$, so part (c) for $p=0$ is just a
reformulation of part (b). Let us consider the rest of the diagrams $\mu$
contributing to $H^{\bullet}(\mathbb{E}(\lambda))$. According to part (a) they
are of height $\mathrm{ht}(\lambda)+1$, so they have $\mathrm{ht}(\lambda)+1$
boxes in the first column. This is only possible if in the process of
obtaining $\mu$ from $\lambda$ no box was removed from the first column and
exactly one box was added to it. This implies that for any diagram $\mu$ of
height $\mathrm{ht}(\lambda)+1$ appearing in the universal complex
$\mathbb{E}(\lambda)$ we have the truncated diagram $\overline{\mu}$ appearing
in $\mathbb{E}(\overline{\lambda})$. More precisely, for any such $\mu$ and
$p\geqslant 2$ we have
$\mathbb{E}(\lambda)^{p}_{\mu}\simeq\mathbb{E}(\overline{\lambda})^{p-2}_{\overline{\mu}}$
and therefore

$H^{p}(\mathbb{E}(\lambda)_{\mu})\simeq
H^{p-2}(\mathbb{E}(\overline{\lambda})_{\overline{\mu}}).$

Thus, we reduced the question to the structure of the universal complex
$\mathbb{E}(\overline{\lambda})$, for a diagram $\overline{\lambda}$ that
contains one less column than $\lambda$. Using induction on the number of
columns we immediately see that this identification gives us the required
isomorphism of part (c). The base of induction is the diagram $\lambda$ of
size zero, in which case the statement follows from the decomposition (2.0.1).

$\square$

We are now ready to prove the main theorem of this section.

######  Theorem 2.4.

Let $\mathfrak{g}=\mathfrak{gl}(m,n)$ with $m\geqslant n\geqslant 0$, and

$\lambda\in\mathcal{H}_{m-n+k,k}-\mathcal{H}_{m-n+k-1,k-1}$

for some $0\leqslant k\leqslant n$. Then

$H^{\bullet}(\mathfrak{g},\Sigma^{\lambda}(V)\mathop{\otimes}\limits\Sigma^{\lambda}(V^{*}))\
\simeq\
\mathbf{k}[e_{1},\ldots,e_{2m-1}]\mathop{\otimes}\limits\mathbf{k}[e^{\prime}_{2(n-k)+1},\ldots,e^{\prime}_{2n-1}],$

where $\mathop{\mathrm{deg}}\nolimits e_{i}=\mathop{\mathrm{deg}}\nolimits
e^{\prime}_{i}=i$. The generators $e_{1},\ldots,e_{2m-1}$ are the images of
the standard generators of $H^{\bullet}(\mathfrak{gl}(m),\mathbf{k})$ under
the composition

(2.4.1)
${H^{\bullet}(\mathfrak{gl}(m),\mathbf{k})}$${H^{\bullet}(\mathfrak{g},\mathbf{k})}$${H^{\bullet}(\mathfrak{g},\Sigma^{\lambda}(V)\mathop{\otimes}\limits\Sigma^{\lambda}(V^{*})).}$$\scriptstyle{\simeq}$$\scriptstyle{\mathrm{res}}$$\scriptstyle{\mathrm{coev}}$

Here $\mathrm{res}$ is the map induced by restriction to the Lie subalgebra
$\mathfrak{gl}(m)\hookrightarrow\mathfrak{g}$, and $\mathrm{coev}$ is induced
by the coevaluation map
$\mathbf{k}\hookrightarrow\Sigma^{\lambda}(V)\mathop{\otimes}\limits\Sigma^{\lambda}(V^{*})$.

Proof: We consider Lie subalgebra
$\mathfrak{h}=\mathfrak{gl}(m,n-1)\oplus\mathfrak{gl}(1)\hookrightarrow\mathfrak{gl}(m,n)$
and prove the theorem by induction on $n$. When $n=0$ this is the classical
purely even case and the result is well known. When $n=1$ the statement of the
theorem is a reformulation of theorem 1.1. From now on, we will assume that
$n\geqslant 2$ and the theorem holds for all $\mathfrak{gl}(m,n^{\prime})$
with $n^{\prime}<n$. According to lemma 2.1 we have the spectral sequence

$E_{2}\ \simeq\
\bigoplus_{\mu}H^{\bullet}(\mathfrak{gl}(m,n-1),\Sigma^{\mu}(W)\mathop{\otimes}\limits\Sigma^{\mu}(W^{*}))\mathop{\otimes}\limits\mathbf{k}[e^{\prime\prime}_{1}]\mathop{\otimes}\limits
H^{\bullet}(\mathbb{E}_{\mu})\Rightarrow
H^{\bullet}(\mathfrak{g},\Sigma^{\lambda}(V)\mathop{\otimes}\limits\Sigma^{\lambda}(V^{*})).$

[Case $k=0$.] First, we consider the case when $k=0$ separately. The condition
$\lambda\in\mathcal{H}_{m-n,0}$ is equivalent to
$\mathrm{ht}(\lambda)\leqslant m-n$. From lemma 2.3 we see that all diagrams
$\mu$ contributing to the second layer of the spectral sequence have
$\mathrm{ht}(\mu)\leqslant m-n+1$, i.e. $\mu\in\mathcal{H}_{m-(n-1),0}$, hence
by the inductive assumption

$H^{\bullet}(\mathfrak{gl}(m,n-1),\Sigma^{\mu}(W)\mathop{\otimes}\limits\Sigma^{\mu}(W^{*}))\
\simeq\ \mathbf{k}[e_{1},\ldots,e_{2m-1}].$

Furthermore, again by lemma 2.3 we have

$E_{2}\ \simeq\
\mathbf{k}[e_{1},\ldots,e_{2m-1}]\mathop{\otimes}\limits\mathbf{k}[e^{\prime\prime}_{1}]\mathop{\otimes}\limits\mathbf{k}[c_{1}]$

and the differential on $E_{2}$ sends $e^{\prime}_{1}$ to $c_{1}$. Therefore,
we find that the spectral sequence converges to

$H^{\bullet}(\mathfrak{g},\Sigma^{\lambda}(V)\mathop{\otimes}\limits\Sigma^{\lambda}(V^{*}))\
\simeq\ \mathbf{k}[e_{1},\ldots,e_{2m-1}].$

The statement regarding classes $e_{i}$ is a tautology for $n=0$. Assume that
it hold for all $n^{\prime}<n$. The coevaluation map
$\mathbf{k}\to\Sigma^{\lambda}(V)\mathop{\otimes}\limits\Sigma^{\lambda}(V^{*})$
induces a morphism of spectral sequences. By inductive assumption this
morphism is an isomorphism on the second layer, therefore it also induces
isomorphism $H^{\bullet}(\mathfrak{g},\mathbf{k})\simeq
H^{\bullet}(\mathfrak{g},\Sigma^{\lambda}(V)\mathop{\otimes}\limits\Sigma^{\lambda}(V^{*}))$.

This completes the proof in the case $k=0$.

[Case $k>0$.] According to lemma 2.3(c) the contribution to the second layer
of the spectral sequence in the column $2p$ comes from the diagram
$(\lambda/p+1)$. To simplify notation, let us write

$\mathcal{H}^{\circ}(n,k)=\mathcal{H}_{m-n+k,k}-\mathcal{H}_{m-n+k-1,k-1}.$

We omit $m$ in the notation since within the scope of this proof the number of
even variables $m$ never changes. It is straightforward to check that since
$\lambda\in\mathcal{H}^{\circ}(n,k)$ for $0\leqslant p\leqslant k-1$ we have

$(\lambda/p+1)\in\mathcal{H}^{\circ}(n-1,k-1),$

and for $p\geqslant k$

$(\lambda/p+1)\in\mathcal{H}^{\circ}(n-1,k).$

Therefore, by inductive assumption for $0\leqslant p\leqslant k-1$ the $2p$’th
column of $E_{2}$ is isomorphic to

$E_{2}^{2p,\bullet}\simeq\mathbf{k}[e_{1},\ldots,e_{2m-1}]\mathop{\otimes}\limits\mathbf{k}[e^{\prime}_{2(n-k)+1},\ldots,e^{\prime}_{2n-3}]\mathop{\otimes}\limits\mathbf{k}[e^{\prime\prime}_{1}].$

For $p\geqslant k$ we consider two cases. First, if $k<n$, then again by
inductive assumption we have

$E_{2}^{2p,\bullet}\simeq\mathbf{k}[e_{1},\ldots,e_{2m-1}]\mathop{\otimes}\limits\mathbf{k}[e^{\prime}_{2(n-k)-1},\ldots,e^{\prime}_{2n-3}]\mathop{\otimes}\limits\mathbf{k}[e^{\prime\prime}_{1}].$

If on the other hand, $k=n$, then
$(\lambda/p+1)\in\mathcal{H}^{\circ}(n-1,n)$, hence the Schur functor
$\Sigma^{(\lambda/p+1)}(W)=0$, and all columns in $E_{2}$ starting from column
$2k$ vanish.

Since the differential on $E_{2}$ sends generator $e^{\prime\prime}_{1}\in
E_{2}^{2p,1}$ to the basis element in $E_{2}^{2(p+1),0}$ we see that on the
third layer the spectral sequence has only two non-zero columns: for $p=0$ and
either for $p=2k$ if $k<n$ or for $p=2k-2$ if $k=n$. Specifically,

$E_{3}^{0,\bullet}\simeq\mathbf{k}[e_{1},\ldots,e_{2m-1}]\mathop{\otimes}\limits\mathbf{k}[e^{\prime}_{2(n-k)+1},\ldots,e^{\prime}_{2n-3}],$

and if $0<k<n$, then

$E_{3}^{2k,\bullet}\simeq\mathbf{k}[e_{1},\ldots,e_{2m-1}]\mathop{\otimes}\limits\mathbf{k}[e^{\prime}_{2(n-k)+1},\ldots,e^{\prime}_{2n-3}]\mathop{\otimes}\limits\mathbf{k}e^{\prime}_{2(n-k)-1},$

and finally if $k=n$, then

$E_{3}^{2(k-1),\bullet}\simeq\mathbf{k}[e_{1},\ldots,e_{2m-1}]\mathop{\otimes}\limits\mathbf{k}[e^{\prime}_{1},\ldots,e^{\prime}_{2n-3}]\mathop{\otimes}\limits\mathbf{k}e^{\prime\prime}_{1}.$

Let us show that starting from $E_{3}$ all differentials in the spectral
sequence vanish. First, consider generators $e_{i}$. The coevaluation map
$\mathbf{k}\to\Sigma^{\lambda}(V)\mathop{\otimes}\limits\Sigma^{\lambda}(V^{*})$
induces a map from the spectral sequence for the trivial coefficients
$F_{\bullet}$ to our spectral sequence $E_{\bullet}$. By inductive assumption
the classes $e_{i}$ in the first column of $F_{\bullet}$ map to corresponding
classes $e_{i}$ in $E_{\bullet}$. As we have seen for the trivial coefficients
all the differentials vanish on $e_{i}$, hence they must also vanish in
$E_{\bullet}$.

Now consider generators $e^{\prime}_{j}$, and assume first that $k<n$. The
differential can only be non-zero on the layer $E_{2k}$ and send generator
$e^{\prime}_{j}$ to $E_{2k}^{2k,j-2k+1}$. However, since $j\leqslant 2n-3$ we
have $j-2k+1\leqslant 2(n-k)-2$ and column $2k$ has non-zero terms only for
$q\geqslant 2(n-k)-1$.

Finally, for $k=n$, the differential can only be non-zero on the layer
$E_{2(n-1)}$ and send generator $e^{\prime}_{j}$ to
$E_{2(n-1)}^{2(n-1),j-2n+3}$. Again, since $j\leqslant 2n-3$ we have
$j-2n+3\leqslant 0$ but all non-zero terms are in degree $q\geqslant 1$.

Observe, that for $k<n$ the total degree of the generator
$e^{\prime}_{2(n-k)-1}$ in the column $2k$ is $(2n-1)$, and similarly for
$k=n$ the total degree of $e^{\prime\prime}_{1}$ in the column $2(n-1)$ is
again $(2n-1)$. By renaming this generator $e^{\prime}_{2n-1}$ we obtain the
required isomorphism. The identification (2.4.1) immediately follows from the
previous discussion.

This concludes the proof of the theorem.

$\square$

##  3 Cohomology of $\mathcal{V}_{m,n}$

The calculation of cohomology $H^{\bullet}(\mathcal{V}_{m,n},\mathbf{k})$
follows the general argument originally developed for the classical case of
$\mathcal{V}_{m,0}$ by Gelfand and Fuchs with some refinements that were
needed to apply it to $\mathcal{V}_{m,1}$. Here we briefly recall the major
steps of this procedure, for details we refer to [Fuk] and [Pi1].

Consider Lie subalgebra $\mathfrak{gl}(m,n)\hookrightarrow\mathcal{V}_{m,n}$,
and let $V$ be the standard representation of $\mathfrak{gl}(m,n)$. The
continuous dual space

$\mathrm{Hom}(\mathcal{V}_{m,n},\mathbf{k})\ \simeq\ \bigoplus_{i\geqslant
0}\left(S^{i}(V^{*})\mathop{\otimes}\limits V\right).$

Therefore, in the spectral sequence for the Lie subalgebra
$\mathfrak{gl}(m,n)\hookrightarrow\mathcal{V}_{m,n}$

$E_{1}^{pq}=H^{q}(\mathfrak{gl}(m,n),\mathrm{Hom}(\Lambda^{p}(\mathcal{V}_{m,n}/\mathfrak{gl}(m,n)),\mathbf{k}))\Rightarrow
H^{p+q}(\mathcal{V}_{m,n},\mathbf{k}),$

the coefficients of the cohomology groups of $\mathfrak{gl}(m,n)$ can be
written as

$\bigoplus_{\sum
p_{i}=p}\Lambda^{p_{i}}\left(S^{i}(V^{*})\mathop{\otimes}\limits V\right),$

where $i\geqslant 0$ and $i\neq 1$. This can be simplified by observing that
the contributions to the first layer of the spectral sequence can only come
from terms of the form

$\Lambda^{p}(V)\mathop{\otimes}\limits\Lambda^{p}(S^{2}(V^{*})\mathop{\otimes}\limits
V).$

By expanding the second exterior power and using calculus of Schur functors
one then shows that the first layer of the spectral sequence has the form

(3.0.1) $E_{1}^{2p,q}\ \simeq\
\bigoplus_{|\lambda|=p}H^{\bullet}(\mathfrak{gl}(m,n),\Sigma^{\widetilde{\lambda}}(V)\mathop{\otimes}\limits\Sigma^{\widetilde{\lambda}}(V^{*})),$

where $\widetilde{\lambda}$ is obtained from $\lambda$ by adding to it one
more column with $|\lambda|$ boxes in it, in other words
$\widetilde{\lambda}_{i}=\lambda_{i}+1$ for $1\leqslant i\leqslant p$.

######  Theorem 3.1.

For any $m\geqslant n\geqslant 0$ we have an isomorphism

$H^{\bullet}(\mathcal{V}_{m,n},\mathbf{k})\ \simeq\
H^{\bullet}(\SS^{2n}X_{2(m-n)},\mathbf{k}).$

Proof: First of all notice that if $\lambda$ is a diagram in (3.0.1) then its
transposed $\lambda^{\prime}$ is a diagram appearing in the similar spectral
sequence for the Lie subalgebra
$\mathfrak{gl}(n-1,m+1)\hookrightarrow\mathcal{V}_{n-1,m+1}$. To simplify
notation we put

$\mathcal{H}^{\circ}(m,n,k)=\mathcal{H}_{m-n+k,k}-\mathcal{H}_{m-n+k-1,k-1}.$

Clearly, if $\lambda\in\mathcal{H}^{\circ}(m,n-1,k)$ for $k\geqslant 1$, then
$\widetilde{\lambda}\in\mathcal{H}^{\circ}(m,n,k+1)$. If $k=0$, then there are
two possibilities: if $|\lambda|\leqslant m-n$, then
$\widetilde{\lambda}\in\mathcal{H}(m,n,0)$, otherwise
$\widetilde{\lambda}\in\mathcal{H}^{\circ}(m,n,1)$.

Denote by $\widehat{\lambda}$ the diagram obtained from $\lambda$ by adding
one more row with $|\lambda|$ boxes, in other words

$\widehat{\lambda}=(|\lambda|,\lambda_{1},\lambda_{2},\ldots)=\left(\widetilde{\lambda^{\prime}}\right)^{\prime}.$

If $\lambda\in\mathcal{H}^{\circ}(m,n-1,k)$ for any $k\geqslant 0$, then
$\widehat{\lambda}\in\mathcal{H}^{\circ}(m+1,n-1,k)$.

Let us denote by $F_{\bullet}$ the spectral sequence (3.0.1) for
$\mathcal{V}_{n-1,m+1}$. On the first layer $F_{1}$ the term corresponding to
the diagram $\lambda\in\mathcal{H}^{\circ}(m,n-1,k)$ for $0\leqslant
k\leqslant n-1$ is isomorphic to

$(F_{1})_{\lambda}\simeq\mathbf{k}[e_{1},\ldots
e_{2m+1}]\mathop{\otimes}\limits\mathbf{k}[e^{\prime}_{2(n-k)-1},\ldots
e^{\prime}_{2n-3}].$

And the corresponding term in the spectral sequence $E_{1}$ is isomorphic to

$(E_{1})_{\lambda}\simeq\begin{cases}\mathbf{k}[e_{1},\ldots
e_{2m-1}]\mathop{\otimes}\limits\mathbf{k}[e^{\prime}_{2(n-k)-1},\ldots
e^{\prime}_{2n-1}],&\text{if $|\lambda|>m-n$},\\\ \mathbf{k}[e_{1},\ldots
e_{2m-1}],&\text{if $|\lambda|\leqslant m-n$}.\end{cases}$

Since $n-1<m+1$ the cohomology of $\mathcal{V}_{n-1,m+1}$ is covered by
theorem 1.2(b). Therefore, the spectral sequence $F_{\bullet}$ converges to
$H^{\bullet}(S^{2m+1},\mathbf{k})=\mathbf{k}[e_{2m+1}]$. Moreover, as was
shown in [AF] this class $e_{2m+1}$ maps to the corresponding class in
$H^{\bullet}(\mathcal{V}_{0,m+1},\mathbf{k})$ under the restriction map to the
Lie subalgebra $\mathcal{V}_{0,m+1}\hookrightarrow\mathcal{V}_{n-1,m+1}$. And
from the discussion in [Pi1] section 3, it follows that this class further
maps to $e_{2m+1}\in H^{\bullet}(\mathfrak{gl}(n-1,m+1),\mathbf{k})$ under the
restriction map to the Lie subalgebra
$\mathfrak{gl}(n-1,m+1)\hookrightarrow\mathcal{V}_{n-1,m+1}$. Hence, all the
differentials in $F_{\bullet}$ vanish on the generator $e_{2m+1}$ and from the
degree considerations $e_{2m+1}$ doesn’t appear in the image of the
differentials of any other generator $e_{i}$ or $e^{\prime}_{j}$. So we have
the sub-spectral sequence of $F_{\bullet}$ that we will denote by
$G_{\bullet}$, such that

$F_{\bullet}\simeq G_{\bullet}\mathop{\otimes}\limits\mathbf{k}[e_{2m+1}].$

The component of $G_{1}$ corresponding to a diagram
$\lambda\in\mathcal{H}^{\circ}(m,n-1,k)$ for $0\leqslant k\leqslant n-1$ is

$(G_{1})_{\lambda}\simeq\mathbf{k}[e_{1},\ldots
e_{2m-1}]\mathop{\otimes}\limits\mathbf{k}[e^{\prime}_{2(n-k)-1},\ldots
e^{\prime}_{2n-3}],$

and $G_{\bullet}$ converges to $\mathbf{k}$ (in degree $0$).

Let us compare spectral sequences $E_{\bullet}$ and $G_{\bullet}$. We
introduce an intermediate spectral sequence $\widetilde{E}_{\bullet}$ by
adding to $E_{\bullet}$ the “missing” classes $e_{2n-1}$ to all the small
diagrams $\lambda$ with $|\lambda|\leqslant m-n$. The differentials in the
spectral sequence $E_{\bullet}$ can be described as follows. For generators
$e_{2i-1}$ the differentials $d_{r}$ vanish up to layer $r=2i$, and on the
layer $E_{2i}$ they send

$(e_{2i-1}z_{\lambda})\mapsto\sum_{\mu\in\lambda\cdot c_{i}}z_{\mu},$

where sum is taken over diagrams $\mu$ in the decomposition of the product of
$\lambda$ and the Chern class $c_{i}$, i.e. $\mu$ is obtained from $\lambda$
by adding $i$ boxes such that no more than one box added in each row. Here
$z_{\lambda}$ denotes the generator of the component $(E_{1})_{\lambda}$.
Similarly, for generators $e^{\prime}_{2j-1}$ differentials vanish up to layer
$r=2j$ and on that layer they send

$e^{\prime}_{2j-1}z_{\lambda}\mapsto\sum_{\nu\in\lambda\cdot s_{j}}z_{\nu},$

where $\nu$ is a diagram in the decomposition of the product of $\lambda$ and
Segre class $s_{j}$, i.e. $\nu$ is obtained from $\lambda$ by adding $j$
boxes, such that no more than one box is added in each column.

We define spectral sequence $\widetilde{E}_{\bullet}$ by setting

$(\widetilde{E}_{1})_{\lambda}=\begin{cases}(E_{1})_{\lambda},&\text{if
$|\lambda|>m-n$},\\\
(E_{1})_{\lambda}\mathop{\otimes}\limits\mathbf{k}[e_{2n-1}],&\text{if
$|\lambda|\leqslant m-n$}.\end{cases}$

The differentials are defined as described above.

In fact one can construct a filtered complex for this spectral sequence
$\widetilde{E}_{\bullet}$. We start from the cochain complex
$C^{\bullet}=C^{\bullet}(\mathcal{V}_{m,n},\mathbf{k})$ with the filtration
induced by the Lie subalgebra $\mathfrak{gl}(m,n)$ as described in 1.2. In
every degree $p\geqslant 0$ this is a bounded filtration of $C^{p}$, therefore
the filtration of the entire complex is both complete and cocomplete. In such
case we can construct a bicomplex $B^{\bullet\bullet}$, so that its
totalization equipped with one of the natural filtrations of the bicomplex
(say in the vertical direction) is filtered quasi-isomorphic to the cochain
complex $C^{\bullet}$. This can be seen as a special case of Koszul duality
between filtered complexes, that are identified via Rees construction with
complexes of (flat) $\mathbf{k}[u]$-modules, and on the dual side complexes of
$\mathbf{k}[\varepsilon]$-modules, that are identified with bicomplexes, where
horizontal differential is given by $d$ and vertical differential by the
action of $\varepsilon$ (for a brief summary we refer to [Pi2] section 1.2,
for a detailed discussion see for example [Po]).

Now, since the totalization $\mathop{\mathrm{Tot}}\nolimits
B^{\bullet\bullet}$ is filtered quasi-isomorphic to $C^{\bullet}$ the first
layers of the corresponding spectral sequences are isomorphic. We construct
bicomplex $\widetilde{B}^{\bullet\bullet}$ by putting

$\widetilde{B}^{p\bullet}\simeq\begin{cases}B^{p\bullet},&\text{if
$p>m-n$},\\\ B^{p\bullet}\mathop{\otimes}\limits\mathbf{k}[e_{2n-1}],&\text{if
$p\leqslant m-n$}.\end{cases}$

We put both horizontal and vertical differentials to be zero on $e_{2n-1}$. We
will denote $\widetilde{C}^{\bullet}$ the totalization
$\mathop{\mathrm{Tot}}\nolimits\widetilde{B}^{\bullet\bullet}$ equipped with
the filtration in the vertical direction.

Notice that $\widetilde{E}_{1}\simeq
G_{1}\mathop{\otimes}\limits\mathbf{k}[e_{2n-1}]$. In the spectral sequence
$G_{\bullet}$ the roles of generator classes $e$ and $e^{\prime}$ are
reversed, however, the diagrams appearing in $G_{\bullet}$ are the transposes
of those appearing in $E_{\bullet}$, therefore the differentials in
$G_{\bullet}$ have the same description as above. Therefore, since the
spectral sequence $G_{\bullet}$ converges to $\mathbf{k}$ all potential
targets for differential starting from the new class $e_{2n-1}$ are already
killed in $G_{\bullet}$. Hence, we find that $\widetilde{E}_{\bullet}$
converges to $\mathbf{k}[e_{2n-1}]$.

Finally, consider the short exact sequence
$C^{\bullet}\to\widetilde{C}^{\bullet}\to
Q^{\bullet}\mathop{\otimes}\limits\mathbf{k}e_{2n-1}$. Here the spectral
sequence for $Q^{\bullet}$ has only contributions from diagrams $\lambda$ with
$|\lambda|\leqslant m-n$ and

$(Q_{1})_{\lambda}\simeq\mathbf{k}[e_{1},\ldots e_{2m-1}].$

This, as in the classical case, is isomorphic to the spectral sequence for the
fiber product

$X_{2(m-n)}=\mathop{\mathrm{sk}}\nolimits_{2(m-n)}BGL(m)\times_{BGL(m)}EGL(m).$

Since $\widetilde{E}_{\bullet}$ converges to $\mathbf{k}[e_{2n-1}]$, from the
long exact sequence we find that

$\displaystyle H^{0}(\mathcal{V}_{m,n},\mathbf{k})$
$\displaystyle=\mathbf{k},$ $\displaystyle
H^{i}(\mathcal{V}_{m,n},\mathbf{k})$ $\displaystyle=0,\quad\text{for
$1\leqslant i\leqslant 2n$}$ $\displaystyle
H^{i}(\mathcal{V}_{m,n},\mathbf{k})$ $\displaystyle\simeq
H^{i-2n}(X_{2(m-n)},\mathbf{k}),\quad\text{for $i>2n$}.$

$\square$

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