# Localization theorem for homological vector fields

Vera Serganova, Alexander Sherman

###### Abstract.

We present a general theorem which computes the cohomology of a homological
vector field on global sections of vector bundles over smooth affine
supervarieties. The hypotheses and results have the clear flavor of a
localization theorem.

##  1\. Introduction

Let $X$ be an affine supervariety, and let $Q$ be an odd vector field on $X$;
that is, an odd derivation of its algebra of functions, $\Bbbk[X]$. We call
$Q$ _homological_ if $Q^{2}$ acts semisimply on $\Bbbk[X]$. As the name
suggests, one may take the cohomology of a homological vector field, namely
$\operatorname{ker}(Q:\Bbbk[X]^{Q^{2}})/\operatorname{Im}(Q:\Bbbk[X]^{Q^{2}})$.
Here $\Bbbk[X]^{Q^{2}}$ denotes the $Q^{2}$-invariants of $\Bbbk[X]$. We write
this cohomology, which is a super vector space, as $DS_{Q}\Bbbk[X]$, because
in the case when $Q$ comes from the action of a Lie superalgebra
$\mathfrak{g}$, it is exactly the value of the Duflo-Serganova functor, from
the theory of Lie superalgebras. The goal of the present paper is to
understand conditions under which we may localize the computation of $DS_{Q}$
to a smooth, closed subvariety $Y$ of $X$.

The Duflo-Serganova functor has played a prominent role in the representation
theory of Lie superalgebras since being introduced in [DS05] (see, for
instance, [HW21], [Ser11A], [GH20], [EAS19], [EAHS18]). For an up-to-date
survey dealing with the Duflo-Serganova functor in the representation theory
of Lie superalgebras, we refer to [GHSS22].

On the other hand, localization theorems have proven to be powerful tools in
computations of integrals and cohomology, providing vast generalizations of
previously understood results. As an important example for us, in [SZ97] a
localization theorem was proven for computing Berezin integrals on a
supermanifold when the volume form admits an odd symmetry satisfying
sufficiently nice properties. This was recently used in [SS22] to prove a
‘splitting ’property of distinguished subgroups of certain (almost) simple
supergroups, which analogizes the definition of Sylow subgroups in the finite
characteristic setting. In this paper, our hypotheses are not too far from the
ones of [SZ97], except that we work in the algebraic category.

More precisely, suppose that $X$ is a smooth affine supervariety with odd
vector field $Q$, such that $Q^{2}$ acts semisimply on $\Bbbk[X]$. Then if $Q$
vanishes on a smooth subvariety $Y$, such that it acts in a certain
nondegenerate way on the normal bundle of $Y$ in $X$, we show that
$DS_{Q}\Bbbk[X]=\Bbbk[Y]$. One can view this theorem as a generalization of a
result which is already known: that if $Q$ is non-vanishing on all of $X$,
then $DS_{Q}\Bbbk[X]=0$.

The precise theorem (4.1) is slightly more general, both in that it begins
with a vector bundle on $X$, and it weakens the assumption that $Q$ needs to
vanish on all of $Y$. In any case, the result provides a powerful tool for
computations, many of which are of great interest. We attempt to survey some
of the applications in Sections 5 and 6.

As will be seen in the proof, the main technical input is from the theory of
commutative algebra; namely, the vector fields we consider have the property
that locally they define a Koszul complex for a sequence of regular elements
on a regular commutative ring. Thus by using what is already known about the
Koszul complex in such a setting, we are able to prove the vanishing results
for the cohomology of our operator.

###  1.1. Relation to physics

The Duflo-Serganova functor has appeared under different guises in theoretical
physics. Odd operators $Q$ in a supersymmetric field theory that satisfy
$Q^{2}=0$ are examples of BRST operators, and allow one to employ the so-
called BRST formalism, which includes taking cohomology in $Q$. Such a
situation arises in several places in the literature. In [W], it was used as a
key part in topological and holomorphic twists of supersymmetric field
theories. Twisting a supersymmetric field theory gives rise to simpler field
theories that can be topological (giving a TQFT), holomorphic, or something in
between, depending on properties of the chosen $Q$. A mathematically rigorous
approach to the twisting of supersymmetric field theories is explained in
[Cos].

Note that our setting is more flexible in that we do not impose the
requirement that $Q^{2}=0$; however we show in 4.4 that one can always reduce
to the case when $Q^{2}=0$.

In [CCMV], the authors also consider the cohomology of homological vector
fields (referred to as cohomological reductions in their paper) on the smooth
(or $L^{2}$) sections of certain vector bundles on the target spaces of sigma
models, which are homogeneous superspaces. On the contrary in the algebraic
setting, we find unexpected extra components of these cohomological
reductions; see for instance the table in LABEL:thm_table for the case of
symmetric spaces. In fact, when our space is a group $G$ and we consider the
action of an adjoint vector field, we find extra odd infinitesimal symmetries
in the cohomological reduction, which will themselves act on the Duflo-
Serganova functor (see 5.4).

We finally highlight the connection of our work to [GK], where the cohomology
of a vector field on $Q$-bundles is applied to the study of gauge PDEs. There,
they define a notion of equivalent reduction of supermanifolds with a BRST
operator $Q$ as ones which differ from one another by a ‘$Q$-contractible’
fiber bundle, i.e. a $Q$-bundle with trivial $Q$-cohomology along the fibers.
In this case they obtain that the $Q$-cohomology of supermanifolds with
equivalent reductions is the same.

###  1.2. Outlook for non-affine varieties

Our localization theorem can be extended to the non-affine case, although one
must work in the derived category to obtain the correct framework. This will
be left to a future work, where we will in particular study applications to
vector bundles on flag varieties.

###  1.3. Summary of sections

In Section 2 we recall the basic supergeometric language we will use, and some
preliminary results. In Section 3 we discuss $\mathfrak{q}$-supervarieties and
introduce the fundamental tools we need for our main theorem. Section 4
contains the main theorem and its proof. Section 5 discusses the first
applications, to the computation of the supergroup $\widetilde{G_{u}}$. Here
we also prove an important criterion for identifying a supervariety as the odd
tangent bundle of an even variety. In Section 6 we discuss computations on
certain homogeneous superspaces, in particular supersymmetric spaces. Section
7, an appendix, computes $\widetilde{G_{u}}$ for a certain root subgroup of
$D(2,1;\alpha)$.

###  1.4. Acknowledgments

The authors are grateful to I. Entova-Aizendud, V. Hinich, and D. Vaintrob for
many helpful discussions. The first author was supported in part by NSF grant
2001191. The second author was partially supported by ISF grant 711/18 and
NSF-BSF grant 2019694.

##  2\. Supergeometric preliminaries

We work throughout over an algebraically closed field $\Bbbk$ of
characteristic zero. For a super vector space $V$ we write
$V=V_{\overline{0}}\oplus V_{\overline{1}}$ for its parity decomposition. If
$v\in V$ is homogeneous, we write
$\overline{v}\in\\{\overline{0},\overline{1}\\}=\mathbb{Z}/2\mathbb{Z}$ for
its parity. If $R$ is a supercommutative ring then by $R^{m|n}$ we denote a
free $R$-module of rank $m|n$.

###  2.1. Notation for supervarieties

We will use the symbols $X,Y,\dots$ for supervarieties with even subschemes
$X_{0},Y_{0},\dots.$ For a supervariety $X$ we write $\mathcal{O}_{X}$ for the
structure sheaf and $\Bbbk[X]:=\Gamma(X,\mathcal{O}_{X})$ for the superalgebra
of global functions on $X$. We will be considering supervarieties in the sense
of [She21], however all spaces of interest will be smooth and affine. A smooth
affine supervariety is always given by the exterior algebra of a finite rank
vector bundle on a smooth affine variety $X_{0}$ (see [VMP90]). For several
equivalent conditions to being smooth, which we will use at various points
without elaboration, see the appendix of [She21].

Note that affine supervarieties and morphisms between them are entirely
determined by their algebras of global functions and maps between them, just
as with affine varieties. Further we note that the space of global sections of
an affine supervariety is finitely generated as a $\Bbbk$-superalgebra. See
[CCF11] for more on the basics of algebraic supergeometry.

We will employ the convention that for functions on a supervariety the symbols
$t,s$ will denote even functions, $\xi,\eta$ odd functions, and $f,g$
functions of ambiguous parity.

If $X$ is a supervariety, there is a canonical closed embedding
$i_{X}:X_{0}\to X$ which is a homeomorphism of underlying topological spaces.
The closed points of $X$ are the $\Bbbk$-points, which we write as $X(\Bbbk)$,
and they are canonically identified with the closed points of $X_{0}$ via
$i_{X}$. If $x$ is a closed point of $X$ and $\mathcal{F}$ is coherent a sheaf
on $X$, we write $\mathcal{F}_{x}$ for the stalk of $\mathcal{F}$ at $x$, and
$\mathcal{F}|_{x}$ for the fiber of $\mathcal{F}$ at $x$. We will write
$\mathfrak{m}_{x}$ for the maximal ideal of $\Bbbk[X]$ corresponding to $x$.
Then the cotangent space at $x$ is given by
$T_{x}^{*}X:=\mathfrak{m}_{x}/\mathfrak{m}_{x}^{2}$, and the tangent space by
$T_{x}X=\left(T_{x}^{*}X\right)^{*}$.

Given a function $f\in\Bbbk[X]$, we will use the notation $D(f)$ of [Har13]
for the open subvariety on which $f$ is non-vanishing.

If $Y$ is a closed subvariety of $X$, we write $\mathcal{I}_{Y}$ for the ideal
sheaf of $Y$ in $X$, and
$\mathcal{N}_{Y}^{\vee}=\mathcal{I}_{Y}/\mathcal{I}_{Y}^{2}$ for the conormal
bundle of $Y$ in $X$.

###  2.2. Coordinates

Let $X$ be a smooth supervariety. We say functions
$t_{1},\dots,t_{m},\xi_{1},\dots,\xi_{n}$ define a coordinate system on an
open set $U$ in $X$ if they are regular on $U$ and
$dt_{1},\dots,dt_{m},d\xi_{1},\dots,d\xi_{n}$ trivialize the cotangent bundle
on $U$. In this case, the tangent bundle will be trivialized on $U$ by the
sections
$\partial_{t_{1}},\dots\partial_{t_{m}},\partial_{\xi_{1}},\dots,\partial_{\xi_{n}}$
which are derivations acting in the obvious way.

By general facts about sections of vector bundles, if we have functions
$t_{1},\dots,t_{m},\xi_{1},\dots,\xi_{n}$ which define linearly independent
differentials at a point, they will define linearly independent differentials
in a neighborhood of that point; in particular if they define a coordinate
system at a point, then they define a coordinate system on a neighborhood of
said point.

We will use the following lemma later on:

######  Lemma 2.1.

Suppose that $X=\operatorname{Spec}A$ is a smooth affine variety and
$\xi_{1},\dots,\xi_{n}\in A_{\overline{1}}$ are global odd coordinates on $X$.
Then the derivations $\partial_{\xi_{1}},\dots,\partial_{\xi_{n}}$ are well-
defined, and if we set
$A_{0}=\operatorname{ker}\partial_{\xi_{1}}\cap\cdots\cap\partial_{\xi_{n}}$,
the natural map

$A_{0}[\xi_{1},\dots,\xi_{n}]\to A$

is an isomorphism.

###### Proof.

Because $X$ is smooth, the sheaf of relative Kähler differentials $\Omega_{X}$
is locally free. The global odd coordinates $\xi_{1},\dots,\xi_{n}$ define a
map $\mathcal{O}_{X}^{0|n}\to\Omega_{X}$ which is injective on fibers, and
thus splits. This splitting provides for us our derivations
$\partial_{\xi_{i}}$ for all $i$. Now let
$A_{0}=\operatorname{ker}\partial_{\xi_{1}}\cap\cdots\cap\partial_{\xi_{n}}$
and consider the natural map

$A_{0}[\xi_{1},\dots,\xi_{n}]\to A.$

To see this map is injective, suppose that $\sum\limits_{I}f_{I}\xi_{I}=0$
where $f_{I}\in A_{0}$. Then if we apply $\partial_{I}:=\prod_{i\in
I}\partial_{\xi_{i}}$ to $f$ we learn that $f_{I}=0$ for all $I$.

To show surjectivity, first order the subsets $I\subseteq\\{1,\dots,n\\}$
using a lexicographical ordering, meaning that
$I=\\{i_{1},\dots,i_{r}\\}<J=\\{j_{1},\dots,j_{s}\\}$ if $|I|<|J|$, and if
$|I|=|J|$ then $I<J$ if for some $k$ we have $i_{k}<j_{k}$ while
$i_{\ell}=j_{\ell}$ for $\ell<k$.

Now let $f\in A$ and let $I$ be the largest subset under this ordering such
that $\partial_{I}(f)\neq 0$. Then we must have that
$\partial_{\xi_{i}}(\partial_{I}(f))=0$ for all $i$, i.e.
$\partial_{I}(f)=g_{I}\in A_{0}$. Consider

$f^{\prime}:=f-\partial_{I}(\xi_{I})g_{I}\xi_{I}$

Then $\partial_{I}(f^{\prime})=0$ by construction, and thus we may conclude by
induction that $f^{\prime}\in A_{0}[\xi_{1},\dots,\xi_{n}]$, completing the
proof. ∎

###  2.3. Subvarieties

By abuse of language we will refer to subsupervarieties as subvarieties. If
$Y\subseteq X$ is a closed subvariety of $X$, then we write $\mathcal{I}_{Y}$
for the ideal sheaf of $Y$, and
$\mathcal{N}_{Y}^{\vee}:=\mathcal{I}_{Y}/\mathcal{I}_{Y}^{2}$ for the conormal
sheaf of $Y$. Then $\mathcal{N}_{Y}^{\vee}$ is a coherent sheaf on $Y$, and if
$Y$ is smooth then $\mathcal{N}_{Y}^{\vee}$ is a vector bundle of rank given
by the codimension of $Y$. In this case, every point on $Y$ has an open
neighborhood in $X$ where we may choose local functions
$t_{1},\dots,t_{m},\xi_{1},\dots,\xi_{n}\in\mathcal{I}_{Y}$ such that they
will trivialize $\mathcal{N}_{Y}^{\vee}$, and they define linearly independent
differentials.

##  3\. $\mathfrak{q}$-supervarieties

We explain several lemmas and definitions we will use repeatedly for the proof
of the general theorem.

###  3.1. $\mathfrak{q}$-modules

Let $\mathfrak{q}$ be the $(1|1)$-dimensional Lie superalgebra with basis
$Q,Q^{2}:=\frac{1}{2}[Q,Q]$, where $Q$ is odd. We will work in
$\operatorname{Rep}_{Q^{2}}\mathfrak{q}(1)$, i.e. the category of
$\mathfrak{q}$-modules (not necessarily finite-dimensional) on which $Q^{2}$
acts semisimply. Thus when we refer to a $\mathfrak{q}$-module we will always
assume that $Q^{2}$ acts semisimply.

######  Definition 3.1 (Duflo-Serganova functor).

For a $\mathfrak{q}$-module $V$ (recall that as explained above, this mean
$Q^{2}$ acts semisimply), define $DS_{Q}V$ to be the super vector space given
by the cohomology of $Q$ when restricted to $V^{Q^{2}}$, the invariants of
$Q^{2}$ on $V$.

The Duflo-Serganova functor was originally introduced in [DS05], and defines a
tensor functor from $\operatorname{Rep}_{Q^{2}}\mathfrak{q}$ to the category
of super vector spaces. For a more recent and thorough survey on this functor,
we refer to [GHSS22].

We recall the following results which will be used throughout.

######  Lemma 3.2.

Let $V$ be a $\mathfrak{q}$-module.

  1. (1)

$V$ is projective if and only if $DS_{Q}V=0$;

  2. (2)

if $V$ admits a finite-length projective resolution, then $V$ is projective;

  3. (3)

if the restriction $Q:V_{\overline{1}}\to V_{\overline{0}}$ is an isomorphism,
then $V$ is projective.

###  3.2. $\mathfrak{q}$-supervarieties

We recall that on an affine supervariety $X=\operatorname{Spec}\Bbbk[X]$, a
vector field is nothing but a derivation (in the super sense) of $\Bbbk[X]$.

######  Definition 3.3.

We say that an affine supervariety $X$ is a $\mathfrak{q}$-supervariety if
there exists an odd vector field $Q$ on $X$ such that $Q^{2}$ acts semisimply
on $\Bbbk[X]:=\Gamma(X,\mathcal{O}_{X})$. In other words $\Bbbk[X]$ is a
$\mathfrak{q}$-module, where $Q$ acts by derivations.

Let $X$ be a $\mathfrak{q}$-supervariety. Then since $\Bbbk[X]$ is finitely
generated, the $\mathfrak{q}$-action action necessarily integrates to the
action of an algebraic supergroup $G_{Q}$ on $X$, where
$(\operatorname{Lie}G_{Q})_{\overline{1}}$ is spanned by $Q$, and
$(G_{Q})_{0}$ is a central torus. However because of the ambiguity in this
choice of torus, it is better to avoid fixing a lift to $G_{Q}$ and instead
only doing so when it is needed in an argument.

######  Definition 3.4.

We will say that an open subvariety $U\subseteq X$ is $\mathfrak{q}$-stable if
$Q^{2}$ acts semisimply on $\Bbbk[U]$. In this case $U$ is itself naturally a
$\mathfrak{q}$-supervariety.

######  Definition 3.5.

Given a vector field $Q$ on $X$ and a point $x\in X(\Bbbk)$, we may evaluate
$Q$ at $x$ to obtain a tangent vector $Q(x)\in T_{x}X$. We say $Q$ is non-
vanishing at $x$ if $Q(x)\neq 0$. We say $Q$ is non-vanishing on $X$ if it is
non-vanishing at every point $x\in X(\Bbbk)$. We write $Z:=Z(Q)$ for the
closed subvariety of $X_{0}$ given by the vanishing set of $Q$.

Observe that if $\mathcal{I}_{Z}$ denotes the ideal sheaf defined by
$Z\subseteq X$, we have $Q(\mathcal{O}_{X})\subseteq\mathcal{I}_{Z}$.

####  3.2.1. Equivariant sheaves

Let $\mathcal{F}$ be a quasi-coherent sheaf on a $\mathfrak{q}$-supervariety
$X$. We say that $\mathcal{F}$ is $\mathfrak{q}$-equivariant if it admits an
action of $Q$ such that $Q^{2}$ acts semisimply on global sections, and

$Q(fs)=Q(f)s+(-1)^{\overline{f}}fQ(s),$

where $s$ is a section and $f$ a function.

######  Lemma 3.6.

Let $X$ be an affine $\mathfrak{q}$-supervariety; then the following are
equivalent:

  1. (1)

$Z(Q)=\emptyset$;

  2. (2)

there exists $\xi\in\mathbb{C}[X]$ with $Q(\xi)=1$;

  3. (3)

$DS_{Q}\mathbb{C}[X]=0$;

  4. (4)

if $\mathcal{F}$ is any quasi-coherent $\mathfrak{q}$-equivariant sheaf on
$X$, then $DS_{Q}\Gamma(X,\mathcal{F})=0$.

###### Proof.

Clearly $(4)\Rightarrow(3)$, $(3)\Rightarrow(2)$, and $(2)\Rightarrow(1)$.
Further $(2)\Rightarrow(4)$ because if $s\in\Gamma(X,\mathcal{F})$ with
$Q(s)=0$, then $Q(\xi s)=s$.

Thus it suffices to show $(1)\Rightarrow(2)$. Since $Q$ is everywhere non-
vanishing and $X$ is affine, for each closed point $x\in X(\Bbbk)$ there exist
an odd function $\xi_{x}\in\Bbbk[X]$ such that $(Q\xi_{x})(x)\neq 0$. Thus the
ideal generated by $\\{Q\xi_{x}\\}_{x\in X(\Bbbk)}$ is equal to $\Bbbk[X]$, so
we may choose finitely many functions in this set, say
$Q\xi_{1},\dots,Q\xi_{k}$, and even functions $g_{1},\dots,g_{k}$ such that

$g_{1}Q\xi_{1}+\dots+g_{k}Q\xi_{k}=1.$

Set $\eta=g_{1}\xi_{1}+\dots+g_{k}\xi_{k}$. We have

$Q\eta=g_{1}Q(\xi_{1})+\dots+g_{k}Q(\xi_{k})+Q(g_{1})\xi_{1}+\dots+Q(g_{k})\xi_{k}=1+r$

where $r=Q(g_{1})\xi_{1}+\dots+Q(g_{k})\xi_{k}$. Since $Q(g_{i})$ is odd,
$r\in(\Bbbk[X]_{\overline{1}})^{2}$ and thus is nilpotent. We may then write
$\eta=\sum\limits_{\lambda}\eta_{\lambda}$ and
$r=\sum\limits_{\lambda}r_{\lambda}$ according the decomposition of $\Bbbk[X]$
into eigenspaces of $Q^{2}$. Since $Q^{2}$ preserves
$(\Bbbk[X]_{\overline{1}})^{2}$, we have
$r_{\lambda}\in(\Bbbk[X]_{\overline{1}})^{2}$ for all $\lambda$. Now we see
that

$Q(p)=\sum\limits_{\lambda}Q(p_{\lambda})=1+\sum\limits_{\lambda}r_{\lambda}.$

Since $Q^{2}(1)=0$ and $[Q^{2},Q]=0$, we find that
$\alpha:=Q(\eta_{0})=1+r_{0}$ is a unit in $\Bbbk[X]$. Further,
$Q\alpha=Q^{2}(p_{0})=0$, so if we set $\xi:=p_{0}/\alpha$, we find that

$Q(\xi)=Q(p_{0}/\alpha)=\alpha/\alpha=1.$

∎

The following lemma is one of the main technical inputs in the proof of the
main theorem in the next section.

######  Lemma 3.7.

Let $\mathcal{W}$ be a $\mathfrak{q}$-equivariant vector bundle on a smooth
affine $\mathfrak{q}$-supervariety $X$, and suppose that $Q$ vanishes at $x\in
X(\Bbbk)$. Then there exists an affine open $\mathfrak{q}$-stable neighborhood
$U$ of $x$ such that we have a $Q^{2}$-equivariant isomorphism of sheaves

$\mathcal{W}|_{U}\cong\mathcal{O}_{U}\otimes_{\Bbbk}\mathcal{W}|_{x}.$

If we suppose further that $\mathcal{W}\to\mathcal{W}|_{x}$ admits a
$\mathfrak{q}$-splitting, then the isomorphism can be made to be
$\mathfrak{q}$-equivariant. In particular in this case,
$\Gamma(U,\mathcal{W})$ is a projective $\mathfrak{q}$-module if and only if
$\mathcal{W}|_{x}$ is.

###### Proof.

Since $Q^{2}$ acts semisimply we may split
$\Gamma(X,\mathcal{W})\to\mathcal{W}|_{x}$ as a $Q^{2}$-module in any case;
choose such a splitting, and if a $\mathfrak{q}$-equivariant splitting exists
then choose the splitting to be $\mathfrak{q}$-equivariant. Then we have a
homogeneous basis $v_{1},\dots,v_{m},w_{1},\dots,w_{n}$ of our assumed
splitting, i.e. coming from the image of $\mathcal{W}|_{x}$ under the
splitting map, such that each $v_{i},w_{j}$ is a global section and a
$Q^{2}$-eigenvector.

Let $U$ be the maximal open subset on which
$v_{1},\dots,v_{m},w_{1},\dots,w_{n}$ trivialize $\mathcal{W}$; then $U$ is
affine, as it is the complement of the divisor determined by a rational,
$Q^{2}$-eigensection of $\operatorname{Ber}(\mathcal{W})$. Since the rational
section is a $Q^{2}$-eigenvector, $\Bbbk[U]$ will admit a semisimple action of
$Q^{2}$. Finally, since $\mathcal{W}$ is trivialized on $U$, the natural map
$\mathcal{O}_{U}\otimes_{\Bbbk}\mathcal{W}|_{x}\to\mathcal{W}|_{U}$ is an
isomorphism. ∎

######  Lemma 3.8.

Suppose that $\mathcal{F}$ is a $\mathfrak{q}$-equivariant quasicoherent sheaf
such that every point $x\in X(\Bbbk)$ has a $\mathfrak{q}$-stable, affine open
neighborhood $U$ such that for every $\mathfrak{q}$-stable affine open
$V\subseteq U$, we have that $\Gamma(V,\mathcal{F})$ is projective over
$\mathfrak{q}$. Then $\Gamma(X,\mathcal{F})$ is a projective
$\mathfrak{q}$-module.

###### Proof.

Using our assumptions we may find a $Q$-stable affine Cech cover $\\{U_{i}\\}$
of $X$ such that $\Gamma(U_{i_{1}}\cap\cdots\cap U_{i_{k}},\mathcal{F})$ is
projective for all $i$. Since $X$ is affine, the Cech complex

$0\to\Gamma(X,\mathcal{F})\to\prod\limits_{i}\Gamma(U_{i},\mathcal{F})\to\prod\limits_{i,j}\Gamma(U_{i}\cap
U_{j},\mathcal{F})\to\cdots$

will be exact and finite, so we can conclude by Lemma 3.2. ∎

######  Corollary 3.9.

If $\mathcal{W}$ is a $\mathfrak{q}$-equivariant vector bundle on a smooth
affine $\mathfrak{q}$-variety $X$ such that at every point $x\in Z(Q)$,
$\mathcal{W}|_{x}$ is a projective $\mathfrak{q}$-module. Then
$\Gamma(X,\mathcal{W})$ is a projective $\mathfrak{q}$ module.

###### Proof.

We apply Lemma 3.8. If $x\in X(\Bbbk)\setminus Z(Q)$, then we may find a
$\mathfrak{q}$-stable, affine open neighborhood $U$ of $x$ such that $Q$ is
nonvanishing on $U$. Thus by Lemma 3.6, $DS_{x}\Gamma(U,\mathcal{W})=0$. If on
the other hand $x\in Z(Q)$, then by Lemma 3.7 there exists a
$\mathfrak{q}$-stable, affine open neighborhood $U$ of $x$ such that
$\Gamma(U,\mathcal{W})$ is projective. ∎

In the following example we show that the converse of 3.9 fails.

######  Example 3.10.

Let $X=\mathbb{A}^{1|1}$ and consider it with a trivial action of
$\mathfrak{q}$; in particular $Z(Q)=X(\Bbbk)$. Present
$\Bbbk[X]=\Bbbk[t,\xi]$, and give $\mathcal{O}_{X}$ the structure of a
$\mathfrak{q}$-equivariant sheaf via $Q(f)=\xi f$. Then we have
$DS_{Q}\Gamma(X,\mathcal{O}_{X})=0$, even though the fibers of this vector
bundle, which are all $\mathfrak{q}$-modules, are not projective.

##  4\. Main theorem

######  Theorem 4.1.

Let $\mathcal{V}$ be a $\mathfrak{q}$-equivariant vector bundle on a smooth
affine $\mathfrak{q}$-supervariety $X$. Write $Z=Z(Q)\subseteq X_{0}$ for the
vanishing subvariety of $Q$, and suppose that $Y$ is a smooth, closed
subvariety of $X$ such that:

  1. (1)

$Z(\Bbbk)\subseteq Y(\Bbbk)$;

  2. (2)

$Q(\mathcal{I}_{Y})\subseteq\mathcal{I}_{Y}$;

  3. (3)

$Q:(\mathcal{N}_{Y}^{\vee}|_{z})_{\overline{1}}\to(\mathcal{N}_{Y}^{\vee}|_{z})_{\overline{0}}$
is an isomorphism for all $z\in Z(\Bbbk)$.

  4. (4)

For all points $z\in Z(\Bbbk)$, there exists a $\mathfrak{q}$-stable affine
open $U_{z}$ of $z$ such that the map
$\Gamma(U_{z},\mathcal{V})\to\mathcal{V}|_{z}$ splits over $\mathfrak{q}$.

Then $Y$ is naturally a $\mathfrak{q}$-supervariety, $\mathcal{V}|_{Y}$ is a
$\mathfrak{q}$-equivariant vector bundle on $Y$, and we have:

$DS_{Q}\Gamma(X,\mathcal{V})=DS_{Q}\Gamma(Y,\mathcal{V}|_{Y}).$

Further this identification is induced by the natural map
$\Gamma(X,\mathcal{V})\to\Gamma(Y,\mathcal{V}|_{Y})$.

###  4.1. Corollaries of 4.1

Note that if $\mathcal{V}=\mathcal{O}_{X}$ is the trivial vector bundle with
the canonical $\mathfrak{q}$-equivariant structure, then condition (4) always
holds. Thus we obtain:

######  Corollary 4.2.

Under hypotheses (1)-(3) of Theorem 4.1, we have that the restriction map
$\Bbbk[X]\to\Bbbk[Y]$ induces an identification:

$DS_{Q}\Bbbk[X]=DS_{Q}\Bbbk[Y],$

######  Corollary 4.3.

Suppose that in the hypotheses of 4.1 we have
$Q(\mathcal{O}_{X})\subseteq\mathcal{I}_{Y}$. Then the restriction map
$\Bbbk[X]\to\Bbbk[Y]$ induces an identification

$DS_{Q}\Bbbk[X]=\Bbbk[Y].$

Further, suppose that $\mathcal{V}$ is a $\mathfrak{q}$-equivariant vector
bundle on $X$ satisfying (4) of 4.1. Then $DS_{Q}\Gamma(X,\mathcal{V})$ will
be the sections of a vector bundle on $Y$ with fibers isomorphic to
$DS_{Q}(\mathcal{V}|_{y})$.

###### Proof.

The first statement follows immediately from 4.1. For the second statement, we
may apply Lemma 3.7 to show that $Q$ defines a constant rank vector bundle
endomorphism of $\mathcal{V}|_{Y}$, whose cohomology will have fibers as
claimed. ∎

By the following corollary, 4.1 allows us to always reduce to the case when
$Q^{2}=0$.

######  Corollary 4.4.

If $Y=\operatorname{Spec}\Bbbk[X]/(\operatorname{Im}Q^{2})$, then $Y$ is a
smooth subvariety of $X$ and we have

$DS_{Q}\mathbb{C}[X]=DS_{Q}\mathbb{C}[Y].$

###### Proof.

We need to check that $Y$ is smooth and that
$Q:(\mathcal{N}_{Y}^{\vee}|_{z})_{\overline{1}}\to(\mathcal{N}_{Y}^{\vee}|_{z})_{\overline{0}}$
is an isomorphism for all $z\in Z(\Bbbk)$. Let $I=(\operatorname{Im}Q^{2})$,
and let $y\in Y(\Bbbk)$. Choose a coordinate system
$t_{1},\dots,t_{m},\xi_{1},\dots,\xi_{n}$ around $y$ such that these
coordinates are eigenvectors. Then we claim that $I_{y}$, the localization of
$I$ at $y$, is generated by those coordinates with nonzero eigenvalues. The
proof of this statement is identical to the one given in Lem. 2.1 of [I72],
via Prop. 3.2 of [Sch89]. Thus we obtain that $Y$ is smooth, and further we
have that $Q^{2}$ is an automorphism of $\mathcal{N}_{Y}^{\vee}|_{y}$ for all
$y\in Y(\Bbbk)$, which is sufficient for condition (3) of 4.1. ∎

###  4.2. Proof of 4.1

The proof of 4.1 will occupy the remainder of this section. In fact, 4.1
follows as corollary of the following result, which is what we will actually
show.

######  Proposition 4.5.

Retain the hypotheses of Theorem 4.1. Let $\mathcal{V}_{Y}$ denote the sheaf
of sections of $\mathcal{V}$ which vanish upon restriction to $Y$. Then
$\Gamma(X,\mathcal{V}_{Y})$ is a projective $\mathfrak{q}$-module; in
particular $DS_{Q}\Gamma(X,\mathcal{V}_{Y})=0$.

Indeed, Proposition 4.5 is enough because then if we apply $DS_{Q}$ to

$0\to\Gamma(X,\mathcal{V}_{Y})\to\Gamma(X,\mathcal{V})\to\Gamma(Y,\mathcal{V}|_{Y})\to
0,$

we obtain that $DS_{Q}\Gamma(X,\mathcal{V})\to
DS_{Q}\Gamma(Y,\mathcal{V}|_{Y})$ is an isomorphism, see Lemma 2.7 of
[GHSS22].

###### Proof of Proposition 4.5.

We use Lemma 3.8: if $x\in X(\Bbbk)\setminus Z(\Bbbk)$, then by affinity and
the semisimplicity of the action of $Q^{2}$, there exists a
$Q^{2}$-eigenfunction $f\in\Gamma(X,\mathcal{I}_{Z})$ such that $f(x)\neq 0$.
Thus $D(f)$ will be a $\mathfrak{q}$-stable affine open subvariety containing
$x$. Further, by Lemma 3.6, $\Gamma(V,\mathcal{V}_{Y})$ is projective for all
$\mathfrak{q}$-stable affine open subsets $V\subseteq D(f)$.

It remains to deal with the points in $Z(\Bbbk)$. For this step we begin by
assuming that $\mathcal{V}=\mathcal{O}_{X}$. Choose $z\in Z(\Bbbk)$. We want
to find a ‘nice’ splitting of
$\mathfrak{m}_{z}\to\mathfrak{m}_{z}/\mathfrak{m}_{z}^{2}$; to expect a full
$\mathfrak{q}$-module splitting would be too much, but we can obtain enough as
follows. Recall that $\mathcal{N}_{Y}^{\vee}|_{z}$ is a subspace of
$\mathfrak{m}_{z}/\mathfrak{m}_{z}^{2}$.

We have a natural, $\mathfrak{q}$-equivariant surjective map
$\Gamma(X,\mathcal{I}_{Y})\to\mathcal{N}^{\vee}_{Y}|_{z}$; by our assumptions
on $\mathcal{N}^{\vee}_{Y}|_{z}$ it is a projective $\mathfrak{q}$-module, and
so we may split this map $\mathfrak{q}$-equivariantly; in particular from this
splitting we may choose sections
$t_{1},\dots,t_{n},\xi_{1},\dots,\xi_{n}\in\Gamma(X,\mathcal{I}_{Y})$
satisfying:

$Q(\xi_{i})=t_{i},\ \ \ \ Q(t_{i})=c_{i}\xi_{i},$

where $c_{i}\in\Bbbk$. Further, by the discussion in Section 2, these
functions define linearly independent differentials in a neighborhood of $z$.
Write $V$ for the vector space spanned by
$t_{1},\dots,t_{n},\xi_{1},\dots,\xi_{n}$. Then the natural map $V\to
T_{z}^{*}X$ is injective and $\mathfrak{q}$-invariant. Choose a
$Q^{2}$-invariant splitting $T^{*}_{z}X=V\oplus W$, and a $Q^{2}$-splitting of
$W$ off $\mathfrak{m}_{z}\to T_{z}^{*}X$, thus giving us functions

$t_{1},\dots,t_{n},s_{1},\dots,s_{r},\xi_{1},\dots,\xi_{n},\eta_{1},\dots,\eta_{\ell}\in\mathfrak{m}_{z}$

each of which is a $Q^{2}$-eigenvector. These functions define linearly
independent differentials in a neighborhood of $z$; we may now apply the proof
of Lemma 3.7 using this splitting, to the case where $\mathcal{W}$ is the
cotangent bundle, to find a $\mathfrak{q}$-stable, affine open neighborhood
$U^{\prime}$ of $z$ on which these functions continue to define linearly
independent differentials.

Consider the ideal
$J:=(t_{1},\dots,t_{n},\xi_{1},\dots,\xi_{n})\subseteq\Bbbk[U^{\prime}]$; then
we need further that $J=I_{Y\cap U^{\prime}}$, i.e. that the ideal $J$ exactly
cuts out $Y\cap U^{\prime}$. A priori we only have $J\subseteq I_{Y\cap
U^{\prime}}$. However since $t_{1},\dots,t_{n},\xi_{1},\dots,\xi_{n}$ are part
of a coordinate system, $J$ will cut out a smooth variety of the same
dimension as $Y$, and thus the zero set of $J$ can be written as $Y\sqcup
Y^{\prime}$, i.e. $Y$ will be connected component of its zero set.

Consider the ideal $I_{Y^{\prime}}$ of $Y^{\prime}$ in $\Bbbk[U^{\prime}]$.
Because $J$ is $Q^{2}$-stable, $I_{Y^{\prime}}$ is also $Q^{2}$-stable; by
affinity there must exist a $Q^{2}$-eigenfunction $f\in I_{Y^{\prime}}$ such
that $f(z)\neq 0$. Now we set $U:=D(f)\subseteq U^{\prime}$; then $U$ is a
$Q^{2}$-stable affine open neighborhood of $z$, and $I_{Y\cap
U}=(t_{1},\dots,t_{n},\xi_{1},\dots,\xi_{n})$.

Now on $U$ we have a trivialization of the tangent bundle defined by the
sections
$\partial_{t_{1}},\dots,\partial_{t_{n}},\partial_{s_{1}},\dots,\partial_{s_{r}},\partial_{\xi_{1}},\dots,\partial_{\xi_{n}},\partial_{\eta_{1}},\dots,\partial_{\eta_{\ell}}$.
In this trivialization $Q$ takes the local form:

$Q=\sum\limits_{i}t_{i}\partial_{\xi_{i}}+\sum\limits_{i}c_{i}\xi_{i}\partial_{t_{i}}+\sum\limits_{i}f_{i}\partial_{s_{i}}+\sum\limits_{i}g_{i}\partial_{\eta_{i}}.$
(4.1)

where $f_{1},\dots,f_{r},g_{1},\dots,g_{\ell}$ are some functions. Further, by
our choice of coordinates we have

$Q^{2}=\sum_{i=1}^{n}c_{i}(t_{i}\partial_{t_{i}}+\xi_{i}\partial_{\xi_{i}})+\sum\limits_{i}d_{i}s_{i}\partial_{s_{i}}+\sum\limits_{i}e_{i}\eta_{i}\partial_{\eta_{i}}.$

where $d_{i},e_{i}\in\Bbbk$. Define the following vector field on $U$:

$h=\xi_{1}\partial_{\xi_{1}}+\dots+\xi_{n}\partial_{\xi_{n}}.$

Then $h$ commutes with $Q^{2}$ and has integral eigenvalues bounded between 0
and $n$. We may decompose $Q$ as

$Q=Q_{-1}+Q_{0}+Q_{1}+\dots+Q_{n}$

according to the eigenvalues of $h$. By Eq. 4.1 we have

$Q_{-1}=t_{1}\partial_{\xi_{1}}+\dots+t_{k}\partial_{\xi_{k}}.$

Write $A=\Bbbk[U]$; then since $U$ is affine and smooth, it is split. By Lemma
2.1, we may split $A$ as

$A=A_{0}\otimes{\bigwedge}^{\bullet}\langle\xi_{1},\dots,\xi_{n},\eta_{1},\dots,\eta_{\ell}\rangle.$

where
$A_{0}=\operatorname{ker}\partial_{\xi_{1}}\cap\cdots\cap\operatorname{ker}\partial_{\xi_{n}}\cap\operatorname{ker}\partial_{\eta_{1}}\cap\cdots\cap\operatorname{ker}\partial_{\eta_{\ell}}$.
In particular $t_{1},\dots,t_{n}\in A_{0}$. Then we have

$I_{Y\cap U}=\left((t_{1},\dots,t_{n})_{0}\oplus
A_{0}\otimes{\bigwedge}^{+}\langle\xi_{1},\dots,\xi_{k}\rangle\right)\otimes{\bigwedge}^{\bullet}\langle\eta_{1},\dots,\eta_{\ell}\rangle.$

Here $(t_{1},\dots,t_{n})_{0}$ denotes the ideal generated by
$t_{1},\dots,t_{n}$ in $A_{0}$, which is exactly the ideal defining $Y_{0}\cap
U_{0}$ in $U_{0}$. Now one sees that $Q_{-1}$ is defining the Koszul complex
on

$A_{0}\otimes{\bigwedge}^{\bullet}\langle\xi_{1},\dots,\xi_{n}\rangle$

and the cohomology is $A_{0}/(t_{1},\dots,t_{n})_{0}$ because
$t_{1},\dots,t_{n}$ define a regular sequence in $A_{0}$. Thus the cohomology
of $Q_{-1}$ on

$(t_{1},\dots,t_{n})_{0}\oplus
A_{0}\otimes{\wedge}^{+}\langle\xi_{1},\dots,\xi_{k}\rangle$

is trivial, implying that its cohomology on $I_{Y\cap U}$ is also trivial,
i.e. $DS_{Q_{-1}}I_{Y\cap U}=0$. To show that $DS_{Q}I_{Y\cap U}=0$, we appeal
to the following lemma.

######  Lemma 4.6.

Suppose that $h,Q_{-1},Q_{0},\dots,Q_{n}$ are operators on a super vector
space $V$ such that

  1. (1)

$h$ is even and semisimple with bounded, integral eigenvalues;

  2. (2)

each $Q_{i}$ is an odd operator;

  3. (3)

$[h,Q_{i}]=iQ_{i}$.

If $DS_{Q_{-1}}V=0$, then $DS_{Q}V=0$, where $Q=Q_{-1}+Q_{0}+\dots+Q_{n}$.

###### Proof.

Let $v\in V$ be such that $Qv=0$, and write $v=v_{i}+v_{i+1}+\dots+v_{j}$ for
the $h$-eigenvector decomposition of $v$, where we assume that $i$ is such
that $v_{i}\neq 0$. Then clearly $Q_{-1}v_{i}=0$, so by assumption there
exists $w=w_{i+1}+\dots$ such that $v-Qw$ only has nonzero components in
eigenspaces of $h$ with eigenvalue strictly larger than $i$. Because $h$ has
bounded eigenvalues on $V$, we may thus continue inductively to construct
$v^{\prime}\in V$ such that $Qv^{\prime}=v$, completing the proof. ∎

Now if $V\subseteq U$ is an affine $\mathfrak{q}$-stable open subvariety of
$U$, then we will still have our coordinates above and so the same arguments
imply that $DS_{Q}I_{Y\cap V}=0$, which completes the proof in the case when
$\mathcal{V}=\mathcal{O}_{X}$.

Now we need to deal with the case of general $\mathcal{V}$. However we use
hypothesis (4) and apply Lemma 3.7 to the case of $X=U_{z}\cap U$ and
$\mathcal{W}=\mathcal{V}$ to find a $\mathfrak{q}$-stable, affine open
neighborhood $U^{\prime}$ of $z$ on which
$\mathcal{V}|_{U^{\prime}}\cong\mathcal{O}_{X}\otimes\mathcal{V}|_{z}$. Now on
$U^{\prime}$ we have $\mathfrak{q}$-equivariant isomorphisms

$(\mathcal{V}_{Y})|_{U^{\prime}}=(\mathcal{I}_{Y}\otimes_{\mathcal{O}_{X}}\mathcal{V})|_{U^{\prime}}\cong(\mathcal{I}_{Y}|_{U^{\prime}})\otimes_{\Bbbk}\mathcal{V}|_{z}.$

In particular we have a $\mathfrak{q}$-module isomorphism

$\Gamma(U^{\prime},\mathcal{V}_{Y})=\Gamma(U^{\prime},\mathcal{I}_{Y})\otimes_{\Bbbk}\mathcal{V}|_{z}.$

Since $DS_{Q}\Gamma(V,\mathcal{I}_{Y})=0$ for any affine open subvariety
$V\subseteq U^{\prime}$, we have finished the proof of 4.5, and thus also of
4.1. ∎

##  5\. The supergroup $\widetilde{G_{u}}$

In this section, $G$ will denote an affine algebraic supergroup over $\Bbbk$
with Lie superalgebra $\mathfrak{g}=\operatorname{Lie}G$. Then $\Bbbk[G]$ is a
Hopf superalgebra, and we write $\Delta$ for its coproduct map.

The supergroup $G$ acts on itself by conjugation, inducing an action of
$\mathfrak{g}$ on $G$ by vector fields. For $u_{e}\in T_{e}G$, the vector
field induced by this action is:

$u_{ad}=u_{R}+u_{L}=(1\otimes u_{e}-u_{e}\otimes 1)\circ\Delta.$

where $u_{R}=(1\otimes u_{e})\circ\Delta$ and $u_{L}=-(u_{e}\otimes
1)\circ\Delta$ are the infinitesimal right and left translations in the
$u_{e}$ direction. We refer to $u_{ad}$ as the adjoint vector field of
$u=u_{e}$.

This adjoint action of $\mathfrak{g}$ on $\Bbbk[G]$ respects the Hopf
superalgebra structure, i.e. all morphisms coming from the Hopf superalgebra
structure are morphisms of $\mathfrak{g}$-modules. Let
$u\in\mathfrak{g}_{\overline{1}}$ be such that $c=u^{2}$ has a semisimple
adjoint vector field. Then since $DS_{u}$ is a tensor functor,
$\Bbbk[G]_{u}=DS_{u_{ad}}\Bbbk[G]$ is a supercommutative Hopf superalgebra.

######  Definition 5.1.

We write $\widetilde{G_{u}}$ for the algebraic supergroup with
$\Bbbk[\widetilde{G_{u}}]=\Bbbk[G]_{u}$, and $\widetilde{\mathfrak{g}_{u}}$
for its Lie superalgebra.

If $V$ is a $G$-module, then we have morphism $V\to V\otimes\Bbbk[G]$, so when
we take $DS_{u}$ we obtain a morphism $V_{u}\to
V_{u}\otimes\Bbbk[G]_{u}=V_{u}\otimes\Bbbk[\widetilde{G_{u}}]$, which gives
$V_{u}$ the natural structure of a $\widetilde{G_{u}}$-module; in this way
$\widetilde{G_{u}}$ acts on the functor $DS_{u}$.

The supergroup $\widetilde{G_{u}}$ was introduced in [She22], and was computed
in several cases when $G=GL(m|n),Q(n)$, or when $G$ is ’split’, meaning that
$\mathfrak{g}_{\overline{1}}$ is an ideal of $\mathfrak{g}$. However the
computations heavily relied on being able to work in coordinates and perform
explicit computations. We will see that with the localization theorem we can
compute the supergroup $\widetilde{G_{u}}$ for a wide range of $u$. We begin
with the following:

######  Lemma 5.2.

The vanishing set of the adjoint vector field $u_{ad}$ is $C(u)_{0}\subseteq
G_{0}$.

###### Proof.

Let $g\in G_{0}(\Bbbk)$; then by left/right translation invariance, we have
$(u_{R})(g)=L_{g^{-1}}^{*}(u)$, and $(u_{L})(g)=R_{g^{-1}}^{*}(-u)$, where
$L_{g^{-1}}$, resp. $R_{g^{-1}}$ denote the left, resp. right translation by
$g^{-1}$. Therefore $u_{ad}(g)=0$ exactly if

$L_{g^{-1}}^{*}(u)+R_{g^{-1}}^{*}(-u)=0\iff\operatorname{Ad}(g)(u_{e})=u_{e}.$

∎

######  Proposition 5.3.

Let $u\in\mathfrak{g}_{\overline{1}}$ such that $c=u^{2}$ has that $c_{ad}$ is
semisimple. Let $C(c)\subseteq G$ be the centralizer of $c$ in $G$. Then the
map $\Bbbk[G]\to\Bbbk[C(c)]$ induces a natural isomorphism of Hopf
superalgebras

$DS_{u}\Bbbk[G]\cong DS_{u}\Bbbk[C(c)],$

i.e. a natural isomorphism of supergroups
$\widetilde{C(c)_{u}}\cong\widetilde{G_{u}}$.

###### Proof.

This follows from 4.4, using that
$C(c)=\operatorname{Spec}\Bbbk[G]/(\operatorname{Im}c)$. ∎

###  5.1. Computation of $\widetilde{G_{u}}$ for Kac-Moody supergroups,
$Q(n)$, and $P(n)$

As a corollary, we obtain the following table of computations of
$\widetilde{G_{u}}$. For the Kac-Moody supergroups and for $P(n)$, choose a
maximal torus, and let
$\mathfrak{s}\mathfrak{l}(1|1)^{k}\subseteq\mathfrak{g}$ be a subalgebra
corresponding to a choice of odd roots $\alpha_{1},\dots,\alpha_{k}$ such that
$-\alpha_{1},\dots,-\alpha_{k}$ are also roots, and
$\pm\alpha_{i}\pm\alpha_{j}$ is not a root whenever $i\neq j$. Now choose
$u\in\mathfrak{s}\mathfrak{l}(1|1)^{k}$ generic in the sense that
$c=u^{2}=(c_{1},\dots,c_{k})$ is generic, meaning that $c_{i}\neq 0$ for all
$i$ and so that we have, apart from the case of $D(2,1;\alpha)$,

$C(c)=G^{\prime}\times GL(1|1)^{k},$

where $G^{\prime}$ is the group in the first factor of the below list. Such an
element $u$ will be said to have rank $k$.

For $D(2,1;\alpha)$, choose an element $u$ of nonzero square in
$\mathfrak{s}\mathfrak{l}(1|1)$, and let $c=u^{2}$. Then $C(c)$ will be a
supergroup with Lie superalgebra $\mathfrak{s}\mathfrak{l}(1|1)+\mathfrak{t}$,
where $\mathfrak{t}$ is maximal torus; it is computed explicitly in the
appendix.

For $G=Q(n)$, we choose a maximal torus; then its centralizer in $G$ is given
by $Q(1)^{n}$. Choose a factor subgroup $Q(1)^{k}$, and let
$u\in\mathfrak{q}(1)_{\overline{1}}^{k}$ be generic, in the sense that if
$u=(u_{1},\dots,u_{k})$ then $u_{i}\neq 0$ for all $i$, and we have:

$C(c)=Q(n-k)\times Q(1)^{k}.$

Again, such an element $u$ will be said to have rank $k$.

Before giving the following theorem we recall that $\mathbb{G}_{m}$ denotes
the one-dimensional complex algebraic torus, and $\mathbb{G}_{a}^{0|n}$
denotes the purely odd abelian Lie supergroup of dimension $(0|n)$.

######  Theorem 5.4.

For $u\in\mathfrak{g}_{\overline{1}}$ generic of rank $k$ as described above,
we have the following table describing $\widetilde{G_{u}}$:

$\begin{array}[]{|c|c|}\hline\cr G&\tilde{G_{u}}\\\ \hline\cr
GL(m|n)&GL(m-k|n-k)\times\mathbb{G}_{a}^{0|k}\\\ \hline\cr
SOSp(m|2n)&SOSp(m-2k|2n-2k)\times\mathbb{G}_{a}^{0|k}\\\ \hline\cr
D(2,1;\alpha),\alpha\in\mathbb{Q}&\mathbb{G}_{m}\times\mathbb{G}_{a}^{0|1}\\\
\hline\cr D(2,1;\alpha),\alpha\notin\mathbb{Q}&\mathbb{G}_{a}^{0|1}\\\
\hline\cr G(2,1)&SL(2)\times\mathbb{G}_{a}^{0|1}\\\ \hline\cr
AB(1,3)&SL(3)\times\mathbb{G}_{a}^{0|1}\\\ \hline\cr
Q(n)&Q(n-k)\times\mathbb{G}_{a}^{0|k}\\\ \hline\cr
P(n)&P(n-2k)\times\mathbb{G}_{a}^{0|k}\\\ \hline\cr\end{array}$

###### Proof.

For $D(2,1;\alpha)$, the explicit computation is done in appendix. For the
rest of the cases when $G\neq Q(n)$, we have by 5.3 that

$DS_{u}\Bbbk[G]\cong DS_{u}\Bbbk[C(c)]=DS_{u}\Bbbk[G^{\prime}\times
GL(1|1)^{k}]=\Bbbk[G^{\prime}]\otimes DS_{u}\Bbbk[GL(1|1)^{k}].$

Now the computation follows from the fact that for any nonzero
$v\in\mathfrak{g}\mathfrak{l}(1|1)_{\overline{1}}^{hom}$, we have
$\widetilde{GL(1|1)_{v}}=\mathbb{G}_{a}^{0|1}$, and this was proven in
[She22]. We will also provide another proof of this statement in 5.10.

When $G=Q(n)$ we use the same ideas, only now we use that
$\widetilde{Q(1)_{v}}=\mathbb{G}_{a}^{0|1}$ for nonzero
$v\in\mathfrak{q}(1)_{\overline{1}}$, which was proven in [She22] and will
also be shown in 5.10. ∎

####  5.1.1. Realization of the Lie superalgebra

Let $G$ and $u$ be as in the setup of Theorem 5.4, except we exclude now the
case $G=Q(n)$. We give an explicit realization of
$\operatorname{Lie}\widetilde{G_{u}}$ as coming from $\mathfrak{g}$, according
to a natural construction introduced in [She22].

In particular let us assume that $u\in\mathfrak{s}\mathfrak{l}(1|1)^{k}$ takes
the form

$u=e_{\alpha_{1}}+\dots+e_{\alpha_{k}}+c_{1}e_{-\alpha_{1}}+\dots+c_{k}e_{-\alpha_{k}},$

where $c_{1},\dots,c_{k}\in\Bbbk^{\times}$ are generic and
$e_{\pm\alpha_{i}}\in\mathfrak{g}_{\pm\alpha_{i}}$. Let
$\mathfrak{h}^{\prime}=\sum\limits_{i}[\mathfrak{g}_{\alpha_{i}},\mathfrak{g}_{-\alpha_{i}}]\subseteq\mathfrak{g}_{\overline{0}}$;
then $\mathfrak{h}^{\prime}\subseteq[u,\mathfrak{g}]$ and is a toral
subalgebra. By Lemma 3.1 of [She22], if $V$ is any representation on which
$\mathfrak{h}^{\prime}$ acts semisimply, we have a canonical isomorphism
$DS_{u}V\cong DS_{u}V^{\mathfrak{h}^{\prime}}$. Therefore, if we define the
Lie superalgebra

$\mathfrak{g}(u,\mathfrak{h}^{\prime}):=DS_{u}\left(\mathfrak{c}(\mathfrak{h}^{\prime})/\mathfrak{h}^{\prime}\right),$

where
$\mathfrak{c}(\mathfrak{h}^{\prime})=\mathfrak{g}^{\mathfrak{h}^{\prime}}$ is
the centralizer of $\mathfrak{h}^{\prime}$ in $\mathfrak{g}$, we will obtain a
natural action of $\mathfrak{g}(u,\mathfrak{h}^{\prime})$ on
$DS_{u}V^{\mathfrak{h}^{\prime}}$. By [She22], we have

$\mathfrak{g}(u,\mathfrak{h}^{\prime})\cong\mathfrak{g}^{\prime}\times\Bbbk\langle
e_{-\alpha_{1}},\dots,e_{-\alpha_{k}}\rangle,$

where $\mathfrak{g}^{\prime}=\operatorname{Lie}G^{\prime}$, and $G^{\prime}$
is the corresponding first factor in our table from Theorem 5.4. Now
$\mathfrak{g}(u,\mathfrak{h}^{\prime})$ will have a natural map to
$\operatorname{Lie}\widetilde{G_{u}}$, as follows.

We have a natural action of $G\times G$ on $G$ by left and right translation,
and thus a map $\mathfrak{g}\times\mathfrak{g}\to\operatorname{Vect}(G)$.
Recall that $u_{ad}$ is the image of $(u,u)\in\mathfrak{g}\times\mathfrak{g}$
under this map. We see then that
$\mathfrak{h}^{\prime}\times\mathfrak{h}^{\prime}\subseteq[(u,u),\mathfrak{g}\times\mathfrak{g}]$,
and thus we obtain an action of

$DS_{(u,u)}\left(\mathfrak{c}(\mathfrak{h}^{\prime}\times\mathfrak{h}^{\prime})/(\mathfrak{h}^{\prime}\times\mathfrak{h}^{\prime})\right)=\mathfrak{g}(u,\mathfrak{h}^{\prime})\times\mathfrak{g}(u,\mathfrak{h}^{\prime})$

on $\Bbbk[G]$. Further, given
$(v,0)\in\mathfrak{g}(u,\mathfrak{h}^{\prime})\times\mathfrak{g}(u,\mathfrak{h}^{\prime})$,
the action on $\mathbb{C}[G]_{u}$ is given exactly by $(v\otimes
1)\circ\Delta$, where we are using that the coproduct on
$\Bbbk[\widetilde{G}_{u}]=\Bbbk[G]_{u}$ is induced by $\Delta$. From this
formula it is clear that $\mathfrak{g}(u,\mathfrak{h}^{\prime})$ will define
right-invariant vector fields on $\widetilde{G_{u}}$, which gives our desired
map
$\mathfrak{g}(u,\mathfrak{h}^{\prime})\to\operatorname{Lie}\widetilde{G_{u}}$.

######  Theorem 5.5.

For $G\neq D(2,1;\alpha)$, the map
$\mathfrak{g}(u,\mathfrak{h}^{\prime})\to\operatorname{Lie}\widetilde{G_{u}}$
is an isomorphism.

###### Proof.

We trace through the constructions of each space. First of all, by Proposition
5.3, we have an isomorphism of Hopf superalgebras

$DS_{u}\Bbbk[C(c)]\cong\Bbbk[\widetilde{G_{u}}].$

Here $c$ is generic in the sense that it satisfies
$C(c)=C(\mathfrak{h}^{\prime})$. Clearly
$\mathfrak{c}(\mathfrak{h}^{\prime})=\mathfrak{c}(c)$ is the Lie superalgebra
of $C(c)$, and now $C(c)=G^{\prime}\times GL(1|1)^{k}$, where
$u\in\mathfrak{g}\mathfrak{l}(1|1)^{k}$ is as written above. Thus it suffices
to prove the statement in the case when
$\mathfrak{g}=\mathfrak{g}\mathfrak{l}(1|1)$ and $u=e_{\alpha}+ce_{-\alpha}$
for $c\in\Bbbk$; this is done in [She22]. ∎

####  5.1.2. Split supergroups

We present below a special case of the general result on split supergroups
given in [She22]. By a split supergroup $G$ we mean a supergroup for which
$[\mathfrak{g}_{\overline{1}},\mathfrak{g}_{\overline{1}}]=0$.

######  Proposition 5.6.

Suppose that $G$ is a split supergroup, and $u\in\mathfrak{g}_{\overline{1}}$
has a reductive stabilizer $C(u)_{0}$ in $G_{0}$. Then $\widetilde{G_{u}}$ is
the Lie supergroup with $(\widetilde{G_{u}})_{0}=C(u)_{0}$ and Lie
superalgebra $\mathfrak{c}(u)/[u,\mathfrak{g}_{\overline{0}}]$.

###### Proof.

Since $C(u)_{0}$ is reductive and $[u,\mathfrak{g}_{\overline{0}}]$ is a
$C(u)_{0}$-module, we may find a complimentary $C(u)_{0}$-module $V$ so that
$\mathfrak{g}_{\overline{1}}=[u,\mathfrak{g}_{\overline{0}}]\oplus V$. In
particular, we have the subgroup $K$ of $G$ with $K_{0}=C(u)_{0}$ and Lie
superalgebra $\mathfrak{k}=\mathfrak{c}(u)_{\overline{0}}\oplus V$. We apply
now Theorem 4.1 with $Y=K$ and $Z=Y_{0}$; notice that to check condition (3),
we need only look at the identity $e\in K(\Bbbk)$ since we may translate by
$C(u)_{0}$. Now our map on the fiber of the conormal bundle at $e$ is exactly
dual to
$[u,-]:\mathfrak{g}_{\overline{0}}/\mathfrak{c}(u)_{\overline{0}}\to[u,\mathfrak{g}_{\overline{0}}]$,
which is clearly an isomorphism. Thus we have

$DS_{u}\Bbbk[G]=DS_{u}\Bbbk[K]=\Bbbk[K].$

∎

###  5.2. Identifying the odd tangent bundle

We are able to compute several more supergroups $\widetilde{G_{u}}$, but in
order to do so we need another tool. For an affine variety $X_{0}$ we define
the supervariety $\Pi\mathcal{T}_{X_{0}}$, the odd cotangent bundle of
$X_{0}$, to have functions given by $\Omega^{\bullet}_{X_{0}}$, the algebra of
de Rham differentials on $X_{0}$. The supervariety $\Pi\mathcal{T}_{X_{0}}$
admits a canonical odd, everywhere vanishing vector field $d$ given by the de
Rham differential. Observe further that if $U_{0}$ is an affine open
subvariety of $X_{0}$, then the open subvariety it determines of
$\Pi\mathcal{T}_{X_{0}}$ is exactly $\Pi\mathcal{T}_{U_{0}}$.

######  Lemma 5.7.

Let $X$ be a smooth affine supervariety, and let $u$ be an odd, everywhere
vanishing vector field on $X$ such that for all $x\in X(\Bbbk)$ we have:

  1. (1)

$u(T_{x}^{*}X)_{\overline{1}}=0$;

  2. (2)

the map $u:(T^{*}_{x}X)_{\overline{0}}\to(T_{x}^{*}X)_{\overline{1}}$ is an
isomorphism.

Then there exists an isomorphism $X\cong\Pi\mathcal{T}_{X_{0}}$ such that $u$
corresponds to the de Rham differential.

###### Proof.

We define an isomorphism $X\to\Pi\mathcal{T}_{X_{0}}$ locally and prove that
it glues.

First of all, because $X$ is smooth and affine, it is split, so we can split
$\Bbbk[X]\to\Bbbk[X_{0}]$, allowing us to identify (non-canonically)
$\Bbbk[X_{0}]$ as a subalgebra of $\Bbbk[X]$. Thus we do so, i.e. we choose
once and for all such a subalgebra $\Bbbk[X_{0}]\subseteq\Bbbk[X]$.

Suppose that $U_{0}$ is open subset of $X_{0}$ on which $X_{0}$ has
coordinates $x_{1},\dots,x_{n}$. Write $U$ for the corresponding open subset
of $X$. Let $\xi_{i}:=u(x_{i})\in\Bbbk[U]$; then by our assumption,
$\xi_{1},\dots,\xi_{n}$ define a full set of odd coordinates on $U$.

We now define an isomorphism $\varphi_{U}:U\to\Pi\mathcal{T}_{U_{0}}$ on
functions by mapping
$\Bbbk[U_{0}]={\bigwedge}^{0}\Omega_{X}\subseteq\Bbbk[\Pi\mathcal{T}_{U_{0}}]$
identically to our copy of $\Bbbk[U_{0}]$ determined by our global splitting,
and sending $dx_{i}$ to $\xi_{i}$. It is clear that under this local
isomorphism $u$ corresponds to the de Rham differential $d$; in particular, on
$U$ we have $u=\sum\limits\xi_{i}\partial_{x_{i}}$.

Now we need to check these maps glue. However this follows from showing that
the above isomorphism is independent of the coordinates we chose. Thus suppose
on $U_{0}$ we have another set of coordinates $y_{1},\dots,y_{n}$. Then if we
set $\eta_{j}:=u(y_{j})$, we have

$\eta_{j}=u(y_{j})=\sum\limits_{i}\xi_{i}\partial_{x_{i}}(y_{j})=\sum\limits_{i}\varphi_{U}^{*}(dx_{i}\partial_{x_{i}}(y_{j}))=\varphi_{U}^{*}(dy_{j}).$

It follows that $\varphi_{U}$ is indeed independent of the choice of
coordinates; it is only dependent on our choice of splitting
$\Bbbk[X_{0}]\subseteq\Bbbk[X]$, but this was global and fixed from the start.
∎

######  Remark 5.8.

Note that Lemma 5.7 was proven in [Vai96] in greater generality, although they
work in the smooth setting and use different tools.

With Lemma 5.7 in hand we now prove several fun results. For context, we
recall that it was originally Hopf who observed in [Hop64] that the de Rham
cohomology, $H_{dR}^{\bullet}(G)$, of a Lie group $G$ admits the structure of
a graded commutative, cocommutative Hopf algebra, and therefore has spectrum
given by an odd abelian supergroup.

######  Proposition 5.9.

Let $G=Q(n)$, and let

$u=\begin{bmatrix}0&I_{n}\\\ I_{n}&0\end{bmatrix}$

Then $\widetilde{G_{u}}=\mathbb{G}_{a}^{0|n}$.

###### Proof.

In this case the adjoint vector field of $u$ vanishes everywhere, and we have
$[u,\mathfrak{g}_{\overline{0}}]=0$ and
$[u,-]:\mathfrak{g}_{\overline{1}}\to\mathfrak{g}_{\overline{0}}$ is an
isomorphism. Thus Lemma 5.7 applies, and we obtain that

$\Bbbk[Q(n)]_{u}=H_{dR}^{\bullet}(GL(n)).$

∎

######  Proposition 5.10.

  1. (1)

Let $G=GL(n|n)$, and let

$u=\begin{bmatrix}0&I_{n}\\\ \lambda I_{n}&0\end{bmatrix}$

in block form, where $\lambda\in\Bbbk$. Then

$\widetilde{GL(n|n)_{u}}=\mathbb{G}_{a}^{0|n}.$

  2. (2)

Let $G=P(n)$, and let

$u=\begin{bmatrix}0&A\\\ 0&0\end{bmatrix}$

in block form, where $A$ is any invertible symmetric matrix. Then

$\widetilde{P(n)_{u}}=\mu_{2}\times\mathbb{G}_{a}^{0|\lfloor n/2\rfloor}.$

  3. (3)

Let $G=P(2n)$, and let

$u=\begin{bmatrix}0&0\\\ B&0\end{bmatrix}$

in block form, where $B$ is any invertible skew-symmetric matrix. Then

$\widetilde{P(2n)_{u}}=\mathbb{G}_{a}^{0|n}.$

###### Proof.

The proofs of (1), (2), and (3) are identical. Namely in each case we have a
decomposition of our Lie superalgebra
$\mathfrak{g}=\mathfrak{g}_{-1}\oplus\mathfrak{g}_{0}\oplus\mathfrak{g}_{1}$
where we can assume for (2) and (3) that $u\in\mathfrak{g}_{1}$, and for (1)
we write $u=u_{+}+\lambda u_{-}$, where $u_{\pm}\in\mathfrak{g}_{\pm 1}$.

Let $K_{0}\subseteq G_{0}$ be the stabilizer of $u$ in $G_{0}$; then we have
in each case that $[u,\mathfrak{g}_{-1}]=\mathfrak{k}=\operatorname{Lie}K$.
Further in all cases $u$ defines an isomorphism:

$[u,-]:(\mathfrak{g}_{0}\oplus\mathfrak{g}_{-1})/(\mathfrak{k}\oplus\mathfrak{g}_{-1})\to\mathfrak{g}_{1}.$

Let $Y\subseteq G$ be the subgroup with $Y_{0}=K_{0}$ and with Lie
superalgebra $\mathfrak{k}\oplus\mathfrak{g}_{-1}$. Then the hypotheses of
Theorem 4.1 are satisfied with $Z=Y_{0}$, and so we learn that

$DS_{u}\Bbbk[G]=DS_{u}\Bbbk[Y].$

However on $Y$ we have that $u$ is everywhere vanishing and satisfies all the
hypotheses of Lemma 5.7, so that we obtain

$DS_{u}\Bbbk[Y]\cong H_{dR}^{\bullet}(K).$

Thus to finish the proof we see that for (1) we have $K=GL(n)$, for $(2)$ we
have $K=O(n)$, and (3) we have $K=Sp(2n)$. ∎

######  Remark 5.11.

We may realize the Lie superalgebra of $\widetilde{G_{u}}$ for the three cases
presented in 5.10 according to the construction described in Section 5 of
[She22]. Namely, if we set
$\mathcal{U}(\mathfrak{g},\mathfrak{k}):=(\mathcal{U}\mathfrak{g})^{\mathfrak{k}}/(\mathfrak{k}\mathcal{U}\mathfrak{g}+\mathcal{U}\mathfrak{g}\mathfrak{k})^{\mathfrak{k}}$,
then $\operatorname{ad}(u)$ preserves $\mathcal{U}(\mathfrak{g},\mathfrak{k})$
and we obtain a natural map

$\mathcal{U}(\mathfrak{g},\mathfrak{k})_{u}\to\mathcal{U}\operatorname{Lie}\widetilde{G_{u}}.$

Then the map
$\mathcal{U}\mathfrak{g}_{-1}^{\mathfrak{k}}\to\mathcal{U}(\mathfrak{g},\mathfrak{k})$
lands in $\operatorname{ker}(u)$, and in fact maps injectively into
$\mathcal{U}(\mathfrak{g},\mathfrak{k})_{u}$, inducing an isomorphism:

$\mathcal{U}\mathfrak{g}_{-1}^{\mathfrak{k}}\to\mathcal{U}\operatorname{Lie}\widetilde{G_{u}}.$

###  5.3. Determination of $\widetilde{G_{u}}$ for all $u$ when $G=GL(m|n)$ or
$Q(n)$

In the following corollary, we use the notion of rank introduced in Section 3
of [She22].

######  Corollary 5.12.

  1. (1)

Let $u\in\mathfrak{g}\mathfrak{l}(m|n)_{\overline{1}}$ with $u^{2}$
semisimple, and such that $u$ is of rank $r$. Then

$\widetilde{GL(m|n)_{u}}\cong GL(m-r|n-r)\times\mathbb{G}_{a}^{0|r}$

  2. (2)

Let $u\in\mathfrak{q}(n)_{\overline{1}}^{hom}$ with $u^{2}$ semisimple, and
such that $u$ is of rank $r$ and $u=T_{0,B}$, where $B$ is of rank $s$. Then

$\widetilde{Q(n)_{u}}\cong Q(n-r)\times\mathbb{G}_{a}^{0|s}.$

###### Proof.

For both of the cases above, we apply Proposition 5.3. In the $GL$ case,
$C(c)$ will be a product of $GL$’s of smaller rank, say
$G_{1}\times\cdots\times G_{\ell}$, where $G_{i}=GL(a_{i}|b_{i})$; then we can
decompose $u=u_{1}+\dots+u_{\ell}$, and in each case
$u_{i}\in\mathfrak{g}_{i}$ will have that $u_{i}^{2}$ is central in
$\mathfrak{g}_{i}$. Therefore either $u_{i}^{2}=0$ and we can conclude from
the results of [She22], or $a_{i}=b_{i}$ and $u_{i}$ is conjugate to the
matrix given in (1) of Proposition 5.10.

For the $Q$ case, $C(c)$ will be a product of $Q$’s of smaller rank,
$Q(b_{1})\times\cdots\times Q(b_{\ell})$, and we can again decompose
$u=u_{1}+\dots+u_{\ell}$. Then once again either $u_{i}\in\mathfrak{q}(b_{i})$
will satisfy $u_{i}^{2}=0$ allowing us to conclude from [She22], or it will be
a multiple of the matrix in Proposition 5.9. ∎

##  6\. Homogeneous spaces

Recall that a supergroup $G$ is called quasireductive if $G_{0}$ is reductive.
We will always assume in this section that $G$ is quasireductive, and write
$\operatorname{Rep}G$ for the category of $G$-modules. We refer to [Ser11B]
for more on the theory of such supergroups. Write $\mathfrak{g}$ its Lie
superalgebra, which will be reductive, and set

$\mathfrak{g}_{\overline{1}}^{hom}:=\\{u\in\mathfrak{g}_{\overline{1}}:u^{2}\text{
is semisimple}\\}.$

An element $u\in\mathfrak{g}_{\overline{1}}^{hom}$ has that $u^{2}$ acts
semisimply on any $G$-module, and thus the functor $DS_{u}$ is defined on
$\operatorname{Rep}G$.

We will consider homogeneous spaces $G/K$ for $K$ a quasireductive subgroup of
$G$; in this case $G/K$ is a smooth affine supervariety. See [MaT] for more on
the construction of this quotient and proof of affinity of $G/K$.

The Lie superalgebra $\mathfrak{g}$ acts by vector fields on $G/K$ via
infinitesimal left translation; for an element $v\in\mathfrak{g}$ we will, by
abuse of notation, write $v$ for the vector field it defines on $G/K$. We
continue to write $Z(v)$ for the vanishing set of $v$ on $G/K$.

######  Lemma 6.1.

For $v\in\mathfrak{g}$, and $g\in G(\Bbbk)$, $v$ vanishes at the point $gK$ on
$G/K$ if and only if $\operatorname{Ad}(g^{-1})(v)\in\mathfrak{k}$.

###### Proof.

Indeed, the stabilizer of $gK$ is $gKg^{-1}$. ∎

For a homogeneous element $v\in\mathfrak{g}$, we write $N_{G}(v)$ for the
normalizer of $v$ in $G$, and $C_{G}(v)$ for the centralizer of $v$ in $G$.
Notice that for a homogeneous space $G/K$, if $v\in\mathfrak{k}$ then
$N_{G_{0}}(v)\cdot v\subseteq\mathfrak{k}$, and thus we have

$N_{G_{0}}(v)K\subseteq Z(v).$

######  Proposition 6.2.

Let $G$ be quasireductive and $K\subseteq G$ a closed quasireductive subgroup.
Let $u\in\mathfrak{k}\cap\mathfrak{g}_{\overline{1}}^{hom}$ with $c=u^{2}$,
and suppose that

$Z(u)\subseteq N_{G_{0}}(c)K.$

Then

$DS_{u}\Bbbk[G/K]=DS_{u}\Bbbk[N_{G}(c)/(N_{G}(c)\cap K)].$

###### Proof.

Let $Y=N_{G}(c)/(N_{G}(c)\cap K)$; then because $c$ acts with non-zero
eigenvalues on $\mathfrak{g}/\operatorname{Lie}N_{G}(c)$, it will act by an
automorphism on $\mathcal{N}^{\vee}_{Y}|_{y}$ for any $y\in Y(\Bbbk)$. Thus if
$Z(u)\subseteq Y_{0}$, $u$ will act by an automorphism on
$\mathcal{N}^{\vee}_{Y}|_{z}$ for any $z\in Z(u)$, and the hypotheses of
Theorem 4.1 will hold. ∎

######  Remark 6.3.

We remind for future use that if $G_{0}$ is a reductive algebraic group and
$c$ is a semisimple element of its Lie algebra, then $C(c)$, the centralizer
of $c$ in $G_{0}$, is connected.

Before discussing symmetric spaces, we give an example in which condition (4)
holds for any finite-dimensional equivariant vector bundle.

######  Example 6.4.

Consider $G=GL(n|n)$, $K=GL(1|1)\times\cdots\times GL(1|1)$. Let
$u\in\mathfrak{k}_{\overline{1}}$ be generic, i.e. so that if $c:=u^{2}$ then
$N_{G}(c)=K$. Then we have $Z(u)=S_{n}\cdot K$, where $S_{n}\subseteq
GL(n|n)_{0}$ consists of the diagonally embedded permutation matrices. We see
that the hypotheses of 6.2 do not apply, however we can still compute
$DS_{u}\Bbbk[G/K]$ as follows.

Let $Y=Z(u)=S_{n}\cdot K$; i.e. $Y$ is a purely even variety consisting of
$n!$ points. Thus for $\sigma\in S_{n}$, $\mathcal{N}^{\vee}_{Y}|_{\sigma
K}=T_{\sigma K}^{*}(G/K)$; as a $\mathfrak{k}$-module, this cotangent space is
isomorphic to $(\mathfrak{g}/\mathfrak{k})^{\sigma}$, i.e. the $\sigma$ twist
of the $\mathfrak{k}$-module $\mathfrak{g}/\mathfrak{k}$. Thus it is clear
that $c$ acts by an automorphism on these fibers, and therefore $u$ does as
well. Therefore the hypotheses of 4.1 apply, giving:

$DS_{u}\Bbbk[G/K]=\prod\limits_{\sigma\in S_{n}}\Bbbk.$

Now let $V$ be a $K$-module, and write $\mathcal{V}$ for $G\times_{K}V$, i.e.
the $G$-equivariant vector bundle on $G/K$ determined by $V$. We may write
$\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{m}$ for a $\mathfrak{k}$-stable
subspace $\mathfrak{m}$, and consider $W:=\\{A\in\mathfrak{m}:I_{n|n}+A\text{
is invertible}\\}\subseteq GL(n|n)$, so that $W$ is an open,
$\operatorname{Ad}(K)$-stable subset of $\mathfrak{m}$ containing $0$. We have
the map

$\phi:W\to GL(n|n)/K,\ \ \ \ A\mapsto(I_{n|n}+A)K.$

One may check that $\phi$ is $K$-equivariant, injective on closed points, and
has that $d_{0}\phi$ is an isomorphism. Thus we may choose a $K$-stable open
subvariety $U\subseteq\mathfrak{m}$, containing $0$, such that $\phi$
restricts to an open embedding $U\hookrightarrow GL(n|n)/K$. Further, by
Sumihiro’s Theorem and the fact that $K_{0}$ is torus, we may assume that $U$
is affine.

Observe now that if we write $\pi:G\to G/K$ for the canonical projection, then
we obtain a $K$-equivariant isomorphism $\pi^{-1}(\phi(U))\cong K\times U$,
where $K$ acts on the right on $\pi^{-1}(U)$ and $K$, and by the adjoint
action on $U$.

It follows that we have
$\Gamma(U,\mathcal{V})\cong(\Bbbk[K]\otimes\Bbbk[U]\otimes V)^{K}$, which in
particular contains the $K$-submodule $(\Bbbk[K]\otimes\Bbbk\langle
1\rangle\otimes V)^{K}\cong V$. This $K$-submodule of $\Gamma(U,\mathcal{V})$
defines a local, $K$-equivariant trivialization of $\mathcal{V}$. Since $G/K$
is a homogeneous space, we may translate the above trivialization to obtain
similar local trivializations around each point $\sigma K$ for $\sigma\in
S_{n}$. It follows that the hypothesis of 4.1 hold, and so we obtain that

$DS_{u}\Gamma(G/K,\mathcal{V})\cong\bigoplus\limits_{\sigma\in
S_{n}}DS_{u}V^{\sigma^{-1}}.$

Before moving to more general computations on supersymmetric spaces, we give a
specific example which is of interest on its own for the simplicity of its
answer.

######  Example 6.5 (Supersphere).

Let $\mathbb{S}^{m-1|2n}=OSp(m|2n)/OSp(m-1|2n)=G/K$ denote the supersphere.
Explicitly, $G/K$ is the closed subvariety of $\Bbbk^{m|2n}$ cut out by the
equation $q=1$, where $q$ is a nondegenerate, supersymmetric quadratic form
preserved by $OSp(m|2n)$ on $\Bbbk^{m|2n}$. In particular if $m=1$,
$(G/K)_{0}=\mathbb{S}^{0}$ consists of two points.

Note that if $m>1$ then we have $\mathbb{S}^{m-1|2n}=SOSp(m|2n)/SOSp(m-1|2n)$.
Let us assume that $m\geq 3$ and $n\geq 1$.

Choose a copy of $SL(1|1)\subseteq K$ corresponding to an odd isotropic root
of $K$ (with respect to some maximal torus), and let
$u\in\mathfrak{s}\mathfrak{l}(1|1)_{\overline{1}}$ be such that $c=u^{2}\neq
0$. Then there exists an odd isotropic root (with respect to some maximal
torus) of $G$ with corresponding subgroup $SL(1|1)$ such that $u$ is conjugate
to an element in this copy of $SL(1|1)$ in $G$. We show in Section 6.1 that

$Z(u)\subseteq N_{G}(c)_{0}K,$

so we obtain from 6.2 that

$DS_{u}\Bbbk[\mathbb{S}^{m-1|2n}]=DS_{u}\Bbbk[N_{G}(c)/(N_{G}(c)\cap K)].$

Now our copy of $SL(1|1)$ is normalized by $N_{G}(c)$, and so since it is
contained in $K$ it will act trivially on $N_{G}(c)/N_{G}(c)\cap K$, meaning
that in particular $u$ does too, so that

$DS_{u}\Bbbk[\mathbb{S}^{m-1|2n}]=\Bbbk[N_{G}(c)/(N_{G}(c)\cap K)].$

For $g\in N_{G}(c)(\Bbbk)$, we have $\operatorname{Ad}(g)c=\pm c$; if $m>3$
then $K(\Bbbk)$ contains an element $g$ (coming from the Weyl group of $K$)
such that $\operatorname{Ad}(g)c=-c$, and we have in this case that

$N_{G}(c)/(N_{G}(c)\cap K)=C_{G}(c)/(C_{G}(c)\cap K).$

Here $C(c)=SOSp(m-2|2n-2)\times GL(1|1)$ and $C(c)\cap K=SOSp(m-3|2n-2)\times
GL(1|1)$, so $C_{G}(c)/(C_{G}(c)\cap K)=\mathbb{S}^{m-3|2n-2}$. If $m=3$, then
the even part of $N_{G}(c)/(N_{G}(c)\cap K)$ has two points, with one point
being $C_{G}(c)/(C_{G}(c)\cap K)$. From this we obtain that
$N_{G}(c)/(N_{G}(c)\cap K)=OSp(1|2n-2)/Sp(2n)=\mathbb{S}^{m-3|2n-2}$. Thus in
general we have the beautiful formula:

$DS_{u}\Bbbk[\mathbb{S}^{m-1|2n}]=\Bbbk[\mathbb{S}^{m-3|2n-2}].$

###  6.1. Supersymmetric spaces

We apply Theorem 4.1 to several spaces of the form $G/K$ where $K$ is a
symmetric subgroup of $G$. By definition, a symmetric subgroup $K$ is one such
that there exists an involution $\theta$ of $G$ with
$(G^{\theta})^{\circ}\subseteq K\subseteq G^{\theta}$, where $G^{\theta}$ are
the fixed points of $\theta$, and $(G^{\theta})^{\circ}$ is the connected
component of the identity.

Note that the diagonally embedded subgroup $G\subseteq G\times G$ is a
symmetric subgroup, and we have $G\cong(G\times G)/G$, so in fact our study of
$\widehat{G_{u}}$ in Section 5 is a special case of the following examples.

Let $G$ be one of $GL(m|n)$, $SOSp(m|2n)$, $P(n)$, or the simply connected
supergroup of an exceptional basic simple Lie superalgebra. Then we will
consider symmetric subgroups $K$ of $G$ with an element
$u\in\mathfrak{k}_{\overline{1}}^{hom}$ which is generic of minimal rank, and
satisfies the hypothesis of Proposition 6.2, namely that

$Z(u)\subseteq N_{G}(c)_{0}K.\ \ \ \ \ (*)$

This condition will hold exactly in the cases for which $\mathfrak{k}$ admits
at most one factor subalgebra $\mathfrak{k}^{\prime}$ which has that
$(\mathfrak{k}^{\prime}_{\overline{1}})^{hom}\neq 0$. One can easily go
through the classification of supersymmetric pairs, found in [Ser83], to check
which ones these are.

Let us explain how we compute in each case. First, choose a maximal torus
$\mathfrak{t}^{\prime}\subseteq\mathfrak{k}_{\overline{0}}$ and extend it to a
maximal torus $\mathfrak{t}$ of $\mathfrak{g}_{\overline{0}}$. For the cases
when $\mathfrak{k}\neq\mathfrak{q}(n)$, choose a root subalgebra
$\mathfrak{s}\mathfrak{l}(1|1)\subseteq\mathfrak{k}$, and let
$u\in\mathfrak{s}\mathfrak{l}(1|1)_{\overline{1}}$ such that $c=u^{2}\neq 0$.
First suppose that we have

$\operatorname{Lie}C_{G}(c)\cong\mathfrak{g}\mathfrak{l}(1|1)\times\mathfrak{g}_{u}.$

The symmetric pairs which will satisfy this along with (*) are:

$(\mathfrak{g}\mathfrak{l}(m|n),\mathfrak{g}\mathfrak{l}(m|n-r)\times\mathfrak{g}\mathfrak{l}(r)),\
\
(\mathfrak{o}\mathfrak{s}\mathfrak{p}(m|2n),\mathfrak{o}\mathfrak{s}\mathfrak{p}(m|2n-2r)\times\mathfrak{s}\mathfrak{p}(2r)),$
$(\mathfrak{o}\mathfrak{s}\mathfrak{p}(m|2n),\mathfrak{o}\mathfrak{s}\mathfrak{p}(m-1|2n-2r)\times\mathfrak{o}\mathfrak{s}\mathfrak{p}(1|2r)),\
\
(\mathfrak{o}\mathfrak{s}\mathfrak{p}(m|2n),\mathfrak{o}\mathfrak{s}\mathfrak{p}(m-r|2n)\times\mathfrak{o}(r)),$
$(\mathfrak{o}\mathfrak{s}\mathfrak{p}(2m|2n),\mathfrak{g}\mathfrak{l}(m|n)),\
\ (\mathfrak{p}(n),\mathfrak{g}\mathfrak{l}(r|n-r)),$
$(\mathfrak{d}(1|2;\alpha),\mathfrak{o}\mathfrak{s}\mathfrak{p}(2|2)\times\mathfrak{s}\mathfrak{o}(2)),\
\
(\mathfrak{a}\mathfrak{b}(1|3),\mathfrak{g}\mathfrak{o}\mathfrak{s}\mathfrak{p}(2|4)),\
\ (\mathfrak{a}\mathfrak{b}(1,3),\mathfrak{s}\mathfrak{l}(1|4)),$
$(\mathfrak{a}\mathfrak{b}(1|3),\mathfrak{d}(1|2,2)\times\mathfrak{s}\mathfrak{l}(2)),\
\ (\mathfrak{g}(1|2),\mathfrak{d}(1|2;3)),\ \
(\mathfrak{g}(1|2),\mathfrak{o}\mathfrak{s}\mathfrak{p}(3|2)\times\mathfrak{s}\mathfrak{l}(2)).$

In all of the above cases, an element $c$ with this centralizer (up to
isomorphism) is unique in $\mathfrak{t}^{\prime}$ up to: adding a central
element in $\mathfrak{g}$, acting by the Weyl group of $\mathfrak{t}^{\prime}$
in $K$, and scaling. Further, either we have $N_{G}(c)=C_{G}(c)$ or
$N_{G}(c)/C_{G}(c)\cong\mu_{2}$, and the action of $\mu_{2}$ on $c$ is
$c\mapsto-c$.

We now show that (*) holds for the pairs listed above. Suppose that in one of
the listed cases we have $u$ vanishes at $gK$; then in particular $c$ vanishes
here, and we have $\operatorname{Ad}(g^{-1})(c)\in\mathfrak{k}$. Choose $k\in
K_{0}$ such that
$c^{\prime}=\operatorname{Ad}(gk)^{-1}(c)\in\mathfrak{t}^{\prime}$. Clearly we
have
$\operatorname{Lie}C(c^{\prime})\cong\mathfrak{g}\mathfrak{l}(1|1)\times\mathfrak{g}_{u}$.
Now to obtain $c$ from $c^{\prime}$, the only case where we could add a
central element in $\mathfrak{g}$ is
$\mathfrak{g}=\mathfrak{g}\mathfrak{l}(m|n)$, and by rank considerations we
see that this couldn’t happen. Thus there exists $w\in K_{0}$, corresponding
to a lift of a Weyl group element, such that $w^{-1}c^{\prime}$ is a scalar
multiple of $c$. Therefore $gkw\in N_{G}(c)_{0}$, showing that the condition
(*) indeed holds. Therefore we obtain by 6.2 for the above cases that

$DS_{u}\Bbbk[G/K]=DS_{u}\Bbbk[N_{G}(c)/(N_{G}(c)\cap K)].$

Further, in all of the above cases we have that our copy of $SL(1|1)$ lies in
$K$ and is normalized by $N_{G}(c)$, so that $u$ will in fact act trivially on
$N_{G}(c)/N_{G}(c)\cap K$; it follows that in the above cases we have:

$DS_{u}\Bbbk[G/K]=\Bbbk[N_{G}(c)/(N_{G}(c)\cap K)],$

and from this we obtain our explicit results given in the table below.

We now consider three other cases:

$(\mathfrak{g}\mathfrak{l}(m|2n),\mathfrak{o}\mathfrak{s}\mathfrak{p}(m|2n)),\
\ \ (\mathfrak{g}\mathfrak{l}(n|n),\mathfrak{p}(n)),\ \ \
(\mathfrak{g}\mathfrak{l}(n|n),\mathfrak{q}(n)).$

In the first two cases, we again choose a root subalgebra
$\mathfrak{s}\mathfrak{l}(1|1)\subseteq\mathfrak{k}$, and let
$u\in\mathfrak{s}\mathfrak{l}(1|1)_{\overline{1}}$ with $c=u^{2}\neq 0$ . Then
we have that

$C_{G}(c)=GL(1|1)\times GL(1|1)\times G^{\prime}.$

where $G^{\prime}=GL(m-2|2n-2)$ for the first pair, and
$G^{\prime}=GL(n-2|n-2)$ for the second pair. For
$(\mathfrak{g}\mathfrak{l}(n|n),\mathfrak{q}(n))$ we choose a copy of
$\mathfrak{q}(1)\subseteq\mathfrak{q}(n)$ which is a factor subalgebra of a
Cartan subalgebra, and let $u\in\mathfrak{q}(1)_{\overline{1}}$ such that
$c=u^{2}\neq 0$. Then we have that

$C_{G}(u)=GL(1|1)\times GL(n-1|n-1).$

For the above three cases, any element $c\in\mathfrak{t}^{\prime}$ with such a
centralizer is unique up to action of the Weyl group of $K$ on
$\mathfrak{t}^{\prime}$, scaling, and multiples of the center. Using the same
argument as above, we find that the vanishing set of $u$ is contained in
$N_{G_{0}}(c)K$, and thus

$DS_{u}\Bbbk[G/K]=DS_{u}\Bbbk[N_{G}(c)/(N_{G}(c)\cap K)].$

Now in all the three above cases we in fact have $N_{G}(c)/(N_{G}(c)\cap
K)=C_{G}(c)/(C_{G}(c)\cap K)$; we thus obtain the following:

$\displaystyle DS_{u}\Bbbk[GL(m|2n)/OSp(m|2n)]$ $\displaystyle=$
$\displaystyle\Bbbk[GL(m-2|2n-2)/OSp(m-2|2n-2)]$ $\displaystyle\otimes$
$\displaystyle DS_{u}\Bbbk[GL(1|1)^{2}/GL(1|1)],$ $\displaystyle
DS_{u}\Bbbk[GL(n|n)/P(n)]$ $\displaystyle=$
$\displaystyle\Bbbk[GL(n-2|n-2)/P(n-2)]$ $\displaystyle\otimes$ $\displaystyle
DS_{u}\Bbbk[GL(1|1)^{2}/GL(1|1)],$

and

$DS_{u}\Bbbk[GL(n|n)/Q(n)]=\Bbbk[GL(n-1|n-1)/Q(n-1)]\otimes
DS_{u}\Bbbk[GL(1|1)/Q(1)].$

Now one may compute explicitly that
$DS_{u}\Bbbk[GL(1|1)^{2}/GL(1|1)]\cong\mathbb{A}^{0|1}$, where
$\mathbb{A}^{0|1}=\operatorname{Spec}\Bbbk[\xi]$ for some odd variable $\xi$,
and $DS_{u}\Bbbk[GL(1|1)/Q(1)]\cong\Bbbk[x]/(x^{2}=1)=\Bbbk[\mathbb{S}^{0}]$.

In the last row of the above table, $z$ is the element of $D(1,2;2)\times
SL(2)$ given by $(x,-1)$, where $-1\in SL_{2}$, and $x\in D(1,2;2)_{0}$ is the
order 2 central element corresponding to $-3$. As a matter of explanation,
$D(2,1;\alpha)$ has a center which is given by a Klein-4 subgroup, where the
nontrivial elements are naturally indexed by $\alpha,1,$ and $-\alpha-1$.

######  Remark 6.6.

For a smooth affine $G$-supervariety $X$, write $D^{G}(X)$ for the algebra of
$G$-equivariant differential operators on $X$. Then if
$u\in\mathfrak{g}_{\overline{1}}^{hom}$ satisfies the hypothesis of 4.3, $Y$
will have a natural action by $\widetilde{G_{u}}$, and we will obtain a
natural algebra morphism:

$D^{G}(X)\to D^{\widetilde{G_{u}}}(Y).$

This map would be especially interesting to study in one of the cases above,
when $G/K$ is a supersymmetric space.

######  Remark 6.7.

To see why the condition (*) is important, consider the supersymmetric space
$G/K=GL(3|3)/(GL(1|1)\times GL(2|2))$, and let
$u\in\mathfrak{g}\mathfrak{l}(1|1)\subseteq\mathfrak{k}$ be a generic rank one
element lying in the $\mathfrak{g}\mathfrak{l}(1|1)_{\overline{1}}$ factor of
$\mathfrak{k}$. Then there is an element $g$ of $G(\Bbbk)$, coming from the
Weyl group, which has that $\operatorname{Ad}(g^{-1})(u)$ lies in the
$\mathfrak{g}\mathfrak{l}(2|2)$ factor of $\mathfrak{k}$. Thus $gK\in Z(u)$;
however one can check that $u$ does not satisfy condition (3) at $g$.

###  6.2. Quotients by Levi subgroups of $GL(n|n)$, $P(n)$, and $Q(n)$

######  Theorem 6.8.

Let $u=\begin{bmatrix}0&I_{n}\\\ \lambda
I_{n}&0\end{bmatrix}\in\mathfrak{g}\mathfrak{l}(n|n)$ where $\lambda\in\Bbbk$,
and consider the subgroup

$K=GL(r_{1}|r_{1})\times\cdots\times GL(r_{k}|r_{k}),$

where $\sum\limits_{i}r_{i}=n$. Then we have an isomorphism

$\displaystyle DS_{u}[GL(n|n)/GL(r_{1}|r_{1})\times\cdots\times
GL(r_{k}|r_{k})]$ $\displaystyle\cong$ $\displaystyle
H_{dR}^{\bullet}(GL(n)/GL(r_{1})\times\cdots\times GL(r_{k})),$

where $H_{dR}^{\bullet}(-)$ denotes the de Rham cohomology.

###### Proof.

Let $H\subseteq GL(n|n)$ be the subgroup with
$\mathfrak{h}=\mathfrak{g}_{-1}\oplus\mathfrak{g}_{0}$ and $H_{0}\cong GL(n)$
embedded in $GL(n|n)$ as the matrices of the form

$\begin{bmatrix}A&0\\\ 0&A\end{bmatrix}$

for $A\in GL(n)$. Let $Y=H/(K\cap H)$; then we claim that the vanishing set of
$u$ is exactly $Y_{0}$. Indeed, suppose that $u$ vanishes at $g^{-1}K$, i.e.
$\operatorname{Ad}(g)(u)\in\mathfrak{k}$, and write $g=(g_{1},g_{2})\in
GL(n)\times GL(n)$. Then we can write $g=(k,I_{n})(g_{2},g_{2})$ where
$k=g_{1}g_{2}^{-1}$, and now $(g_{2},g_{2})\in H_{0}$. Observe that

$\operatorname{Ad}(k,I_{n})(u)=\begin{bmatrix}0&k\\\ \lambda
k^{-1}&0\end{bmatrix}.$

Thus we must have $k\in GL(r_{1})\times\cdots\times GL(r_{k})$ so that
$(k,I_{n})\in K$. Therefore $g^{-1}K=h^{-1}K$, where $h=(g_{2},g_{2})$. Now
from the 4.1, we obtain that

$DS_{u}\Bbbk[GL(n|n)/K]=DS_{u}\Bbbk[Y].$

We finish by observing that $u$ defines an everywhere vanishing vector field
on $Y$, and satisfies the hypotheses of 5.7. Since
$Y_{0}=GL(n)/GL(r_{1})\times\cdots\times GL(r_{k})$, this completes the proof.
∎

######  Theorem 6.9.

Let $G=P(n)$, $K=P(r_{1})\times\cdots\times P(r_{k})$, where
$\sum\limits_{i}r_{i}=n$. Let

$u=\begin{bmatrix}0&I_{n}\\\
0&0\end{bmatrix}\in\mathfrak{p}(n)_{\overline{1}};$

if $n$ is even, then further set

$v=\begin{bmatrix}0&0\\\ A&0\end{bmatrix}\in\mathfrak{p}(n)_{\overline{1}},$

where $A$ is an invertible skew-symmetric matrix. Then

$DS_{u}\Bbbk[G/K]\cong H^{\bullet}_{dR}(O(n)/O(r_{1})\times\cdots\times
O(r_{k})),$

and when $n,r_{1},\dots,r_{k}$ are all even,

$DS_{v}\Bbbk[G/K]\cong H^{\bullet}_{dR}(Sp(n)/Sp(r_{1})\times\cdots\times
Sp(r_{k})).$

###### Proof.

In the first case, consider the subgroup $H\subseteq G$ with
$H_{0}=O(n)\subseteq GL(n)$, and
$\mathfrak{h}_{\overline{1}}=\mathfrak{g}_{-1}$. Set $Y=H/H\cap K\subseteq
G/K$. In the second case, consider the subgroup $H^{\prime}\subseteq G$ with
$H^{\prime}_{0}=Sp(\Bbbk^{n},A)\subseteq GL(n)$, the matrices preserving $A$,
and $\mathfrak{h}_{\overline{1}}=\mathfrak{g}_{1}$. Then set $Y=H/H\cap
K\subseteq G/K$. Now using the equivalence of all symmetric, resp. alternating
nondegenerate bilinear forms on $\Bbbk^{n}$, we can prove that the vanishing
set of $u$, resp. $v$ in $G/K$ is given by $Y_{0}$. From here, we use that
$[u,-]:\mathfrak{h}_{\overline{1}}\to\mathfrak{h}_{\overline{0}}$, resp.
$[v,-]:\mathfrak{h}^{\prime}_{\overline{1}}\to\mathfrak{h}^{\prime}_{\overline{0}}$
are isomorphisms, and Lemma 5.7. ∎

######  Theorem 6.10.

Consider the element in $\mathfrak{q}(n)$ given by:

$u=\begin{bmatrix}0&I_{n}\\\ I_{n}&0\end{bmatrix}.$

Let $r_{1},\dots,r_{k}$ be positive integers with $\sum\limits r_{i}=n$. Then
we have an isomorphism of algebras:

###### Proof.

We check the hypotheses of Lemma 5.7; clearly $u^{2}$ acts trivially, and it
commutes with $Q(n)_{0}$. Since it normalizes
$\mathfrak{k}=\mathfrak{q}(r_{1})\times\dots\times\mathfrak{q}(r_{k})$, and
defines an isomorphism
$\mathfrak{q}(n)_{\overline{1}}\to\mathfrak{q}(n)_{\overline{0}}$, the map

$(\mathfrak{q}(n)/\mathfrak{k})_{\overline{1}}\to(\mathfrak{q}(n)/\mathfrak{k})_{\overline{0}}$

is an isomorphism. ∎

##  7\. Appendix: A root subgroup of $D(2,1;\alpha)$

Consider the subgroup $H$ of $D(2,1;\alpha)$ with Lie superalgebra given by
$\mathfrak{g}_{\beta}\oplus\mathfrak{g}_{-\beta}\oplus\mathfrak{t}$, where
$\mathfrak{t}$ is a maximal torus and $\beta$ is an isotropic odd root. Its
Hopf superalgebra structure is given as follows: it has functions
$\Bbbk[x_{1}^{\pm 1},x_{2}^{\pm 1},x_{3}^{\pm 1},\xi,\eta]$, where the
coproduct $\Delta$ is given by:

$\Delta(x_{i})=x_{i}\otimes x_{i}+c_{i}x_{i}\eta\otimes x_{i}\xi,\ \ \
\Delta(\xi)=\xi\otimes 1+x_{1}x_{2}x_{3}\otimes\xi,\ \ \
\Delta(\eta)=\eta\otimes x_{1}x_{2}x_{3}+1\otimes\eta.$

where $c_{1}+c_{2}+c_{3}=0$, and explicitly we have $c_{1}=\alpha$, $c_{2}=1$,
$c_{3}=-1-\alpha$.

Let $x\in\mathfrak{g}_{\beta}$ correspond to the derivation $\partial_{\xi}$
at the identity, and $y\in\mathfrak{g}_{-\beta}$ correspond to the derivation
$\partial_{\eta}$ at the identity. Then we obtain that the adjoint vector
field of $u=x+y$ is given by:

$(1-x_{1}x_{2}x_{3})(\partial_{\eta}-\partial_{\xi})-(\xi+\eta)\sum\limits_{i}c_{i}x_{i}\partial_{x_{i}}.$

We can compute its cohomology on $\Bbbk[H]$ as follows: set $\zeta=\xi+\eta$
and $\gamma=(\eta-\xi)/2$. Then we may rewrite the above as

$D_{1}+D_{2}=(1-x_{1}x_{2}x_{3})\partial_{\gamma}-\zeta\sum\limits_{i}c_{i}x_{i}\partial_{x_{i}}.$

We obtain $[D_{1},D_{2}]=0,$ so we can view this as giving a spectral
sequence. Cohomology with respect to $D_{1}$ will give a Koszul complex, and
taking its cohomology we obtain:

$\Bbbk[x_{1}^{\pm},x_{2}^{\pm},x_{3}^{\pm},\zeta]/(1-x_{1}x_{2}x_{3}).$

The spectral sequence collapses on the second page, and since it is
concentrated along a single column, it will in fact exactly compute the
cohomology for us.

Thus it remains to compute the cohomology of $D_{2}$; from the relation, we
can forget about $x_{3}$, and so we obtain
$\Bbbk[x_{1}^{\pm},x_{2}^{\pm},\zeta]$ with operator
$D_{2}=\zeta(c_{1}x_{1}\partial_{x_{2}}+c_{2}x_{2}\partial_{x_{2}})$. Recall
that $c_{1}=\alpha$ and $c_{2}=1$. Thus if $\alpha\notin\mathbb{Q}$, we obtain
that the cohomology is $\Bbbk[\zeta]$, i.e. we have
$\widetilde{H_{u}}\cong\mathbb{G}^{0|1}$.

On the other hand, suppose that $\alpha=m/n\in\mathbb{Q}$ in reduced form.
Then the cohomology is given by $\Bbbk[z^{\pm 1},\zeta]$, where
$z=x_{1}^{n}x_{2}^{-m}$. One can check that the Hopf algebra structure gives
$\widetilde{H_{u}}\cong\mathbb{G}_{m}\times\mathbb{G}^{0|1}$.

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Vera Serganova, Dept. of Mathematics, University of California at Berkeley,
Berkeley, CA 94720

Email address<EMAIL_ADDRESS>
Alexander Sherman, Dept. of Mathematics, Ben Gurion University, Beer-Sheva,
Israel

Email address<EMAIL_ADDRESS>