# Finite $W$-superalgebras and the dimensional lower bounds for the
representations of basic Lie superalgebras

Yang Zeng and Bin Shu  School of Science, Nanjing Audit University, Nanjing,
Jiangsu Province 211815, China<EMAIL_ADDRESS>Department of Mathematics,
East China Normal University, Shanghai 200241, China<EMAIL_ADDRESS>
###### Abstract.

In this paper we formulate a conjecture about the minimal dimensional
representations of the finite $W$-superalgebra
$U(\mathfrak{g}_{\mathbb{C}},e)$ over the field of complex numbers and
demonstrate it with examples including all the cases of type $A$. Under the
assumption of this conjecture, we show that the lower bounds of dimensions in
the modular representations of basic Lie superalgebras are attainable. Such
lower bounds, as a super-version of Kac-Weisfeiler conjecture, were formulated
by Wang-Zhao in [35] for the modular representations of a basic Lie
superalgebra ${\mathfrak{g}}_{{{\mathds{k}}}}$ over an algebraically closed
field ${\mathds{k}}$ of positive characteristic $p$.

Mathematics Subject Classification (2000 revision) Primary 17B35. Secondary
17B81. This work is supported partially by the NSF of China (No. 11271130;
11201293; 111126062), the Innovation Program of Shanghai Municipal Education
Commission (No. 12zz038).

###### Contents

  1. 1 Basic Lie superalgebras and finite $W$-superalgebras
  2. 2 Transition subalgebras of finite $W$-superalgebras in prime characteristic
  3. 3 Conjectural one-dimensional representations for finite $W$-superalgebras when $d_{1}$ is even
  4. 4 Conjectural two-dimensional representations for finite $W$-superalgebras when $d_{1}$ is odd
  5. 5 The realization of minimal-dimensional representations for reduced enveloping algebra $U_{\xi}({\mathfrak{g}}_{{\mathds{k}}})$

## Introduction

This paper is a sequel to [40]. On the basis of the structure theory of finite
$W$-superalgebras developed there, we study the modular representations of
basic Lie superalgebras, as a remarkable application of finite
$W$-superalgebras.

### 0.1.

A finite $W$-algebra $U(\mathfrak{g},e)$ is a certain associative algebra
associated to a complex semisimple Lie algebra $\mathfrak{g}$ and a nilpotent
element $e\in{\mathfrak{g}}$. The study of finite $W$-algebras can be traced
back to Kostant’s work in the case when $e$ is regular [15], then the further
study was done by Lynch in the case when $e$ is an arbitrary even nilpotent
element (cf. [19]). Premet developed the finite $W$-algebras in full
generality in [27]. On his way of proving the celebrated Kac-Weisfeiler
conjecture for Lie algebras of reductive groups in [26], Premet first
constructed the modular version of finite $W$-algebras (they will be called
the reduced $W$-algebras in this paper). By means of a complicated but natural
“admissible” procedure, the finite $W$-algebras over the field of complex
numbers were introduced in [27], which showed that they are filtrated
deformations of the coordinate rings of Slodowy slices.

Aside from the advances in finite $W$-algebras over the field of complex
numbers, the modular theory of finite $W$-algebras is also developed
excitingly. It is remarkable that in [29] Premet proved that if the
$\mathbb{C}$-algebra $U(\mathfrak{g},e)$ has a one-dimensional representation,
then under the assumption $p\gg 0$ for the positive characteristic field
${{\mathds{k}}}=\overline{\mathbb{F}}_{p}$, the reduced enveloping algebra
$U_{\chi}(\mathfrak{g}_{\mathds{k}})$ of the modular counterpart
$\mathfrak{g}_{\mathds{k}}$ of $\mathfrak{g}$ possesses an irreducible module
of dimension $d(e)$ (where $\chi$ is the linear function on
$\mathfrak{g}_{{{\mathds{k}}}}$ corresponding to $e$, and $d(e)$ is half of
the dimension of the orbit $G_{\mathds{k}}\cdot\chi$ for the simple, simply-
connected algebraic group $G_{\mathds{k}}$ with
$\mathfrak{g}_{\mathds{k}}=\text{Lie}(G_{\mathds{k}})$), which is a lower
bound predicted by Kac-Weisfeiler conjecture mentioned above.

The existence of one-dimensional representations for $U(\mathfrak{g},e)$
associated to a classical Lie algebra over $\mathbb{C}$ was obtained by Losev
in [17, Theorem 1.2.3(1)] (see also [16, §6]). Goodwin-Röhrle-Ubly [8] proved
that the $W$-algebras associated to exceptional Lie algebras
$E_{6},E_{7},F_{4},G_{2}$, or $E_{8}$ with $e$ not rigid, admit one-
dimensional representations (see also [29]).

### 0.2.

The theory of finite $W$-superalgebras was developed in the same time. In the
work of De Sole and Kac [34], finite $W$-superalgebras were defined in terms
of BRST cohomology under the background of vertex algebras and quantum
reduction. The topics on finite $W$-superalgebras attracted many researchers
(cf. [2], [21], [22], [23], [24], [25], [36] and [43]).

In the work of Wang-Zhao [35], they initiated the study of modular
representations of basic Lie superalgebras over an algebraically closed field
of positive characteristic, formulating the super Kac-Weisfeiler property for
those Lie superalgebras as well as presenting the definition of modular
$W$-superalgebras.

### 0.3.

Based on Premet’s and Wang-Zhao’s work as mentioned above, our previous paper
[40] presents the PBW structure theorem for the finite $W$-superalgebras
(along with the reduced $W$-superalgebras), which shows that the construction
of finite $W$-superalgebras (also the reduced $W$-superalgebras) can be
divided into two cases in virtue of the parity of
$\text{dim}\,\mathfrak{g_{\mathbb{F}}}(-1)_{\bar{1}}$ (where $\mathbb{F}$ is
an algebraically closed field of any characteristic). To some extent, the
situation of finite $W$-superalgebras is significantly different from that of
finite $W$-algebras.

Explicitly speaking, let
${\mathfrak{g}}={\mathfrak{g}}_{\bar{0}}+{\mathfrak{g}}_{\bar{1}}$ be a basic
Lie superalgebra over $\mathbb{C}$ excluding type $D(2,1;a)$ ($a\in\mathbb{C}$
is not an algebraic number), and $e\in{\mathfrak{g}}_{\bar{0}}$ a nilpotent
element. Fix an $\mathfrak{sl}_{2}$-triple $f,h,e$, and denote by
${\mathfrak{g}}^{e}:=\text{Ker}(\text{ad}\,e)$ in ${\mathfrak{g}}$. The linear
operator ad $h$ defines a ${\mathbb{Z}}$-grading
${\mathfrak{g}}=\bigoplus\limits_{i\in{\mathbb{Z}}}{\mathfrak{g}}(i)$. Define
the Kazhdan degree on ${\mathfrak{g}}$ by declaring that
$x\in{\mathfrak{g}}(j)$ is of $(j+2)$. A finite $W$-superalgebra is defined by

$U({\mathfrak{g}},e)=(\text{End}_{\mathfrak{g}}Q_{\chi})^{\text{op}},$

where $Q_{\chi}$ is the generalized Gelfand-Graev ${\mathfrak{g}}$-module
associated to $e$. In [40] we showed that

######  Theorem 0.1.

([40]) Under the Kazhdan grading, we have

  * (1)

$\text{gr}\,U(\mathfrak{g},e)\cong S(\mathfrak{g}^{e})$ as
$\mathbb{C}$-algebras when $\dim\mathfrak{g}(-1)_{\bar{1}}$ is even;

  * (2)

$\text{gr}\,U(\mathfrak{g},e)\cong
S(\mathfrak{g}^{e})\otimes\mathbb{C}[\Theta]$ as vector spaces over
$\mathbb{C}$ when $\dim\mathfrak{g}(-1)_{\bar{1}}$ is odd,

where $\mathbb{C}[\Theta]$ is the exterior algebra generated by one element
$\Theta$.

### 0.4.

The main purpose of this paper is to further develop the construction and
representation theory of finite $W$-superalgebras both over the field of
complex numbers and over a field in prime characteristic. The most important
part is the accessibility of lower bounds in the super Kac-Weisfeiler
property. Our approach is roughly generalizing the “reduction modulo $p$”
method introduced by Premet for the finite $W$-algebra case in [29], with
careful analysis and examination on the variation of structural features
arising from the parity of $\dim\mathfrak{g}(-1)_{\bar{1}}$. Let us explain it
roughly as below.

Let ${\mathfrak{g}}_{{\mathds{k}}}$ and $Q_{\chi,{{\mathds{k}}}}$ be the
modular counterparts of $\mathfrak{g}$, and of the generalized Gelfand-Graev
$\mathfrak{g}$-module $Q_{\chi}$ respectively. To ease notation, we will
identify the nilpotent element $e\in\mathfrak{g}$ over $\mathbb{C}$ with the
element $\bar{e}=e\otimes 1$ in ${\mathfrak{g}}_{{\mathds{k}}}$ by “reduction
modulo $p$” in the following. Define the finite $W$-superalgebra over
${{\mathds{k}}}$ by

$U({\mathfrak{g}}_{{\mathds{k}}},e):=(\text{End}_{{\mathfrak{g}}_{{\mathds{k}}}}Q_{\chi,{{\mathds{k}}}})^{\text{op}}.$

Let $T({\mathfrak{g}}_{{\mathds{k}}},e)$ be the transition subalgebra of
$U({\mathfrak{g}}_{{\mathds{k}}},e)$ which is derived from the
${\mathbb{C}}$-algebra $U({\mathfrak{g}},e)$ by “reduction modulo $p$”. In our
arguments, the transition subalgebras will play some medium role between the
theory of finite $W$-superalgebras over $\mathbb{C}$ and that of reduced
enveloping algebras over ${\mathds{k}}$. So in the first part of the paper, we
will investigate the structure of the transition subalgebra
$T(\mathfrak{g}_{\mathds{k}},e)$ and the finite $W$-superalgebra
$U(\mathfrak{g}_{\mathds{k}},e)$ over ${\mathds{k}}$. Explicitly speaking, for
any real number $a\in\mathbb{R}$, let $\lceil a\rceil$ denote the largest
integer lower bound of $a$, and $\lfloor a\rfloor$ the least integer upper
bound of $a$. It is notable that these notations also work for the numbers in
the prime field $\mathbb{F}_{p}$. Set
$d_{i}:=\text{dim}{\mathfrak{g}}_{i}-\text{dim}{\mathfrak{g}}^{e}_{i}$ for
$i\in\mathbb{Z}_{2}=\\{{\bar{0}},{\bar{1}}\\}$, then we have

######  Theorem 0.2.

There is a subspace $\mathfrak{a}_{\mathds{k}}$ of $\mathfrak{g}_{\mathds{k}}$
with
$\underline{\dim}\,\mathfrak{a}_{\mathds{k}}=({\frac{d_{0}}{2}}|{\lceil\frac{d_{1}}{2}\rceil})$
such that $U({\mathfrak{g}}_{{\mathds{k}}},e)\cong
T({\mathfrak{g}}_{{\mathds{k}}},e)\otimes_{{\mathds{k}}}Z_{p}(\mathfrak{a}_{{\mathds{k}}})$
as ${{\mathds{k}}}$-algebras, where $Z_{p}(\mathfrak{a}_{\mathds{k}})$ is the
$p$-center as usually defined, with respect to the subspace
$\mathfrak{a}_{\mathds{k}}$.

In the second part, we exploit some remarkable applications of finite
$W$-superalgebras to the modular representations of basic Lie superalgebras.
We provide a super version of Premet’s work, as afore-mentioned, on the
accessibility of lower-bounds of dimensions in the modular representations of
reductive Lie algebras predicted by Kac-Weisfeiler conjecture. For this, we
will formulate a conjecture about the “small representations” of
$U({\mathfrak{g}},e)$ over $\mathbb{C}$ (in the paper, we call a
representation of an algebra “small” if its dimension is minimal among all the
representations of this algebra). By [40, Remark 2.7] we know that $d_{1}$ has
the same parity with $\text{dim}\,{\mathfrak{g}}(-1)_{\bar{1}}$. As seen
before, the variation of the parity of $d_{1}$ gives rise to the change of the
structure of finite $W$-superalgebras, and we will see furthermore, also of
the representations of finite $W$-superalgebras. Firstly, we provide the
following highly-plausible conjecture, generalizing a conjecture proposed by
Premet on the representations of finite $W$-algebras which has been already
confirmed (cf. [29] and [31]).

######  Conjecture 0.3.

Let ${\mathfrak{g}}$ be a basic Lie superalgebra over ${\mathbb{C}}$. Then the
following statements hold:

  * (1)

when $d_{1}$ is even, the finite $W$-superalgebra $U({\mathfrak{g}},e)$
affords a one-dimensional representation;

  * (2)

when $d_{1}$ is odd, the finite $W$-superalgebra $U({\mathfrak{g}},e)$ affords
a two-dimensional representation.

For the case ${\mathfrak{g}}$ is of type $A$, Conjecture 0.3 is confirmed in
the present paper, which is accomplished by conversion from the verification
of the attainableness of lower-bounds of modular dimensions for basic Lie
superalgebras of the same type by some direct computation; see [41] for more
details. However, our final result on the lower bounds of modular dimensions
for basic Lie superalgebras is generally dependent on the above conjecture.

### 0.5.

Under the assumption of Conjecture 0.3, we finally accomplish a super version
of Premet’s work on classical Lie algebras. Explicitly speaking, let
$\xi\in({\mathfrak{g}}_{{\mathds{k}}})^{*}_{\bar{0}}$ be any $p$-character of
${\mathfrak{g}}_{{\mathds{k}}}$ corresponding to an element
$\bar{x}\in({\mathfrak{g}}_{{\mathds{k}}})_{\bar{0}}$ such that
$\xi(\bar{y})=(\bar{x},\bar{y})$ for any
$\bar{y}\in{\mathfrak{g}}_{{\mathds{k}}}$. Now let
$d_{0}=\text{dim}\,({\mathfrak{g}}_{{\mathds{k}}})_{\bar{0}}-\text{dim}\,({\mathfrak{g}}_{{\mathds{k}}}^{\bar{x}})_{\bar{0}}$
and
$d_{1}=\text{dim}\,({\mathfrak{g}}_{{\mathds{k}}})_{\bar{1}}-\text{dim}\,({\mathfrak{g}}_{{\mathds{k}}}^{\bar{x}})_{\bar{1}}$,
where ${\mathfrak{g}}_{{\mathds{k}}}^{\bar{x}}$ denotes the centralizer of
$\bar{x}$ in ${\mathfrak{g}}_{{\mathds{k}}}$. Recall that the dimension of any
irreducible representation of $\mathfrak{g}_{{\mathds{k}}}$ is divisible by
$p^{\frac{d_{0}}{2}}2^{\lfloor\frac{d_{1}}{2}\rfloor}$ (cf. [35, Theorem
5.6]). The main result of the present paper is the following theorem.

######  Theorem 0.4.

Let ${\mathfrak{g}}_{{\mathds{k}}}$ be a basic Lie superalgebra over
${{\mathds{k}}}=\overline{\mathbb{F}}_{p}$, and let
$\xi\in({\mathfrak{g}}_{{\mathds{k}}})^{*}_{\bar{0}}$. If Conjecture 0.3
establishes for all the basic Lie superalgebras over $\mathbb{C}$, then for
$p\gg 0$ the reduced enveloping algebra
$U_{\xi}({\mathfrak{g}}_{{\mathds{k}}})$ admits irreducible representations of
dimension $p^{\frac{d_{0}}{2}}2^{\lfloor\frac{d_{1}}{2}\rfloor}$.

The main body of the proof of the above theorem will be to deal with the
situation when $\xi$ is nilpotent. The arguments in Premet’s work will be
exploited in the super case in the present work. The greatest challenge here
is to deal with the structural change arising from the variation of the parity
of $\dim\mathfrak{g}(-1)_{\bar{1}}$. We sketch here some main ingredients in
our proof, beyond exploiting Premet’s arguments. Based on the PBW structure
theorems of finite $W$-superalgebras established in [40], we first prove that
when $\dim\mathfrak{g}(-1)_{\bar{1}}$ is odd, the $\mathbb{C}$-algebra
$U(\mathfrak{g},e)$ doesn’t admit one-dimensional representations (Proposition
2.7). The possible two-dimensional modules of $U(\mathfrak{g},e)$ will turn to
be of type $Q$, with parity involution arising from some special odd element
$\Theta_{l+q+1}\in U(\mathfrak{g},e)$ only appearing in the case of
$\dim\mathfrak{g}(-1)_{\bar{1}}$ being odd (Proposition 4.2), while what we
need to deal with more are lots of arguments involving the generators and
defining relations of $U(\mathfrak{g},e)$ associated with $\Theta_{l+q+1}$
(see §4.2).

As for the case when the $p$-character
$\chi\in(\mathfrak{g}_{\mathds{k}})^{*}_{\bar{0}}$ is nilpotent, corresponding
to a nilpotent element $e\in(\mathfrak{g}_{\mathds{k}})_{\bar{0}}$ with
respect to the non-degenerate bilinear form on $\mathfrak{g}_{\mathds{k}}$
(see §1.2.1) such that $\chi(\bar{x})=(e,\bar{x})$ for any
$\bar{x}\in\mathfrak{g}_{\mathds{k}}$, the following theorem releases the
condition in Theorem 0.4.

######  Theorem 0.5.

Let ${\mathfrak{g}}_{{\mathds{k}}}$ be a basic Lie superalgebra over
${{\mathds{k}}}=\overline{\mathbb{F}}_{p}$, and let
$\chi\in({\mathfrak{g}}_{{\mathds{k}}})^{*}_{\bar{0}}$ be a nilpotent
$p$-character, with respect to the element
$e\in(\mathfrak{g}_{\mathds{k}})_{\bar{0}}$ as described above. If the
corresponding finite $W$-superalgebra $U({\mathfrak{g}},e)$ over
${\mathbb{C}}$ affords a one-dimensional (resp. two-dimensional)
representation when $d_{1}$ is even (resp. odd), then for $p\gg 0$ the reduced
enveloping algebra $U_{\chi}({\mathfrak{g}}_{{\mathds{k}}})$ admits
irreducible representations of dimension
$p^{\frac{d_{0}}{2}}2^{\lfloor\frac{d_{1}}{2}\rfloor}$, where
$d_{0}=\text{dim}\,({\mathfrak{g}}_{{\mathds{k}}})_{\bar{0}}-\text{dim}\,({\mathfrak{g}}_{{\mathds{k}}}^{e})_{\bar{0}}$
and
$d_{1}=\text{dim}\,({\mathfrak{g}}_{{\mathds{k}}})_{\bar{1}}-\text{dim}\,({\mathfrak{g}}_{{\mathds{k}}}^{e})_{\bar{1}}$.

### 0.6.

The paper is organized as follows. In §1 some basics on algebraic supergroups,
Lie superalgebras and finite $W$-superalgebras are recalled. In §2 the
transition subalgebra $T({\mathfrak{g}}_{{\mathds{k}}},e)$ over ${\mathds{k}}$
is introduced and studied, then follows the structure relation among the
finite $W$-superalgebras, the transition subalgebras and the $p$-central
subalgebras of some subspaces. In §3 and §4, the minimal dimensions for the
representations of $U({\mathfrak{g}},e)$ over ${\mathbb{C}}$ are estimated and
conjectured. We demonstrate that the conjecture is true for some cases,
including the whole case of type $A$. The concluding section will be devoted
to the proof of our main theorems. In §5.2 we first complete the proof of
Theorem 0.5. In subsection §5.3, we improve the result on the dimensional
lower-bounds of modular representations for a direct sum of basic Lie
superalgebras with nilpotent $p$-characters in Proposition 5.5 and Remark 5.6
(note that this conclusion does not depend on Conjecture 0.3), which was
originally discussed by Wang-Zhao in [35, Remark 4.5]. Then the accessibility
of the lower bounds for the refined version is obtained under Conjecture 0.3.
The subsection §5.4 is devoted to the proof Theorem 0.4. In virtue of the
results obtained in §5.3, we further show that the lower bounds in the super
Kac-Weisfeiler property with arbitrary $p$-characters in [35] are also
reachable under Conjecture 0.3. The main tool applied in this section is the
method of nilpotent orbit theory, and also the techniques for the modular
representation theory of restricted Lie superalgebras.

### 0.7.

Throughout we work with the field of complex numbers ${\mathbb{C}}$, or the
algebraically closed field ${{\mathds{k}}}=\overline{\mathbb{F}}_{p}$ of
positive characteristic $p$ as the ground field.

Let ${\mathbb{Z}}_{+}$ be the set of all the non-negative integers in
${\mathbb{Z}}$, and denote by ${\mathbb{Z}}_{2}$ the residue class ring modulo
$2$ in ${\mathbb{Z}}$. A superspace is a ${\mathbb{Z}}_{2}$-graded vector
space $V=V_{\bar{0}}\oplus V_{\bar{1}}$, in which we call elements in
$V_{\bar{0}}$ and $V_{\bar{1}}$ even and odd, respectively. Write
$|v|\in{\mathbb{Z}}_{2}$ for the parity (or degree) of $v\in V$, which is
implicitly assumed to be ${\mathbb{Z}}_{2}$-homogeneous. We will use the
notations

$\text{\text@underline{dim}}V=(\text{dim}V_{\bar{0}},\text{dim}V_{\bar{1}}),\quad\text{dim}V=\text{dim}V_{\bar{0}}+\text{dim}V_{\bar{1}}.$

All Lie superalgebras ${\mathfrak{g}}$ will be assumed to be finite-
dimensional.

Recall that a superalgebra analogue of Schur’s Lemma states that the
endomorphism ring of an irreducible module of a superalgebra is either one-
dimensional or two-dimensional (in the latter case it is isomorphic to a
Clifford algebra), cf. for example, Kleshchev [14, Chapter 12]. An irreducible
module is of type $M$ if its endomorphism ring is one-dimensional and it is of
type $Q$ otherwise.

By vector spaces, subalgebras, ideals, modules, and submodules etc., we mean
in the super sense unless otherwise specified, throughout the paper.

##  1\. Basic Lie superalgebras and finite $W$-superalgebras

In this section, we will recall some knowledge on basic classical Lie
superalgebras along with the corresponding algebraic supergroups, and finite
$W$-(super)algebras for use in the sequel. We refer the readers to [5], [11]
and [20] for Lie superalgebras, [6] and [32] for algebraic supergroups, and
[27], [29], [37] and [40] for finite $W$-(super)algebras.

###  1.1. Basic classical Lie superalgebras and the corresponding algebraic
supergroups

####  1.1.1. Basic classical Lie superalgebras

Following [5, §1], [11, §2.3-§2.4], [12, §1] and [35, §2], we recall the list
of basic classical Lie superalgebras over $\mathbb{F}$ for
$\mathbb{F}=\mathbb{C}$ or $\mathbb{F}={\mathds{k}}$. These Lie superalgebras,
with even parts being Lie algebras of reductive algebraic groups, are simple
over $\mathbb{F}$ (the general linear Lie superalgebras, though not simple,
are also included), and they admit an even non-degenerate supersymmetric
invariant bilinear form in the following sense.

######  Definition 1.1.

Let $V=V_{\bar{0}}\oplus V_{\bar{1}}$ be a $\mathbb{Z}_{2}$-graded space and
$(\cdot,\cdot)$ be a bilinear form on $V$.

  * (1)

If $(a,b)=0$ for any $a\in V_{\bar{0}},b\in V_{\bar{1}}$, then $(\cdot,\cdot)$
is called even;

  * (2)

if $(a,b)=(-1)^{|a||b|}(b,a)$ for any homogeneous elements $a,b\in V$, then
$(\cdot,\cdot)$ is called supersymmetric;

  * (3)

if $([a,b],c)=(a,[b,c])$ for any homogeneous elements $a,b,c\in V$, then
$(\cdot,\cdot)$ is called invariant;

  * (4)

if one can conclude from $(a,V)=0$ that $a=0$, then $(\cdot,\cdot)$ is called
non-degenerate.

Note that when $\mathbb{F}={\mathds{k}}$ is a field of characteristic $p>0$,
there are restrictions on $p$, for example, as shown in [35, Table 1]. So we
have the following list

(Table 1): basic classical Lie ${\mathds{k}}$-superalgebras  $\mathfrak{g}$
$\mathfrak{g}_{\bar{0}}$ the restriction of $p$ when $\mathbb{F}={\mathds{k}}$
$\mathfrak{gl}(m|n$) $\mathfrak{gl}(m)\oplus\mathfrak{gl}(n)$ $p>2$
$\mathfrak{sl}(m|n)$
$\mathfrak{sl}(m)\oplus\mathfrak{sl}(n)\oplus{\mathds{k}}$ $p>2,p\nmid(m-n)$
$\mathfrak{osp}(m|n)$ $\mathfrak{so}(m)\oplus\mathfrak{sp}(n)$ $p>2$
$\text{D}(2,1,\bar{a})$
$\mathfrak{sl}(2)\oplus\mathfrak{sl}(2)\oplus\mathfrak{sl}(2)$ $p>3$
$\text{F}(4)$ $\mathfrak{sl}(2)\oplus\mathfrak{so}(7)$ $p>15$ $\text{G}(3)$
$\mathfrak{sl}(2)\oplus\text{G}_{2}$ $p>15$

Throughout the paper, we will simply call all $\mathfrak{g}$ listed above
“basic Lie superalgebras”.

####  1.1.2. Algebraic supergroups and restricted Lie superalgebras

For a given basic Lie superalgebra listed in §1.1.1, there is an algebraic
supergroup $G_{\mathds{k}}$ satisfying
$\text{Lie}(G_{\mathds{k}})=\mathfrak{g}_{\mathds{k}}$ such that

  * (1)

$G_{\mathds{k}}$ has a subgroup scheme $(G_{\mathds{k}})_{\text{ev}}$ which is
an ordinary connected reductive group with
$\text{Lie}((G_{\mathds{k}})_{\text{ev}})=(\mathfrak{g}_{\mathds{k}})_{\bar{0}}$;

  * (2)

there is a well-defined action of $(G_{\mathds{k}})_{\text{ev}}$ on
$\mathfrak{g}_{\mathds{k}}$, reducing to the adjoint action of
$(\mathfrak{g}_{\mathds{k}})_{\bar{0}}$.

The above algebraic supergroup can be constructed as a Chevalley supergroup in
[6]. The pair ($(G_{\mathds{k}})_{\text{ev}},\mathfrak{g}_{\mathds{k}})$
constructed in this way is called a Chevalley super Harish-Chandra pair (cf.
[6, Theorem 5.35] and [7, §3.3]). Partial results on $G_{\mathds{k}}$ and
$(G_{\mathds{k}})_{\text{ev}}$ can be found in [1, Ch. 2.2], [6], [7, §3.3]
etc.. In the present paper, we will call $(G_{\mathds{k}})_{\text{ev}}$ the
purely even subgroup of $G_{\mathds{k}}$. When the ground field is
${\mathds{k}}$ of odd prime characteristic $p$, one easily knows that
$\mathfrak{g}_{\mathds{k}}$ is a restricted Lie superalgebra (cf. [32,
Definition 2.1] and [33]) in the following sense.

######  Definition 1.2.

A Lie superalgebra
${\mathfrak{g}}_{{\mathds{k}}}=({\mathfrak{g}}_{{\mathds{k}}})_{\bar{0}}\oplus({\mathfrak{g}}_{{\mathds{k}}})_{\bar{1}}$
over ${{\mathds{k}}}$ is called a restricted Lie superalgebra, if there is a
$p$-th power map
$({\mathfrak{g}}_{{\mathds{k}}})_{\bar{0}}\rightarrow({\mathfrak{g}}_{{\mathds{k}}})_{\bar{0}}$,
denoted as $(-)^{[p]}$, satisfying

(a) $(k\bar{x})^{[p]}=k^{p}\bar{x}^{[p]}$ for all $k\in{{\mathds{k}}}$ and
$\bar{x}\in({\mathfrak{g}}_{{\mathds{k}}})_{\bar{0}}$;

(b) $[\bar{x}^{[p]},\bar{y}]=(\text{ad}\bar{x})^{p}(\bar{y})$ for all
$\bar{x}\in({\mathfrak{g}}_{{\mathds{k}}})_{\bar{0}}$ and
$\bar{y}\in{\mathfrak{g}}_{{\mathds{k}}}$;

(c)
$(\bar{x}+\bar{y})^{[p]}=\bar{x}^{[p]}+\bar{y}^{[p]}+\sum\limits_{i=1}^{p-1}s_{i}(\bar{x},\bar{y})$
for all $\bar{x},\bar{y}\in({\mathfrak{g}}_{{\mathds{k}}})_{\bar{0}}$, where
$is_{i}(\bar{x},\bar{y})$ is the coefficient of $\lambda^{i-1}$ in
$(\text{ad}(\lambda\bar{x}+\bar{y}))^{p-1}(\bar{x})$.

Let $\mathfrak{g}_{\mathds{k}}$ be a restricted Lie superalgebra. For each
$\bar{x}\in(\mathfrak{g}_{\mathds{k}})_{\bar{0}}$, the element
$\bar{x}^{p}-\bar{x}^{[p]}\in U(\mathfrak{g}_{\mathds{k}})$ is central by
Definition 1.2, and all of which generate a central subalgebra of
$U(\mathfrak{g}_{\mathds{k}})$. Let $\\{\bar{w}_{1},\cdots,\bar{w}_{c}\\}$ and
$\\{\bar{w}^{\prime}_{1},\cdots,\bar{w}^{\prime}_{d}\\}$ be the basis of
$(\mathfrak{g}_{\mathds{k}})_{\bar{0}}$ and
$(\mathfrak{g}_{\mathds{k}})_{\bar{1}}$ respectively. For a given
$\chi\in(\mathfrak{g}_{\mathds{k}})_{\bar{0}}^{*}$, let $J_{\chi}$ be the
ideal of the universal enveloping algebra $U(\mathfrak{g}_{\mathds{k}})$
generated by the even central elements
$\bar{w}^{p}-\bar{w}^{[p]}-\chi(\bar{w})^{p}$ for all
$\bar{w}\in(\mathfrak{g}_{\mathds{k}})_{\bar{0}}$. The quotient algebra
$U_{\chi}(\mathfrak{g}_{\mathds{k}}):=U(\mathfrak{g}_{\mathds{k}})/J_{\chi}$
is called the reduced enveloping algebra with $p$-character $\chi$. We often
regard $\chi\in\mathfrak{g}_{\mathds{k}}^{*}$ by letting
$\chi((\mathfrak{g}_{\mathds{k}})_{\bar{1}})=0$. If $\chi=0$, then
$U_{0}(\mathfrak{g}_{\mathds{k}})$ is called the restricted enveloping
algebra. It is a direct consequence from the PBW theorem that the
${\mathds{k}}$-algebra $U_{\chi}(\mathfrak{g}_{\mathds{k}})$ is of dimension
$p^{c}2^{d}$, and has a basis

$\\{\bar{w}_{1}^{a_{1}}\cdots\bar{w}_{c}^{a_{c}}(\bar{w}^{\prime}_{1})^{b_{1}}\cdots(\bar{w}^{\prime}_{d})^{b_{d}}\mid
0\leqslant a_{i}<p,b_{j}\in\\{0,1\\}\mbox{ for all }1\leqslant i\leqslant
c,1\leqslant j\leqslant d\\}.$

###  1.2. Finite $W$-superalgebras over the field of complex numbers

#### 1.2.1.

Let ${\mathfrak{g}}$ be a basic Lie superalgebra over ${\mathbb{C}}$ and
$\mathfrak{h}$ be a typical Cartan subalgebra of ${\mathfrak{g}}$. Let $\Phi$
be a root system of ${\mathfrak{g}}$ relative to $\mathfrak{h}$ whose simple
roots system $\Delta=\\{\alpha_{1},\cdots,\alpha_{l}\\}$ is distinguished (cf.
[12, Proposition 1.5]). Let $\Phi^{+}$ be the corresponding positive system in
$\Phi$, and put $\Phi^{-}:=-\Phi^{+}$. Let
${\mathfrak{g}}=\mathfrak{n}^{-}\oplus\mathfrak{h}\oplus\mathfrak{n}^{+}$ be
the corresponding triangular decomposition of ${\mathfrak{g}}$. By [6, §3.3],
we can choose a Chevalley basis
$B=\\{e_{\gamma}\mid\gamma\in\Phi\\}\cup\\{h_{\alpha}\mid\alpha\in\Delta\\}$
of ${\mathfrak{g}}$ excluding the case $D(2,1;a)(a\notin{\mathbb{Z}}$). Let
${\mathfrak{g}}_{\mathbb{Z}}$ denote the Chevalley ${\mathbb{Z}}$-form in
${\mathfrak{g}}$ and $U_{\mathbb{Z}}$ the Kostant ${\mathbb{Z}}$-form of
$U({\mathfrak{g}})$ associated to $B$. Given a ${\mathbb{Z}}$-module $V$ and a
${\mathbb{Z}}$-algebra $A$, we write $V_{A}:=V\otimes_{\mathbb{Z}}A$.

Let $G$ be the algebraic supergroup of ${\mathfrak{g}}$, then the even part of
$G$ is reductive and denote it by $G_{\text{ev}}$, with
$(G_{\text{ev}},\mathfrak{g})$ being a super Harish-Chandra pair. Let
$e\in{\mathfrak{g}}_{\bar{0}}$ be a nilpotent element in ${\mathfrak{g}}$. By
the Dynkin-Kostant theory, $\text{ad}\,G_{\text{ev}}.e$ interacts with
$({\mathfrak{g}}_{\mathbb{Z}})_{\bar{0}}$ nonempty. Without loss of
generality, one can assume that the nilpotent $e$ is in
$({\mathfrak{g}}_{\mathbb{Z}})_{\bar{0}}$. By the same discussion as [27,
§4.2], for any given nilpotent element
$e\in({\mathfrak{g}}_{\mathbb{Z}})_{\bar{0}}$ we can find
$f,h\in({\mathfrak{g}}_{\mathbb{Q}})_{\bar{0}}$ such that $(e,h,f)$ is an
$\mathfrak{sl}_{2}$-triple in ${\mathfrak{g}}$. [40, Proposition 2.1] shows
that there exists an even non-degenerate supersymmetric invariant bilinear
form $(\cdot,\cdot)$, under which the Chevalley basis $B$ of ${\mathfrak{g}}$
takes value in $\mathbb{Q}$, and $(e,f)=1$. Define $\chi\in{\mathfrak{g}}^{*}$
by letting $\chi(x)=(e,x)$ for all $x\in{\mathfrak{g}}$, then we have
$\chi({\mathfrak{g}}_{\bar{1}})=0$.

Following [40, Definition 2.4] we call a commutative ring $A$ admissible if
$A$ is a finitely generated ${\mathbb{Z}}$-subalgebra of ${\mathbb{C}}$,
$(e,f)\in A^{\times}(=A\backslash\\{0\\})$ and all bad primes of the root
system of ${\mathfrak{g}}$ and the determinant of the Gram matrix of
($\cdot,\cdot$) relative to a Chevalley basis of ${\mathfrak{g}}$ are
invertible in $A$. It is clear by the definition that every admissible ring is
a Noetherian domain. Given a finitely generated ${\mathbb{Z}}$-subalgebra $A$
of ${\mathbb{C}}$, denote by $\text{Specm}A$ the maximal spectrum of $A$. It
is well known that for every element $\mathfrak{P}\in\text{Specm}A$, the
residue field $A/\mathfrak{P}$ is isomorphic to $\mathbb{F}_{q}$, where $q$ is
a $p$-power depending on $\mathfrak{P}$. We denote by $\Pi(A)$ the set of all
primes $p\in\mathbb{N}$ that occur in this way. Since the choice of $A$ does
not depend on the super structure of ${\mathfrak{g}}$, it follows from the
arguments in the proof of [29, Lemma 4.4] that the set $\Pi(A)$ contains
almost all primes in $\mathbb{N}$.

Let ${\mathfrak{g}}(i)=\\{x\in{\mathfrak{g}}\mid[h,x]=ix\\}$, then
${\mathfrak{g}}=\bigoplus_{i\in{\mathbb{Z}}}{\mathfrak{g}}(i)$. By the
$\mathfrak{sl}_{2}$-theory, all subspaces ${\mathfrak{g}}(i)$ are defined over
$\mathbb{Q}$. Also, $e\in{\mathfrak{g}}(2)_{\bar{0}}$ and
$f\in{\mathfrak{g}}(-2)_{\bar{0}}$. There exists a symplectic (resp.
symmetric) bilinear form $\langle\cdot,\cdot\rangle$ on the
${\mathbb{Z}}_{2}$-graded subspace ${\mathfrak{g}}(-1)_{\bar{0}}$ (resp.
${\mathfrak{g}}(-1)_{\bar{1}}$) given by $\langle
x,y\rangle:=(e,[x,y])=\chi([x,y])$ for all
$x,y\in{\mathfrak{g}}(-1)_{\bar{0}}~{}(\text{resp.}\,x,y\in{\mathfrak{g}}(-1)_{\bar{1}})$.
There exist bases $\\{u_{1},\cdots,u_{2s}\\}$ of
${\mathfrak{g}}(-1)_{\bar{0}}$ and $\\{v_{1},\cdots,v_{r}\\}$ of
${\mathfrak{g}}(-1)_{\bar{1}}$ contained in ${\mathfrak{g}}_{\mathbb{Q}}$ such
that $\langle u_{i},u_{j}\rangle=i^{*}\delta_{i+j,2s+1}$ for $1\leqslant
i,j\leqslant 2s$, where
$i^{*}=\left\\{\begin{array}[]{ll}-1&\text{if}~{}1\leqslant i\leqslant s;\\\
1&\text{if}~{}s+1\leqslant i\leqslant 2s\end{array}\right.$, and $\langle
v_{i},v_{j}\rangle=\delta_{i+j,r+1}$ for $1\leqslant i,j\leqslant r$.

Set
$\mathfrak{m}:=\bigoplus_{i\leqslant-2}{\mathfrak{g}}(i)\oplus{\mathfrak{g}}(-1)^{\prime}$
with
${\mathfrak{g}}(-1)^{\prime}={\mathfrak{g}}(-1)^{\prime}_{\bar{0}}\oplus{\mathfrak{g}}(-1)^{\prime}_{\bar{1}}$,
where ${\mathfrak{g}}(-1)^{\prime}_{\bar{0}}$ is the ${\mathbb{C}}$-span of
$u_{s+1},\cdots,u_{2s}$ and ${\mathfrak{g}}(-1)^{\prime}_{\bar{1}}$ is the
${\mathbb{C}}$-span of $v_{\frac{r}{2}+1},\cdots,v_{r}$ (resp.
$v_{\frac{r+3}{2}},\cdots,v_{r}$) when
$r=\text{dim}\,{\mathfrak{g}}(-1)_{\bar{1}}$ is even (resp. odd), then $\chi$
vanishes on the derived subalgebra of $\mathfrak{m}$. Define
${\mathfrak{p}}:=\bigoplus_{i\geqslant 0}{\mathfrak{g}}(i)$,
$\mathfrak{m}^{\prime}:=\left\\{\begin{array}[]{ll}\mathfrak{m}&\text{if}~{}r~{}\text{is
even;}\\\
\mathfrak{m}\oplus{\mathbb{C}}v_{\frac{r+1}{2}}&\text{if}~{}r~{}\text{is
odd.}\end{array}\right.$

Write ${\mathfrak{g}}^{e}$ for the centralizer of $e$ in ${\mathfrak{g}}$ and
denote by
$d_{i}:=\text{dim}\,{\mathfrak{g}}_{i}-\text{dim}\,{\mathfrak{g}}^{e}_{i}$ for
$i\in{\mathbb{Z}}_{2}$, then $r:=\dim\mathfrak{g}(-1)_{\bar{1}}$ and $d_{1}$
always have the same parity by [35, Theorem 4.3]. This parity is a crucial
factor deciding the structure of the finite $W$-superalgebras (cf. [40,
Theorem 4.5]), which is called the judging parity in [40]. We further have

$\text{\text@underline{dim}}\,\mathfrak{m}=\begin{cases}(\frac{d_{0}}{2},\frac{d_{1}}{2})&\mbox{
if }d_{1}\mbox{ is even;}\cr(\frac{d_{0}}{2},\frac{d_{1}-1}{2})&\mbox{ if
}d_{1}\mbox{ is odd.}\end{cases}$

After enlarging $A$ one can assume that
${\mathfrak{g}}_{A}=\bigoplus_{i\in{\mathbb{Z}}}{\mathfrak{g}}_{A}(i)$, and
each ${\mathfrak{g}}_{A}(i):={\mathfrak{g}}_{A}\cap{\mathfrak{g}}(i)$ is
freely generated over $A$ by a basis of the vector space ${\mathfrak{g}}(i)$.
Then $\\{u_{1},\cdots,u_{2s}\\}$ and $\\{v_{1},\cdots,v_{r}\\}$ are free basis
of $A$-module ${\mathfrak{g}}_{A}(-1)_{\bar{0}}$ and
${\mathfrak{g}}_{A}(-1)_{\bar{1}}$, respectively. It is obvious that
$\mathfrak{m}_{A}:={\mathfrak{g}}_{A}\cap\mathfrak{m}$,
$\mathfrak{m}^{\prime}_{A}:={\mathfrak{g}}_{A}\cap\mathfrak{m}^{\prime}$ and
${\mathfrak{p}}_{A}:={\mathfrak{g}}_{A}\cap{\mathfrak{p}}$ are free
$A$-modules and direct summands of ${\mathfrak{g}}_{A}$. Moreover, one can
assume $e,f\in({\mathfrak{g}}_{A})_{\bar{0}}$ after enlarging $A$ possibly;
$[e,{\mathfrak{g}}_{A}(i)]$ and $[f,{\mathfrak{g}}_{A}(i)]$ are direct
summands of ${\mathfrak{g}}_{A}(i+2)$ and ${\mathfrak{g}}_{A}(i-2)$
respectively, and ${\mathfrak{g}}_{A}(i+2)=[e,{\mathfrak{g}}_{A}(i)]$ for each
$i\geqslant-1$ by the $\mathfrak{sl}_{2}$-theory.

As in [40, §2] we can choose a basis
$x_{1},\cdots,x_{l},x_{l+1},\cdots,x_{m}\in({\mathfrak{p}}_{A})_{\bar{0}},y_{1},\cdots,y_{q},$
$y_{q+1},\cdots,y_{n}\in({\mathfrak{p}}_{A})_{\bar{1}}$ of the free $A$-module
${\mathfrak{p}}_{A}=\bigoplus_{i\geqslant 0}{\mathfrak{g}}_{A}(i)$ such that

(a)
$x_{i}\in{\mathfrak{g}}_{A}(k_{i})_{\bar{0}},y_{j}\in{\mathfrak{g}}_{A}(k^{\prime}_{j})_{\bar{1}}$,
where $k_{i},k^{\prime}_{j}\in{\mathbb{Z}}_{+}$ with $1\leqslant i\leqslant m$
and $1\leqslant j\leqslant n$;

(b) $x_{1},\cdots,x_{l}$ is a basis of $({\mathfrak{g}}_{A})^{e}_{\bar{0}}$
and $y_{1},\cdots,y_{q}$ is a basis of $({\mathfrak{g}}_{A})^{e}_{\bar{1}}$;

(c) $x_{l+1},\cdots,x_{m}\in[f,({\mathfrak{g}}_{A})_{\bar{0}}]$ and
$y_{q+1},\cdots,y_{n}\in[f,({\mathfrak{g}}_{A})_{\bar{1}}]$.

#### 1.2.2.

Recall that a Gelfand-Graev ${\mathfrak{g}}$-module associated to $\chi$ is
defined by

$Q_{\chi}:=U({\mathfrak{g}})\otimes_{U(\mathfrak{m})}{\mathbb{C}}_{\chi},$

where ${\mathbb{C}}_{\chi}={\mathbb{C}}1_{\chi}$ is a one-dimensional
$\mathfrak{m}$-module such that $x.1_{\chi}=\chi(x)1_{\chi}$ for all
$x\in\mathfrak{m}$. For $k\in\mathbb{Z}_{+}$, define

$\begin{array}[]{llllll}\mathbb{Z}_{+}^{k}&:=&\\{(i_{1},\cdots,i_{k})\mid
i_{j}\in\mathbb{Z}_{+}\\},&\Lambda^{\prime}_{k}&:=&\\{(i_{1},\cdots,i_{k})\mid
i_{j}\in\\{0,1\\}\\}\end{array}$

with $1\leqslant j\leqslant k$. For $\mathbf{i}=(i_{1},\cdots,i_{k})$ in
$\mathbb{Z}_{+}^{k}$ or $\Lambda^{\prime}_{k}$, set
$|\mathbf{i}|=i_{1}+\cdots+i_{k}$. For any real number $a\in\mathbb{R}$, let
$\lceil a\rceil$ denote the largest integer lower bound of $a$, and $\lfloor
a\rfloor$ the least integer upper bound of $a$. Given
$(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})\in{\mathbb{Z}}^{m}_{+}\times\Lambda^{\prime}_{n}\times{\mathbb{Z}}^{s}_{+}\times\Lambda^{\prime}_{t}$
(where
$t:=\lfloor\frac{r}{2}\rfloor=\lfloor\frac{\text{dim}\,{\mathfrak{g}}(-1)_{\bar{1}}}{2}\rfloor$),
let $x^{\mathbf{a}}y^{\mathbf{b}}u^{\mathbf{c}}v^{\mathbf{d}}$ denote the
monomial $x_{1}^{a_{1}}\cdots x_{m}^{a_{m}}y_{1}^{b_{1}}\cdots
y_{n}^{b_{n}}u_{1}^{c_{1}}\cdots u_{s}^{c_{s}}v_{1}^{d_{1}}\cdots
v_{t}^{d_{t}}$ in $U({\mathfrak{g}})$. Set
$Q_{\chi,A}:=U({\mathfrak{g}}_{A})\otimes_{U(\mathfrak{m}_{A})}A_{\chi}$,
where $A_{\chi}=A1_{\chi}$. By the definition $Q_{\chi,A}$ is a
${\mathfrak{g}}_{A}$-stable $A$-lattice in $Q_{\chi}$ with
$\\{x^{\mathbf{a}}y^{\mathbf{b}}u^{\mathbf{c}}v^{\mathbf{d}}\otimes
1_{\chi}\mid(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})\in{\mathbb{Z}}^{m}_{+}\times\Lambda^{\prime}_{n}\times{\mathbb{Z}}^{s}_{+}\times\Lambda^{\prime}_{t}\\}$
as a free basis. Given
$(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})\in{\mathbb{Z}}_{+}^{m}\times\Lambda^{\prime}_{n}\times{\mathbb{Z}}_{+}^{s}\times\Lambda^{\prime}_{t}$,
set

$|(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})|_{e}:=\sum_{i=1}^{m}a_{i}(k_{i}+2)+\sum_{i=1}^{n}b_{i}(k^{\prime}_{i}+2)+\sum_{i=1}^{s}c_{i}+\sum_{i=1}^{t}d_{i}.$

Set

$Y_{i}:=\left\\{\begin{array}[]{ll}x_{i}&\text{if}~{}1\leqslant i\leqslant
l;\\\ y_{i-l}&\text{if}~{}l+1\leqslant i\leqslant l+q;\\\
v_{\frac{r+1}{2}}&\text{if}~{}i=l+q+1,\end{array}\right.$

where $Y_{i}\in{\mathfrak{g}}(m_{i})$ with $m_{i}\in{\mathbb{Z}}$ and the term
$Y_{l+q+1}$ occurs only when $r=\dim\mathfrak{g}(-1)_{\bar{1}}$ is odd. Then
there are Lie superalgebra operator identities in the setting of
$[Y_{i},Y_{j}]=\sum\limits_{k=1}^{l+q}\alpha_{ij}^{k}Y_{k}$ in
${\mathfrak{g}}^{e}$ for $1\leqslant i,j\leqslant l+q$. Set $q^{\prime}=q$ if
$r$ (or equivalently, $d_{1}$) is even, and $q^{\prime}=q+1$ if $r$ is odd. By
[40, Theorem 4.7], the finite $W$-superalgebra
$U({\mathfrak{g}},e):=(\text{End}_{\mathfrak{g}}Q_{\chi})^{\text{op}}$ can be
determined by a data of generators and defining relations. Those generators
are $\Theta_{1},\cdots,\Theta_{l}\in U(\mathfrak{g},e)_{\bar{0}}$ and
$\Theta_{l+1},\cdots,\Theta_{l+q^{\prime}}\in U(\mathfrak{g},e)_{\bar{1}}$
with

$\begin{array}[]{ll}&\Theta_{k}(1_{\chi})\\\
=&(Y_{k}+\sum\limits_{\mbox{\tiny$\begin{array}[]{c}|\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}|_{e}=m_{k}+2,\\\
|\mathbf{a}|+|\mathbf{b}|+|\mathbf{c}|+|\mathbf{d}|\geqslant
2\end{array}$}}\lambda^{k}_{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}}x^{\mathbf{a}}y^{\mathbf{b}}u^{\mathbf{c}}v^{\mathbf{d}}+\sum\limits_{|\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}|_{e}<m_{k}+2}\lambda^{k}_{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}}x^{\mathbf{a}}y^{\mathbf{b}}u^{\mathbf{c}}v^{\mathbf{d}})\otimes
1_{\chi}\end{array}$ (1.1)

for $1\leqslant k\leqslant l+q$, where
$\lambda^{k}_{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}}\in\mathbb{Q}$, and
$\lambda^{k}_{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}}=0$ if
$a_{l+1}=\cdots=a_{m}=b_{q+1}=\cdots=b_{n}=c_{1}=\cdots=c_{s}=d_{1}=\cdots=d_{\lceil\frac{r}{2}\rceil}=0$.
When $r$ is odd, set $\Theta_{l+q+1}(1_{\chi})=Y_{l+q+1}\otimes
1_{\chi}=v_{\frac{r+1}{2}}\otimes 1_{\chi}$.

By [40, Theorem 4.5], the monomials
$\Theta_{1}^{a_{1}}\cdots\Theta_{l}^{a_{l}}\Theta_{l+1}^{b_{1}}\cdots\Theta_{l+q^{\prime}}^{b_{q^{\prime}}}$
with $a_{i}\in{\mathbb{Z}}_{+},b_{j}\in\\{0,1\\}$ for $1\leqslant i\leqslant
l$ and $1\leqslant j\leqslant q^{\prime}$ form a basis of the vector space
$U({\mathfrak{g}},e)$. The monomial
$\Theta_{1}^{a_{1}}\cdots\Theta_{l}^{a_{l}}\Theta_{l+1}^{b_{1}}\cdots\Theta_{l+q^{\prime}}^{b_{q^{\prime}}}$
is said to have Kazhdan degree
$\sum\limits_{i=1}^{l}a_{i}(m_{i}+2)+\sum\limits_{i=1}^{q^{\prime}}b_{i}(m_{l+i}+2)$.
For $k\in{\mathbb{Z}}_{+}$, let $\text{F}_{k}U({\mathfrak{g}},e)$ denote the
${\mathbb{C}}$-span of all monomials
$\Theta_{1}^{a_{1}}\cdots\Theta_{l}^{a_{l}}\Theta_{l+1}^{b_{1}}\cdots\Theta_{l+q^{\prime}}^{b_{q^{\prime}}}$
of Kazhdan degree $\leqslant k$. The subspaces
$\text{F}_{k}U({\mathfrak{g}},e)$ with $k\geqslant 0$ form an increasing
exhaustive filtration of the algebra $U({\mathfrak{g}},e)$, which is called
the Kazhdan filtration. The corresponding graded algebra gr
$U({\mathfrak{g}},e)$ is a polynomial superalgebra in
$\text{gr}\,\Theta_{1},\cdots,\text{gr}\,\Theta_{l+q^{\prime}}$. Recall [40,
Theorem 4.7] shows that there are super-polynomials $F_{ij}$’s with
$i,j=1,\cdots,l+q^{\prime}$ such that the defining relations on those
generators can be described as

$\displaystyle[\Theta_{i},\Theta_{j}]=F_{ij}(\Theta_{1},\cdots,\Theta_{l+q^{\prime}}),\quad
i,j=1,\cdots,l+q^{\prime},$ (1.2)

while

$\displaystyle
F_{ij}(\Theta_{1},\cdots,\Theta_{l+q^{\prime}})\equiv\sum_{k=1}^{l+q}\alpha_{ij}^{k}\Theta_{k}+q_{ij}(\Theta_{1},\cdots,\Theta_{l+q^{\prime}})\,(\mbox{mod
}\text{F}_{m_{i}+m_{j}+1}U({\mathfrak{g}},e))$ (1.3)

for $i,j=1,\cdots,l+q$. For the case when one of the indices $i,j$ equals
$l+q+1$, it follows from [40, Remark 3.8] that there are no obvious formulas
for the super-polynomials $F_{ij}$’s as shown in (1.3). However, by the same
discussion as [40, Theorem 4.5], one can still choose the super-polynomials
$F_{ij}$’s properly such that the Kazhdan degree for all the monomials in the
$\mathbb{C}$-span of $F_{ij}$’s is less than $m_{i}+m_{j}+2$. Moreover,

$\displaystyle F_{l+q+1,l+q+1}(\Theta_{1},\cdots,\Theta_{l+q+1})=1\otimes
1_{\chi}$ (1.4)

when $r$ is odd.

In fact, some of the defining relations in (1.2) are equivalent to each other.
By the same discussion as [40, Remark 3.8(4)], after deleting all the
redundant commutating relations in (1.2), the remaining ones are with indices
$i,j$ satisfying $1\leqslant i<j\leqslant l$, $l+1\leqslant i\leqslant
j\leqslant l+q^{\prime}$, and $1\leqslant i\leqslant l<j\leqslant
l+q^{\prime}$.

In the sequent arguments, when we consider the corresponding counterparts of
all above over the algebraic closured field
${{\mathds{k}}}=\overline{\mathbb{F}}_{p}$ of positive characteristic $p$, we
assume that the prime $p$ is large enough such that the admissible ring $A$
contains all $\lambda^{k}_{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}}$ in
(1.1) and all coefficients of the $F_{ij}$’s in (1.2), thereby the “admissible
procedure” developed by Premet for finite $W$-algebras can be reproduced in
the super case.

For ${\bf a}=(a_{1},\cdots,a_{l})\in\mathbb{Z}_{+}^{l}$ and ${\bf
b}=(b_{1},\cdots,b_{q^{\prime}})\in\Lambda^{\prime}_{q^{\prime}}$, let
$U({\mathfrak{g}}_{A},e)$ be the $A$-span of the monomials

$\\{\Theta_{1}^{a_{1}}\cdots\Theta_{l}^{a_{l}}\cdot\Theta_{l+1}^{b_{1}}\cdots\Theta_{l+q^{\prime}}^{b_{q^{\prime}}}\mid({\bf{a}},{\bf{b}})\in\mathbb{Z}_{+}^{l}\times\Lambda^{\prime}_{q^{\prime}}\\}.$

#### 1.2.3.

Let $I_{\chi}$ denote the ${\mathbb{Z}}_{2}$-graded ideal in
$U({\mathfrak{g}})$ generated by all $x-\chi(x)$ with
$x\in\mathfrak{m}_{i},i\in{\mathbb{Z}}_{2}$. By construction, $I_{\chi}$ is a
$(U({\mathfrak{g}}),U(\mathfrak{m}))$-bimodule. The fixed point space
$(U({\mathfrak{g}})/I_{\chi})^{\text{ad}\,\mathfrak{m}}$ carries a natural
algebra structure given by $(x+I_{\chi})\cdot(y+I_{\chi}):=(xy+I_{\chi})$ for
all $x,y\in U({\mathfrak{g}})$. Then $Q_{\chi}\cong
U({\mathfrak{g}})/I_{\chi}$ as ${\mathfrak{g}}$-modules via the
${\mathfrak{g}}$-module map sending $1+I_{\chi}$ to $1_{\chi}$, and
$Q_{\chi}^{\text{ad}\,\mathfrak{m}}\cong U({\mathfrak{g}},e)$ as
$\mathbb{C}$-algebras. Any element of $U({\mathfrak{g}},e)$ is uniquely
determined by its effect on the generator $1_{\chi}\in Q_{\chi}$, and it
follows from [40, Theorem 2.12] that the canonical isomorphism between
$Q_{\chi}^{\text{ad}\,\mathfrak{m}}$ and $U({\mathfrak{g}},e)$ is given by
$u\mapsto u(1_{\chi})$ for any $u\in Q_{\chi}^{\text{ad}\,\mathfrak{m}}$. In
what follows we will often identify $Q_{\chi}$ with
$U({\mathfrak{g}})/I_{\chi}$ and $U({\mathfrak{g}},e)$ with
$Q_{\chi}^{\text{ad}\,\mathfrak{m}}$.

Let $w_{1},\cdots,w_{c}$ be a basis of $\mathfrak{g}$ over $\mathbb{C}$. Let
$U({\mathfrak{g}})=\bigcup_{i\in{\mathbb{Z}}}\text{F}_{i}U({\mathfrak{g}})$ be
the Kazhdan filtration of $U({\mathfrak{g}})$, where
$\text{F}_{i}U({\mathfrak{g}})$ is the ${\mathbb{C}}$-span of all $w_{1}\cdots
w_{c}$ with
$w_{1}\in{\mathfrak{g}}(j_{1}),\cdots,w_{c}\in{\mathfrak{g}}(j_{c})$ and
$(j_{1}+2)+\cdots+(j_{c}+2)\leqslant i$. The Kazhdan filtration on $Q_{\chi}$
is defined by $\text{F}_{i}Q_{\chi}:=\pi(\text{F}_{i}U({\mathfrak{g}}))$ with
$\pi:U({\mathfrak{g}})\twoheadrightarrow U({\mathfrak{g}})/I_{\chi}$ being the
canonical homomorphism, which makes $Q_{\chi}$ into a filtrated
$U({\mathfrak{g}})$-module. The Kazhdan filtration of $Q_{\chi}$ has no
negative components, and the Kazhdan filtration of $U({\mathfrak{g}},e)$
defined above is nothing but the filtration of
$U({\mathfrak{g}},e)=Q_{\chi}^{\text{ad}\,\mathfrak{m}}$ induced from the
Kazhdan filtration of $Q_{\chi}$ through the embedding
$Q_{\chi}^{\text{ad}\,\mathfrak{m}}\hookrightarrow Q_{\chi}$ (see [40, §2.3]).

###  1.3. Finite $W$-superalgebras in positive characteristic

#### 1.3.1.

Pick a prime $p\in\Pi(A)$ and denote by
${{\mathds{k}}}=\overline{\mathbb{F}}_{p}$ the algebraic closure of
$\mathbb{F}_{p}$. Since the bilinear form $(\cdot,\cdot)$ is $A$-valued on
${\mathfrak{g}}_{A}$, it induces a bilinear form on the Lie superalgebra
${\mathfrak{g}}_{{\mathds{k}}}\cong{\mathfrak{g}}_{A}\otimes_{A}{{\mathds{k}}}$.
In the following we still denote this bilinear form by $(\cdot,\cdot)$. If we
denote by $G_{{\mathds{k}}}$ the algebraic ${{\mathds{k}}}$-supergroup of
distribution algebra
$U_{{\mathds{k}}}=U_{\mathbb{Z}}\otimes_{\mathbb{Z}}{{\mathds{k}}}$, then
${\mathfrak{g}}_{{\mathds{k}}}=\text{Lie}(G_{{\mathds{k}}})$ (cf. [9, §I.7.10]
and [32, §2.2]). Note that the bilinear form $(\cdot,\cdot)$ is non-degenerate
and $\text{Ad}\,(G_{{\mathds{k}}})_{\text{ev}}$-invariant. For
$x\in{\mathfrak{g}}_{A}$, set $\bar{x}:=x\otimes 1$, an element of
${\mathfrak{g}}_{{\mathds{k}}}$. To ease notation we identify $e,f,h$ with the
nilpotent elements $\bar{e}=e\otimes 1,~{}\bar{f}=f\otimes 1$ and
$\bar{h}=h\otimes 1$ in ${\mathfrak{g}}_{{\mathds{k}}}$, and $\chi$ with the
linear function $(e,\cdot)$ on ${\mathfrak{g}}_{{\mathds{k}}}$.

The Lie superalgebra
${\mathfrak{g}}_{{\mathds{k}}}=\text{Lie}\,(G_{{{\mathds{k}}}})$ carries a
natural $p$-mapping $x\mapsto x^{[p]}$ for all
$x\in({\mathfrak{g}}_{{\mathds{k}}})_{\bar{0}}$, equivariant under the adjoint
action of $(G_{{{\mathds{k}}}})_{\text{ev}}$. The subalgebra of
$U({\mathfrak{g}}_{{\mathds{k}}})$ generated by all
$\bar{x}^{p}-\bar{x}^{[p]}$ with
$\bar{x}\in({\mathfrak{g}}_{{\mathds{k}}})_{\bar{0}}$ is called the $p$-center
of $U({\mathfrak{g}}_{{\mathds{k}}})$ and we denote it by
$Z_{p}({\mathfrak{g}}_{{\mathds{k}}})$ for short. It follows from the PBW
theorem of $U({\mathfrak{g}}_{{\mathds{k}}})$ that
$Z_{p}({\mathfrak{g}}_{{\mathds{k}}})$ is isomorphic to a polynomial algebra
(in the usual sense) in
$\text{dim}\,({\mathfrak{g}}_{{\mathds{k}}})_{\bar{0}}$ variables. For every
maximal ideal $H$ of $Z_{p}({\mathfrak{g}}_{{\mathds{k}}})$ there is a unique
linear function
$\eta=\eta_{H}\in({\mathfrak{g}}_{{\mathds{k}}})_{\bar{0}}^{*}$ such that

$H=\langle\bar{x}^{p}-\bar{x}^{[p]}-\eta(\bar{x})^{p}\mid\bar{x}\in({\mathfrak{g}}_{{\mathds{k}}})_{\bar{0}}\rangle.$

Since the Frobenius map of ${{\mathds{k}}}$ is bijective, this enables us to
identify the maximal spectrum
$\text{Specm}(Z_{p}({\mathfrak{g}}_{{\mathds{k}}}))$ of
$Z_{p}({\mathfrak{g}}_{{\mathds{k}}})$ with
$({\mathfrak{g}}_{{\mathds{k}}})_{\bar{0}}^{*}$.

For any $\xi\in({\mathfrak{g}}_{{\mathds{k}}})_{\bar{0}}^{*}$ we write
$J_{\xi}$ the two-sided ideal of $U({\mathfrak{g}}_{{\mathds{k}}})$ generated
by the even central elements
$\\{\bar{x}^{p}-\bar{x}^{[p]}-\xi(\bar{x})^{p}\mid\bar{x}\in({\mathfrak{g}}_{{\mathds{k}}})_{\bar{0}}\\}$.
Then the quotient algebra
$U_{\xi}({\mathfrak{g}}_{{\mathds{k}}}):=U({\mathfrak{g}}_{{\mathds{k}}})/J_{\xi}$
is called the reduced enveloping algebra with $p$-character $\xi$. We have
$\dim
U_{\xi}({\mathfrak{g}}_{{\mathds{k}}})=p^{\text{dim}\,({\mathfrak{g}}_{{\mathds{k}}})_{\bar{0}}}2^{\text{dim}\,({\mathfrak{g}}_{{\mathds{k}}})_{\bar{1}}}$
by construction. It follows from the Schur Lemma that any irreducible
${\mathfrak{g}}_{{\mathds{k}}}$-module $V$ is of
$U_{\xi}({\mathfrak{g}}_{{\mathds{k}}})$ for a unique
$\xi=\xi_{V}\in({\mathfrak{g}}_{{\mathds{k}}})_{\bar{0}}^{*}$. We often regard
$\xi\in{\mathfrak{g}}_{{\mathds{k}}}^{*}$ by letting
$\xi(({\mathfrak{g}}_{{\mathds{k}}})_{\bar{1}})=0$.

#### 1.3.2.

For $i\in{\mathbb{Z}}$, set
${\mathfrak{g}}_{{\mathds{k}}}(i):={\mathfrak{g}}_{A}(i)\otimes_{A}{{\mathds{k}}}$
and put
$\mathfrak{m}_{{\mathds{k}}}:=\mathfrak{m}_{A}\otimes_{A}{{\mathds{k}}}$, then
[40, Lemma 2.18] shows that $\mathfrak{m}_{{\mathds{k}}}$ is a restricted
subalgebra of $\mathfrak{g}_{{\mathds{k}}}$. Denote by
$\mathfrak{m}^{\prime}_{{\mathds{k}}}:=\mathfrak{m}^{\prime}_{A}\otimes_{A}{{\mathds{k}}}$
and $\mathfrak{p}_{{\mathds{k}}}:=\mathfrak{p}_{A}\otimes_{A}{{\mathds{k}}}$.
Due to our assumptions on $A$, the elements $\bar{x}_{1},\cdots,\bar{x}_{l}$
and $\bar{y}_{1},\cdots,\bar{y}_{q}$ form a basis of the centralizer
$({\mathfrak{g}}^{e}_{{\mathds{k}}})_{\bar{0}}$ and
$({\mathfrak{g}}^{e}_{{\mathds{k}}})_{\bar{1}}$ of $e$ in
${\mathfrak{g}}_{{\mathds{k}}}$, respectively. It follows from [35, §4.1] that
the subalgebra $\mathfrak{m}_{{\mathds{k}}}$ is $p$-nilpotent, and the linear
function $\chi$ vanishes on the $p$-closure of
$[\mathfrak{m}_{{\mathds{k}}},\mathfrak{m}_{{\mathds{k}}}]$. Set
$Q_{\chi,{{\mathds{k}}}}:=U({\mathfrak{g}}_{{\mathds{k}}})\otimes_{U(\mathfrak{m}_{{\mathds{k}}})}{{\mathds{k}}}_{\chi}$,
where
${{\mathds{k}}}_{\chi}=A_{\chi}\otimes_{A}{{\mathds{k}}}={{\mathds{k}}}1_{\chi}$.
Clearly, ${{\mathds{k}}}1_{\chi}$ is a one-dimensional
$\mathfrak{m}_{{\mathds{k}}}$-module with the property
$\bar{x}.1_{\chi}=\chi(\bar{x})1_{\chi}$ for all
$\bar{x}\in\mathfrak{m}_{{\mathds{k}}}$, and it is obvious that
$Q_{\chi,{{\mathds{k}}}}\cong Q_{\chi,A}\otimes_{A}{{\mathds{k}}}$ as
${\mathfrak{g}}_{{\mathds{k}}}$-modules. Define the finite $W$-superalgebra
over ${{\mathds{k}}}$ by
$U({\mathfrak{g}}_{{\mathds{k}}},e):=(\text{End}_{{\mathfrak{g}}_{{\mathds{k}}}}Q_{\chi,{{\mathds{k}}}})^{\text{op}}$.

Let ${\mathfrak{g}}_{A}^{*}$ be the $A$-module dual to ${\mathfrak{g}}_{A}$
and let $(\mathfrak{m}_{A}^{\perp})_{\bar{0}}$ denote the set of all linear
functions on $({\mathfrak{g}}_{A})_{\bar{0}}$ vanishing on
$(\mathfrak{m}_{A})_{\bar{0}}$. By the assumptions on $A$,
$(\mathfrak{m}_{A}^{\perp})_{\bar{0}}$ is a free $A$-submodule and a direct
summand of ${\mathfrak{g}}^{*}_{A}$. Note that
$(\mathfrak{m}_{A}^{\perp}\otimes_{A}{\mathbb{C}})_{\bar{0}}$ and
$(\mathfrak{m}_{A}^{\perp}\otimes_{A}{{\mathds{k}}})_{\bar{0}}$ can be
identified with the annihilators
$\mathfrak{m}^{\perp}_{\bar{0}}:=\\{f\in{\mathfrak{g}}^{*}_{\bar{0}}\mid
f(\mathfrak{m}_{\bar{0}})=0\\}$ and
$(\mathfrak{m}_{{\mathds{k}}}^{\perp})_{\bar{0}}:=\\{f\in({\mathfrak{g}}_{{\mathds{k}}})^{*}_{\bar{0}}\mid
f((\mathfrak{m}_{{\mathds{k}}})_{\bar{0}})=0\\}$, respectively. Let
$I_{\chi,A}$ denote the $A$-span of the left ideal of $U(\mathfrak{g}_{A})$
generated by all $x-\chi(x)$ with $x\in\mathfrak{m}_{A}$.

Given a linear function
$\eta\in\chi+(\mathfrak{m}_{{\mathds{k}}}^{\perp})_{\bar{0}}$, set
${\mathfrak{g}}_{{\mathds{k}}}$-module
$Q_{\chi}^{\eta}:=Q_{\chi,{{\mathds{k}}}}/J_{\eta}Q_{\chi,{{\mathds{k}}}}$.
Each ${\mathfrak{g}}_{{\mathds{k}}}$-endomorphism $\bar{\Theta}_{i}$ of
$Q_{\chi,{\mathds{k}}}$ preserves $J_{\eta}Q_{\chi,{{\mathds{k}}}}$, hence
induces a ${\mathfrak{g}}_{{\mathds{k}}}$-endomorphism of $Q_{\chi}^{\eta}$
which is denoted by $\theta_{i}$. As in [40], we set
$U_{\eta}({\mathfrak{g}}_{{\mathds{k}}},e)=(\text{End}_{{\mathfrak{g}}_{{\mathds{k}}}}Q_{\chi}^{\eta})^{\text{op}}$,
and call it a reduced $W$-superalgebra.

Since the restriction of $\eta$ to $\mathfrak{m}_{{\mathds{k}}}$ coincides
with that of $\chi$, the left ideal of $U({\mathfrak{g}}_{{\mathds{k}}})$
generated by all $x-\eta(x)$ with $x\in\mathfrak{m}_{{\mathds{k}}}$ equals
$I_{\chi,{{\mathds{k}}}}:=I_{\chi,A}\otimes_{A}{{\mathds{k}}}$ and
${{\mathds{k}}}_{\chi}={{\mathds{k}}}_{\eta}$ as
$\mathfrak{m}_{{\mathds{k}}}$-modules. We denote by
$I_{\mathfrak{m}_{{\mathds{k}}}}$ the left ideal of
$U_{\eta}({\mathfrak{g}}_{{\mathds{k}}})$ generated by all $x-\eta(x)$ with
$x\in\mathfrak{m}_{{\mathds{k}}}$.

For a (left) $U_{\eta}({\mathfrak{g}}_{{\mathds{k}}})$-module $M$, define

$M^{\mathfrak{m}_{{\mathds{k}}}}:=\\{v\in M\mid
I_{\mathfrak{m}_{{\mathds{k}}}}.v=0\\}.$

It follows from [40, Proposition 2.21] that
$U_{\eta}({\mathfrak{g}}_{{\mathds{k}}},e)$ can be identified with the
${\mathds{k}}$-algebra
$U_{\eta}({\mathfrak{g}}_{{\mathds{k}}})^{\text{ad}\,{\mathfrak{m}}_{{\mathds{k}}}}/U_{\eta}({\mathfrak{g}}_{{\mathds{k}}})^{\text{ad}\,{\mathfrak{m}}_{{\mathds{k}}}}\cap
I_{{\mathfrak{m}}_{{\mathds{k}}}}$. Therefore, any left
$U_{\eta}({\mathfrak{g}}_{{\mathds{k}}})^{\text{ad}\,{\mathfrak{m}}_{{\mathds{k}}}}$-module
can be considered as a $U_{\eta}({\mathfrak{g}}_{{\mathds{k}}},e)$-module with
the trivial action of the ideal
$U_{\eta}({\mathfrak{g}}_{{\mathds{k}}})^{\text{ad}\,{\mathfrak{m}}_{{\mathds{k}}}}\cap
I_{{\mathfrak{m}}_{{\mathds{k}}}}$. Recall [40, Theorem 2.24] shows that

######  Theorem 1.3.

The correspondence $M$ to $M^{\mathfrak{m}_{\mathds{k}}}$ gives rise to a
category equivalence between the category of
$U_{\eta}({\mathfrak{g}}_{{\mathds{k}}})$-modules and the category of
$U_{\eta}({\mathfrak{g}}_{{\mathds{k}}},e)$-modules:

$U_{\eta}({\mathfrak{g}}_{{\mathds{k}}})\text{-mod}\longrightarrow
U_{\eta}({\mathfrak{g}}_{{\mathds{k}}},e)\text{-mod}$

with the inverse

$U_{\eta}({\mathfrak{g}}_{{\mathds{k}}},e)\text{-mod}\longrightarrow
U_{\eta}({\mathfrak{g}}_{{\mathds{k}}})\text{-mod},\qquad V\mapsto
U_{\eta}({\mathfrak{g}}_{{\mathds{k}}})\otimes_{U_{\eta}({\mathfrak{g}}_{{\mathds{k}}})^{\text{ad}\,\mathfrak{m}_{{\mathds{k}}}}}V.$

Furthermore, for any $U_{\eta}({\mathfrak{g}}_{{\mathds{k}}})$-module $M$,
[40, Lemma 2.22] shows that $M^{\mathfrak{m}_{{\mathds{k}}}}$ is a free
$U_{\eta}(\mathfrak{m}_{{\mathds{k}}})$-module. By the same discussion as [35,
Proposition 4.2], one can conclude that there is an isomorphism of vector
spaces $M\cong
U_{\eta}(\mathfrak{m}_{{\mathds{k}}})^{*}\otimes_{{{\mathds{k}}}}M^{\mathfrak{m}_{{\mathds{k}}}}$.
It is immediate that

$\text{dim}\,M^{\mathfrak{m}_{{\mathds{k}}}}=\frac{\text{dim}\,M}{\text{dim}\,U_{\eta}(\mathfrak{m}_{{\mathds{k}}})}.$
(1.5)

In particular, we have
$\text{dim}\,U_{\eta}({\mathfrak{g}}_{{\mathds{k}}})=\text{dim}\,U_{\eta}(\mathfrak{m}_{{\mathds{k}}})\cdot\text{dim}\,U_{\eta}({\mathfrak{g}}_{{\mathds{k}}})^{\text{ad}\,\mathfrak{m}_{{\mathds{k}}}}$.

##  2\. Transition subalgebras of finite $W$-superalgebras in prime
characteristic

Throughout the paper, we will maintain the notations and conventions as in §1.
Especially, $\mathfrak{g}$ is a given basic Lie superalgebra over
$\mathbb{C}$, $A$ is an associated admissible ring, and ${\mathds{k}}$ is an
algebraically closed field of characteristic $p\in\Pi(A)$. Throughout this
section, fix a nilpotent element $e\in\mathfrak{g}_{\bar{0}}$, then we can
define
$d_{i}=\text{dim}\,{\mathfrak{g}}_{i}-\text{dim}\,{\mathfrak{g}}^{e}_{i}$ for
$i\in{\mathbb{Z}}_{2}$. Further recall that both
$\text{dim}\,{\mathfrak{g}}(-1)_{\bar{1}}$ and
$d_{1}=\text{dim}\,{\mathfrak{g}}_{\bar{1}}-\text{dim}\,{\mathfrak{g}}^{e}_{\bar{1}}$
have the same parity by §1.2.1. We always set $q^{\prime}=q$ if $d_{1}$ is
even, and $q^{\prime}=q+1$ if $d_{1}$ is odd.

In this section we will introduce a so-called transition subalgebra of the
finite $W$-superalgebra $U({\mathfrak{g}}_{{\mathds{k}}},e)$ over
${{\mathds{k}}}$. Some structure relations between
$U(\mathfrak{g}_{\mathds{k}},e)$ and its transition subalgebra are presented,
which enables us to connect the information of finite $W$-superalgebras over
$\mathbb{C}$ with the modular representations of reduced enveloping algebras
of basic Lie superalgebras over ${\mathds{k}}$ in the following sections.

This section is somewhat a generalization of the Lie algebra case by Premet in
[29, §2], with a few modifications. One can find that the emergence of odd
parts in the Lie superalgebra ${\mathfrak{g}}_{{\mathds{k}}}$ makes the
situation complicated.

###  2.1. Transition subalgebras

Recall in §1.2.2 we have defined

$\begin{array}[]{llllll}\mathbb{Z}_{+}^{k}&:=&\\{(i_{1},\cdots,i_{k})\mid
i_{j}\in\mathbb{Z}_{+}\\},&\Lambda^{\prime}_{k}&:=&\\{(i_{1},\cdots,i_{k})\mid
i_{j}\in\\{0,1\\}\\}\end{array}$

for $k\in\mathbb{Z}_{+}$ with $1\leqslant j\leqslant k$. For ${\bf
a}=(a_{1},\cdots,a_{l})\in\mathbb{Z}_{+}^{l}$ and ${\bf
b}=(b_{1},\cdots,b_{q^{\prime}})\in\Lambda^{\prime}_{q^{\prime}}$,
$U({\mathfrak{g}}_{A},e)$ denotes the $A$-span of the monomials

$\\{\Theta_{1}^{a_{1}}\cdots\Theta_{l}^{a_{l}}\cdot\Theta_{l+1}^{b_{1}}\cdots\Theta_{l+q^{\prime}}^{b_{q^{\prime}}}\mid({\bf{a}},{\bf{b}})\in\mathbb{Z}_{+}^{l}\times\Lambda^{\prime}_{q^{\prime}}\\}$

in §1.2.2. Our assumption on $A$ guarantees $U({\mathfrak{g}}_{A},e)$ to be an
$A$-subalgebra of $U({\mathfrak{g}},e)$ contained in
$(\text{End}_{{\mathfrak{g}}_{A}}Q_{\chi,A})^{\text{op}}$. By the definition
of $Q_{\chi,A}$ in §1.2.2 and $I_{\chi,A}$ in §1.3.2 we know that $Q_{\chi,A}$
can be identified with the ${\mathfrak{g}}_{A}$-module
$U({\mathfrak{g}}_{A})/I_{\chi,A}$. Hence $U({\mathfrak{g}}_{A},e)$ embeds
into the $A$-algebra
$(U({\mathfrak{g}}_{A})/I_{\chi,A})^{\text{ad}\,\mathfrak{m}_{A}}\cong(Q_{\chi,A})^{\text{ad}\,\mathfrak{m}_{A}}$.
As $Q_{\chi,A}$ is a free $A$-module with basis
$\\{x^{\mathbf{a}}y^{\mathbf{b}}u^{\mathbf{c}}v^{\mathbf{d}}\otimes
1_{\chi}\mid(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})\in{\mathbb{Z}}^{m}_{+}\times\Lambda^{\prime}_{n}\times{\mathbb{Z}}^{s}_{+}\times\Lambda^{\prime}_{t}\\}$,
an easy induction on Kazhdan degree (based on [40, Lemma 4.3] and the formulas
displayed in [40, Lemma 4.2, Theorem 4.5]) shows that

$U({\mathfrak{g}}_{A},e)=(\text{End}_{{\mathfrak{g}}_{A}}Q_{\chi,A})^{\text{op}}\cong
Q_{\chi,A}^{\text{ad}\,\mathfrak{m}_{A}}.$

######  Definition 2.1.

Set the ${\mathds{k}}$-algebra
$T({\mathfrak{g}}_{{\mathds{k}}},e):=U({\mathfrak{g}}_{A},e)\otimes_{A}{{\mathds{k}}}$.
In the following we will call $T(\mathfrak{g}_{\mathds{k}},e)$ a transition
subalgebra.

It is notable that by the definition, $T({\mathfrak{g}}_{{\mathds{k}}},e)$ can
be naturally identified with a subalgebra of the finite $W$-superalgebra
$U({\mathfrak{g}}_{{\mathds{k}}},e)=(\text{End}_{{\mathfrak{g}}_{{\mathds{k}}}}Q_{\chi,{{\mathds{k}}}})^{\text{op}}$
over ${{\mathds{k}}}$. Moreover, $T({\mathfrak{g}}_{{\mathds{k}}},e)$ has a
${{\mathds{k}}}$-basis consisting of all monomials
$\bar{\Theta}_{1}^{a_{1}}\cdots\bar{\Theta}_{l+q^{\prime}}^{b_{q^{\prime}}}$,
where $\bar{\Theta}_{i}:=\Theta_{i}\otimes 1\in
U({\mathfrak{g}}_{A},e)\otimes_{A}{{\mathds{k}}}$ for $1\leqslant i\leqslant
l+q^{\prime}$.

Since all the coefficients of polynomials $F_{ij}$’s for $1\leqslant
i,j\leqslant l+q^{\prime}$ in §1.2.2 are in $\mathbb{Q}$, one can assume the
$F_{ij}$’s are over $A$ after enlarging $A$ if needed. Given a super-
polynomial $g\in A[T_{1},\cdots T_{n}]$, let ${}^{p}{\hskip-1.42271pt}g$
denote the image of $g$ in the polynomial superalgebra
${{\mathds{k}}}[T_{1},\cdots T_{n}]=A[T_{1},\cdots
T_{n}]\otimes_{A}{{\mathds{k}}}$. By the same discussion as [40, Theorem 4.7],
we know that there exist super-polynomials ${}^{p}{\hskip-1.42271pt}F_{ij}$’s
of $l+q^{\prime}$ indeterminants over ${\mathds{k}}$
($i,j=1,\cdots,l+q^{\prime}$) with the first $l$ indeterminants being even,
and the others being odd, such that

$[\bar{\Theta}_{i},\bar{\Theta}_{j}]=^{p}{\hskip-5.69046pt}F_{ij}(\bar{\Theta}_{1},\cdots,\bar{\Theta}_{l+q^{\prime}}),~{}~{}i,j=1,\cdots,l+q^{\prime},$

while the
${}^{p}{\hskip-1.42271pt}F_{ij}(\bar{\Theta}_{1},\cdots,\bar{\Theta}_{l+q^{\prime}})$’s
satisfy the same relations as (1.3) and (1.4).

######  Theorem 2.2.

Maintain the notations as above. Then the $\bar{\Theta}_{i}$’s and
${}^{p}{\hskip-1.42271pt}F_{ij}$’s for $i,j=1,\cdots,l+q^{\prime}$ constitute
a data of generators and defining relations of
$T(\mathfrak{g}_{{\mathds{k}}},e)$, with
$\bar{\Theta}_{1},\cdots,\bar{\Theta}_{l}\in
T({\mathfrak{g}}_{{\mathds{k}}},e)_{\bar{0}}$ and
$\bar{\Theta}_{l+1},\cdots,\bar{\Theta}_{l+q^{\prime}}\in
T({\mathfrak{g}}_{{\mathds{k}}},e)_{\bar{1}}$ as the generators of
$T({\mathfrak{g}}_{{\mathds{k}}},e)$ subject to the relations

$[\bar{\Theta}_{i},\bar{\Theta}_{j}]=^{p}{\hskip-5.69046pt}F_{ij}(\bar{\Theta}_{1},\cdots,\bar{\Theta}_{l+q^{\prime}}),$

where $1\leqslant i,j\leqslant l+q^{\prime}$.

###  2.2. Revisit to reduced $W$-superalgebras with $p$-characters
$\eta\in\chi+(\mathfrak{m}_{{\mathds{k}}}^{\bot})_{\bar{0}}$

Recall in §1.3.2 we have defined the ${\mathfrak{g}}_{{\mathds{k}}}$-module
$Q_{\chi}^{\eta}=Q_{\chi,{{\mathds{k}}}}/J_{\eta}Q_{\chi,{{\mathds{k}}}}$ and
the reduced $W$-superalgebra
$U_{\eta}({\mathfrak{g}}_{{\mathds{k}}},e)=(\text{End}_{{\mathfrak{g}}_{{\mathds{k}}}}Q_{\chi}^{\eta})^{\text{op}}$,
and §1.2.1 shows that $\\{x_{1},\cdots,x_{m},y_{1},\cdots,$
$y_{n}\\}$ is an $A$-basis of ${\mathfrak{p}}_{A}$. Set

$X_{i}:=\left\\{\begin{array}[]{ll}x_{i+l}&\text{if}~{}1\leqslant i\leqslant
m-l;\\\ y_{l+q-m+i}&\text{if}~{}m-l+1\leqslant i\leqslant m+n-l-q;\\\
u_{l+q-m-n+i}&\text{if}~{}m+n-l-q+1\leqslant i\leqslant m+n-l-q+s;\\\
v_{l+q-m-n-s+i}&\text{if}~{}m+n-l-q+s+1\leqslant i\leqslant
m+n-l-q+s+t^{\prime},\end{array}\right.$

where
$t^{\prime}:=\lceil\frac{r}{2}\rceil=\lceil\frac{\text{dim}\,\mathfrak{g}_{\mathds{k}}(-1)_{\bar{1}}}{2}\rceil$.

######  Conventions 2.3.

We will denote $\lceil\frac{r}{2}\rceil$ by $t^{\prime}$ once and for all. It
follows from [35, Theorem 4.3] that
$\text{dim}\,U_{\chi}(\mathfrak{m}_{{\mathds{k}}})=p^{\frac{d_{0}}{2}}2^{\lceil\frac{d_{1}}{2}\rceil}$
and we denote it by $\delta$ afterwards. By the assumption of the notations we
have $\frac{d_{0}}{2}=m-l+s$ and $\lceil\frac{d_{1}}{2}\rceil=n-q+t^{\prime}$.

For
$(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})\in{\mathbb{Z}}_{+}^{m-l}\times\Lambda^{\prime}_{n-q}\times{\mathbb{Z}}_{+}^{s}\times\Lambda^{\prime}_{t^{\prime}}$,
define

$\begin{array}[]{ccl}X^{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}}:&=&X_{1}^{a_{1}}\cdots
X_{m-l}^{a_{m-l}}X_{m-l+1}^{b_{1}}\cdots
X_{m+n-l-q}^{b_{n-q}}X_{m+n-l-q+1}^{c_{1}}\cdots\\\ &&\cdot
X_{m+n-l-q+s}^{c_{s}}X_{m+n-l-q+s+1}^{d_{1}}\cdots
X_{m+n-l-q+s+t^{\prime}}^{d_{t^{\prime}}}\end{array}$

and

$\begin{array}[]{ccl}\bar{X}^{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}}:&=&\bar{X}_{1}^{a_{1}}\cdots\bar{X}_{m-l}^{a_{m-l}}\bar{X}_{m-l+1}^{b_{1}}\cdots\bar{X}_{m+n-l-q}^{b_{n-q}}\bar{X}_{m+n-l-q+1}^{c_{1}}\cdots\\\
&&\cdot\bar{X}_{m+n-l-q+s}^{c_{s}}\bar{X}_{m+n-l-q+s+1}^{d_{1}}\cdots\bar{X}_{m+n-l-q+s+t^{\prime}}^{d_{t^{\prime}}},\end{array}$

elements of $U({\mathfrak{g}}_{A})$ and $U({\mathfrak{g}}_{{\mathds{k}}})$,
respectively. Denote by $\bar{1}_{\chi}$ the image of $1_{\chi}\in
Q_{\chi,{{\mathds{k}}}}$ in $Q^{\eta}_{\chi}$. For $k\in{\mathbb{Z}}_{+}$,
define

$\Lambda_{k}:=\\{(i_{1},\cdots,i_{k})\mid
i_{j}\in{\mathbb{Z}}_{+},~{}0\leqslant i_{j}\leqslant p-1\\}$

with $1\leqslant j\leqslant k$.

######  Lemma 2.4.

The right modules $Q_{\chi,A}$ and $Q_{\chi}^{\eta}$ with
$\eta\in\chi+(\mathfrak{m}_{{\mathds{k}}}^{\bot})_{\bar{0}}$ are free over
$U({\mathfrak{g}}_{A},e)$ and $U_{\eta}({\mathfrak{g}}_{{\mathds{k}}},e)$
respectively. More precisely,

  * (1)

the set $\\{X^{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}}\otimes
1_{\chi}\mid(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})\in{\mathbb{Z}}_{+}^{m-l}\times\Lambda^{\prime}_{n-q}\times{\mathbb{Z}}_{+}^{s}\times\Lambda^{\prime}_{t^{\prime}}\\}$
is a free basis of the $U({\mathfrak{g}}_{A},e)$-module $Q_{\chi,A}$;

  * (2)

the set
$\\{\bar{X}^{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}}\otimes\bar{1}_{\chi}\mid(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})\in\Lambda_{m-l}\times\Lambda^{\prime}_{n-q}\times\Lambda_{s}\times\Lambda^{\prime}_{t^{\prime}}\\}$
is a free basis of the $U_{\eta}({\mathfrak{g}}_{{\mathds{k}}},e)$-module
$Q_{\chi}^{\eta}$.

###### Proof.

The proof is the same as the finite $W$-algebra case, thus will be omitted
here (see [28, Lemma 4.2] and [29, Lemma 2.3]). ∎

###  2.3. Transition for finite $W$-superalgebras

Let $\rho_{{\mathds{k}}}$ denote the representation of
$U({\mathfrak{g}}_{{\mathds{k}}})$ in
$\text{End}_{{\mathds{k}}}Q_{\chi,{{\mathds{k}}}}$. Given a subspace $V$ in
${\mathfrak{g}}_{{\mathds{k}}}$ we denote by $Z_{p}(V)$ the subalgebra of
$p$-center $Z_{p}({\mathfrak{g}}_{{\mathds{k}}})$ generated by all
$\bar{x}^{p}-\bar{x}^{[p]}$ with $\bar{x}\in V_{\bar{0}}$. Clearly, $Z_{p}(V)$
is isomorphic to an (ordinary) polynomial algebra in $\text{dim}\,V_{\bar{0}}$
variables. We will denote $Z_{p}({\mathfrak{g}}_{{\mathds{k}}})$ by $Z_{p}$
for short.

Let $\mathfrak{a}_{{\mathds{k}}}$ be the ${{\mathds{k}}}$-span of
$\bar{X}_{1},\cdots,\bar{X}_{m+n-l-q+s+t^{\prime}}$ in
${\mathfrak{g}}_{{\mathds{k}}}$ and put
$\widetilde{{\mathfrak{p}}}_{{\mathds{k}}}=\mathfrak{a}_{{\mathds{k}}}\oplus{\mathfrak{g}}_{{\mathds{k}}}^{e}$
(resp.
$\widetilde{{\mathfrak{p}}}_{{\mathds{k}}}=\mathfrak{a}_{{\mathds{k}}}\oplus{\mathfrak{g}}_{{\mathds{k}}}^{e}\oplus{{\mathds{k}}}v_{\frac{r+1}{2}}$)
when $d_{1}$ is even (resp. odd), then
$\mathfrak{g}_{\mathds{k}}=\mathfrak{m}_{\mathds{k}}\oplus\widetilde{\mathfrak{p}}_{\mathds{k}}$.

By our assumptions on $x_{l+1},\cdots,x_{m},y_{q+1},\cdots,y_{n}$ and the
inclusion
${\mathfrak{g}}_{{\mathds{k}}}^{f}\subseteq\bigoplus\limits_{i\leqslant
0}{\mathfrak{g}}_{{\mathds{k}}}(i)$, we have that

$\mathfrak{a}_{{\mathds{k}}}=\\{\bar{x}\in\widetilde{{\mathfrak{p}}}_{{\mathds{k}}}\mid(\bar{x},{\mathfrak{g}}_{{\mathds{k}}}^{f})=0\\}\qquad(\text{resp.}~{}\mathfrak{a}_{{\mathds{k}}}\oplus{{\mathds{k}}}v_{\frac{r+1}{2}}=\\{\bar{x}\in\widetilde{{\mathfrak{p}}}_{{\mathds{k}}}\mid(\bar{x},{\mathfrak{g}}_{{\mathds{k}}}^{f})=0\\})$

when $d_{1}$ is even (resp. odd).

######  Theorem 2.5.

For any even nilpotent element
$e\in({\mathfrak{g}}_{{\mathds{k}}})_{\bar{0}}$, we have

  * (1)

$\rho_{{\mathds{k}}}(Z_{p})\cong
Z_{p}(\widetilde{{\mathfrak{p}}}_{{\mathds{k}}})$ as
${{\mathds{k}}}$-algebras.

  * (2)

$U({\mathfrak{g}}_{{\mathds{k}}},e)$ is a free
$\rho_{{\mathds{k}}}(Z_{p})$-module of rank $p^{l}2^{q^{\prime}}$. Especially,
$U({\mathfrak{g}}_{{\mathds{k}}},e)$ is generated by its subalgebras
$T({\mathfrak{g}}_{{\mathds{k}}},e)$ and $\rho_{{\mathds{k}}}(Z_{p})$.

  * (3)

Furthermore, $U({\mathfrak{g}}_{{\mathds{k}}},e)\cong
T({\mathfrak{g}}_{{\mathds{k}}},e)\otimes_{{\mathds{k}}}Z_{p}(\mathfrak{a}_{{\mathds{k}}})$
as ${{\mathds{k}}}$-algebras.

This theorem is a generalization of the finite $W$-algebra case in [29,
Theorem 2.1]. Compared with finite $W$-algebras, the construction of finite
$W$-superalgebras is much more complicated. In particular, some new phenomenon
occurs when $d_{1}$ is odd. Now we will prove the theorem in detail.

###### Proof.

(1) It follows from
${\mathfrak{g}}_{{\mathds{k}}}=\mathfrak{m}_{{\mathds{k}}}\oplus\widetilde{{\mathfrak{p}}}_{{\mathds{k}}}$
that $Z_{p}({\mathfrak{g}}_{{\mathds{k}}})\cong
Z_{p}(\mathfrak{m}_{{\mathds{k}}})\otimes_{{\mathds{k}}}Z_{p}(\widetilde{{\mathfrak{p}}}_{{\mathds{k}}})$
as ${{\mathds{k}}}$-algebras. As
$Z_{p}(\mathfrak{m}_{{\mathds{k}}})\cap\text{Ker}\rho_{{\mathds{k}}}$ is an
ideal of codimension $1$ in $Z_{p}(\mathfrak{m}_{{\mathds{k}}})$, one can
conclude that
$\rho_{{\mathds{k}}}(Z_{p})=\rho_{{\mathds{k}}}(Z_{p}(\widetilde{{\mathfrak{p}}}_{{\mathds{k}}}))$.
As the monomials
$\bar{x}^{\mathbf{a}}\bar{y}^{\mathbf{b}}\bar{u}^{\mathbf{c}}\bar{v}^{\mathbf{d}}\otimes
1_{\chi}$ with
$(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})\in{\mathbb{Z}}_{+}^{m}\times\Lambda^{\prime}_{n}\times{\mathbb{Z}}_{+}^{s}\times\Lambda^{\prime}_{t}$
(recall that
$t=\lfloor\frac{\text{dim}\,{\mathfrak{g}}_{{\mathds{k}}}(-1)_{\bar{1}}}{2}\rfloor$)
form a basis of $Q_{\chi,{{\mathds{k}}}}$, and
$Z_{p}(\widetilde{{\mathfrak{p}}}_{{\mathds{k}}})$ is a polynomial algebra in
$\\{\bar{x}_{i}^{p}-\bar{x}_{i}^{[p]}\mid 1\leqslant i\leqslant
m\\}\cup\\{\bar{u}_{j}^{p}-\bar{u}_{j}^{[p]}\mid 1\leqslant j\leqslant s\\}$,
we have
$Z_{p}(\widetilde{{\mathfrak{p}}}_{{\mathds{k}}})\cap\text{Ker}\rho_{{\mathds{k}}}=\\{0\\}$.
It follows that $\rho_{{\mathds{k}}}(Z_{p})\cong
Z_{p}(\widetilde{{\mathfrak{p}}}_{{\mathds{k}}})$ as
${{\mathds{k}}}$-algebras. This completes the proof of statement (1).

(2) As the proof of the remaining statements are the same for both cases
corresponding to $d_{1}$ being even and odd respectively, it is sufficient for
us to make arguments under the assumption that $d_{1}$ is odd.

First recall that
$S((\widetilde{{\mathfrak{p}}}_{{\mathds{k}}})_{\bar{0}})\cong{{\mathds{k}}}[\chi+(\mathfrak{m}_{{\mathds{k}}}^{\bot})_{\bar{0}}]$
by the discussion of [40, §2.3], hence
$Z_{p}(\widetilde{{\mathfrak{p}}}_{{\mathds{k}}})\cong{{\mathds{k}}}[(\chi+(\mathfrak{m}_{{\mathds{k}}}^{\bot})_{\bar{0}})^{(1)}]$,
where
$(\chi+(\mathfrak{m}_{{\mathds{k}}}^{\bot})_{\bar{0}})^{(1)}\subseteq({\mathfrak{g}}_{{\mathds{k}}}^{*})_{\bar{0}}^{(1)}$
is the Frobenius twist of
$\chi+(\mathfrak{m}_{{\mathds{k}}}^{\bot})_{\bar{0}}$. Then we will manage to
construct a set of free basis of $U(\mathfrak{g}_{\mathds{k}},e)$ as a
$\rho_{{\mathds{k}}}(Z_{p})$-module, via the reduced $W$-superalgebra
$U_{\eta}(\mathfrak{g}_{\mathds{k}},e)$ with
$\eta\in\chi+(\mathfrak{m}_{{\mathds{k}}}^{\bot})_{\bar{0}}$. Now we proceed
by steps.

(2-i) Let us begin with understanding the
$Z_{p}(\widetilde{{\mathfrak{p}}}_{{\mathds{k}}})$-module
$Q_{\chi,{{\mathds{k}}}}$.

As an immediate consequence of [40, Theorem 4.5(1)], we have

$\bar{\Theta}_{k}^{p}(1_{\chi})-(\bar{x}_{k}^{p}+\sum\limits_{|(\mathbf{a},\mathbf{0},\mathbf{c},\mathbf{0})|_{e}=m_{k}+2}\mu^{k}_{\mathbf{a},\mathbf{0},\mathbf{c},\mathbf{0}}\bar{x}^{p\mathbf{a}}\bar{u}^{p\mathbf{c}})\otimes
1_{\chi}\in(Q_{\chi,{{\mathds{k}}}})_{p(m_{k}+2)-1}$ (2.1)

for $1\leqslant k\leqslant l$, where
$\mu^{k}_{\mathbf{a},\mathbf{0},\mathbf{c},\mathbf{0}}\in\mathbb{F}_{p}$.
Since $\bar{x}^{[p]}\in{\mathfrak{g}}_{{\mathds{k}}}(pi)$ whenever
$\bar{x}\in{\mathfrak{g}}_{{\mathds{k}}}(i)$ for all $i\in{\mathbb{Z}}$ (see
the proof of [40, Lemma 2.18]), discussing in the graded algebra
$\text{gr}(U({\mathfrak{g}}_{{\mathds{k}}}))$ under the Kazhdan filtration we
can obtain that

$\text{gr}(\bar{x}_{i}^{p}-\bar{x}_{i}^{[p]})=\text{gr}(\bar{x}_{i})^{p},~{}~{}~{}~{}~{}~{}~{}~{}\text{gr}(\bar{u}_{j}^{p}-\bar{u}_{j}^{[p]})=\text{gr}(\bar{u}_{j})^{p}~{}~{}~{}~{}(1\leqslant
i\leqslant m;~{}~{}1\leqslant j\leqslant s).$ (2.2)

On the other hand, Lemma 2.4(1) implies that the vectors
$\bar{X}^{(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})}\otimes 1_{\chi}$ with

$\begin{array}[]{ccl}(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})&=&(a_{1},\cdots,a_{m-l};b_{1},\cdots,b_{n-q};c_{1},\cdots,c_{s};d_{1},\cdots,d_{t^{\prime}})\\\
&\in&{\mathbb{Z}}_{+}^{m-l}\times\Lambda^{\prime}_{n-q}\times{\mathbb{Z}}_{+}^{s}\times\Lambda^{\prime}_{t^{\prime}}\end{array}$

form a free basis of the right $T({\mathfrak{g}}_{{\mathds{k}}},e)$-module
$Q_{\chi,{{\mathds{k}}}}$. As $Q_{\chi,{{\mathds{k}}}}$ is a Kazhdan-filtrated
$T({\mathfrak{g}}_{{\mathds{k}}},e)$-module, straightforward induction on
filtration degree based on (2.1) and (2.2) shows that
$Q_{\chi,{{\mathds{k}}}}$ is generated as a
$Z_{p}(\widetilde{{\mathfrak{p}}}_{{\mathds{k}}})$-module by the set

$\\{\bar{X}^{(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})}\bar{\Theta}^{(\mathbf{i},\mathbf{j})}\otimes
1_{\chi}\mid(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d},\mathbf{i},\mathbf{j})\in\Lambda_{m-l}\times\Lambda^{\prime}_{n-q}\times\Lambda_{s}\times\Lambda^{\prime}_{t^{\prime}}\times\Lambda_{l}\times\Lambda^{\prime}_{q+1}\\}.$

(2-ii) Let $h$ be an arbitrary element of
$U({\mathfrak{g}}_{{\mathds{k}}},e)$. Then by the above discussion we can
assume that

$\begin{array}[]{cccl}h(1_{\chi})&=&\sum
f_{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d},\mathbf{i},\mathbf{j}}&\bar{X}_{1}^{a_{1}}\cdots\bar{X}_{m-l}^{a_{m-l}}\bar{X}_{m-l+1}^{b_{1}}\cdots\bar{X}_{m+n-l-q}^{b_{n-q}}\bar{X}_{m+n-l-q+1}^{c_{1}}\cdots\\\
&&&\bar{X}_{m+n-l-q+s}^{c_{s}}\bar{X}_{m+n-l-q+s+1}^{d_{1}}\cdots\bar{X}_{m+n-l-q+s+t^{\prime}}^{d_{t^{\prime}}}\cdot\bar{\Theta}_{1}^{i_{1}}\cdots\bar{\Theta}_{l}^{i_{l}}\\\
&&&\bar{\Theta}_{l+1}^{j_{1}}\cdots\bar{\Theta}_{l+q}^{j_{q}}\bar{\Theta}_{l+q+1}^{j_{q+1}}(1_{\chi}),\end{array}$

where
$f_{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d},\mathbf{i},\mathbf{j}}\in
Z_{p}(\widetilde{{\mathfrak{p}}}_{{\mathds{k}}})$ with
$(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d},\mathbf{i},\mathbf{j})$ in the
set
$\Lambda_{m-l}\times\Lambda^{\prime}_{n-q}\times\Lambda_{s}\times\Lambda^{\prime}_{t^{\prime}}\times\Lambda_{l}\times\Lambda^{\prime}_{q+1}$.
For any $\eta\in\chi+(\mathfrak{m}_{{\mathds{k}}}^{\bot})_{\bar{0}}$, the
image of
$f_{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d},\mathbf{i},\mathbf{j}}$ in
$U_{\eta}({\mathfrak{g}}_{{\mathds{k}}})$ is a scalar in ${{\mathds{k}}}$,
which shall be denoted by
$\eta(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d},\mathbf{i},\mathbf{j})$.

Suppose
$f_{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d},\mathbf{i},\mathbf{j}}\neq 0$
for a nonzero
$(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})\in\Lambda_{m-l}\times\Lambda^{\prime}_{n-q}\times\Lambda_{s}\times\Lambda^{\prime}_{t^{\prime}}$
and some $(\mathbf{i},\mathbf{j})\in\Lambda_{l}\times\Lambda^{\prime}_{q+1}$.
Then there exists $\eta\in\chi+(\mathfrak{m}_{{\mathds{k}}}^{\bot})_{\bar{0}}$
such that
$\eta(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d},\mathbf{i},\mathbf{j})\neq
0$. Let $h_{\eta}$ be the image of $h\in U({\mathfrak{g}}_{{\mathds{k}}},e)$
in
$U_{\eta}({\mathfrak{g}}_{{\mathds{k}}},e)=(\text{End}_{{\mathfrak{g}}_{{\mathds{k}}}}Q_{\chi}^{\eta})^{\text{op}}$.
[40, Remark 3.9] shows that there exist even elements
$\theta_{1},\cdots,\theta_{l}\in
U_{\eta}({\mathfrak{g}}_{{\mathds{k}}},e)_{\bar{0}}$ and odd elements
$\theta_{l+1},\cdots,\theta_{l+q+1}\in
U_{\eta}({\mathfrak{g}}_{{\mathds{k}}},e)_{\bar{1}}$ in the same sense as in
[40, Corollary 3.6], such that the monomials

$\theta_{1}^{a_{1}}\cdots\theta_{l}^{a_{l}}\theta_{l+1}^{b_{1}}\cdots\theta_{l+q+1}^{b_{q+1}}$

with $0\leqslant a_{k}\leqslant p-1$ for $1\leqslant k\leqslant l$ and
$0\leqslant b_{k}\leqslant 1$ for $1\leqslant k\leqslant q+1$ form a
${{\mathds{k}}}$-basis of $U_{\eta}({\mathfrak{g}}_{{\mathds{k}}},e)$.
Therefore, $h_{\eta}(\bar{1}_{\chi})$ is a ${{\mathds{k}}}$-linear combination
of
$\theta_{1}^{i_{1}}\cdots\theta_{l}^{i_{l}}\theta_{l+1}^{j_{1}}\cdots\theta_{l+q}^{j_{q}}\theta_{l+q+1}^{j_{q+1}}(\bar{1}_{\chi})$
with

$(i_{1},\cdots,i_{l};j_{1},\cdots,j_{q};j_{q+1})\in\Lambda_{l}\times\Lambda^{\prime}_{q}\times\Lambda^{\prime}_{1}.$

By Lemma 2.4(2), the set

$\\{\bar{X}^{(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})}\otimes\bar{1}_{\chi}\mid(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})\in\Lambda_{m-l}\times\Lambda^{\prime}_{n-q}\times\Lambda_{s}\times\Lambda^{\prime}_{t^{\prime}}\\}$

is a free basis of the right
$U_{\eta}({\mathfrak{g}}_{{\mathds{k}}},e)$-module $Q_{\chi}^{\eta}$. Since
$\eta(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d},\mathbf{i},\mathbf{j})\neq
0$ and
$\theta_{1}^{i_{1}}\cdots\theta_{l}^{i_{l}}\theta_{l+1}^{j_{1}}\cdots\theta_{l+q}^{j_{q}}\theta_{l+q+1}^{j_{q+1}}$
is the image of
$\bar{\Theta}_{1}^{i_{1}}\cdots\bar{\Theta}_{l}^{i_{l}}\bar{\Theta}_{l+1}^{j_{1}}\cdots\bar{\Theta}_{l+q}^{j_{q}}\bar{\Theta}_{l+q+1}^{j_{q+1}}$
in $U_{\eta}({\mathfrak{g}}_{{\mathds{k}}},e)$, it is now evident that
$h_{\eta}(\bar{1}_{\chi})$ cannot be a ${{\mathds{k}}}$-linear combination of
$\theta_{1}^{i_{1}}\cdots\theta_{l}^{i_{l}}\theta_{l+1}^{j_{1}}\cdots$
$\theta_{l+q}^{j_{q}}\theta_{l+q+1}^{j_{q+1}}(\bar{1}_{\chi})$ with

$(i_{1},\cdots,i_{l};j_{1},\cdots,j_{q};j_{q+1})\in\Lambda_{l}\times\Lambda^{\prime}_{q}\times\Lambda^{\prime}_{1}.$

This contradiction shows that
$f_{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d},\mathbf{i},\mathbf{j}}=0$
unless $(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})=\mathbf{0}$. As a
consequence,

$\displaystyle\\{\bar{\Theta}_{1}^{i_{1}}\cdots\bar{\Theta}_{l}^{i_{l}}\bar{\Theta}_{l+1}^{j_{1}}\cdots\bar{\Theta}_{l+q}^{j_{q}}\bar{\Theta}_{l+q+1}^{j_{q+1}}\mid(\mathbf{i},\mathbf{j})\in\Lambda_{l}\times\Lambda^{\prime}_{q+1}\\}$
(2.3)

generates $U({\mathfrak{g}}_{{\mathds{k}}},e)$ as a
$Z_{p}(\widetilde{{\mathfrak{p}}}_{{\mathds{k}}})$-module. Specialising at a
suitable $\eta\in\chi+(\mathfrak{m}_{{\mathds{k}}}^{\bot})_{\bar{0}}$ and
applying [40, Remark 3.9] we further deduce that the set

$\displaystyle\\{\bar{\Theta}_{1}^{i_{1}}\cdots\bar{\Theta}_{l}^{i_{l}}\bar{\Theta}_{l+1}^{j_{1}}\cdots\bar{\Theta}_{l+q}^{j_{q}}\bar{\Theta}_{l+q+1}^{j_{q+1}}\mid(\mathbf{i},\mathbf{j})\in\Lambda_{l}\times\Lambda^{\prime}_{q+1}\\}$
(2.4)

is a free basis of the
$Z_{p}(\widetilde{{\mathfrak{p}}}_{{\mathds{k}}})$-module
$U({\mathfrak{g}}_{{\mathds{k}}},e)$. Then
$U({\mathfrak{g}}_{{\mathds{k}}},e)$ is a free
$\rho_{{\mathds{k}}}(Z_{p})$-module of rank $p^{l}2^{q+1}$. Note that the
elements of (2.4) are in the ${\mathds{k}}$-algebra
$T(\mathfrak{g}_{\mathds{k}},e)$, then the second part of statement (2)
follows. We complete the proof of statement (2).

(3) We first claim that

$\displaystyle u\in T({\mathfrak{g}}_{{\mathds{k}}},e)\cdot
Z_{p}(\mathfrak{a}_{{\mathds{k}}})~{}\text{for each}~{}u\in
U({\mathfrak{g}}_{{\mathds{k}}},e).$ (2.5)

We proceed the proof of Claim (2.5) by steps, starting with some necessary
preparations.

(3-i) Note that every ${\mathfrak{g}}_{{\mathds{k}}}$-endomorphism of
$Q_{\chi,{{\mathds{k}}}}$ is uniquely determined by its value at $1_{\chi}$.
For a nonzero $u\in U({\mathfrak{g}}_{{\mathds{k}}},e)$ with highest Kazhdan
degree $n(u)$, we can write

$u(1_{\chi})=\sum\limits_{|(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})|_{e}\leqslant
n(u)}\lambda_{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}}\bar{x}^{\mathbf{a}}\bar{y}^{\mathbf{b}}\bar{u}^{\mathbf{c}}\bar{v}^{\mathbf{d}}\otimes
1_{\chi},$

where $\lambda_{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}}\neq 0$ for at
least one $(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})$ with
$|(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})|_{e}=n(u)$. For
$k\in{\mathbb{Z}}_{+}$ put

$\Lambda^{k}(u):=\\{(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})\in{\mathbb{Z}}_{+}^{m}\times\Lambda^{\prime}_{n}\times{\mathbb{Z}}_{+}^{s}\times\Lambda^{\prime}_{t}\mid\lambda_{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}}\neq
0~{}\&~{}|(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})|_{e}=k\\},$

and denote by $\Lambda^{\text{max}}(u)$ the set of all
$(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})\in\Lambda^{n(u)}(u)$ for which
the quantity $n(u)-|\mathbf{a}|-|\mathbf{b}|-|\mathbf{c}|-|\mathbf{d}|$
assumes its maximum value. This maximum value will be denoted by $N(u)$. For
each $(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})\in\Lambda^{\text{max}}$,
let
$\bar{x}_{i}\in{\mathfrak{g}}_{{\mathds{k}}}(k_{i})_{\bar{0}},~{}\bar{y}_{j}\in{\mathfrak{g}}_{{\mathds{k}}}(k^{\prime}_{j})_{\bar{1}}$
for $1\leqslant i\leqslant m$ and $1\leqslant j\leqslant n$ with
$k_{i},k_{j}^{\prime}\in{\mathbb{Z}}_{+}$, then we have

$\begin{array}[]{cl}&|(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})|_{e}-|\mathbf{a}|-|\mathbf{b}|-|\mathbf{c}|-|\mathbf{d}|\\\
=&\sum\limits_{i=1}^{m}(k_{i}+2)a_{i}+\sum\limits_{i=1}^{n}(k^{\prime}_{i}+2)b_{i}+\sum\limits_{i=1}^{s}c_{i}+\sum\limits_{i=1}^{t}d_{i}-|\mathbf{a}|-|\mathbf{b}|-|\mathbf{c}|-|\mathbf{d}|\geqslant
0.\end{array}$

Consequently, $n(u),~{}N(u)\in{\mathbb{Z}}_{+}$ and $n(u)\geqslant N(u)$.

(3-ii) The previous arguments in step (2) along with [40, Theorem 4.5(1)] show
that

$\begin{array}[]{ll}\Lambda^{\text{max}}(\bar{\Theta}_{i})=\\{(\mathbf{e}_{i},\mathbf{0},\mathbf{0},\mathbf{0})\\}&\text{for}~{}1\leqslant
i\leqslant l;\\\
\Lambda^{\text{max}}(\rho_{{\mathds{k}}}(\bar{x}_{i}^{p}-\bar{x}_{i}^{[p]}))=\\{(p\mathbf{e}_{i},\mathbf{0},\mathbf{0},\mathbf{0})\\}&\text{for}~{}1\leqslant
i\leqslant m;\\\
\Lambda^{\text{max}}(\rho_{{\mathds{k}}}(\bar{u}_{j}^{p}-\bar{u}_{j}^{[p]}))=\\{(\mathbf{0},\mathbf{0},p\mathbf{e}_{j},\mathbf{0})\\}&\text{for}~{}1\leqslant
j\leqslant s;\\\
\Lambda^{\text{max}}(\bar{\Theta}_{k})=\\{(\mathbf{0},\mathbf{e}_{k-l},\mathbf{0},\mathbf{0})\\}&\text{for}~{}l+1\leqslant
k\leqslant l+q;\\\
\Lambda^{\text{max}}(\bar{\Theta}_{l+q+1})=\\{(\mathbf{0},\mathbf{0},\mathbf{0},\mathbf{e}_{t})\\}.&\end{array}$

Here $\mathbf{e}_{i}=(\delta_{i1},\delta_{i2},\cdots,\delta_{ij})$ is a tuple
with $j$ entries for $j\in{\mathbb{Z}}_{+}$, and $\delta_{ik}$ is the
Kronecker function for $k=1,\cdots,j$.

Since $Q_{\chi,{{\mathds{k}}}}$ is a Kazhdan filtrated
$U({\mathfrak{g}}_{{\mathds{k}}})$-module, this implies that

$\begin{array}[]{ll}&\Lambda^{\text{max}}(\prod\limits_{i=1}^{m}\rho_{{\mathds{k}}}(\bar{x}_{i}^{p}-\bar{x}_{i}^{[p]})^{a_{i}}\cdot\prod\limits_{i=1}^{s}\rho_{{\mathds{k}}}(\bar{u}_{i}^{p}-\bar{u}_{i}^{[p]})^{b_{i}}\cdot\bar{\Theta}_{1}^{c_{1}}\cdots\bar{\Theta}_{l}^{c_{l}}\bar{\Theta}_{l+1}^{d_{1}}\cdots\bar{\Theta}_{l+q}^{d_{q}}\bar{\Theta}_{l+q+1}^{d_{q+1}})\\\
=&\\{(\sum\limits_{i=1}^{m}pa_{i}\mathbf{e}_{i}+\sum\limits_{j=1}^{l}c_{j}\mathbf{e}_{j},\sum\limits_{i=1}^{q}d_{i}\mathbf{e}_{i},\sum\limits_{i=1}^{s}pb_{i}\mathbf{e}_{i},d_{q+1}\mathbf{e}_{t})\\}\end{array}$

for all
$(a_{1},\cdots,a_{m};b_{1},\cdots,b_{s};c_{1},\cdots,c_{l};d_{1},\cdots,d_{q+1})\in{\mathbb{Z}}_{+}^{m}\times{\mathbb{Z}}_{+}^{s}\times\Lambda_{l}\times\Lambda^{\prime}_{q+1}$.

Thanks to statement (2), $U({\mathfrak{g}}_{{\mathds{k}}},e)$ is generated as
a $Z_{p}(\widetilde{{\mathfrak{p}}}_{{\mathds{k}}})$-module by the set

$\\{\bar{\Theta}_{1}^{i_{1}}\cdots\bar{\Theta}_{l}^{i_{l}}\bar{\Theta}_{l+1}^{j_{1}}\cdots\bar{\Theta}_{l+q}^{j_{q}}\bar{\Theta}_{l+q+1}^{j_{q+1}}\mid(\mathbf{i},\mathbf{j})\in\Lambda_{l}\times\Lambda^{\prime}_{q+1}\\}.$

It follows that for every $u\in U({\mathfrak{g}}_{{\mathds{k}}},e)$ with
$(n(u),N(u))=(d,d^{\prime})$ there exists a ${{\mathds{k}}}$-linear
combination $u^{\prime}$ of the endomorphisms

$u(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}):=\prod\limits_{i=1}^{m}\rho_{{\mathds{k}}}(\bar{x}_{i}^{p}-\bar{x}_{i}^{[p]})^{a_{i}}\cdot\prod\limits_{i=1}^{s}\rho_{{\mathds{k}}}(\bar{u}_{i}^{p}-\bar{u}_{i}^{[p]})^{b_{i}}\cdot\bar{\Theta}_{1}^{c_{1}}\cdots\bar{\Theta}_{l}^{c_{l}}\bar{\Theta}_{l+1}^{d_{1}}\cdots\bar{\Theta}_{l+q}^{d_{q}}\bar{\Theta}_{l+q+1}^{d_{q+1}}$

for all
$(a_{1},\cdots,a_{m};b_{1},\cdots,b_{s};c_{1},\cdots,c_{l};d_{1},\cdots,d_{q+1})\in{\mathbb{Z}}_{+}^{m}\times{\mathbb{Z}}_{+}^{s}\times\Lambda_{l}\times\Lambda^{\prime}_{q+1}$
with
$\Lambda^{\text{max}}(u(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}))\subseteq\Lambda^{\text{max}}(u)$
such that either $n(u-u^{\prime})<d$, or $n(u-u^{\prime})=d$ and
$N(u-u^{\prime})<d^{\prime}$.

(3-iii) Let $\Omega:=\\{(n_{1},n_{2})\in\mathbb{Z}_{+}^{2}\mid n_{1}\geq
n_{2}\\}$ be a totally-ordered set with tuples ordered lexicographically. Due
to the arguments in (3-i), $(n(u),N(u))\in\Omega$ for all $u\in
U(\mathfrak{g}_{\mathds{k}},e)$. Now we prove Claim (2.5) by induction on
$(n(u),N(u))$ in the totally-ordered set $\Omega$. The claim is clearly valid
when $(n(u),N(u))=(0,0)$. Assume that $u\in
T({\mathfrak{g}}_{{\mathds{k}}},e)\cdot Z_{p}(\mathfrak{a}_{{\mathds{k}}})$
for all nonzero $u\in U({\mathfrak{g}}_{{\mathds{k}}},e)$ with
$(n(u),N(u))\prec(d,d^{\prime})$. Now let $u\in
U({\mathfrak{g}}_{{\mathds{k}}},e)$ be such that $(n(u),N(u))=(d,d^{\prime})$.
By the preceding remark we know that there exists
$u^{\prime}=\sum\limits_{(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})}\lambda_{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}}u(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})$
with
$\Lambda^{\text{max}}(u(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}))\subseteq\Lambda^{\text{max}}(u)$
for all $(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})$ with
$\lambda_{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}}\neq 0$ such that
$(n(u-u^{\prime}),N(u-u^{\prime}))\prec(d,d^{\prime})$. Set

$\begin{array}[]{lll}v(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})&:=&u((0,\cdots,0,a_{l+1},\cdots,a_{m}),\mathbf{b},\mathbf{0},\mathbf{0})\cdot\\\
&&\prod\limits_{i=1}^{l}\bar{\Theta}_{i}^{pa_{i}}\cdot(\bar{\Theta}_{1}^{c_{1}}\cdots\bar{\Theta}_{l}^{c_{l}}\bar{\Theta}_{l+1}^{d_{1}}\cdots\bar{\Theta}_{l+q}^{d_{q}}\bar{\Theta}_{l+q+1}^{d_{q+1}}).\end{array}$

Using (2.1) it is easy to observe that
$\Lambda^{\text{max}}(u(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}))=\Lambda^{\text{max}}(v(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}))$
and

$\begin{array}[]{lll}&(n(u(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})-v(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})),N(u(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})-v(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})))\\\
\prec&(n(u(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})),N(u(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}))).\end{array}$

We now put
$u^{{}^{\prime\prime}}:=\sum\limits_{(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})}\lambda_{\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}}v(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})$,
an element of $T({\mathfrak{g}}_{{\mathds{k}}},e)\cdot
Z_{p}(\mathfrak{a}_{{\mathds{k}}})$. Because
$(n(u-u^{{}^{\prime\prime}}),N(u-u^{{}^{\prime\prime}}))\prec(n(u),N(u))$, the
equality
$U({\mathfrak{g}}_{{\mathds{k}}},e)=T({\mathfrak{g}}_{{\mathds{k}}},e)\cdot
Z_{p}(\mathfrak{a}_{{\mathds{k}}})$ follows by induction on the length of
$(d,d^{\prime})$ in the linearly ordered set $(\Omega,\prec)$. We complete the
proof of Claim (2.5).

Next we will finish the proof of statement (3). By Lemma 2.4(1) and the
procedure of “modular $p$ reduction” we know that the vectors
$\bar{X}^{(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})}\otimes 1_{\chi}$ with

$\begin{array}[]{lll}(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})&=&(a_{1},\cdots,a_{m-l};b_{1},\cdots,b_{n-q};c_{1},\cdots,c_{s};d_{1},\cdots,d_{t^{\prime}})\\\
&\in&{\mathbb{Z}}_{+}^{m-l}\times\Lambda^{\prime}_{n-q}\times{\mathbb{Z}}_{+}^{s}\times\Lambda^{\prime}_{t^{\prime}}\end{array}$

form a free basis of the right $T({\mathfrak{g}}_{{\mathds{k}}},e)$-module
$Q_{\chi,{{\mathds{k}}}}$. Since (2.2) shows that $\bar{X}_{i}^{p}$ and
$\bar{X}_{i}^{p}-\bar{X}_{i}^{[p]}$ have the same Kazhdan degree in
$U({\mathfrak{g}}_{{\mathds{k}}})$ for $1\leqslant i\leqslant m-l$ and
$m+n-l-q+1\leqslant i\leqslant m+n-l-q+s$ respectively, and
$Q_{\chi,{{\mathds{k}}}}$ is a Kazhdan filtered
$U({\mathfrak{g}}_{{\mathds{k}}})$-module, it follows that the vectors

$\prod\limits_{i=1}^{m-l}\prod\limits_{j=m+n-l-q+1}^{m+n-l-q+s}\rho_{{\mathds{k}}}(\bar{X}_{i}^{p}-\bar{X}_{i}^{[p]})^{a_{i}}\rho_{{\mathds{k}}}(\bar{X}_{j}^{p}-\bar{X}_{j}^{[p]})^{b_{j}}\cdot\bar{\Theta}_{1}^{c_{1}}\cdots\bar{\Theta}_{l}^{c_{l}}\bar{\Theta}_{l+1}^{d_{1}}\cdots\bar{\Theta}_{l+q}^{d_{q}}\bar{\Theta}_{l+q+1}^{d_{q+1}}$

are linearly independent, where
$(\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d})\in{\mathbb{Z}}_{+}^{m-l}\times{\mathbb{Z}}_{+}^{s}\times{\mathbb{Z}}_{+}^{l}\times\Lambda^{\prime}_{q+1}$.

Combining all above, we have an isomorphism between ${{\mathds{k}}}$-algebras

$U({\mathfrak{g}}_{{\mathds{k}}},e)\cong
T({\mathfrak{g}}_{{\mathds{k}}},e)\otimes_{{\mathds{k}}}Z_{p}(\mathfrak{a}_{{\mathds{k}}}).$

By the analysis at the beginning of step (2), we complete the proof. ∎

With the above theorem, we can define a “reduced” quotient algebra of
$U(\mathfrak{g}_{\mathds{k}},e)$, analogous to the reduced enveloping algebra
of a restricted Lie (super)algebra, by

$\widetilde{U}_{\eta}({\mathfrak{g}}_{{\mathds{k}}},e):=U({\mathfrak{g}}_{{\mathds{k}}},e)\otimes_{Z_{p}(\widetilde{{\mathfrak{p}}}_{{\mathds{k}}})}{{\mathds{k}}}_{\eta}.$

######  Lemma 2.6.

For any given $\eta\in\chi+(\mathfrak{m}_{{\mathds{k}}}^{\bot})_{\bar{0}}$,
the above “reduced” quotient algebras are isomorphic to the reduced
$W$-superalgebras, i.e.
$\widetilde{U}_{\eta}({\mathfrak{g}}_{{\mathds{k}}},e)\cong
U_{\eta}(\mathfrak{g}_{\mathds{k}},e)$.

###### Proof.

The canonical projection $Q_{\chi,{{\mathds{k}}}}\twoheadrightarrow
Q_{\chi,{{\mathds{k}}}}/J_{\eta}Q_{\chi,{{\mathds{k}}}}=Q_{\chi}^{\eta}$ gives
rise to an algebra homomorphism
$\rho_{\eta}:\widetilde{U}_{\eta}({\mathfrak{g}}_{{\mathds{k}}},e)\rightarrow(\text{End}_{{\mathfrak{g}}_{{\mathds{k}}}}Q_{\chi}^{\eta})^{\text{op}}=U_{\eta}({\mathfrak{g}}_{{\mathds{k}}},e)$.
As $\text{dim}\,\widetilde{U}_{\eta}({\mathfrak{g}}_{{\mathds{k}}},e)\leqslant
p^{l}2^{q^{\prime}}$ by Theorem 2.5(2), [40, Remark 3.9] yields that
$\rho_{\eta}$ is an algebra isomorphism. We complete the proof. ∎

###  2.4. Transition for minimal dimensional representations

At the beginning of this subsection, first notice the following important
fact:

######  Proposition 2.7.

Assume that $d_{1}$ is odd. Then the finite $W$-superalgebra
$U({\mathfrak{g}},e)$ over ${\mathbb{C}}$ can not afford a one-dimensional
representation.

###### Proof.

Recall that when $d_{1}$ is odd, §1.2.2 shows that there is an element
$\Theta_{l+q+1}=v_{\frac{r+1}{2}}\otimes 1_{\chi}\in
U({\mathfrak{g}},e)_{\bar{1}}$, then

$\Theta_{l+q+1}^{2}(1_{\chi})=v_{\frac{r+1}{2}}^{2}\otimes
1_{\chi}=\frac{1}{2}[v_{\frac{r+1}{2}},v_{\frac{r+1}{2}}]\otimes
1_{\chi}=\frac{1}{2}\chi([v_{\frac{r+1}{2}},v_{\frac{r+1}{2}}])\otimes
1_{\chi}=\frac{1}{2}\otimes 1_{\chi}.$

Thus $\Theta_{l+q+1}^{2}=\frac{1}{2}\text{id}$.

For any $U({\mathfrak{g}},e)$-module $M$, let $0\neq v\in M$ be a
${\mathbb{Z}}_{2}$-homogeneous element. We claim that $\Theta_{l+q+1}.v\neq
0$. If not, i.e. $\Theta_{l+q+1}.v=0$, then $\Theta_{l+q+1}^{2}.v=0$. However,
by the preceding remark we have $\Theta_{l+q+1}^{2}.v=\frac{1}{2}v$, a
contradiction. Therefore, $\Theta_{l+q+1}.v$ is a nonzero element in $M$,
which obviously shares the different parity from that of $v$. Thus the
dimension of any $U({\mathfrak{g}},e)$-module (as a vector space) is at least
two, and the algebra $U({\mathfrak{g}},e)$ can not afford a one-dimensional
representation in this case. ∎

######  Remark 2.8.

Recall that in §1.2.3 we have obtained an algebra isomorphism

$\begin{array}[]{lcll}\phi:&(\text{End}_{\mathfrak{g}}Q_{\chi})^{\text{op}}&\cong&Q_{\chi}^{\text{ad}\,\mathfrak{m}}\\\
&\Theta&\mapsto&\Theta(1_{\chi}).\end{array}$

In the paper we will often identify
$U({\mathfrak{g}},e)=(\text{End}_{\mathfrak{g}}Q_{\chi})^{\text{op}}$ with
$Q_{\chi}^{\text{ad}\,\mathfrak{m}}$ as ${\mathbb{C}}$-algebras, and which
will cause no confusion. For any ${\mathbb{Z}}_{2}$-homogeneous elements
$\Theta_{1},\Theta_{2}\in Q_{\chi}^{\text{ad}\,\mathfrak{m}}$, since
$\phi(\Theta_{1}\cdot\Theta_{2})=\phi(\Theta_{1})\phi(\Theta_{2})$, we can
identify $\Theta_{1}\cdot\Theta_{2}$ in $U({\mathfrak{g}},e)$ with
$\Theta_{1}(1_{\chi})\cdot\Theta_{2}(1_{\chi})$ in
$Q_{\chi}^{\text{ad}\,\mathfrak{m}}$. When $d_{1}$ is odd, the element
$\Theta_{l+q+1}$ in $(\text{End}_{\mathfrak{g}}Q_{\chi})^{\text{op}}$ can be
considered as the element $v_{\frac{r+1}{2}}\otimes 1_{\chi}$ in
$Q_{\chi}^{\text{ad}\,\mathfrak{m}}$, and for any
${\mathbb{Z}}_{2}$-homogeneous element $v$ in a $U({\mathfrak{g}},e)$-module
$M$, we have $\Theta_{l+q+1}^{2}.v=\frac{1}{2}v$ by Proposition 2.7.

Now we are in the position to talk about the transiting role of the transition
subalgebras for the minimal dimensions of modular representations of basic Lie
superalgebras.

######  Proposition 2.9.

Keep the above notations. If $p\gg 0$ for the field
${\mathds{k}}=\overline{\mathbb{F}}_{p}$, the following hold.

(1) Assume that $d_{1}$ is even. Then the following items are equivalent:

  * (1-i)

the transition subalgebra $T(\mathfrak{g}_{\mathds{k}},e)$ admits one-
dimensional representations;

  * (1-ii)

there exists $\eta\in\chi+(\mathfrak{m}_{{\mathds{k}}}^{\bot})_{\bar{0}}$ such
that $U_{\eta}(\mathfrak{g}_{\mathds{k}})$ admits irreducible representations
of dimension $p^{\frac{d_{0}}{2}}2^{\frac{d_{1}}{2}}$.

(2) Assume that $d_{1}$ is odd. Then the following items are equivalent:

  * (2-i)

the transition subalgebra $T(\mathfrak{g}_{\mathds{k}},e)$ admits two-
dimensional representations;

  * (2-ii)

there exists $\eta\in\chi+(\mathfrak{m}_{{\mathds{k}}}^{\bot})_{\bar{0}}$ such
that $U_{\eta}(\mathfrak{g}_{\mathds{k}})$ admits irreducible representations
of dimension $p^{\frac{d_{0}}{2}}2^{\frac{d_{1}+1}{2}}$.

###### Proof.

Let us first prove statement (2).

“(2-i)$\Rightarrow$ (2-ii)”: Recall that there is a ${{\mathds{k}}}$-algebra
isomorphism $U({\mathfrak{g}}_{{\mathds{k}}},e)\cong
T({\mathfrak{g}}_{{\mathds{k}}},e)\otimes_{{\mathds{k}}}Z_{p}(\mathfrak{a}_{{\mathds{k}}})$
by Theorem 2.5(3). Thus the assumption (2-i) implies that the
${{\mathds{k}}}$-algebra $U({\mathfrak{g}}_{{\mathds{k}}},e)$ affords a two-
dimensional representation too; we denote it by $\nu$ with the representation
space $V$.

Let $v_{\bar{0}}\in V_{\bar{0}}$ be a nonzero even vector in $V$. The proof of
Proposition 2.7 shows that $\Theta_{l+q+1}.v_{\bar{0}}\in V_{\bar{1}}$ is a
nonzero odd vector, and we denote it by $v_{\bar{1}}$. Then $V$ is
${\mathds{k}}$-spanned by $v_{\bar{0}}$ and $v_{\bar{1}}$ as a vector space.
For any $\bar{x}\in({\mathfrak{g}}_{{\mathds{k}}})_{\bar{0}}$, since
$\bar{x}^{p}-\bar{x}^{[p]}\in Z_{p}({\mathfrak{g}}_{{\mathds{k}}})$ is central
in $U({\mathfrak{g}}_{{\mathds{k}}})$, we have
$[\rho(\bar{x}^{p}-\bar{x}^{[p]}),\Theta_{l+q+1}]=0$. Therefore, both
${{\mathds{k}}}v_{\bar{0}}$ and ${{\mathds{k}}}v_{\bar{1}}$ are one-
dimensional representations of $\rho_{{\mathds{k}}}(Z_{p})$, decided by the
same function on $(\mathfrak{g}_{\mathds{k}})_{\bar{0}}$. By Theorem 2.5,
$\rho_{{\mathds{k}}}(Z_{p})\cap\text{ker}\nu$ is a maximal ideal of the
algebra $\rho_{{\mathds{k}}}(Z_{p})\cong
Z_{p}(\widetilde{{\mathfrak{p}}}_{{\mathds{k}}})\cong{{\mathds{k}}}[(\chi+(\mathfrak{m}_{{\mathds{k}}}^{\bot})_{\bar{0}})^{(1)}]$.
Then there exists $\eta\in\chi+(\mathfrak{m}_{{\mathds{k}}}^{\bot})_{\bar{0}}$
such that
$\rho_{{\mathds{k}}}(\bar{x}^{p}-\bar{x}^{[p]}-\eta(\bar{x})^{p})\in\text{Ker}\nu$
for all $\bar{x}\in({\mathfrak{g}}_{{\mathds{k}}})_{\bar{0}}$. Our choice of
$\eta$ ensures that the “reduced” quotient algebra
$\widetilde{U}_{\eta}({\mathfrak{g}}_{{\mathds{k}}},e)$ affords a two-
dimensional representation. It follows from Lemma 2.6 and Theorem 1.3 that the
${\mathds{k}}$-algebra $U_{\eta}({\mathfrak{g}}_{{\mathds{k}}})$ has an
irreducible representation of dimension
$p^{\frac{d_{0}}{2}}2^{\frac{d_{1}+1}{2}}$.

“(2-ii)$\Rightarrow$(2-i)”: Conversely, under the assumption (2-ii) it follows
from Theorem 1.3 that the reduced $W$-superalgebra
$U_{\eta}(\mathfrak{g}_{\mathds{k}},e)$ admits a two-dimensional
representation, thereby does the ${\mathds{k}}$-algebra
$\widetilde{U}_{\eta}(\mathfrak{g}_{\mathds{k}},e)$, then the “reduced”
quotient of $U(\mathfrak{g}_{\mathds{k}},e)$ by Lemma 2.6. Since the
transition subalgebra $T({\mathfrak{g}}_{{\mathds{k}}},e)$ is a subalgebra of
$U({\mathfrak{g}}_{{\mathds{k}}},e)$ by the definition,
$T(\mathfrak{g}_{\mathds{k}},e)$ also affords a two-dimensional
representation.

The same arguments also go through for the proof of statement (1), which will
be omitted here. We complete the proof. ∎

##  3\. Conjectural one-dimensional representations for finite
$W$-superalgebras when $d_{1}$ is even

In this and the next sections we proceed to investigate small representations
for the finite $W$-superalgebra $U({\mathfrak{g}}_{\mathbb{F}},e)$ both over
the field of complex numbers $\mathbb{F}={\mathbb{C}}$ and over a field
$\mathbb{F}={{\mathds{k}}}$ of positive characteristic. We will find that the
parity of $d_{1}$ plays a key role for the dimensions of the small
representations of $U({\mathfrak{g}}_{\mathbb{F}},e)$, which is significantly
different from the finite $W$-algebras. We will present a plausible conjecture
for such dimensions, and demonstrate the conjecture with some examples. Based
on these results, we will discuss the accessibility of the lower bounds of
dimensions predicted in the super Kac-Weisfeiler property [35, Theorems 4.3
and 5.6] in §5. For simplicity we will always assume that the characteristic
of the field ${{\mathds{k}}}=\overline{\mathbb{F}}_{p}$ satisfies $p\gg 0$
unless otherwise specified, in this and the next sections.

For the ordinary finite $W$-algebra counterpart of the above issue, there are
some remarkable work and exciting progress (cf. [29]). However, when we turn
to the study of the finite $W$-superalgebra case, the tool available is very
limited. Therefore, the issue of minimal dimensions for the representations of
finite $W$-superalgebras over ${\mathbb{C}}$ is in a position of reasonably
estimating, but far from the solution.

###  3.1. On the minimal-dimension conjecture when $d_{1}$ is even

Recall that the parity of $d_{1}$ plays the key role for the construction of
finite $W$-superalgebra $U({\mathfrak{g}},e)$ in [40, Theorem 4.5]. Based on
the different parity of $d_{1}$, for each case we will consider separately.
This section is devoted to the case when $d_{1}$ is even.

######  Conjecture 3.1.

When $d_{1}$ is even, the $\mathbb{C}$-algebra $U({\mathfrak{g}},e)$ affords a
one-dimensional representation.

Under the assumption that Conjecture 3.1 holds, we can assume that this one-
dimensional representation $\textsf{O}:={\mathbb{C}}\textsf{o}$ is generated
by a nonzero vector o. Now we will go on investigating the consequences.

###  3.2. The analogue of communicative quotients for finite $W$-superalgebras
in the even case

Recall that the ${\mathbb{C}}$-algebra $U({\mathfrak{g}},e)$ is generated by
${\mathbb{Z}}_{2}$-homogeneous elements $\Theta_{1},\cdots,\Theta_{l}\in
U({\mathfrak{g}},e)_{\bar{0}}$ and $\Theta_{l+1},\cdots,\Theta_{l+q}\in
U({\mathfrak{g}},e)_{\bar{1}}$ in [40, Theorem 4.5]. Let $M$ be any
$U({\mathfrak{g}},e)$-module. For a given odd element $u\in
U({\mathfrak{g}},e)_{\bar{1}}$ and a homogeneous vector $m\in M$, we know that
$m$ and $u.m$ have different parity. As $\mathsf{O}$ is a one-dimensional
superspace, then $\Theta_{i}.\textsf{o}=0$ for $l+1\leqslant i\leqslant l+q$
by parity consideration. Set $\Theta_{i}.\textsf{o}=c_{i}\textsf{o}$ for
$1\leqslant i\leqslant l$ with $c_{i}\in{\mathbb{C}}$. Recall [40, Theorem
4.7] shows that the algebra $U({\mathfrak{g}},e)$ is completely determined by
the commuting relations of $\Theta_{1},\cdots,\Theta_{l+q}$ (see also §1.2.2).
Based on the parity of these generators, for each case we will consider
separately.

(i) For $1\leqslant i<j\leqslant l$, the element $[\Theta_{i},\Theta_{j}]$ is
even since $\Theta_{i},\Theta_{j}\in U({\mathfrak{g}},e)_{\bar{0}}$. It is
immediate from
$[\Theta_{i},\Theta_{j}].\textsf{o}=(\Theta_{i}\cdot\Theta_{j}-\Theta_{j}\cdot\Theta_{i}).\textsf{o}=(c_{i}c_{j}-c_{j}c_{i}).\textsf{o}=0$
that the polynomial superalgebra $F_{ij}(\Theta_{1},\cdots,\Theta_{l+q})$ in
$l+q$ variables acts on O trivially. When we put each polynomial
$F_{ij}(\Theta_{1},\cdots,\Theta_{l+q})$ as a ${\mathbb{C}}$-linear
combination of
$\Theta_{1}^{a_{1}}\cdots\Theta_{l}^{a_{l}}\Theta_{l+1}^{b_{1}}\cdots\Theta_{l+q}^{b_{q}}$,
deleting all the terms for which any of the odd elements
$\Theta_{l+1},\cdots,\Theta_{l+q}$ occurs, one can obtain an (ordinary)
polynomial in $l$ variables, and denote it by
$F^{\prime}_{ij}(\Theta_{1},\cdots,\Theta_{l})$. Since
$\Theta_{i}.\textsf{o}=0$ for $l+1\leqslant i\leqslant l+q$ by the preceding
remark, then $F^{\prime}_{ij}(\Theta_{1},\cdots,\Theta_{l}).\textsf{o}=0$ for
$1\leqslant i<j\leqslant l$.

(ii) For $l+1\leqslant i\leqslant j\leqslant l+q$, the element
$[\Theta_{i},\Theta_{j}]$ is still even since $\Theta_{i},\Theta_{j}$ are both
odd. As $\Theta_{i}.\textsf{o}=0$ for $l+1\leqslant i\leqslant l+q$, we have
$[\Theta_{i},\Theta_{j}].\textsf{o}=(\Theta_{i}\cdot\Theta_{j}+\Theta_{j}\cdot\Theta_{i}).\textsf{o}=0$.
By the same discussion as (i) we can also get polynomials
$F^{\prime}_{ij}(\Theta_{1},\cdots,\Theta_{l})$ for $l+1\leqslant i\leqslant
j\leqslant l+q$, and the one-dimensional property of O entails that
$F^{\prime}_{ij}(\Theta_{1},\cdots,\Theta_{l}).\textsf{o}=0$.

(iii) For $1\leqslant i\leqslant l<j\leqslant l+q$, the element
$[\Theta_{i},\Theta_{j}]$ is odd since $\Theta_{i}\in
U({\mathfrak{g}},e)_{\bar{0}}$ and $\Theta_{j}\in
U({\mathfrak{g}},e)_{\bar{1}}$. As $\Theta_{i}.\textsf{o}=0$ for $l+1\leqslant
i\leqslant l+q$, we have
$[\Theta_{i},\Theta_{j}].\textsf{o}=(\Theta_{i}\cdot\Theta_{j}-\Theta_{j}\cdot\Theta_{i}).\textsf{o}=0$,
which entails that $F_{ij}(\Theta_{1},\cdots,\Theta_{l+q})$ acts on O
trivially. It is immediate from
$[\Theta_{i},\Theta_{j}]=F_{ij}(\Theta_{1},\cdots,\Theta_{l+q})$ that all
$F_{ij}(\Theta_{1},\cdots,\Theta_{l+q})$’s are odd elements in
$U(\mathfrak{g},e)$. Therefore, when we put each polynomial
$F_{ij}(\Theta_{1},\cdots,\Theta_{l+q})$ as a ${\mathbb{C}}$-linear
combination of monomials
$\Theta_{1}^{a_{1}}\cdots\Theta_{l}^{a_{l}}\Theta_{l+1}^{b_{1}}\cdots\Theta_{l+q}^{b_{q}}$,
in each given monomial some odd element $\Theta_{k}$ with $l+1\leqslant
k\leqslant l+q$ will occur at least once. Since $\Theta_{k}.\textsf{o}=0$ for
$l+1\leqslant k\leqslant l+q$, the equations
$F_{ij}(\Theta_{1},\cdots,\Theta_{l+q}).\textsf{o}=0$ are trivial for
$1\leqslant i\leqslant l<j\leqslant l+q$. In this case no new equations are
obtained.

Keep in mind all polynomials $F^{\prime}_{ij}$ from the above arguments.
Actually, since the polynomials $F_{ij}(\Theta_{1},\cdots,\Theta_{l+q})$’s
give rise to the defining relations of $U(\mathfrak{g},e)$, from all above one
can conclude that the one-dimensional modules of $U(\mathfrak{g},e)$ are
completely determined by the polynomials
$F^{\prime}_{ij}(\Theta_{1},\cdots,\Theta_{l})$’s. Set
$U({\mathfrak{g}},e)^{\text{ab}}$ to be the quotient algebra of
$U({\mathfrak{g}},e)$ by $R$, where $R$ is the ideal of $U({\mathfrak{g}},e)$
generated by all the odd generators $\Theta_{l+1},\cdots,\Theta_{l+q}$ and all
commutators $[a,b]$ with $a,b\in U({\mathfrak{g}},e)$. Then
$U({\mathfrak{g}},e)^{\text{ab}}$ is isomorphic to the algebra
${\mathbb{C}}[T_{1},\cdots,T_{l}]/\Lambda$, where
${\mathbb{C}}[T_{1},\cdots,T_{l}]$ is an (ordinary) polynomial algebra in $l$
variables, and $\Lambda$ the ideal of ${\mathbb{C}}[T_{1},\cdots,T_{l}]$
generated by all $F^{\prime}_{ij}(T_{1},\cdots,T_{l})$’s for $1\leqslant
i<j\leqslant l$ and $l+1\leqslant i\leqslant j\leqslant l+q$. Such a
commutative quotient in the Lie algebra case is studied by Premet in [29],
which leads to a lot of understanding on the small representations of finite
$W$-algebras. Now we exploit this machinery in the super case. In fact,
combining with all the discussions above, Hilbert’s Nullstellensatz shows that
the maximal spectrum
$\mathscr{E}:=\text{Specm}\,U({\mathfrak{g}},e)^{\text{ab}}$ parameterizes the
one-dimensional representations of $U({\mathfrak{g}},e)$. Denoting by
$\mathscr{E}(\mathbb{C})$ the set of all common zeros of the polynomials
$F^{\prime}_{ij}$ for $1\leqslant i<j\leqslant l$ and $l+1\leqslant i\leqslant
j\leqslant l+q$ in the affine space $\mathbb{A}^{l}_{\mathbb{C}}$, we have

######  Lemma 3.2.

When $d_{1}$ is even, the Zariski closed set $\mathscr{E}(\mathbb{C})$
parameterizes the one-dimensional representations of finite $W$-superalgebra
$U({\mathfrak{g}},e)$.

Note that for $1\leqslant i<j\leqslant l$, or $l+1\leqslant i\leqslant
j\leqslant l+q$, all the coefficients of $F_{ij}$’s are over the admissible
ring $A$. Thus all the coefficients of $F^{\prime}_{ij}$’s are also over $A$
by the definition. Set
${}^{p}{\hskip-1.42271pt}F^{\prime}_{ij}:=F^{\prime}_{ij}\otimes_{A}{{\mathds{k}}}$,
the polynomials over ${{\mathds{k}}}$, and denote by
$\mathscr{E}({{\mathds{k}}})$ the set of all common zeros of the polynomials
${}^{p}{\hskip-1.42271pt}F_{ij}^{\prime}$ in the affine space
$\mathbb{A}^{l}_{{\mathds{k}}}$ with $1\leqslant i<j\leqslant l$, or
$l+1\leqslant i\leqslant j\leqslant l+q$. Since the transition subalgebra
$T({\mathfrak{g}}_{{\mathds{k}}},e)$ over ${{\mathds{k}}}$ is induced from the
$\mathbb{C}$-algebra $U({\mathfrak{g}},e)$ by “modular $p$ reduction”, Theorem
2.2 and Lemma 3.2 show that the Zariski closed set
$\mathscr{E}({{\mathds{k}}})$ parametrises the one-dimensional representations
of the ${{\mathds{k}}}$-algebra $T({\mathfrak{g}}_{{\mathds{k}}},e)$. By the
same arguments as in [29, Theorem 2.2(a)], one can verify that

######  Lemma 3.3.

Assume that $d_{1}$ is even. If the finite $W$-superalgebra
$U({\mathfrak{g}},e)$ over ${\mathbb{C}}$ affords one-dimensional
representations, then the transition subalgebra
$T({\mathfrak{g}}_{{\mathds{k}}},e)$ over
${{\mathds{k}}}=\overline{\mathbb{F}}_{p}$ also admits one-dimensional
representations.

Now we can talk about the small representations for the reduced enveloping
algebra of a basic Lie superalgebra. The following result is an immediate
consequence of Proposition 2.9(1) and Lemma 3.3.

######  Lemma 3.4.

When $d_{1}$ is even, if the finite $W$-superalgebra $U({\mathfrak{g}},e)$
over ${\mathbb{C}}$ affords a one-dimensional representation, then for $p\gg
0$ there exists $\eta\in\chi+(\mathfrak{m}_{{\mathds{k}}}^{\bot})_{\bar{0}}$
such that the reduced enveloping algebra
$U_{\eta}({\mathfrak{g}}_{{\mathds{k}}})$ admits irreducible representations
of dimension $p^{\frac{d_{0}}{2}}2^{\frac{d_{1}}{2}}$.

###  3.3. Confirmation of Conjecture 3.1 for $\mathfrak{gl}(M|N)$ and
$\mathfrak{sl}(M|N)$

In [35] Wang and Zhao gave some explicit description for the Dynkin gradings
of basic Lie superalgebras of all types. In particular, they showed that
$\dim\mathfrak{gl}(M|N)^{e}_{\bar{1}}$ is always an even number for any
nilpotent element $e\in\mathfrak{gl}(M|N)_{\bar{0}}$ (cf. [35, §3.2]). As the
dimension of $\mathfrak{gl}(M|N)_{\bar{1}}$ is also even, then
$d_{1}=\dim\mathfrak{gl}(M|N)_{\bar{1}}-\dim\mathfrak{gl}(M|N)^{e}_{\bar{1}}$
is always an even number. It is notable that although Wang-Zhao’s origin
arguments are carried over the field ${\mathds{k}}$ in positive characteristic
(see Table 1 in §1.1.1), as showed in [35, Remark 3.2], all the discussions
there are still valid for the case of complex numbers. Actually, in order to
confirm Conjecture 3.1 for these complex finite $W$-superalgebras of
$\mathfrak{gl}(M|N)$ and $\mathfrak{sl}(M|N)$, we will firstly consider whose
transition subalgebra $T({\mathfrak{g}}_{{\mathds{k}}},e)$ in positive
characteristic.

######  Lemma 3.5.

Let ${\mathfrak{g}}_{{{\mathds{k}}}}$ be a Lie superalgebra of type
$\mathfrak{gl}(M|N)$ or $\mathfrak{sl}(M|N)$ with $M,N\in{\mathbb{Z}}_{+}$.
For any nilpotent element $e\in({\mathfrak{g}}_{{{\mathds{k}}}})_{\bar{0}}$,
the ${{\mathds{k}}}$-algebra $T({\mathfrak{g}}_{{\mathds{k}}},e)$ affords a
one-dimensional representation.

###### Proof.

For the Lie superalgebra ${\mathfrak{g}}_{{{\mathds{k}}}}=\mathfrak{gl}(M|N)$
or $\mathfrak{sl}(M|N)$, in [41] the authors showed that the reduced
enveloping algebra $U_{\chi}({\mathfrak{g}}_{{\mathds{k}}})$ admits an
irreducible representation of dimension
$p^{\frac{d_{0}}{2}}2^{\frac{d_{1}}{2}}$ over
${{\mathds{k}}}=\overline{\mathbb{F}}_{p}$ under the assumption that (i) $p>2$
when ${\mathfrak{g}}=\mathfrak{gl}(M|N)$; (ii) $p>2$ and $p\nmid(M-N)$ when
${\mathfrak{g}}=\mathfrak{sl}(M|N)$. Since $d_{1}$ is even in this case, we
have
$\text{\text@underline{dim}}\,\mathfrak{m}=(\frac{d_{0}}{2},\frac{d_{1}}{2})$
by §1.2.1. Hence Proposition 2.9(1) yields that the ${\mathds{k}}$-algebra
$T({\mathfrak{g}}_{{\mathds{k}}},e)$ admits a one-dimensional representation.
∎

The following result is an immediate consequence of field theory.

######  Lemma 3.6.

Let ${\mathfrak{g}}$ be a basic Lie superalgebra over ${\mathbb{C}}$. When
$d_{1}$ is even, if the transition subalgebra
$T({\mathfrak{g}}_{{\mathds{k}}},e)$ with
${{\mathds{k}}}=\overline{\mathbb{F}}_{p}$ affords one-dimensional
representations for infinitely many $p\in\Pi(A)$, then the finite
$W$-superalgebra $U({\mathfrak{g}},e)$ over ${\mathbb{C}}$ has a one-
dimensional representation.

###### Proof.

Recall Lemma 3.2 and what follows show that the one-dimensional
representations of finite $W$-superalgebra $U(\mathfrak{g},e)$ over
$\mathbb{C}$ can be parametrised by the Zariski closed set
$\mathscr{E}({\mathbb{C}})$, and the transition subalgebra
$T(\mathfrak{g}_{\mathds{k}},e)$ by $\mathscr{E}({{\mathds{k}}})$. The lemma
follows the same means as the Lie algebra case [29, Corollary 2.1] by the
knowledge of Galois theory, thus will be omitted here. ∎

Now we are in a position to introduce the main result of this subsection.

######  Proposition 3.7.

Let ${\mathfrak{g}}=\mathfrak{gl}(M|N)$ or $\mathfrak{sl}(M|N)$
($M,N\in{\mathbb{Z}}_{+}$) over ${\mathbb{C}}$. For any nilpotent element
$e\in{\mathfrak{g}}_{\bar{0}}$, the finite $W$-superalgebra
$U({\mathfrak{g}},e)$ affords a one-dimensional representation.

###### Proof.

The proposition readily follows from Lemma 3.5 and Lemma 3.6. ∎

##  4\. Conjectural two-dimensional representations for finite
$W$-superalgebras when $d_{1}$ is odd

In this section we will always assume that $d_{1}$ is odd.

###  4.1. On the minimal-dimension conjecture when $d_{1}$ is odd

In virtue of Proposition 2.7, we can formulate the following conjecture on the
minimal-dimension representations of $U({\mathfrak{g}},e)$ when $d_{1}$ is
odd.

######  Conjecture 4.1.

When $d_{1}$ is odd, the finite $W$-superalgebra $U({\mathfrak{g}},e)$ over
$\mathbb{C}$ affords a two-dimensional representation of minimal dimension.

In the section, we mainly investigate such two-dimensional modules under the
assumption that the above conjecture holds, and also confirm the conjecture
for the regular nilpotent elements of $\mathfrak{osp}(1|2n)$. Take such a
module as in Conjecture 4.1 and denote it by $V$. Recall that in [40, §4.4] we
introduced the refined finite $W$-superalgebra
$W^{\prime}_{\chi}:=Q_{\chi}^{\text{ad}\mathfrak{m}^{\prime}}$ over
$\mathbb{C}$, a proper subalgebra of $U({\mathfrak{g}},e)$. In virtue of this
algebra, we can formulate a more precise description for the two-dimensional
representation $V$ of the ${\mathbb{C}}$-algebra $U({\mathfrak{g}},e)$.

######  Proposition 4.2.

Any irreducible $U({\mathfrak{g}},e)$-module in Conjecture 4.1 can be regarded
as a $W^{\prime}_{\chi}$-module of type $Q$.

###### Proof.

Let $V$ be a two-dimensional representation of the ${\mathbb{C}}$-algebra
$U({\mathfrak{g}},e)$ in Conjecture 4.1. Define the ${\mathbb{C}}$-mapping

$\begin{array}[]{lcll}\tau:&V&\rightarrow&V\\\
&v&\mapsto&\sqrt{2}\Theta_{l+q+1}.v.\end{array}$

It is easy to verify that the mapping $\tau$ is odd and surjective. In fact,
$\tau$ is also injective since $\tau^{2}(v)=2\Theta_{l+q+1}^{2}.v=v$. We claim
that $\tau$ is a homomorphism of $W^{\prime}_{\chi}$-module $V$.

For any ${\mathbb{Z}}_{2}$-homogeneous elements $\Theta\in W^{\prime}_{\chi}$
and $v\in V$, we have $\tau(\Theta.v)=\sqrt{2}\Theta_{l+q+1}.\Theta.v$ by the
definition. Since $\Theta_{l+q+1}\in\mathfrak{m}^{\prime}$ and $\Theta\in
Q_{\chi}^{\text{ad}\mathfrak{m}^{\prime}}$, then $[\Theta_{l+q+1},\Theta]=0$.
Moreover,
$[\Theta_{l+q+1},\Theta]=\Theta_{l+q+1}\cdot\Theta-(-1)^{|\Theta|}\Theta\cdot\Theta_{l+q+1}$.
It is immediate that
$\Theta_{l+q+1}\cdot\Theta=(-1)^{|\Theta|}\Theta\cdot\Theta_{l+q+1}$, then
$\Theta_{l+q+1}.\Theta.v=(-1)^{|\Theta|}\Theta.\Theta_{l+q+1}.v$, i.e.
$\tau(\Theta.v)=(-1)^{|\Theta|}\Theta.\tau(v)$.

As $V$ is also irreducible as a $W^{\prime}_{\chi}$-module, all the
discussions above imply that $V$ is of type $Q$, completing the proof. ∎

###  4.2. The analogue of communicative quotients for finite $W$-superalgebras
in the odd case

Recall that in [40, Theorem 4.7] we have chosen the
${\mathbb{Z}}_{2}$-homogeneous elements $\Theta_{1},\cdots,\Theta_{l+q+1}$ as
a set of generators for the ${\mathbb{C}}$-algebra $U({\mathfrak{g}},e)$
subject to the relations

$[\Theta_{i},\Theta_{j}]=F_{ij}(\Theta_{1},\cdots,\Theta_{l+q+1}),~{}~{}1\leqslant
i,j\leqslant l+q+1.$

To ease notation, we will denote $F_{ij}(\Theta_{1},\cdots,\Theta_{l+q+1})$ by
$F_{ij}$ for short.

If the ${\mathbb{C}}$-algebra $U({\mathfrak{g}},e)$ affords a two-dimensional
representation $V$, Proposition 4.2 yields that $V$ is ${\mathbb{C}}$-spanned
by an even element $v\in V_{\bar{0}}$ and the odd element $\Theta_{l+q+1}.v\in
V_{\bar{1}}$. Hence we can get $4(l+q+1)$ variables
$k_{i}^{0},k_{i}^{1},K_{i}^{0},K_{i}^{1}\in{\mathbb{C}}$ such that

$\Theta_{i}.v=k_{i}^{0}v+k_{i}^{1}\Theta_{l+q+1}.v,\qquad\Theta_{i}.\Theta_{l+q+1}.v=K_{i}^{0}v+K_{i}^{1}\Theta_{l+q+1}.v,$
(4.1)

where $1\leqslant i\leqslant l+q+1$.

Similarly, there exist $4(l+q+1)^{2}$ variables
$(F_{ij})^{0}_{\bar{0}},(F_{ij})^{0}_{\bar{1}},(F_{ij})^{1}_{\bar{0}},(F_{ij})^{1}_{\bar{1}}\in{\mathbb{C}}$
with $1\leqslant i,j\leqslant l+q+1$ such that

$F_{ij}.v=(F_{ij})^{0}_{\bar{0}}v+(F_{ij})^{0}_{\bar{1}}\Theta_{l+q+1}.v,\qquad
F_{ij}.\Theta_{l+q+1}.v=(F_{ij})^{1}_{\bar{0}}v+(F_{ij})^{1}_{\bar{1}}\Theta_{l+q+1}.v.$
(4.2)

It is worth noting that each polynomial $F_{ij}$ is generated by the
${\mathbb{Z}}_{2}$-homogeneous elements $\Theta_{1},\cdots,\Theta_{l+q+1}$ of
$U({\mathfrak{g}},e)$ over $\mathbb{Q}$. After enlarging the admissible ring
$A$ possibly, one can further assume that the $F_{ij}$’s for $1\leqslant
i,j\leqslant l+q+1$ are defined over $A$. Since the action of $F_{ij}$ on $v$
and $\Theta_{l+q+1}.v$ is completely determined by the constants in (4.1),
then
$(F_{ij})^{0}_{\bar{0}},(F_{ij})^{0}_{\bar{1}},(F_{ij})^{1}_{\bar{0}},(F_{ij})^{1}_{\bar{1}}$
can be written as an $A$-linear combination of the monomials in
$k_{i}^{0},k_{i}^{1},K_{i}^{0},K_{i}^{1}$, thus there are no new variables
appear in (4.2).

Since each $\Theta_{i}$ is ${\mathbb{Z}}_{2}$-homogeneous for $1\leqslant
i\leqslant l+q+1$, it follows that $2(l+q+1)$ variables in (4.1) equal zero.
More precisely, $k_{i}^{1}=K_{i}^{0}=0$ for $1\leqslant i\leqslant l$ (in this
case all $\Theta_{i}$’s are even) and $k_{i}^{0}=K_{i}^{1}=0$ for
$l+1\leqslant i\leqslant l+q+1$ (in this case all $\Theta_{i}$’s are odd). By
the definition it is obvious that all $F_{ij}$’s are ${\mathbb{Z}}_{2}$-graded
and have the same parity with $[\Theta_{i},\Theta_{j}]$ for $1\leqslant
i,j\leqslant l+q+1$. Therefore, $2(l+q+1)^{2}$ variables equal zero in (4.2).
More precisely, $(F_{ij})^{0}_{\bar{1}}=(F_{ij})^{1}_{\bar{0}}=0$ if $F_{ij}$
is even, and $(F_{ij})^{0}_{\bar{0}}=(F_{ij})^{1}_{\bar{1}}=0$ if $F_{ij}$ is
odd.

In virtue of (4.1), simple calculation shows that

$\begin{split}\Theta_{i}.\Theta_{j}.v&=(k_{i}^{0}k_{j}^{0}+K_{i}^{0}k_{j}^{1})v+(k_{i}^{1}k_{j}^{0}+K_{i}^{1}k_{j}^{1})\Theta_{l+q+1}.v,\\\
\Theta_{i}.\Theta_{j}.\Theta_{l+q+1}.v&=(k_{i}^{0}K_{j}^{0}+K_{i}^{0}K_{j}^{1})v+(k_{i}^{1}K_{j}^{0}+K_{i}^{1}K_{j}^{1})\Theta_{l+q+1}.v\end{split}$
(4.3)

for $1\leqslant i,j\leqslant l+q+1$.

Recall that the structure of ${\mathbb{C}}$-algebra $U({\mathfrak{g}},e)$ is
completely determined by a data of generators and defining relations (cf. [40,
Theorem 4.7]). Hence, $V$ is completely decided by the following equalities

$\begin{split}(\Theta_{i}.\Theta_{j}-(-1)^{|\Theta_{i}||\Theta_{j}|}\Theta_{j}\Theta_{i}-F_{ij}).v&=0,\\\
(\Theta_{i}.\Theta_{j}-(-1)^{|\Theta_{i}||\Theta_{j}|}\Theta_{j}\Theta_{i}-F_{ij}).\Theta_{l+q+1}.v&=0\end{split}$
(4.4)

for $1\leqslant i,j\leqslant l+q+1$. Simple calculation based on (4.3) and
(4.4) shows that the variables $k_{i}^{0},k_{i}^{1},K_{i}^{0},K_{i}^{1}$ and
$(F_{ij})^{0}_{\bar{0}},(F_{ij})^{0}_{\bar{1}},(F_{ij})^{1}_{\bar{0}},(F_{ij})^{1}_{\bar{1}}$
for $1\leqslant i,j\leqslant l+q+1$ should satisfy the following system of
linear equations:

$\begin{array}[]{rcl}(k_{i}^{0}k_{j}^{0}+K_{i}^{0}k_{j}^{1})-(-1)^{|\Theta_{i}||\Theta_{j}|}(k_{i}^{0}k_{j}^{0}+k_{i}^{1}K_{j}^{0})-(F_{ij})^{0}_{\bar{0}}&=&0;\\\
(k_{i}^{1}k_{j}^{0}+K_{i}^{1}k_{j}^{1})-(-1)^{|\Theta_{i}||\Theta_{j}|}(k_{i}^{0}k_{j}^{1}+k_{i}^{1}K_{j}^{1})-(F_{ij})^{0}_{\bar{1}}&=&0;\\\
(k_{i}^{0}K_{j}^{0}+K_{i}^{0}K_{j}^{1})-(-1)^{|\Theta_{i}||\Theta_{j}|}(K_{i}^{0}k_{j}^{0}+K_{i}^{1}K_{j}^{0})-(F_{ij})^{1}_{\bar{0}}&=&0;\\\
(k_{i}^{1}K_{j}^{0}+K_{i}^{1}K_{j}^{1})-(-1)^{|\Theta_{i}||\Theta_{j}|}(K_{i}^{0}k_{j}^{1}+K_{i}^{1}K_{j}^{1})-(F_{ij})^{1}_{\bar{1}}&=&0.\end{array}$

It is notable that $4(l+q+1)$ variables are involved in above equations
(recall that $F_{ij}$’s ($1\leqslant i,j\leqslant l+q+1$) can be written as an
$A$-linear combination of the monomials in
$k_{i}^{0},k_{i}^{1},K_{i}^{0},K_{i}^{1}$ with $1\leqslant i\leqslant l+q+1$),
and the remark preceding (4.3) shows that $2(l+q+1)$ variables equal zero. Set

${\mathbb{C}}[X_{1}^{0},X_{2}^{0}\cdots,X_{l}^{0},X_{l+1}^{1},\cdots,X_{l+q+1}^{1},Y_{1}^{1},Y_{2}^{1},\cdots,Y_{l}^{1},Y_{l+1}^{0}\cdots,Y_{l+q+1}^{0}]$

to be an (ordinary) polynomial algebra in $2(l+q+1)$ variables over
${\mathbb{C}}$. For $1\leqslant i\leqslant l+q+1$, substitute the constants
$k_{i}^{0},k_{i}^{1},K_{i}^{0},K_{i}^{1}$ for the variables
$X_{i}^{0},X_{i}^{1},Y_{i}^{0},Y_{i}^{1}$ respectively, and define the
polynomials

$\begin{array}[]{rcl}A_{ij}&:=&(X_{i}^{0}X_{j}^{0}+X_{j}^{1}Y_{i}^{0})-(-1)^{|\Theta_{i}||\Theta_{j}|}(X_{i}^{0}X_{j}^{0}+X_{i}^{1}Y_{j}^{0})-S_{ij}^{0};\\\
B_{ij}&:=&(X_{i}^{1}X_{j}^{0}+X_{j}^{1}Y_{i}^{1})-(-1)^{|\Theta_{i}||\Theta_{j}|}(X_{i}^{0}X_{j}^{1}+X_{i}^{1}Y_{j}^{1})-S_{ij}^{1};\\\
C_{ij}&:=&(X_{i}^{0}Y_{j}^{0}+Y_{i}^{0}Y_{j}^{1})-(-1)^{|\Theta_{i}||\Theta_{j}|}(X_{j}^{0}Y_{i}^{0}+Y_{i}^{1}Y_{j}^{0})-T_{ij}^{0};\\\
D_{ij}&:=&(X_{i}^{1}Y_{j}^{0}+Y_{i}^{1}Y_{j}^{1})-(-1)^{|\Theta_{i}||\Theta_{j}|}(X_{j}^{1}Y_{i}^{0}+Y_{i}^{1}Y_{j}^{1})-T_{ij}^{1}\end{array}$

for $1\leqslant i,j\leqslant l+q+1$, where
$S_{ij}^{0},S_{ij}^{1},T_{ij}^{0},T_{ij}^{1}$ stand for the polynomials over
$A$ obtained by substituting the variables
$k_{i}^{0},k_{i}^{1},K_{i}^{0},K_{i}^{1}$ in the polynomials
$(F_{ij})^{0}_{\bar{0}},(F_{ij})^{0}_{\bar{1}},$
$(F_{ij})^{1}_{\bar{0}},(F_{ij})^{1}_{\bar{1}}$ for the indeterminate
$X_{i}^{0},X_{i}^{1},Y_{i}^{0},Y_{i}^{1}$, respectively. By the preceding
remark, for the terms in $A_{ij},B_{ij},C_{ij},D_{ij}$ with $1\leqslant
i,j\leqslant l+q+1$, we have

(1) $X_{i}^{1}=Y_{i}^{0}=0$ for $1\leqslant i\leqslant l$ (in this case
$\Theta_{i}^{\prime}s$ are even);

(2) $X_{i}^{0}=Y_{i}^{1}=0$ for $l+1\leqslant i\leqslant l+q+1$ (in this case
$\Theta_{i}^{\prime}s$ are odd);

(3) $S_{ij}^{1}=T_{ij}^{0}=0$ when $1\leqslant i,j\leqslant l$, or
$l+1\leqslant i,j\leqslant l+q+1$ (in this case $F_{ij}$’s are even);

(4) $S_{ij}^{0}=T_{ij}^{1}=0$ when $1\leqslant i\leqslant l<j\leqslant l+q+1$,
or $1\leqslant j\leqslant l<i\leqslant l+q+1$ (in this case $F_{ij}$’s are
odd).

It follows from (4.3), (4.4) and [40, Theorem 4.7] that there is a $1$-$1$
correspondence between the two-dimensional representations of
${\mathbb{C}}$-algebra $U({\mathfrak{g}},e)$ and the points of all common
zeros of the polynomials $A_{ij},B_{ij},C_{ij},D_{ij}$ in $2(l+q+1)$ variables
for $1\leqslant i,j\leqslant l+q+1$ subject to conditions (1)-(4).

Given a subfield $K$ of ${\mathbb{C}}$ containing $A$ we denote by
$\mathscr{E}(K)$ the set of all common zeros of the polynomials
$A_{ij},B_{ij},C_{ij},D_{ij}$ for $1\leqslant i,j\leqslant l+q+1$ in the
affine space $\mathbb{A}^{2(l+q+1)}_{K}$ subject to conditions (1)-(4).
Clearly, the $A$-defined Zariski closed set $\mathscr{E}(\mathbb{C})$
parameterizes the two-dimensional representations of ${\mathbb{C}}$-algebra
$U({\mathfrak{g}},e)$. More precisely,

######  Lemma 4.3.

Assume that $d_{1}$ is odd. Then the two-dimensional representations of
${\mathbb{C}}$-algebra $U({\mathfrak{g}},e)$ are uniquely determined by all
common zeros of the polynomials $A_{ij},B_{ij},C_{ij},D_{ij}$ ($1\leqslant
i,j\leqslant l+q+1$) subject to conditions (1)-(4) in the affine space
$\mathbb{A}^{2(l+q+1)}_{\mathbb{C}}$.

Similarly, let $\mathscr{E}({{\mathds{k}}})$ be the set of common zeros of the
polynomials
${}^{p}{\hskip-1.42271pt}A_{ij},^{p}{\hskip-1.42271pt}B_{ij},^{p}{\hskip-1.42271pt}C_{ij},$
${}^{p}{\hskip-1.42271pt}D_{ij}$ subject to the “modular $p$” version of the
conditions (1)-(4) in the affine space $\mathbb{A}^{2(l+q+1)}_{{\mathds{k}}}$
with $1\leqslant i,j\leqslant l+q+1$, where
${}^{p}{\hskip-1.42271pt}A_{ij},^{p}{\hskip-1.42271pt}B_{ij},^{p}{\hskip-1.42271pt}C_{ij},^{p}{\hskip-1.42271pt}D_{ij}$
stand for the polynomials over ${{\mathds{k}}}$ obtained from
$A_{ij},B_{ij},C_{ij},D_{ij}$ by “modular $p$ reduction”, i.e.

$\begin{array}[]{rl}&{{\mathds{k}}}[X_{1}^{0},X_{2}^{0}\cdots,X_{l}^{0},X_{l+1}^{1},\cdots,X_{l+q+1}^{1},Y_{1}^{1},Y_{2}^{1},\cdots,Y_{l}^{1},Y_{l+1}^{0}\cdots,Y_{l+q+1}^{0}]\\\
=&A[X_{1}^{0},X_{2}^{0}\cdots,X_{l}^{0},X_{l+1}^{1},\cdots,X_{l+q+1}^{1},Y_{1}^{1},Y_{2}^{1},\cdots,Y_{l}^{1},Y_{l+1}^{0}\cdots,Y_{l+q+1}^{0}]\otimes_{A}{{\mathds{k}}}.\end{array}$

It follows from Theorem 2.2 that the Zariski closed set
$\mathscr{E}({{\mathds{k}}})$ parameterizes the two-dimensional
representations of the ${{\mathds{k}}}$-algebra
$T({\mathfrak{g}}_{{\mathds{k}}},e)$.

Following Premet’s treatment to the Lie algebra case in [29, Theorem 2.2(a)],
we have

######  Lemma 4.4.

When $d_{1}$ is odd, if the ${\mathbb{C}}$-algebra $U({\mathfrak{g}},e)$
affords two-dimensional representations, then the transition subalgebra
$T({\mathfrak{g}}_{{\mathds{k}}},e)$ also admits two-dimensional
representations.

###### Proof.

Taking Lemma 4.3 into account, we can prove the lemma by the same discussion
as Lemma 3.3. The detailed arguments are omitted here. ∎

As a corollary of the above lemma and Proposition 2.9(2), we have

######  Lemma 4.5.

When $d_{1}$ is odd, if the finite $W$-superalgebra $U({\mathfrak{g}},e)$ over
${\mathbb{C}}$ affords a two-dimensional representation, then for $p\gg 0$
there exists $\eta\in\chi+(\mathfrak{m}_{{\mathds{k}}}^{\bot})_{\bar{0}}$
associated to which the reduced enveloping algebra
$U_{\eta}({\mathfrak{g}}_{{\mathds{k}}})$ admits irreducible representations
of dimension $p^{\frac{d_{0}}{2}}2^{\frac{d_{1}+1}{2}}$.

###  4.3. Confirmation of Conjecture 4.1 for
${\mathfrak{g}}=\mathfrak{osp}(1|2n)$ with regular nilpotent elements

We first need the following observation:

######  Lemma 4.6.

Let ${\mathfrak{g}}_{{\mathds{k}}}=\mathfrak{osp}(1|2n)$ be a basic Lie
superalgebra over ${\mathds{k}}$. For any regular nilpotent element
$e\in({\mathfrak{g}}_{{{\mathds{k}}}})_{\bar{0}}$, the transition subalgebra
$T({\mathfrak{g}}_{{\mathds{k}}},e)$ affords a two-dimensional representation.

###### Proof.

First note that $d_{1}$ is odd in this case (cf. [24, Corollary 2.10]). Let
${\mathfrak{g}}_{{\mathds{k}}}=\mathfrak{n}_{{\mathds{k}}}^{+}\oplus\mathfrak{h}_{{\mathds{k}}}\oplus\mathfrak{n}_{{\mathds{k}}}^{-}$
denote the canonical triangular decomposition of Lie superalgebra
$\mathfrak{osp}(1|2n)$. It follows from [35, Corollary 5.8] that

$\text{\text@underline{dim}}\,\mathfrak{n}_{{\mathds{k}}}^{-}=\text{\text@underline{dim}}\,\mathfrak{m}^{\prime}_{{\mathds{k}}}=(\text{dim}\,(\mathfrak{m}_{{\mathds{k}}})_{\bar{0}},\text{dim}\,(\mathfrak{m}_{{\mathds{k}}})_{\bar{1}}+1).$

Moreover, the baby Verma module $Z_{\chi}(\lambda)$ of reduced enveloping
algebra $U_{\chi}({\mathfrak{g}}_{{\mathds{k}}})$ associated to the regular
$p$-character $\chi$ is irreducible (cf. [35, Corollary 5.8]), which has the
same dimension as the vector space
$U_{\chi}(\mathfrak{m}_{{\mathds{k}}}^{\prime})$. Theorem 1.3 shows that
$Z_{\chi}(\lambda)^{\mathfrak{m}_{{\mathds{k}}}}$ is a
$U_{\chi}(\mathfrak{osp}(1|2n),e)$-module, and there is an isomorphism of
$U_{\chi}(\mathfrak{m}_{{\mathds{k}}})$-modules $Z_{\chi}(\lambda)\cong
U_{\chi}(\mathfrak{m}_{{\mathds{k}}})^{*}\otimes_{{\mathds{k}}}Z_{\chi}(\lambda)^{\mathfrak{m}_{{\mathds{k}}}}$
by the proof of [35, Proposition 4.2]. Then we have

$\text{dim}\,Z_{\chi}(\lambda)^{\mathfrak{m}_{{\mathds{k}}}}=\frac{\text{dim}\,Z_{\chi}(\lambda)}{\text{dim}\,U_{\chi}(\mathfrak{m}_{{\mathds{k}}})}=\frac{\text{dim}\,U_{\chi}(\mathfrak{m}_{{\mathds{k}}}^{\prime})}{\text{dim}\,U_{\chi}(\mathfrak{m}_{{\mathds{k}}})}=2.$

Therefore, the algebra $U_{\chi}({\mathfrak{g}}_{{\mathds{k}}},e)$ admits a
two-dimensional representation. By the same discussion as the proof of
Proposition 2.9(2), one can conclude that the transition subalgebra
$T({\mathfrak{g}}_{{\mathds{k}}},e)$ also affords a two-dimensional
representation. ∎

Recall Lemma 4.3 shows that the two-dimensional representations of finite
$W$-superalgebras over ${\mathbb{C}}$ can be parameterized by the Zariski
closed set $\mathscr{E}({\mathbb{C}})$ for the case when $d_{1}$ is odd. By
the same consideration as Lemma 3.6, one can also obtain that

######  Lemma 4.7.

Let ${\mathfrak{g}}$ be a basic Lie superalgebra over ${\mathbb{C}}$. When
$d_{1}$ is odd, if the transition subalgebra
$T({\mathfrak{g}}_{{\mathds{k}}},e)$ affords two-dimensional representations
for infinitely many $p\in\Pi(A)$, then the finite $W$-superalgebra
$U({\mathfrak{g}},e)$ over ${\mathbb{C}}$ has a two-dimensional
representation.

Now we are in a position to introduce the main result of this subsection.

######  Proposition 4.8.

Let $e\in{\mathfrak{g}}_{\bar{0}}$ be a regular nilpotent element in the Lie
superalgebra ${\mathfrak{g}}=\mathfrak{osp}(1|2n)$ over ${\mathbb{C}}$, then
the finite $W$-superalgebra $U({\mathfrak{g}},e)$ affords a two-dimensional
representation.

###### Proof.

The proposition readily follows from Lemma 4.6 and Lemma 4.7. ∎

##  5\. The realization of minimal-dimensional representations for reduced
enveloping algebra $U_{\xi}({\mathfrak{g}}_{{\mathds{k}}})$

###  5.1. Introduction

This section is devoted to the accessibility of dimensional lower bounds for
the irreducible representations of $U_{\xi}({\mathfrak{g}}_{{\mathds{k}}})$
predicted by Wang-Zhao in [35, Theorem 5.6] as below:

######  Proposition 5.1.

([35]) Let ${\mathfrak{g}}_{{\mathds{k}}}$ be a basic Lie superalgebra over
${{\mathds{k}}}=\overline{\mathbb{F}}_{p}$, assuming that the prime $p$
satisfies the restriction imposed in §1.1(Table 1). Let $\xi$ be arbitrary
$p$-character in $({\mathfrak{g}}_{{\mathds{k}}})^{*}_{\bar{0}}$. Then the
dimension of every $U_{\xi}({\mathfrak{g}}_{{\mathds{k}}})$-module $M$ is
divisible by $p^{\frac{d_{0}}{2}}2^{\lfloor\frac{d_{1}}{2}\rfloor}$.

Therefore, the number $p^{\frac{d_{0}}{2}}2^{\lfloor\frac{d_{1}}{2}\rfloor}$
becomes a lower bound of dimensions for the irreducible modules of
${\mathds{k}}$-algebra $U_{\xi}(\mathfrak{g}_{\mathds{k}})$. A natural
question is the accessibility of this number, i.e. whether or not there is any
irreducible module of $U_{\xi}(\mathfrak{g}_{\mathds{k}})$ with dimension
equaling such a lower bound.

For the Lie superalgebra ${\mathfrak{g}}_{{\mathds{k}}}$ of type
$\mathfrak{gl}(M|N)$ or $\mathfrak{sl}(M|N)$ with $M,N\in{\mathbb{Z}}_{+}$, in
[41] the authors showed that every reduced enveloping algebra
$U_{\xi}({\mathfrak{g}}_{{\mathds{k}}})$ has a “small” representation of
dimension equaling the lower bound. The method applied there is to construct
an appropriate parabolic subalgebra which has a one-dimensional module, then
induce it to an irreducible representation of ${\mathfrak{g}}_{{\mathds{k}}}$.
However, this method can not be easily exploited in the general case. Thus the
general attainableness of such lower bounds of dimensions in Proposition 5.1
is an open problem.

Now we first formulate Conjecture 0.3, summarizing the previous two sections:

Conjecture 0.3. Let ${\mathfrak{g}}$ be a basic Lie superalgebra over
${\mathbb{C}}$.

  * (1)

When $d_{1}$ is even, the finite $W$-superalgebra $U({\mathfrak{g}},e)$
affords a one-dimensional representation;

  * (2)

when $d_{1}$ is odd, the finite $W$-superalgebra $U({\mathfrak{g}},e)$ affords
a two-dimensional representation.

In virtue of Conjecture 0.3, we will prove that the lower bounds of the
dimensions in Proposition 5.1 are accessible for $p\gg 0$ in this section. In
the first part, we will deal with the case for nilpotent $p$-character
$\chi\in(\mathfrak{g}_{\mathds{k}})^{*}_{\bar{0}}$, mainly following Premet’s
treatment for the Lie algebra case in [29, §2.8], with a few modifications.
One can observe that the emergence of odd part in the Lie superalgebra
${\mathfrak{g}}_{{\mathds{k}}}$ makes the situation much more complicated. In
the second part, we will deal with the case for arbitrary $p$-character
$\xi\in(\mathfrak{g}_{\mathds{k}})^{*}_{\bar{0}}$, which may be not nilpotent.
A lot of precise analysis on the modular representations of basic Lie
superalgebras has to be done for the second case.

###  5.2. On the dimensional lower bounds for the representations of basic Lie
superalgebras with nilpotent $p$-characters

#### 5.2.1.

Recall that in Lemmas 3.4 and 4.5 we have discussed the representations of
minimal dimensions for the reduced enveloping algebra
$U_{\eta}({\mathfrak{g}}_{{\mathds{k}}})$ associated to some $p$-character
$\eta\in\chi+(\mathfrak{m}_{{\mathds{k}}}^{\bot})_{\bar{0}}$ based on the
parity of $d_{1}$, respectively. It is notable that the $p$-character $\eta$
can only be guaranteed in
$\chi+(\mathfrak{m}_{{\mathds{k}}}^{\bot})_{\bar{0}}$, with no further
information apparently contributing to
$U_{\chi}(\mathfrak{g}_{{\mathds{k}}})$. Owing to Premet’s treatment to finite