Cohomology of $\mathfrak{sl}_{3}$ and $\mathfrak{gl}_{3}$ with Coefficients in
Simple Modules

Cohomology of $\boldsymbol{\mathfrak{sl}_{3}}$ and
$\boldsymbol{\mathfrak{gl}_{3}}$ with Coefficients in Simple
Modules and Weyl Modules in Positive Characteristics

Sherali Sh. IBRAEV

Sh.Sh. Ibraev

Korkyt Ata Kyzylorda University, Aiteke bie St., 29A, 120014, Kzylorda,
Kazakhstan<EMAIL_ADDRESS>
Received August 12, 2021, in final form March 26, 2022; Published online March
30, 2022

We calculate the cohomology of $\mathfrak{sl}_{3}(k)$ over an algebraically
closed field $k$ of characteristic $p>3$ with coefficients in simple modules
and Weyl modules. We also give descriptions of the corresponding cohomology of
$\mathfrak{gl}_{3}(k)$.

Lie algebra; simple module; cohomology

17B20; 17B45; 20G05

##  1 Introduction

There are many remarkable results in the cohomology theory of modular Lie
algebras. Any simple module over the restricted Lie algebra with nontrivial
cohomology is restricted, see [9, Theorem 2]. Modules over a Lie algebra with
nonzero cohomology are called peculiar. For any finite-dimensional Lie
algebra, the number of non-isomorphic peculiar indecomposable modules is
finite, see [10, Theorem 1].

Now, let $\mathfrak{g}=\mathfrak{sl}_{3}(k)$. The cohomology with coefficients
in simple $\mathfrak{g}$ -modules is completely described only in small
characteristics $p=2,3$, see [18, Theorem 1], [19, Theorem 1]. In the case
where $p>3$, the cohomology of simple $\mathfrak{g}$-modules are known in the
following cases: for $H^{1}(\mathfrak{g},M)$, see [21, p. 301], for
$H^{2}(\mathfrak{g},M)$, see [11, Theorem 1.1]. In small degrees, the
cohomology has the following interpretations:
$H^{1}({\mathfrak{g}},{\mathfrak{g}})$ is identified with the space of outer
differentiation and $H^{2}({\mathfrak{g}},{\mathfrak{g}})$ is identified with
the space of local deformations of the Lie algebra $\mathfrak{g}$. It is known
that these spaces are trivial, see [23, p. 124], [6, p. 125], [2, Lemma
2.2.1b], [3, p. 32]. The cohomology of ${\mathfrak{g}}$ with coefficients in
the trivial module is also known, see [3, p. 42]. In other cases, the
interpretation of the cohomology with coefficients in simple modules remains
open. In this paper, we give a complete description of the cohomology of
$\mathfrak{sl}_{3}(k)$ with coefficients in the simple modules for $p>3$.

###  1.1 Notation

Let $\mathfrak{g}=\mathfrak{sl}_{3}(k)$ over an algebraically closed field $k$
of characteristic $p>3$ and $M$ a simple $\mathfrak{g}$-module. Let $L(r,s)$
denote a simple $\mathfrak{g}$-module with the highest weight
$r\omega_{1}+s\omega_{2}$, where $\omega_{1}$, $\omega_{2}$ are fundamental
weights.

Let $G={\rm SL}_{3}(k)$; it is an algebraic group, its Lie algebra is
$\mathfrak{sl}_{3}(k)$. We will consider cohomology of $\mathfrak{sl}_{3}(k)$
as $G$-modules. Let $V$ be a $G$-module and $M$ be a simple $G$-module. We
define a composition coefficient $[V:M]$ for $M$ from the formula

$\displaystyle\operatorname{ch}(V)=\sum_{M\ \text{is
simple}}[V:M]\operatorname{ch}(M),$

where $\operatorname{ch}(V)$ is the formal character of the $G$-module $V$. If
$[V:M]\neq 0$, then we say that $M$ is a composition factor of $V$.

For a vector space $L$ over $k$, we denote by $L^{(1)}$ the vector space over
$k$ that coincides with $L$ as an additive group and with the scalar
multiplication given by

$\displaystyle a\cdot v=\sqrt[p]{a}v\qquad\text{for all}\quad a\in k,\quad
v\in L,$

where the left hand side is the new multiplication and the right hand side the
old one. If $L$ is a $G$-module, then $L^{(1)}$ is also a $G$-module using the
given action of any $g\in G$ on the additive group $L^{(1)}=L$. The new
$G$-module $L^{(1)}$ is called the Frobenius twist of $L$. We define higher
Frobenius twists inductively: $L^{(d+1)}=\big{(}L^{(d)}\big{)}^{(1)}$. To each
weight $\mu$ of the space $L$ there corresponds the weight $p^{d}\mu$ of the
space $L^{(d)}$.

A weight $r\omega_{1}+s\omega_{2}$ is restricted if $0\leq r,s\leq p-1$. The
composition factors of $H^{n}(\mathfrak{g},M)$ are Frobenius twists of some
simple $G$-modules with restricted highest weights.

A $G$-module $L$ is rational if the corresponding representation is a
homomorphism from $G$ to ${\rm GL}(L)$. Suppose $V$ is the Frobenius twist of
some rational $G$-module. Then, there is a unique $d>0$ and rational
$G$-module $L$ such that $L^{(d)}=V$. Denote this module by $V^{(-d)}$.

We will usually denote $H^{n}(\mathfrak{g},k)$ by $H^{n}(\mathfrak{g})$, and
use the following short notation:

$\displaystyle mV:=V\oplus\cdots\oplus V\qquad(m\text{ summands}),$

where $V$ is a $G$-module.

###  1.2 Main result

In this paper, $k$ is always algebraically closed field $k$ of characteristic
$p>3$.

######  Theorem 1.1.

Let $\mathfrak{g}=\mathfrak{sl}_{3}(k)$ and $M$ be a simple
$\mathfrak{g}$-module. Then, the following isomorphisms of $G$-modules hold:

  1. $(a)$

$H^{n}(\mathfrak{g})\cong k$ for $n=0,3,5,8;$

  2. $(b)$

$H^{n}(\mathfrak{g},L(p-2,1))\cong\begin{cases}L(1,0)^{(1)}&\text{if}\
n=1,7,\\\ 2L(1,0)^{(1)}&\text{if}\ n=4;\end{cases}$

  3. $(c)$

$H^{n}(\mathfrak{g},L(1,p-2))\cong\begin{cases}L(0,1)^{(1)}&\text{if}\
n=1,7,\\\ 2L(0,1)^{(1)}&\text{if}\ n=4;\end{cases}$

  4. $(d)$

$H^{n}(\mathfrak{g},L(p-3,0))\cong L(1,0)^{(1)}$ for $n=2,3,5,6;$

  5. $(e)$

$H^{n}(\mathfrak{g},L(0,p-3))\cong L(0,1)^{(1)}$ for $n=2,3,5,6;$

  6. $(f)$

$H^{n}(\mathfrak{g},L(p-2,p-2))\cong\begin{cases}k&\text{if}\ n=1,7,\\\
L(1,1)^{(1)}&\text{if}\ n=3,5,\\\ 2L(1,1)^{(1)}\oplus 2k&\text{if}\
n=4.\end{cases}$

Otherwise, $H^{n}(\mathfrak{g},M)=0$.

This theorem completes the description of the cohomology of
$\mathfrak{sl}_{3}(k)$ with coefficients in simple modules over an
algebraically closed fields of positive characteristics.

###  1.3 Some applications of the main result

Using Theorem 1.1, one can easily describe the cohomology of
$\mathfrak{gl}_{3}(k)$ with coefficients in simple modules. Let $M$ be an
$\mathfrak{sl}_{3}(k)$-module. Since
$\mathfrak{gl}_{3}(k)\cong\mathfrak{sl}_{3}(k)\oplus I$, where $I$ is the
subspace spanned by the identity $3\times 3$ matrix, a
$\mathfrak{gl}_{3}(k)$-module structure on $M$ can be determined by setting

$\displaystyle(x,a)m=xm+\mu(a)m\qquad\text{for
any}\quad(x,a)\in\mathfrak{gl}_{3}(k),\quad
x\in\mathfrak{sl}_{3}(k),\quad\text{and}\quad a\in I,$ (1.1)

where $\mu$ is a linear form on $I$. We denote the obtained
$\mathfrak{gl}_{3}(k)$-module also by $M$. Using Theorem 1.1 and the
isomorphism (see [18, p. 737])

$\displaystyle H^{n}(\mathfrak{gl}_{3}(k),M)\cong
H^{n}(\mathfrak{sl}_{3}(k),M)\oplus H^{n-1}(\mathfrak{sl}_{3}(k),M)$

for $\mu=0$, we obtain for the cohomology of simple
$\mathfrak{gl}_{3}(k)$-modules the following

######  Corollary 1.2.

Let $\mathfrak{g}=\mathfrak{gl}_{3}(k)$, let $M$ be a simple
$\mathfrak{g}$-module defined by the formula (1.1). If $\mu=0$, then the
following isomorphisms of $G$-modules hold:

  1. $(a)$

$H^{n}(\mathfrak{g})\cong k$ for $n=0,1,3,4,5,6,8,9;$

  2. $(b)$

$H^{n}(\mathfrak{g},L(p-2,1))\cong\begin{cases}L(1,0)^{(1)}&\text{if}\
n=1,2,7,8,\\\ 2L(1,0)^{(1)}&\text{if}\ n=4,5;\end{cases}$

  3. $(c)$

$H^{n}(\mathfrak{g},L(1,p-2))\cong\begin{cases}L(0,1)^{(1)}&\text{if}\
n=1,2,7,8,\\\ 2L(0,1)^{(1)}&\text{if}\ n=4,5;\end{cases}$

  4. $(d)$

$H^{n}(\mathfrak{g},L(p-3,0))\cong\begin{cases}L(1,0)^{(1)}&\text{if}\
n=2,4,5,7,\\\ 2L(1,0)^{(1)}&\text{if}\ n=3,6;\end{cases}$

  5. $(e)$

$H^{n}(\mathfrak{g},L(0,p-3))\cong\begin{cases}L(0,1)^{(1)}&\text{if}\
n=2,4,5,7,\\\ 2L(0,1)^{(1)}&\text{if}\ n=3,6;\end{cases}$

  6. $(f)$

$H^{n}(\mathfrak{g},L(p-2,p-2))\cong\begin{cases}k&\text{if}\ n=1,2,7,8,\\\
L(1,1)^{(1)}&\text{if}\ n=3,6,\\\ 3L(1,1)^{(1)}\oplus 2k&\text{if}\
n=4,5.\end{cases}$

Otherwise, $H^{n}(\mathfrak{g},M)=0$.

The results of Corollary 1.2 can be applied to describe the cohomology of the
general Lie algebra of Cartan type $W_{3}(\mathbf{m})$ (for the definition of
$W_{n}(\mathbf{m})$, see [18, Section 1.3]). For example, using in [25,
Theorem 0.2] and the statement $(a)$ of Corollary 1.2, one can easily describe
the cohomology of the restricted Lie algebra of Cartan type
$W_{3}(\mathbf{1})$ with coefficients in the divided power algebra.

Let $V(\lambda)$ be the Weyl module with highest weight
$\lambda=r\omega_{1}+s\omega_{2}$ (for the definition, see Section 2.2) and
$H^{0}(\lambda)=V(-w_{0}(\lambda))^{*}$, where $w_{0}$ is the longest element
of the Weyl group $W$ of the Lie algebra $\mathfrak{g}$. As an $G$-module,
$H^{0}(\lambda)$ is isomorphic to the induced $G$-module
$\operatorname{Ind}_{B}^{G}(k_{\lambda})$, where $B$ is the Borel subgroup of
$G$, corresponding to the negative roots, and $k_{\lambda}$ is a one-
dimensional $B$-module. A module $V$ over $G$ is $G$-acyclic, if
$H^{n}(G,V)=0$ for all $n>0$. The restricted weights $\lambda$ and $\mu$ are
linked if there is $w\in W$ such that

$\displaystyle\lambda+\rho\equiv w(\mu+\rho)\mod pX(T),$

where $\rho$ is the half-sum of positive roots and $X(T)$ is the additive
character group of the maximal torus $T$ of $G$. We say that two $G$-modules
with highest weights are linked if their highest weights are linked. As is
well-known, $H^{0}(\lambda)$ and $V(\lambda)$ are $G$-acyclic, see [8,
Corollary 3.4]. But, from the proof of Theorem 1.1, we will see that the
$\mathfrak{g}$-modules $H^{0}(\lambda)$ and $V(\lambda)$, linked with simple
peculiar modules, are peculiar for $\mathfrak{g}$. For the cohomology of these
modules, the following result occurs:

######  Corollary 1.3.

Let $\mathfrak{g}=\mathfrak{sl}_{3}(k)$ and $V=H^{0}(\lambda)$. Then, the
following isomorphisms of $G$-modules hold:

  1. $(a)$

$H^{n}(\mathfrak{g},H^{0}(0,0))\cong k$ for $n=0,3,5,8;$

  2. $(b)$

$H^{n}(\mathfrak{g},H^{0}(p-2,1))\cong\begin{cases}L(1,0)^{(1)}&\text{if}\
n=1,2,3,5,6,7,\\\ 2L(1,0)^{(1)}&\text{if}\ n=4;\end{cases}$

  3. $(c)$

$H^{n}(\mathfrak{g},H^{0}(1,p-2))\cong\begin{cases}L(0,1)^{(1)}&\text{if}\
n=1,2,3,5,6,7,\\\ 2L(0,1)^{(1)}&\text{if}\ n=4;\end{cases}$

  4. $(d)$

$H^{n}(\mathfrak{g},H^{0}(p-3,0))\cong L(1,0)^{(1)}$ for $n=2,3,5,6;$

  5. $(e)$

$H^{n}(\mathfrak{g},H^{0}(0,p-3))\cong L(0,1)^{(1)}$ for $n=2,3,5,6;$

  6. $(f)$

$H^{n}(\mathfrak{g},H^{0}(p-2,p-2))\cong\begin{cases}L(1,1)^{(1)}&\text{if}\
n=3,\\\ 2L(1,1)^{(1)}\oplus k&\text{if}\ n=4,\\\ L(1,1)^{(1)}\oplus
k&\text{if}\ n=5,\\\ k&\text{if}\ n=7,8.\end{cases}$

Otherwise, $H^{n}(\mathfrak{g},V)=0$.

######  Corollary 1.4.

Let $\mathfrak{g}=\mathfrak{sl}_{3}(k)$ and $V=V(\lambda)$. Then, the
following isomorphisms of $G$-modules hold:

  1. $(a)$

$H^{n}(\mathfrak{g},V(0,0))\cong k$ for $n=0,3,5,8;$

  2. $(b)$

$H^{n}(\mathfrak{g},V(p-2,1))\cong\begin{cases}L(1,0)^{(1)}&\text{if}\
n=1,2,3,5,6,7,\\\ 2L(1,0)^{(1)}&\text{if}\ n=4;\end{cases}$

  3. $(c)$

$H^{n}(\mathfrak{g},V(1,p-2))\cong\begin{cases}L(0,1)^{(1)}&\text{if}\
n=1,2,3,5,6,7,\\\ 2L(0,1)^{(1)}&\text{if}\ n=4;\end{cases}$

  4. $(d)$

$H^{n}(\mathfrak{g},V(p-3,0))\cong L(1,0)^{(1)}$ for $n=2,3,5,6;$

  5. $(e)$

$H^{n}(\mathfrak{g},V(0,p-3))\cong L(0,1)^{(1)}$ for $n=2,3,5,6;$

  6. $(f)$

$H^{n}(\mathfrak{g},V(p-2,p-2))\cong\begin{cases}k&\text{if}\ n=0,1,\\\
L(1,1)^{(1)}\oplus k&\text{if}\ n=3,\\\ 2L(1,1)^{(1)}\oplus k&\text{if}\
n=4,\\\ L(1,1)^{(1)}&\text{if}\ n=5.\end{cases}$

Otherwise, $H^{n}(\mathfrak{g},V)=0$.

##  2 Preliminary facts

###  2.1 Properties of cohomology

In this section, we give some properties of the cohomology for the Lie algebra
$\mathfrak{g}$ that are used to prove the main results. Cohomology

$\displaystyle H^{\bullet}({\mathfrak{g}},M)=\bigoplus_{n\geq
0}H^{n}({\mathfrak{g}},M)$

can be computed using a complex
$\big{(}\bigwedge^{\bullet}{\mathfrak{g}}^{*}\bigotimes M,d\big{)}$, see [22,
Section I.9.17]. Therefore, we can identify the space of cochains
$C^{n}({\mathfrak{g}},M)$ with the space
$\bigwedge^{n}{\mathfrak{g}}^{*}\bigotimes M$ and regard the space
$C^{n}({\mathfrak{g}},M)$ as the $G$-module. We decompose the space of
cochains $C^{n}(\mathfrak{g},M)$ into a direct sum of the eigenspaces with
respect to the maximal torus $T$ of the group $G$:

$\displaystyle C^{n}({\mathfrak{g}},M)=\bigoplus_{\mu\in
X(T)}C^{n}_{\mu}(\mathfrak{g},M),$

where $X(T)$ is the additive character group of the torus $T$. Then,

$\displaystyle H^{n}({\mathfrak{g}},M)=\bigoplus_{\mu\in
X(T)}H^{n}_{\mu}({\mathfrak{g}},M).$

Denote by $\prod(V)$ the set of weights of the subspace $V$ of the $G$-module
$C^{n}(\mathfrak{g},M)$. Since

$\displaystyle\prod(H^{n}(\mathfrak{g},M))\subseteq
pX(T)\bigcap\prod\Big{(}\bigwedge{{}^{n}}\mathfrak{g}^{*}\bigotimes M\Big{)},$

we will consider elements of the subspace $\overline{C}^{n}(\mathfrak{g},M)$
of the space $C^{n}(\mathfrak{g},M)$ with weights from the set

$\displaystyle
pX(T)\bigcap\prod\Big{(}\bigwedge{{}^{n}}\mathfrak{g}^{*}\bigotimes M\Big{)}.$

The corresponding subspaces of cocycles and cohomology are denoted by
$\overline{Z}^{n}\\!(\mathfrak{g},M)$ and
$\overline{H}^{n}\\!(\mathfrak{g},M)$, respectively. Note that

$\displaystyle H^{n}(\mathfrak{g},M)=\overline{H}^{n}(\mathfrak{g},M).$

By the definition of $H^{n}(\mathfrak{g},M)$,

$\displaystyle\dim H^{n}(\mathfrak{g},M)=\dim Z^{n}(\mathfrak{g},M)-\dim
B^{n}(\mathfrak{g},M),$

and by the definition of $B^{n}(\mathfrak{g},M)$,

$\displaystyle\dim B^{n}(\mathfrak{g},M)=\dim C^{n-1}(\mathfrak{g},M)-\dim
Z^{n-1}(\mathfrak{g},M).$

Then, we get

$\displaystyle\dim
H^{n}(\mathfrak{g},M)=\dim\overline{Z}^{n}(\mathfrak{g},M)+\dim\overline{Z}^{n-1}(\mathfrak{g},M)-\dim\overline{C}^{n-1}(\mathfrak{g},M).$
(2.1)

Since $\operatorname{Tr}(\operatorname{ad}_{x})=0$ for all $x\in\mathfrak{g}$,
then, according to the main theorem in [15, p. 639], we get the following
isomorphism:

$\displaystyle
H^{n}(\mathfrak{g},M^{*})\cong\big{(}H^{\dim\mathfrak{g}-n}(\mathfrak{g},M)\big{)}^{*}.$
(2.2)

The weight subspaces are invariant under the coboundary operator. Therefore,
the formula (2.1) also holds for weight subspaces:

$\displaystyle\dim
H^{n}_{\mu}(\mathfrak{g},M)=\dim\overline{Z}^{n}_{\mu}(\mathfrak{g},M)+\dim\overline{Z}^{n-1}_{\mu}(\mathfrak{g},M)-\dim\overline{C}^{n-1}_{\mu}(\mathfrak{g},M).$
(2.3)

###  2.2 Peculiar modules

In this section, we describe simple peculiar $\mathfrak{sl}_{3}(k)$-modules.
Let $\\{e_{1},e_{2},e_{3},h_{1},h_{2},f_{1},f_{2},f_{3}\\}$ be the Chevalley
basis of $\mathfrak{g}$ with the nonzero brackets

$\displaystyle[e_{i},f_{i}]=h_{i},\qquad[h_{i},e_{i}]=2e_{i},\qquad[h_{i},f_{i}]=-2f_{i},\qquad
i=1,2,3,$
$\displaystyle[h_{1},e_{2}]=-e_{2},\qquad[h_{1},e_{3}]=e_{3},\qquad[h_{2},e_{1}]=-e_{1},\qquad[h_{2},e_{3}]=e_{3},$
$\displaystyle[h_{1},f_{2}]=f_{2},\qquad[h_{1},f_{3}]=-f_{3},\qquad[h_{2},f_{1}]=f_{1},\qquad[h_{2},f_{3}]=-f_{3},$
$\displaystyle[e_{1},e_{2}]=e_{3},\qquad[e_{3},f_{1}]=-e_{2},\qquad[e_{3},f_{2}]=e_{1},$
$\displaystyle[f_{1},f_{2}]=-f_{3},\qquad[e_{1},f_{3}]=-f_{2},\qquad[e_{2},f_{3}]=f_{1},$

where $h_{3}=h_{1}+h_{2}$. It is known (see [11, p. 145]) that there are six
simple peculiar $\mathfrak{g}$-modules:

$\displaystyle L(0,0),\quad L(p-2,1),\quad L(1,p-2),\quad L(p-3,0),\quad
L(0,p-3),\quad L(p-2,p-2).$

A linear span of a set $\\{v_{1},\dots,v_{m}\\}$ of vectors of a vector space
$V$ over $k$ is the smallest linear subspace of $V$ that contains the set
$\\{v_{1},\dots,v_{m}\\}$. Let $\langle v_{1},\dots,v_{m}\rangle_{k}$ denote
the linear span of the set $\\{v_{1},\dots,v_{m}\\}$ of vectors of the vector
space $V$ over $k$. For a detailed description of the peculiar simple modules,
consider the restricted Verma module

$\displaystyle
W(r,s):=\bigg{\langle}v_{i,j,t}:=\frac{f_{3}^{t}f_{2}^{j}f_{1}^{i}}{t!j!i!}u_{r,s}\,\bigg{|}\,0\leq
i,j,t\leq p-1\bigg{\rangle}_{k}$

with the following action of $\mathfrak{sl}_{3}(k)$:

$\displaystyle e_{1}v_{i,j,t}=-(j+1)v_{i,j+1,t-1}+(r-i+1)v_{i-1,j,t},$
$\displaystyle e_{2}v_{i,j,t}=(s+i-j-t+1)v_{i,j-1,t}+(i+1)v_{i+1,j,t-1},$
$\displaystyle e_{3}v_{i,j,t}=(r+s-i-j-t+1)v_{i,j,t-1}+(r-i+1)v_{i-1,j-1,t},$
$\displaystyle h_{1}v_{i,j,t}=(r-2i+j-t)v_{i,j,t},\qquad
h_{2}v_{i,j,t}=(s+i-2j-t)v_{i,j,t},$ $\displaystyle
f_{1}v_{i,j,t}=-(t+1)v_{i,j-1,t+1}+(i+1)v_{i+1,j,t},$ $\displaystyle
f_{2}v_{i,j,t}=(j+1)v_{i,j+1,t},\qquad f_{3}v_{i,j,t}=(t+1)v_{i,j,t+1}.$

The restricted Verma module $W(r,s)$ has a submodule $I(r,s)$ generated by the
vectors $v_{r+1,0,0}$ and $v_{0,s+1,0}$. The quotient $V(r,s)=W(r,s)/I(r,s)$
is also restricted; let us call it the Weyl module. In the modular case, the
term “Weyl module” was first used in [26, p. 321], see also [13, p. 59]. For
groups of Lie type over a field of positive characteristic, the term “Weyl
module” has also been used for a long time (see, for example, [5, p. 213],
[17, p. 262], [20, p. 291]). Obviously, for the Weyl module $V(r,s)$, the
following relations hold:

$\displaystyle v_{r+1,0,0}=0,\qquad v_{0,s+1,0}=0.$

The submodule structure of $V(r,s)$ is well-known, see [4, p. 484], [24, pp.
151 and 157]. The results of these papers say that the quotient of $V(r,s)$ by
the maximal submodule is the restricted simple module isomorphic to $L(r,s)$.
In particular, for the peculiar simple $\mathfrak{g}$-modules we get

$\displaystyle L(0,0)=V(0,0),\qquad L(p-3,0)=V(p-3,0),\qquad
L(0,p-3)=V(0,p-3),$ $\displaystyle L(p-2,1)=V(p-2,1)/L(p-3,0),\qquad
L(1,p-2)=V(1,p-2)/L(0,p-3),$ $\displaystyle L(p-2,p-2)=V(p-2,p-2)/L(0,0).$

To describe simple modules, we will use the basis vectors of the corresponding
restricted Verma modules. The maximal submodules of these Weyl modules are
generated by the highest weight vectors

$\displaystyle w_{p-3,0}=v_{1,1,0}-2v_{0,0,1}\qquad\text{for}\quad V(p-2,1),$
$\displaystyle w_{0,p-3}=v_{1,1,0}+v_{0,0,1}\qquad\text{for}\quad V(1,p-2),$
$\displaystyle
w_{0,0}=\sum_{i=0}^{p-2}(p-2-i)i!v_{p-2-i,p-2-i,i}\qquad\text{for}\quad
V(p-2,p-2),$

respectively. Then, for simple non-trivial peculiar modules, we obtain the
following descriptions in terms of the basis vectors of the restricted Verma
module:

$\displaystyle L(p-3,0)=\langle v_{i,j,t}\mid 0\leq i\leq p-3,\,0\leq j\leq
i,\,0\leq t\leq p-3-i\rangle_{k},$ $\displaystyle L(0,p-3)=\langle
v_{i,j,t}\mid 0\leq i\leq j,\,0\leq j\leq p-3,\,0\leq t\leq p-3-j\rangle_{k},$
$\displaystyle L(p-2,1)=\langle v_{i,j,t}\mid 0\leq i\leq p-2,\,0\leq j\leq
i+1,\,0\leq t\leq p-1-i;\,w_{p-3,0}=0\rangle_{k},$ $\displaystyle
L(1,p-2)=\langle v_{i,j,t}\mid 0\leq i\leq j+1,\,0\leq j\leq p-2,\,0\leq t\leq
p-1-j;\,w_{0,p-3}=0\rangle_{k},$ $\displaystyle L(p-2,p-2)=\langle
v_{i,j,t}\mid 0\leq i\leq p-2,\,0\leq j\leq p-2,\,0\leq t\leq
p-2-i-j;\,w_{0,0}=0\rangle_{k}.$

##  3 Proof of Theorem 1.1

As noted above, there are only six peculiar simple modules. Let us prove the
theorem for each peculiar simple module separately.

$(a)$ Since $p>3$, then the Killing form on $\mathfrak{g}$ is non-degenerate.
Then, for the trivial one-dimensional module $M=L(0,0)$, the result obtained
earlier for zero characteristic remains true in our case as well. So, we
consider only non-trivial peculiar simple modules.

$(b)$ Let $M=L(p-2,1)$.

######  Lemma 3.1.

Let $\mathfrak{g}=\mathfrak{sl}_{3}(k)$ and $M=L(p-2,1)$. Then,
$H^{n}(\mathfrak{g},M)=0$, except for the following cases:

  1. $(i)$

$H^{1}(\mathfrak{g},L(p-2,1))\cong H^{7}(\mathfrak{g},L(p-2,1))\cong
L(1,0)^{(1)}$,

  2. $(ii)$

$H^{4}(\mathfrak{g},L(p-2,1))\cong 2L(1,0)^{(1)}$.

###### Proof.

Obviously, $H^{0}(\mathfrak{g},L(p-2,1))=0$. It is also known that
$H^{1}(\mathfrak{g},L(p-2,1))\cong L(1,0)^{(1)}$, see [21, p. 301], and
$H^{2}(\mathfrak{g},L(p-2,1))=0$, see [11, Theorem 1.1].

Now we show that $H^{3}(\mathfrak{g},L(p-2,1))=0$. We get

$\displaystyle\prod\big{(}\overline{C}^{\bullet}(\mathfrak{g},L(p-2,1))\big{)}=\\{p\omega_{1},p(-\omega_{1}+\omega_{2}),-p\omega_{2}\\}.$

The subspace $\overline{C}^{2}(\mathfrak{g},L(p-2,1))$ is $21$-dimensional and
its set of weights consists of three elements $p\omega_{1}$,
$p(-\omega_{1}+\omega_{2})$, $-p\omega_{2}$. We have

$\displaystyle\dim\overline{C}^{2}_{p\omega_{1}}(\mathfrak{g},L(p-2,1))$
$\displaystyle=\dim\overline{C}^{2}_{p(-\omega_{1}+\omega_{2})}(\mathfrak{g},L(p-2,1))$
$\displaystyle=\dim\overline{C}^{2}_{-p\omega_{2}}(\mathfrak{g},L(p-2,1))=7.$

The subspace $\overline{C}^{2}_{p\omega_{1}}(\mathfrak{g},L(p-2,1))$ is
spanned by the $2$-cochains

$\displaystyle\psi_{1}^{2}=h_{1}^{*}\wedge f_{1}^{*}\otimes
v_{0,0,0},\qquad\psi_{2}^{2}=h_{2}^{*}\wedge f_{1}^{*}\otimes
v_{0,0,0},\qquad\psi_{3}^{2}=e_{2}^{*}\wedge f_{3}^{*}\otimes v_{0,0,0},$
$\displaystyle\psi_{4}^{2}=h_{1}^{*}\wedge f_{3}^{*}\otimes
v_{0,1,0},\qquad\psi_{5}^{2}=h_{2}^{*}\wedge f_{3}^{*}\otimes v_{0,1,0},$
$\displaystyle\psi_{6}^{2}=f_{1}^{*}\wedge f_{2}^{*}\otimes
v_{0,1,0},\qquad\psi_{7}^{2}=f_{1}^{*}\wedge f_{3}^{*}\otimes v_{0,0,1}.$

Let $\sum_{i=1}^{7}a_{i}\psi_{i}^{2}\in Z^{2}(\mathfrak{g},L(p-2,1))$, where
$a_{i}\in k$ for all $i$. Then, by the cocycle condition,

$\displaystyle a_{1}=a_{2}=a_{4}=a_{5}=0,\qquad a_{3}=a_{6}=a_{7}.$

This means that $\dim Z^{2}_{p\omega_{1}}(\mathfrak{g},L(p-2,1))=1$.

The subspace $\overline{C}^{3}_{p\omega_{1}}(\mathfrak{g},L(p-2,1))$ is
spanned by the $3$-cochains

$\displaystyle\psi_{1}^{3}=h_{1}^{*}\wedge h_{2}^{*}\wedge f_{1}^{*}\otimes
v_{0,0,0},\qquad$ $\displaystyle\psi_{2}^{3}=h_{1}^{*}\wedge e_{2}^{*}\wedge
f_{3}^{*}\otimes v_{0,0,0},$ $\displaystyle\psi_{3}^{3}=h_{2}^{*}\wedge
e_{2}^{*}\wedge f_{3}^{*}\otimes v_{0,0,0},\qquad$
$\displaystyle\psi_{4}^{3}=e_{2}^{*}\wedge f_{1}^{*}\wedge f_{1}^{*}\otimes
v_{0,0,0},$ $\displaystyle\psi_{5}^{3}=e_{3}^{*}\wedge f_{1}^{*}\wedge
f_{3}^{*}\otimes v_{0,0,0},\qquad$ $\displaystyle\psi_{6}^{3}=e_{2}^{*}\wedge
f_{1}^{*}\wedge f_{3}^{*}\otimes v_{1,0,0},$
$\displaystyle\psi_{7}^{3}=e_{1}^{*}\wedge f_{1}^{*}\wedge f_{3}^{*}\otimes
v_{0,1,0},\qquad$ $\displaystyle\psi_{8}^{3}=h_{1}^{*}\wedge h_{2}^{*}\wedge
f_{3}^{*}\otimes v_{0,1,0},$ $\displaystyle\psi_{9}^{3}=e_{2}^{*}\wedge
f_{2}^{*}\wedge f_{3}^{*}\otimes v_{0,1,0},\qquad$
$\displaystyle\psi_{10}^{3}=h_{2}^{*}\wedge f_{1}^{*}\wedge f_{2}^{*}\otimes
v_{0,1,0},$ $\displaystyle\psi_{11}^{3}=h_{1}^{*}\wedge f_{1}^{*}\wedge
f_{2}^{*}\otimes v_{0,1,0},\qquad$ $\displaystyle\psi_{12}^{3}=h_{1}^{*}\wedge
f_{1}^{*}\wedge f_{3}^{*}\otimes v_{0,0,1},$
$\displaystyle\psi_{13}^{3}=h_{2}^{*}\wedge f_{1}^{*}\wedge f_{3}^{*}\otimes
v_{0,0,1},\qquad$ $\displaystyle\psi_{14}^{3}=f_{1}^{*}\wedge f_{2}^{*}\wedge
f_{3}^{*}\otimes v_{0,1,1}.$

So, $\dim\overline{C}^{3}_{p\omega_{1}}(\mathfrak{g},L(p-2,1))=14$. Suppose
$\sum_{i=1}^{14}b_{i}\psi_{i}^{3}\in Z^{3}(\mathfrak{g},L(p-2,1))$, where
$b_{i}\in k$ for all $i$. Then, using the cocycle condition, we get

$\displaystyle b_{1}=b_{8}=0,\qquad b_{2}+b_{4}-b_{5}-2b_{6}-b_{7}=0,\qquad-
b_{2}-b_{3}+b_{5}+b_{7}+b_{9}-2b_{14}=0,$ $\displaystyle b_{10}-b_{3}=0,\qquad
b_{11}-b_{2}=0,\qquad b_{12}-b_{2}=0,\qquad b_{13}-b_{3}=0.$

Consider these equalities as a system of equations for $b_{i}$, where
$i=1,\dots,14$. The rank of the matrix of this system is equal to $8$.
Therefore, $\dim Z^{3}_{p\omega_{1}}(\mathfrak{g},L(p-2,1))=14-8=6$. Then, by
(2.3),

$\displaystyle\dim H^{3}_{p\omega_{1}}(\mathfrak{g},L(p-2,1))={}$
$\displaystyle\dim Z^{3}_{p\omega_{1}}(\mathfrak{g},L(p-2,1))+\dim
Z^{2}_{p\omega_{1}}(\mathfrak{g},L(p-2,1))$ $\displaystyle-\dim
C^{2}_{p\omega_{1}}(\mathfrak{g},L(p-2,1))=6+1-7=0.$

Thus, all $3$-cocycles with dominant highest weight $p\omega_{1}$ are
coboundaries. Therefore,

$\displaystyle H^{3}(\mathfrak{g},L(p-2,1))=0.$

Now we will calculate $H^{4}(\mathfrak{g},L(p-2,1))$. The subspace
$\overline{C}^{4}_{p\omega_{1}}(\mathfrak{g},L(p-2,1))$ is $18$-dimensional
and is spanned by the $4$-cochains

$\displaystyle\psi_{1}^{4}=h_{1}^{*}\wedge h_{2}^{*}\wedge e_{2}^{*}\wedge
f_{3}^{*}\otimes v_{0,0,0},\qquad$ $\displaystyle\psi_{2}^{4}=h_{1}^{*}\wedge
e_{2}^{*}\wedge f_{1}^{*}\wedge f_{2}^{*}\otimes v_{0,0,0},$
$\displaystyle\psi_{3}^{4}=h_{2}^{*}\wedge e_{2}^{*}\wedge f_{1}^{*}\wedge
f_{2}^{*}\otimes v_{0,0,0},\qquad$ $\displaystyle\psi_{4}^{4}=h_{1}^{*}\wedge
e_{3}^{*}\wedge f_{1}^{*}\wedge f_{3}^{*}\otimes v_{0,0,0},$
$\displaystyle\psi_{5}^{4}=h_{2}^{*}\wedge e_{3}^{*}\wedge f_{1}^{*}\wedge
f_{3}^{*}\otimes v_{0,0,0},\qquad$ $\displaystyle\psi_{6}^{4}=e_{1}^{*}\wedge
e_{2}^{*}\wedge f_{1}^{*}\wedge f_{3}^{*}\otimes v_{0,0,0},$
$\displaystyle\psi_{7}^{4}=h_{1}^{*}\wedge e_{2}^{*}\wedge f_{1}^{*}\wedge
f_{3}^{*}\otimes v_{1,0,0},\qquad$ $\displaystyle\psi_{8}^{4}=h_{2}^{*}\wedge
e_{2}^{*}\wedge f_{1}^{*}\wedge f_{3}^{*}\otimes v_{1,0,0},$
$\displaystyle\psi_{9}^{4}=h_{1}^{*}\wedge h_{2}^{*}\wedge f_{1}^{*}\wedge
f_{2}^{*}\otimes v_{1,0,0},\qquad$ $\displaystyle\psi_{10}^{4}=h_{1}^{*}\wedge
e_{1}^{*}\wedge f_{1}^{*}\wedge f_{3}^{*}\otimes v_{0,1,0},$
$\displaystyle\psi_{11}^{4}=h_{2}^{*}\wedge e_{1}^{*}\wedge f_{1}^{*}\wedge
f_{3}^{*}\otimes v_{0,1,0},\qquad$ $\displaystyle\psi_{12}^{4}=h_{1}^{*}\wedge
e_{2}^{*}\wedge f_{2}^{*}\wedge f_{3}^{*}\otimes v_{0,1,0},$
$\displaystyle\psi_{13}^{4}=h_{2}^{*}\wedge e_{2}^{*}\wedge f_{2}^{*}\wedge
f_{3}^{*}\otimes v_{0,1,0},\qquad$ $\displaystyle\psi_{14}^{4}=e_{3}^{*}\wedge
f_{1}^{*}\wedge f_{2}^{*}\wedge f_{3}^{*}\otimes v_{0,1,0},$
$\displaystyle\psi_{15}^{4}=h_{1}^{*}\wedge h_{2}^{*}\wedge f_{1}^{*}\wedge
f_{3}^{*}\otimes v_{0,0,1},\qquad$ $\displaystyle\psi_{16}^{4}=e_{2}^{*}\wedge
f_{1}^{*}\wedge f_{2}^{*}\wedge f_{3}^{*}\otimes v_{0,0,1},$
$\displaystyle\psi_{17}^{4}=h_{1}^{*}\wedge f_{1}^{*}\wedge f_{2}^{*}\wedge
f_{3}^{*}\otimes v_{0,1,1},\qquad$ $\displaystyle\psi_{18}^{4}=h_{2}^{*}\wedge
f_{1}^{*}\wedge f_{2}^{*}\wedge f_{3}^{*}\otimes v_{0,1,1}.$

Suppose $\sum_{i=1}^{18}c_{i}\psi_{i}^{4}\in Z^{4}(\mathfrak{g},L(p-2,1))$,
where $c_{i}\in k$ for all $i$. Then, using the cocycle condition, we get

$\displaystyle c_{1}=c_{9}=c_{15},\qquad
c_{1}-c_{3}+c_{5}+2c_{8}+c_{11}=0,\qquad
c_{2}+c_{3}+c_{5}-c_{6}-c_{14}-c_{16}=0,$ $\displaystyle
c_{2}-2c_{7}+c_{12}-c_{15}-2c_{17}=0,\qquad c_{3}-2c_{8}+c_{13}-2c_{18}=0,$
$\displaystyle c_{4}-c_{2}+2c_{7}+c_{10}=0,\qquad
c_{4}-c_{9}+c_{10}+c_{12}-2c_{17}=0,$ $\displaystyle
c_{5}+c_{9}+c_{11}+c_{13}-2c_{18}=0,\qquad
c_{6}-c_{11}-c_{12}+c_{14}+c_{16}=0.$

The rank of the matrix of this system is equal to $8$. Therefore,

$\displaystyle\dim Z^{4}_{p\omega_{1}}(\mathfrak{g},L(p-2,1))=18-8=10.$

Then, by (2.3),

$\displaystyle\dim H^{4}_{p\omega_{1}}(\mathfrak{g},L(p-2,1))={}$
$\displaystyle\dim Z^{4}_{p\omega_{1}}(\mathfrak{g},L(p-2,1))+\dim
Z^{3}_{p\omega_{1}}(\mathfrak{g},L(p-2,1))$ $\displaystyle-\dim
C^{3}_{p\omega_{1}}(\mathfrak{g},L(p-2,1))=10+6-14=2.$

Thus, $H^{4}(\mathfrak{g},L(p-2,1))$ is generated by the two cohomological
classes with weight $p\omega_{1}$. So, $H^{4}(\mathfrak{g},L(p-2,1))\cong
2L(1,0)^{(1)}$.

Similar calculations give us

$\displaystyle\dim C^{5}_{p\omega_{1}}(\mathfrak{g},L(p-2,1))=14,\qquad$
$\displaystyle\dim Z^{5}_{p\omega_{1}}(\mathfrak{g},L(p-2,1))=8,$
$\displaystyle\dim C^{6}_{p\omega_{1}}(\mathfrak{g},L(p-2,1))=7,\qquad$
$\displaystyle\dim Z^{6}_{p\omega_{1}}(\mathfrak{g},L(p-2,1))=6,$
$\displaystyle\dim C^{7}_{p\omega_{1}}(\mathfrak{g},L(p-2,1))=2,\qquad$
$\displaystyle\dim Z^{7}_{p\omega_{1}}(\mathfrak{g},L(p-2,1))=2.$

Then, using (2.3), we get

$\displaystyle\dim H^{5}_{p\omega_{1}}(\mathfrak{g},L(p-2,1))=0,\qquad\dim
H^{6}_{p\omega_{1}}(\mathfrak{g},L(p-2,1))=0,$ $\displaystyle\dim
H^{7}_{p\omega_{1}}(\mathfrak{g},L(p-2,1))=1.$

This completes the proof of the lemma. ∎

$(c)$ Let $M=L(1,p-2)$. Obviously, $M$ is dual to $L(p-2,1)$. Then, using
(2.2) and Lemma 3.1, we get the following

######  Lemma 3.2.

Let $\mathfrak{g}=\mathfrak{sl}_{3}(k)$ and $M=L(1,p-2)$. Then,
$H^{n}(\mathfrak{g},M)=0$, except for the following cases:

  1. $(i)$

$H^{1}(\mathfrak{g},L(1,p-2))\cong H^{7}(\mathfrak{g},L(1,p-2))\cong
L(0,1)^{(1)}$,

  2. $(ii)$

$H^{4}(\mathfrak{g},L(1,p-2))\cong 2L(0,1)^{(1)}$.

$(d)$ Let $M=L(p-3,0)$.

######  Lemma 3.3.

Let $\mathfrak{g}=\mathfrak{sl}_{3}(k)$ and $M=L(p-3,0)$. Then,
$H^{n}(\mathfrak{g},M)=0$, except for the following cases:

  1. $(i)$

$H^{2}(\mathfrak{g},L(p-3,0))\cong H^{6}(\mathfrak{g},L(p-3,0))\cong
L(1,0)^{(1)}$,

  2. $(ii)$

$H^{3}(\mathfrak{g},L(p-3,0))\cong H^{5}(\mathfrak{g},L(p-3,0))\cong
L(1,0)^{(1)}$.

###### Proof.

It is easy to see that

$\displaystyle\prod\big{(}\overline{C}^{\bullet}(\mathfrak{g},L(p-3,0))\big{)}=\\{p\omega_{1},p(-\omega_{1}+\omega_{2}),-p\omega_{2}\\}.$

Then, it is obvious that

$\displaystyle\prod\big{(}\overline{C}^{i}(\mathfrak{g},L(p-3,0))\big{)}\bigcap\prod\big{(}\overline{C}^{*}(\mathfrak{g},L(p-3,0))\big{)}=\varnothing\qquad\text{for}\quad
i=1,2.$

Therefore,

$\displaystyle H^{0}(\mathfrak{g},L(p-3,0))=0\qquad\text{and}\qquad
H^{1}(\mathfrak{g},L(p-3,0))=0.$

Further, we get

$\displaystyle\prod\big{(}\overline{C}^{2}(\mathfrak{g},L(p-3,0))\big{)}=\\{p\omega_{1},p(-\omega_{1}+\omega_{2}),-p\omega_{2}\\}.$

Any composition factor of $H^{2}(\mathfrak{g},L(p-3,0))$, as a $G$-module, is
uniquely determined by its highest weight. The highest weight of a simple
$G$-module is dominant, see [17, p. 260]. Recall that the weight
$\lambda=r\omega_{1}+s\omega_{2}$ is dominant if $r\geq 0$ and $s\geq 0$.
Then, $H^{2}(\mathfrak{g},L(p-3,0))$ can be generated only by the classes of
cocycles with dominant weight $p\omega_{1}$. Therefore, it is sufficient to
determine the multiplicity of $p\omega_{1}$. Consider the subspace of
$2$-cochains with dominant weight $p\omega_{1}$. The subspace
$\overline{C}^{2}_{p\omega_{1}}(\mathfrak{g},L(p-3,0))$ is one-dimensional and
is spanned by the 2-cochain $\psi^{2}=f_{1}^{*}\wedge f_{3}^{*}\otimes
v_{0,0,0}$. It is easy to see that $\psi^{2}$ is a 2-cocycle. Since
$\overline{C}^{1}_{p\omega_{1}}(\mathfrak{g},L(p-3,0))\allowbreak=0$, it
follows that $\psi^{2}$ cannot be a coboundary. Therefore,
$H^{2}(\mathfrak{g},L(p-3,0))$, as a $G$-module, is generated by the class
$\big{[}\psi^{2}\big{]}$ of $2$-cocycles with weight $p\omega_{1}$ and is
isomorphic to $L(1,0)^{(1)}$.

The set of weights of the subspace
$\prod\big{(}\overline{C}^{3}(\mathfrak{g},L(p-3,0))\big{)}$ is also equal to
$\\{p\omega_{1},p(-\omega_{1}+\omega_{2}),\allowbreak-p\omega_{2}\\}$. The
subspace $\overline{C}^{3}_{p\omega_{1}}(\mathfrak{g},L(p-3,0))$ is two-
dimensional and is spanned by the $3$-cochains

$\displaystyle\psi^{3}_{1}=h_{1}^{*}\wedge f_{1}^{*}\wedge f_{3}^{*}\otimes
v_{0,0,0},\qquad\psi^{3}_{2}=h_{2}^{*}\wedge f_{1}^{*}\wedge f_{3}^{*}\otimes
v_{0,0,0}.$

If $a_{1}\psi^{3}_{1}+a_{2}\psi^{3}_{2}$ is a $3$-cocycle, then it follows
from the cocycle condition that $a_{2}=0$. Since

$\displaystyle\dim\overline{C}^{2}_{p\omega_{1}}(\mathfrak{g},L(p-3,0))=\dim\overline{Z}^{2}_{p\omega_{1}}(\mathfrak{g},L(p-3,0))=1,$

by (2.3), we see that $\dim
H^{3}_{p\omega_{1}}(\mathfrak{g},L(p-3,0))=1+1-1=1$. Therefore,
$H^{3}(\mathfrak{g},L(p-3,0))$, as a $G$-module, is generated by the class
$\big{[}\psi^{3}_{1}\big{]}$ of $3$-cocycles with weight $p\omega_{1}$ and is
isomorphic to $L(1,0)^{(1)}$.

Now, we will prove that $H^{4}(\mathfrak{g},L(p-3,0))=0$. The weight subspace
$\overline{C}^{4}_{p\omega_{1}}(\mathfrak{g},L(p-3,0))$ is two-dimensional and
is spanned by the $4$-cochains

$\displaystyle\psi^{4}_{1}=h_{1}^{*}\wedge h_{2}^{*}\wedge f_{1}^{*}\wedge
f_{3}^{*}\otimes v_{0,0,0},\qquad\psi^{4}_{2}=e_{2}^{*}\wedge f_{1}^{*}\wedge
f_{2}^{*}\wedge f_{3}^{*}\otimes v_{0,0,0}.$

If $b_{1}\psi^{4}_{1}+b_{2}\psi^{4}_{2}$ is a $4$-cocycle, then it follows
from the cocycle condition that $b_{1}=0$. Since

$\displaystyle\dim\overline{C}^{3}_{p\omega_{1}}(\mathfrak{g},L(p-3,0))=2\qquad\text{and}\qquad\dim\overline{Z}^{3}_{p\omega_{1}}(\mathfrak{g},L(p-3,0))=1,$

by (2.3), it follows that

$\displaystyle\dim H^{4}_{p\omega_{1}}(\mathfrak{g},L(p-3,0))=1+1-2=0.$

Therefore,

$\displaystyle\dim H^{4}(\mathfrak{g},L(p-3,0))=\dim
H^{4}_{p\omega_{1}}(\mathfrak{g},L(p-3,0))=0.$

The weight subspace $\overline{C}^{5}_{p\omega_{1}}(\mathfrak{g},L(p-3,0))$ is
two-dimensional and is spanned by the $5$-cochains

$\displaystyle\psi^{5}_{1}=h_{1}^{*}\wedge e_{2}^{*}\wedge f_{1}^{*}\wedge
f_{2}^{*}\wedge f_{3}^{*}\otimes v_{0,0,0},\qquad\psi^{5}_{2}=h_{2}^{*}\wedge
e_{2}^{*}\wedge f_{1}^{*}\wedge f_{2}^{*}\wedge f_{3}^{*}\otimes v_{0,0,0}.$

It follows from the cocycle condition that
$c_{1}\psi^{5}_{1}+c_{2}\psi^{5}_{2}$ is a $5$-cocycle for any $c_{1},c_{2}\in
k$. So, $\dim\overline{Z}^{5}_{p\omega_{1}}(\mathfrak{g},L(p-3,0))=2$. Since

$\displaystyle\dim\overline{C}^{4}_{p\omega_{1}}(\mathfrak{g},L(p-3,0))=2\qquad\text{and}\qquad\dim\overline{Z}^{4}_{p\omega_{1}}(\mathfrak{g},L(p-3,0))=1,$

then by (2.3), we see that

$\displaystyle\dim H^{5}_{p\omega_{1}}(\mathfrak{g},L(p-3,0))=2+1-2=1.$

Therefore, $H^{5}(\mathfrak{g},L(p-3,0))$, as a $G$-module, is generated by
the class $\big{[}\psi^{5}_{1}\big{]}$ of $5$-cocycles with weight
$p\omega_{1}$ and is isomorphic to $L(1,0)^{(1)}$.

The weight subspace $\overline{C}^{6}_{p\omega_{1}}(\mathfrak{g},L(p-3,0))$ is
one-dimensional and is spanned by the $6$-cochain

$\displaystyle\psi^{6}=h_{1}^{*}\wedge h_{2}^{*}\wedge e_{2}^{*}\wedge
f_{1}^{*}\wedge f_{2}^{*}\wedge f_{3}^{*}\otimes v_{0,0,0}.$

It follows from the cocycle condition that $\psi^{6}$ is a $6$-cocycle. So,
$\dim\overline{Z}^{6}_{p\omega_{1}}(\mathfrak{g},L(p-3,0))=1$. Since

$\displaystyle\dim\overline{C}^{5}_{p\omega_{1}}(\mathfrak{g},L(p-3,0))=2\qquad\text{and}\qquad\dim\overline{Z}^{5}_{p\omega_{1}}(\mathfrak{g},L(p-3,0))=2,$

then by (2.3),

$\displaystyle\dim H^{6}_{p\omega_{1}}(\mathfrak{g},L(p-3,0))=1.$

Therefore, $H^{6}(\mathfrak{g},L(p-3,0))$, as a $G$-module, is generated by
the class $[\psi^{6}]$ of $6$-cocycles with weight $p\omega_{1}$ and is
isomorphic to $L(1,0)^{(1)}$.

Finally, the subspaces $\overline{C}^{7}(\mathfrak{g},L(p-3,0))$ and
$\overline{C}^{8}(\mathfrak{g},L(p-3,0))$ are trivial, therefore,

$H^{7}(\mathfrak{g},L(p-3,0))=0\qquad\text{and}\qquad
H^{8}(\mathfrak{g},L(p-3,0))=0.$

$(e)$ Let $M=L(0,p-3)$. Obviously, $M$ is dual to $L(p-3,0)$. Then, using
(2.2) and Lemma 3.3, we get the following

######  Lemma 3.4.

Let $\mathfrak{g}=\mathfrak{sl}_{3}(k)$ and $M=L(0,p-3)$. Then,
$H^{n}(\mathfrak{g},M)=0$, except for the following cases:

  1. $(i)$

$H^{2}(\mathfrak{g},L(0,p-3))\cong H^{6}(\mathfrak{g},L(0,p-3))\cong
L(0,1)^{(1)}$,

  2. $(ii)$

$H^{3}(\mathfrak{g},L(0,p-3))\cong H^{5}(\mathfrak{g},L(0,p-3))\cong
L(0,1)^{(1)}$.

$(f)$ Finally, let $M=L(p-2,p-2)$. In this case, we will use some properties
of the connection between ordinary and restricted cohomologies. The restricted
cohomolology of a restricted Lie algebra with coefficients in a restricted
module was introduced by Hochschild in [16, p. 561]. The restricted
$n$-cohomology of $\mathfrak{g}$ with coefficients in a restricted
$\mathfrak{g}$-module $V$ is denoted by $H^{n}_{\rm res}(\mathfrak{g},V)$.

For $M=L(p-2,p-2)$, there is the following short exact sequence of
$\mathfrak{g}$-modules:

$\displaystyle 0\longrightarrow M\longrightarrow H^{0}(p-2,p-2)\longrightarrow
k\longrightarrow 0.$ (3.1)

If the cohomology of $H^{0}(p-2,p-2)$ is known, then using the long exact
cohomology sequence

$\displaystyle\cdots\longrightarrow H^{n}(\mathfrak{g},M)\longrightarrow
H^{n}\big{(}\mathfrak{g},H^{0}(p-2,p-2)\big{)}\longrightarrow
H^{n}(\mathfrak{g})\longrightarrow\cdots,$ (3.2)

corresponding to the short exact sequence (3.1), we can obtain information
about $H^{n}(\mathfrak{g},M)$. We calculate
$H^{n}\big{(}\mathfrak{g},H^{0}(p-2,p-2)\big{)}$ in two steps.

First, we calculate the restricted cohomology $H^{n}_{\rm
res}\big{(}\mathfrak{g},H^{0}(p-2,p-2)\big{)}$, using the equivalence of the
cohomologies $H^{n}_{\rm res}\big{(}\mathfrak{g},H^{0}(p-2,p-2)\big{)}$ and
$H^{n}\big{(}G_{1},H^{0}(p-2,p-2)\big{)}$, where $G_{1}$ is the first
Frobenius kernel for $G$, see [22, Section I.9.6], and Andersen–Jantzen
formula on cohomology of $G_{1}$ with coefficients in $H^{0}(\lambda)$, see
[1]. Let $p>3$, and $\lambda=w\cdot 0+p\nu$. Then, see [1, p. 501],

$\displaystyle
H^{i}\big{(}G_{1},H^{0}(\lambda)\big{)}^{(-1)}\cong\begin{cases}\operatorname{Ind}_{B}^{G}\big{(}S^{(i-l(w))/2}(\mathfrak{u}^{*})\otimes
k_{\nu}\big{)}&\text{if}\ i-l(w)\ \text{is even},\\\ 0&\text{if}\ i-l(w)\
\text{is odd},\end{cases}$ (3.3)

where $\mathfrak{u}$ is the maximal nilpotent subalgebra of $\mathfrak{g}$,
corresponding to the negative roots. The Lie algebra $\mathfrak{u}$ is the Lie
algebra of the unipotent radical $U$ of $B$.

Then, to pass to the usual cohomology
$H^{n}\big{(}\mathfrak{g},H^{0}(p-2,p-2)\big{)}$, we use the
Friedlander–Parshall–Farnsteiner spectral sequence, see [14, Section 5] and
[12, Theorem 4.1]. In [12, Theorem 4.1], choosing the zero ideal as an ideal
of a given Lie algebra, we obtain the following spectral sequence for the
cohomology of the Lie algebra $\mathfrak{g}$ with coefficients in the
$\mathfrak{g}$-module $V$:

$\displaystyle\bigoplus_{i+j=n}\operatorname{Hom}_{k}\Big{(}\bigwedge{{}^{i}}(\mathfrak{g}),H^{j}_{\rm
res}(\mathfrak{g},V)\Big{)}\Longrightarrow H^{n}(\mathfrak{g},V).$

In particular, the following lemma can be directly obtained from the last
spectral sequence:

######  Lemma 3.5.

Let $\mathfrak{g}=\mathfrak{sl}_{3}(k)$ $p>3$ and $V$ a $\mathfrak{g}$-module.
Then

  1. $(i)$

if $H^{i}_{\rm res}(\mathfrak{g},V)=0$ for all $i\leq n$, then
$H^{i}(\mathfrak{g},V)=0$ for all $i\leq n$,

  2. $(ii)$

if $H^{i}(\mathfrak{g},V)=0$ for all $i\leq n-2$, then

$\displaystyle H^{n-1}(\mathfrak{g},V)\cong H^{n-1}_{\rm res}(\mathfrak{g},V)$
(3.4)

and the following sequence is exact:

$\displaystyle 0$ $\displaystyle\longrightarrow H^{n}_{\rm
res}(\mathfrak{g},V)\longrightarrow
H^{n}(\mathfrak{g},V)\longrightarrow\operatorname{Hom}_{k}\big{(}\mathfrak{g},H^{n-1}_{\rm
res}(\mathfrak{g},V)\big{)}\longrightarrow H^{n+1}_{\rm res}(\mathfrak{g},V)$
$\displaystyle\longrightarrow H^{n+1}(\mathfrak{g},V).$ (3.5)

We start by calculating the cohomology
$H^{n}\big{(}G_{1},H^{0}(p-2,p-2)\big{)}$ with $n\leq\dim\mathfrak{g}$.

######  Lemma 3.6.

Let $G_{1}$ be the first Frobenius kernel of $G$, and $V=H^{0}(p-2,p-2)$ the
$G_{1}$-module. Then,

  1. $(i)$

$H^{i}(G_{1},V)=0$ for $i=0,1,2,4,6,8$,

  2. $(ii)$

$H^{3}(G_{1},V)\cong L(1,1)^{(1)}$,

  3. $(iii)$

$H^{5}(G_{1},V)\cong L(3,0)^{(1)}\oplus L(0,3)^{(1)}\oplus L(2,2)^{(1)}$.

###### Proof.

$(i)$ Since

$\displaystyle\lambda=(p-2)(\omega_{1}+\omega_{2})=s_{1}s_{2}s_{1}\cdot
0+p(\omega_{1}+\omega_{2}),$

we get

$\displaystyle w=s_{1}s_{2}s_{1},\qquad
l(w)=3,\qquad\text{and}\qquad\nu=\omega_{1}+\omega_{2}.$

Then, by (3.3), $H^{i}(G_{1},V)=0$ for $i=0,1,2,4,6,8$. The statement $(i)$ is
proved.

$(ii)$ We have

$\displaystyle S^{(3-l(w))/2}(\mathfrak{u}^{*})\otimes
k_{\nu}=S^{0}(\mathfrak{u}^{*})\otimes k_{\nu}\cong
k_{\nu}=k_{\omega_{1}+\omega_{2}}$

and

$\displaystyle H^{0}\big{(}S^{(3-l(w))/2}(\mathfrak{u}^{*})\otimes
k_{\nu}\big{)}\cong
H^{0}(k_{\omega_{1}+\omega_{2}})=\operatorname{Ind}_{B}^{G}(k_{\omega_{1}+\omega_{2}})\cong
L(1,1).$

Then, by (3.3),

$\displaystyle H^{3}\big{(}G_{1},H^{0}(p-2,p-2)\big{)}^{(-1)}\cong L(1,1).$

$(iii)$ We have

$\displaystyle S^{(5-l(w))/2}(\mathfrak{u})^{*}\otimes
k_{\nu}\cong(k_{\alpha_{1}}\oplus k_{\alpha_{2}}\oplus
k_{\alpha_{1}+\alpha_{2}})\otimes k_{\omega_{1}+\omega_{2}}\cong
k_{3\omega_{1}}\oplus k_{3\omega_{2}}\oplus k_{2\omega_{1}+2\omega_{2}}$

and

$\displaystyle H^{0}\big{(}S^{(5-l(w))/2}(\mathfrak{u}^{*})\otimes
k_{\nu}\big{)}=\operatorname{Ind}_{B}^{G}(k_{3\omega_{1}}\oplus
k_{3\omega_{2}}\oplus k_{2\omega_{1}+2\omega_{2}})\cong L(3,0)\oplus
L(0,3)\oplus L(2,2).$

Then, by (3.3),

$H^{5}\big{(}G_{1},H^{0}(p-2,p-2)\big{)}^{(-1)}\cong L(3,0)\oplus L(0,3)\oplus
L(2,2).$

Now, we calculate the cohomology
$H^{n}\big{(}\mathfrak{g},H^{0}(p-2,p-2)\big{)}$.

######  Lemma 3.7.

Let $\mathfrak{g}=\mathfrak{sl}_{3}(k)$ and $V=H^{0}(p-2,p-2)$. Then,

  1. $(i)$

$H^{i}(\mathfrak{g},V)=0$ for $i=0,1,2$,

  2. $(ii)$

$H^{3}(\mathfrak{g},V)\cong L(1,1)^{(1)}$,

  3. $(iii)$

$H^{4}(\mathfrak{g},V)\cong 2L(1,1)^{(1)}\oplus k$.

###### Proof.

$(i)$ Follows from the statements $(i)$ of Lemmas 3.5 and 3.6. $(ii)$ Follows
from the statements $(ii)$ of Lemma 3.6 and formula (3.4). $(iii)$ We get

$\displaystyle\operatorname{Hom}_{k}\big{(}\mathfrak{g},H^{3}_{\rm
res}\big{(}\mathfrak{g},H^{0}(p-2,p-2)\big{)}^{(-1)}\big{)}$
$\displaystyle\cong\mathfrak{g}^{*}\otimes
H^{3}\big{(}G_{1},H^{0}(p-2,p-2)\big{)}^{(-1)}$ $\displaystyle\cong
L(1,1)\otimes L(1,1)$ $\displaystyle\cong L(3,0)\oplus L(0,3)\oplus
L(2,2)\oplus 2L(1,1)\oplus k.$

Then, by (3.5) and the statement $(iii)$ of Lemma 3.6,

$H^{4}\big{(}\mathfrak{g},H^{0}(p-2,p-2)\big{)}^{(-1)}\cong 2L(1,1)\oplus k.$

For $H^{n}(\mathfrak{g},M)$, where $M=L(p-2,p-2)$, we obtain the following
result:

######  Lemma 3.8.

Let $\mathfrak{g}=\mathfrak{sl}_{3}(k)$ and $M=L(p-2,p-2)$. Then,
$H^{n}(\mathfrak{g},M)=0$, except for the following cases:

  1. $(i)$

$H^{1}(\mathfrak{g},M)\cong H^{7}(\mathfrak{g},M)\cong k$,

  2. $(ii)$

$H^{3}(\mathfrak{g},M)\cong H^{5}(\mathfrak{g},M)\cong L(1,1)^{(1)}$,

  3. $(iii)$

$H^{4}(\mathfrak{g},M)\cong 2L(1,1)^{(1)}\oplus 2k$.

###### Proof.

Obviously,
$H^{0}\big{(}\mathfrak{g},H^{0}(p-2,p-2)\big{)}=H^{8}\big{(}\mathfrak{g},H^{0}(p-2,p-2)\big{)}=0$.

$(i)$ The initial terms of the exact sequence (3.2) give us the following
exact sequence:

$\displaystyle 0\longrightarrow H^{0}(\mathfrak{g})\longrightarrow
H^{1}(\mathfrak{g},L(p-2,p-2))\longrightarrow
H^{1}\big{(}\mathfrak{g},H^{0}(p-2,p-2)\big{)}.$

By Lemma 3.7,

$\displaystyle H^{1}(\mathfrak{g},L(p-2,p-2))\cong H^{0}(\mathfrak{g})\cong
k.$

Since $L(p-2,p-2)$ is a self-dual module, then by (2.2), we get

$\displaystyle H^{7}(\mathfrak{g},L(p-2,p-2))\cong
H^{1}(\mathfrak{g},L(p-2,p-2))^{*}\cong k.$

Let us prove that $H^{2}(\mathfrak{g},L(p-2,p-2))=0$. Obviously,
$H^{1}(\mathfrak{g})=0$, and by Lemma 3.7,
$H^{2}\big{(}\mathfrak{g},H^{0}(p-2,p-2)\big{)}=0$. Then, it follows from the
exactness of the sequence (3.2) that $H^{2}(\mathfrak{g},L(p-2,p-2))=0$. By
(2.2),

$\displaystyle H^{6}(\mathfrak{g},L(p-2,p-2))\cong
H^{2}(\mathfrak{g},L(p-2,p-2))^{*}=0.$

$(ii)$ Since $H^{2}(\mathfrak{g})=0$, it follows from the exactness of the
sequence (3.2) that the sequence

$\displaystyle 0\longrightarrow H^{3}(\mathfrak{g},L(p-2,p-2))\longrightarrow
H^{3}\big{(}\mathfrak{g},H^{0}(p-2,p-2)\big{)}\longrightarrow
H^{3}(\mathfrak{g}).$ (3.6)

is exact. It is known that $H^{3}(\mathfrak{g})\cong k$, see [7, p. 113].
Moreover, by Lemma 3.7,

$\displaystyle H^{3}\big{(}\mathfrak{g},H^{0}(p-2,p-2)\big{)}\cong
L(1,1)^{(1)}.$

Then, the exactness of the sequence (3.6) implies that

$\displaystyle H^{3}(\mathfrak{g},L(p-2,p-2))\cong
H^{3}\big{(}\mathfrak{g},H^{0}(p-2,p-2)\big{)}\cong L(1,1)^{(1)},$

since there is no $G$-homomorphism between the modules $L(1,1)^{(1)}$ and $k$.
By (2.2),

$\displaystyle H^{5}(\mathfrak{g},L(p-2,p-2))\cong
H^{3}(\mathfrak{g},L(p-2,p-2))^{*}\cong L(1,1)^{(1)}.$

$(iii)$ In the previous statement, we proved that

$\displaystyle H^{3}(\mathfrak{g},L(p-2,p-2))\cong
H^{3}\big{(}\mathfrak{g},H^{0}(p-2,p-2)\big{)}.$

Then, since the sequence (3.2) is exact, the following sequence is exact:

$\displaystyle 0\longrightarrow H^{3}(\mathfrak{g})\longrightarrow
H^{4}(\mathfrak{g},L(p-2,p-2))\longrightarrow
H^{4}\big{(}\mathfrak{g},H^{0}(p-2,p-2)\big{)}\longrightarrow
H^{4}(\mathfrak{g}).$

Since $H^{3}(\mathfrak{g})\cong k$ and $H^{4}(\mathfrak{g})=0$, then the
sequence

$\displaystyle 0\longrightarrow k\longrightarrow
H^{4}(\mathfrak{g},L(p-2,p-2))\longrightarrow
H^{4}\big{(}\mathfrak{g},H^{0}(p-2,p-2)\big{)}\longrightarrow 0$

is exact. By the statement $(iii)$ of Lemma 3.7,

$\displaystyle H^{4}\big{(}\mathfrak{g},H^{0}(p-2,p-2)\big{)}\cong
2L(1,1)^{(1)}\oplus k.$

There is no $G$-homomorphism between the modules $L(1,1)^{(1)}$ and $k$, so
the last exact sequence is split. Then, we get an isomorphism of $G$-modules
of the statement $(iii)$. ∎

Theorem 1.1 follows from Lemmas 3.1–3.4 and 3.8.

##  4 Cohomology with coefficients in Weyl modules

In this section, we prove Corollaries 1.3 and 1.4. Let us start with Corollary
1.3. We use the following linkage principle (see [17, p. 264]): Let $V$ be
indecomposable $G$-module, having $L(\lambda)$ and $L(\mu)$ as composition
factors. Then $\lambda$ and $\mu$ are linked.

Obviously, $H^{0}(\lambda)$ is peculiar, if it contains a peculiar composition
factor. According to the linkage principle, any composition factor of
$H^{0}(\lambda)$ is linked to $L(\lambda)$. By Theorem 1.1, there are only six
peculiar simple modules. Therefore, $H^{0}(\lambda)$ is peculiar only in the
following cases, which appear in Theorem 1.1:

$\displaystyle\lambda=0,\quad(p-2)\omega_{1}+\omega_{2},\quad\omega_{1}+(p-2)\omega_{2},\quad(p-3)\omega_{1},\quad(p-3)\omega_{2},\quad(p-2)(\omega_{1}+\omega_{2}).$

Let us consider each of these cases separately.

$(a)$ Obviously, $H^{0}(0,0)\cong k$. Then, the needed statement follows from
the statement $(a)$ of Theorem 1.1.

Further, we will proceed as in the proof of Lemma 3.8.

$(b)$ There is the short exact sequence

$\displaystyle 0\longrightarrow L(p-2,1)\longrightarrow
H^{0}(p-2,1)\longrightarrow L(p-3,0)\longrightarrow 0.$

Consider the corresponding long cohomological exact sequence

$\displaystyle\cdots\longrightarrow
H^{n}(\mathfrak{g},L(p-2,1))\longrightarrow
H^{n}\big{(}\mathfrak{g},H^{0}(p-2,1)\big{)}\longrightarrow
H^{n}(\mathfrak{g},L(p-3,0))\longrightarrow\cdots.$

According to Lemmas 3.1, 3.3, the last long cohomological exact sequence
splits into the following exact sequences:

$\displaystyle 0\longrightarrow
H^{0}\big{(}\mathfrak{g},H^{0}(p-2,1)\big{)}\longrightarrow 0,$ $\displaystyle
0\longrightarrow L(1,0)^{(1)}\longrightarrow
H^{1}\big{(}\mathfrak{g},H^{0}(p-2,1)\big{)}\longrightarrow 0,$ $\displaystyle
0\longrightarrow H^{2}\big{(}\mathfrak{g},H^{2}(p-2,1)\big{)}\longrightarrow
L(1,0)^{(1)}\longrightarrow 0,$ $\displaystyle 0\longrightarrow
H^{3}\big{(}\mathfrak{g},H^{0}(p-2,1)\big{)}\longrightarrow
H^{3}(\mathfrak{g},L(p-3,0))\longrightarrow 2L(1,0)^{(1)}$
$\displaystyle\hphantom{0}{}\longrightarrow
H^{4}\big{(}\mathfrak{g},H^{0}(p-2,1)\big{)}\longrightarrow 0,$ $\displaystyle
0\longrightarrow H^{5}\big{(}\mathfrak{g},H^{0}(p-2,1)\big{)}\longrightarrow
L(1,0)^{(1)}\longrightarrow 0,$ $\displaystyle 0\longrightarrow
H^{6}(\mathfrak{g},H^{0}(p-2,1))\longrightarrow L(1,0)^{(1)}\longrightarrow
H^{7}(\mathfrak{g},L(p-2,1))$ $\displaystyle\hphantom{0}{}\longrightarrow
H^{7}\big{(}\mathfrak{g},H^{0}(p-2,1)\big{)}\longrightarrow 0,$ $\displaystyle
0\longrightarrow H^{8}\big{(}\mathfrak{g},H^{0}(p-2,1)\big{)}\longrightarrow
0.$

The first and last exact sequences yield
$H^{n}\big{(}\mathfrak{g},H^{0}(p-2,1)\big{)}=0$ for $n=0,8$. The second,
third, and fifth exact sequences yield isomorphisms

$\displaystyle H^{1}\big{(}\mathfrak{g},H^{0}(p-2,1)\big{)}\cong
L(1,0)^{(1)},\qquad H^{2}\big{(}\mathfrak{g},H^{0}(p-2,1)\big{)}\cong
L(1,0)^{(1)},$

and

$\displaystyle H^{5}\big{(}\mathfrak{g},H^{0}(p-2,1)\big{)}\cong
L(1,0)^{(1)},$

respectively. Consider the fourth exact sequence. Composition factors of
$H^{3}\big{(}\mathfrak{g},H^{0}(p-2,1)\big{)}$ can only be $H^{3}$ with
coefficients in either $L(p-2,1)$ or the socle of $H^{0}(p-2,1)/L(p-2,1)$.
According to Lemma 3.1, $H^{3}(\mathfrak{g},L(p-2,1))=0$. The socle of
$H^{0}(p-2,1)/L(p-2,1)$ is isomorphic to the simple module $L(p-3,0)$.
According to Lemma 3.3,

$\displaystyle H^{3}(\mathfrak{g},L(p-3,0))\cong L(1,0)^{(1)}.$

Therefore,

$\displaystyle H^{3}\big{(}\mathfrak{g},H^{0}(p-2,1)\big{)}\cong
L(1,0)^{(1)}.$

Then, the fourth exact sequence yields an isomorphism

$\displaystyle H^{4}\big{(}\mathfrak{g},H^{0}(p-2,1)\big{)}\cong
2L(1,0)^{(1)}.$

According to Lemma 3.1, in the sixth exact sequence, the map

$\displaystyle H^{7}(\mathfrak{g},L(p-2,1))\longrightarrow
H^{7}\big{(}\mathfrak{g},H^{0}(p-2,1)\big{)}$

is an epimorphism. Consequently, there are the isomorphisms

$\displaystyle H^{7}(\mathfrak{g},L(p-2,1))\cong
H^{7}\big{(}\mathfrak{g},H^{0}(p-2,1)\big{)}\cong L(1,0)^{(1)}.$

Then the sixth exact sequence yields an isomorphism

$\displaystyle H^{6}\big{(}\mathfrak{g},H^{0}(p-2,1)\big{)}\cong
L(1,0)^{(1)}.$

$(c)$ The proof is similar to the previous statement.

$(d)$ Since $H^{0}(p-3,0)\cong L(p-3,0)$, the statement follows from the
statement $(d)$ of Theorem 1.1.

$(e)$ Since $H^{0}(0,p-3)\cong L(0,p-3)$, the statement follows from the
statement $(e)$ of Theorem 1.1.

$(f)$ A part of this statement is proved in Lemma 3.7. We will prove only the
rest of the statement. Using the statement $(a)$ of Theorem 1.1 and Lemma 3.8,
and the long cohomological exact sequence (3.2), we obtain the following exact
sequences:

$\displaystyle 0\longrightarrow L(1,1)^{(1)}\longrightarrow
H^{5}\big{(}\mathfrak{g},H^{0}(p-2,p-2)\big{)}\longrightarrow k\longrightarrow
0,$ $\displaystyle 0\longrightarrow
H^{6}\big{(}\mathfrak{g},H^{0}(p-2,p-2)\big{)}\longrightarrow 0,$
$\displaystyle 0\longrightarrow k\longrightarrow
H^{7}\big{(}\mathfrak{g},H^{0}(p-2,p-2)\big{)}\longrightarrow 0,$
$\displaystyle 0\longrightarrow
H^{8}\big{(}\mathfrak{g},H^{0}(p-2,p-2)\big{)}\longrightarrow k\longrightarrow
0.$

These exact sequences yield the isomorphisms

$\displaystyle H^{5}\big{(}\mathfrak{g},H^{0}(p-2,p-2)\big{)}\cong
L(1,1)^{(1)}\oplus k,$ $\displaystyle
H^{6}\big{(}\mathfrak{g},H^{0}(p-2,p-2)\big{)}=0,$ $\displaystyle
H^{7}\big{(}\mathfrak{g},H^{0}(p-2,p-2)\big{)}\cong k,$ $\displaystyle
H^{8}\big{(}\mathfrak{g},H^{0}(p-2,p-2)\big{)}\cong k.$

Using (2.2) and Corollary 1.3 for $H^{n}(\mathfrak{g},V(\lambda))$, we get
Corollary 1.4.

Corollaries 1.3 and 1.4 show that
$H^{n}\big{(}\mathfrak{g},H^{0}(\lambda)\big{)}\cong
H^{n}(\mathfrak{g},V(\lambda))$, except for the case where

$\displaystyle\lambda=(p-2)(\omega_{1}+\omega_{2}).$

### Acknowledgements

This research is funded by the Science Committee of the Ministry of Education
and Science of the Republic of Kazakhstan (grant No AP08855935). The author is
grateful to the editor and referees whose comments greatly improved the
exposition of this paper.

## References

  * [1] Andersen H.H., Jantzen J.C., Cohomology of induced representations for algebraic groups, Math. Ann. 269 (1984), 487–525.
  * [2] Bouarroudj S., Grozman P., Lebedev A., Leites D., Derivations and central extensions of simple modular Lie algebras and superalgebras, arXiv:1307.1858.
  * [3] Bouarroudj S., Grozman P., Leites D., Deformations of symmetric simple modular Lie superalgebras, arXiv:0807.3054.
  * [4] Braden B., Restricted representations of classical Lie algebras of types $A_{2}$ and $B_{2}$, Bull. Amer. Math. Soc. 73 (1967), 482–486.
  * [5] Carter R.W., Lusztig G., On the modular representations of the general linear and symmetric groups, Math. Z. 136 (1974), 193–242.
  * [6] Chebochko N.G., Deformations of classical Lie algebras with homogeneous root system in characteristic two. I, Sb. Math. 196 (2005), 1371–1402.
  * [7] Chevalley C., Eilenberg S., Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63 (1948), 85–124.
  * [8] Cline E., Parshall B., Scott L., van der Kallen W., Rational and generic cohomology, Invent. Math. 39 (1977), 143–163.
  * [9] Dzhumadil’daev A.S., On the cohomology of modular Lie algebras, Math. USSR-Sb. 47 (1984), 127–143.
  * [10] Dzhumadil’daev A.S., Abelian extensions of modular Lie algebras, Algebra Logic 24 (1985), 1–7.
  * [11] Dzhumadil’daev A.S., Ibraev Sh.Sh., Nonsplit extensions of modular Lie algebras of rank 2, Homology Homotopy Appl. 4 (2002), 141–163.
  * [12] Farnsteiner R., Cohomology groups of reduced enveloping algebras, Math. Z. 206 (1991), 103–117.
  * [13] Franklin J., Homomorphisms between Verma modules in characteristic $p$, J. Algebra 112 (1988), 58–85.
  * [14] Friedlander E.M., Parshall B.J., Modular representation theory of Lie algebras, Amer. J. Math. 110 (1988), 1055–1093.
  * [15] Hazewinkel M., A duality theorem for the cohomology of Lie algebras, Math. USSR-Sb. 12 (1970), 638–644.
  * [16] Hochschild G., Cohomology of restricted Lie algebras, Amer. J. Math. 76 (1954), 555–580.
  * [17] Humphreys J.E., Modular representations of finite groups of Lie type, London Mathematical Society Lecture Note Series, Vol. 326, Cambridge University Press, Cambridge, 2006.
  * [18] Ibraev Sh.Sh., Turbayev B.T., Cohomology for the Lie algebra of type $A_{2}$ over a field of characteristic 2, Sib. Electron. Math. Rep. 18 (2021), 729–739.
  * [19] Ibrayeva A.A., Ibraev Sh.Sh., Yeshmurat G.K., Cohomology of simple modules for $\mathfrak{sl}_{3}(k)$ in characteristic $3$, Bull. Karaganda Univ. Math. Ser. (2021), no. 3, 36–43, arXiv:2108.13652.
  * [20] Jantzen J.C., Weyl modules for groups of Lie type, in Finite Simple Groups II, University of Durham, 1980, 291–300.
  * [21] Jantzen J.C., First cohomology groups for classical Lie algebras, in Representation Theory of Finite Groups and Finite-Dimensional Algebras (Bielefeld, 1991), Progr. Math., Vol. 95, Birkhäuser, Basel, 1991, 289–315.
  * [22] Jantzen J.C., Representations of algebraic groups, 2nd ed., Mathematical Surveys and Monographs, Vol. 107, Amer. Math. Soc., Providence, RI, 2003.
  * [23] Permyakov D.S., Derivations of classical Lie algebras over a field of characteristic $2$, Vestnik Lobachevsky State Univ. Nizhni Novgorod Ser. Math. 1 (1978), 123–134, available at http://www.vestnik.unn.ru/en/nomera?anum=1455.
  * [24] Rudakov A.N., Dimensions of certain irreducible representations of semisimple Lie algebras of classical type over fields of finite characteristic, Trudy Sem. Petrovsk. (1978), no. 3, 147–160.
  * [25] Shu B., Yao Y.-F., On cohomology of a class of nonclassical restricted simple Lie algebras, J. Algebra Appl. 16 (2017), 1750157, 13 pages.
  * [26] Sullivan J.B., The second Lie algebra cohomology group and Weyl modules, Pacific J. Math. 86 (1980), 321–326.