2.1 The general linear Lie algebra and reduced enveloping algebras

Let $\Bbbk$ be an algebraically closed field of characteristic $p>0$ and let $N\in \mathbb{Z}_{{\geqslant}1}$ . Throughout this paper $G:=\text{GL}_{N}(\Bbbk )$ and $\mathfrak{g}:=\mathfrak{g}\mathfrak{l}_{N}(\Bbbk )$ is the Lie algebra of $G$ , which is spanned by the matrix units $\{e_{i,j}\mid 1\leqslant i,j\leqslant N\}$ . Let $(\cdot \,,\cdot ):\mathfrak{g}\times \mathfrak{g}\rightarrow \Bbbk$ denote the trace form associated to the natural representation of $G$ , which we use to identify $\mathfrak{g}\cong \mathfrak{g}^{\ast }$ as $G$ -modules. The universal enveloping algebra of $\mathfrak{g}$ is denoted by $U(\mathfrak{g})$ .

We occasionally need to call on some results from characteristic 0 and so we fix some more notation. We let $\mathfrak{g}_{\mathbb{Z}}$ denote the general linear Lie $\mathbb{Z}$ -algebra $\mathfrak{g}\mathfrak{l}_{N}(\mathbb{Z})$ and we write $\mathfrak{g}_{\mathbb{C}}$ for $\mathfrak{g}\mathfrak{l}_{N}(\mathbb{C})$ . Throughout we use the identifications $\mathfrak{g}\cong \mathfrak{g}_{\mathbb{Z}}\otimes _{\mathbb{Z}}\Bbbk$ and $\mathfrak{g}_{\mathbb{C}}\cong \mathfrak{g}_{\mathbb{Z}}\otimes _{\mathbb{Z}}\mathbb{C}$ , and by a slight abuse of notation we view the matrix units $e_{i,j}$ as elements of $\mathfrak{g}_{\mathbb{Z}}$ or $\mathfrak{g}_{\mathbb{C}}$ when it is convenient to do so. We often consider subalgebras of $\mathfrak{g}$ , which are spanned by matrix units, so have analogues inside $\mathfrak{g}_{\mathbb{Z}}$ and $\mathfrak{g}_{\mathbb{C}}$ and we denote them by decorating with subscripts $\mathbb{Z}$ and $\mathbb{C}$ . We mention that since $\mathfrak{g}_{\mathbb{Z}}$ is a free $\mathbb{Z}$ -module the PBW theorem holds for $U(\mathfrak{g}_{\mathbb{Z}})$ , so that $U(\mathfrak{g}_{\mathbb{Z}})$ is a free $\mathbb{Z}$ -module with a basis consisting of ordered monomials in the matrix units with respect to any choice of total order.

Let $g\in G$ , $x\in \mathfrak{g}$ and $\unicode[STIX]{x1D712}\in \mathfrak{g}^{\ast }$ . We write $g\cdot x$ for the image of $x$ under the adjoint action of $g$ , so as matrices $g\cdot x=gxg^{-1}$ ; this action extends to an action on $U(\mathfrak{g})$ by algebra automorphisms. The centralizer of $x$ in $G$ is denoted by $G^{x}:=\{g\in G\mid g\cdot x=x\}$ and the centralizer of $x$ in $\mathfrak{g}$ is denoted by $\mathfrak{g}^{x}:=\{y\in \mathfrak{g}\mid [y,x]=0\}$ ; we note that $\mathfrak{g}^{x}=\operatorname{Lie}(G^{x})$ . We write $G\cdot \unicode[STIX]{x1D712}$ for the coadjoint orbit of $\unicode[STIX]{x1D712}$ . It is well known that $\dim (G\cdot \unicode[STIX]{x1D712})$ is even and we define $d_{\unicode[STIX]{x1D712}}:=\frac{1}{2}\dim (G\cdot \unicode[STIX]{x1D712})$ .

Let $T\subseteq B\subseteq G$ be the maximal torus and Borel subgroup consisting of diagonal matrices and upper triangular matrices respectively, and let $\mathfrak{t}:=\operatorname{Lie}(T)$ , $\mathfrak{b}:=\operatorname{Lie}(B)$ . We use the notation $\operatorname{diag}(d_{1},\ldots ,d_{n})$ to denote the element of $T$ with $d_{i}$ in the $i$ th entry of the diagonal. We write $X^{\ast }(T)$ for the group of characters, and let $\{\unicode[STIX]{x1D700}_{1},\ldots ,\unicode[STIX]{x1D700}_{N}\}$ be the standard basis of $X^{\ast }(T)$ defined by $\unicode[STIX]{x1D700}_{i}(\operatorname{diag}(d_{1},\ldots ,d_{n}))=d_{i}$ . Let $\unicode[STIX]{x1D6F7}\subseteq X^{\ast }(T)$ be the root system of $G$ with respect to $T$ , so $\unicode[STIX]{x1D6F7}=\{\unicode[STIX]{x1D700}_{i}-\unicode[STIX]{x1D700}_{j}\mid 1\leqslant i,j\leqslant n,i\neq j\}$ . We write $e_{i,j}$ for the matrix unit that spans the root space corresponding to $\unicode[STIX]{x1D700}_{i}-\unicode[STIX]{x1D700}_{j}$ . The root subgroup corresponding to $\unicode[STIX]{x1D700}_{i}-\unicode[STIX]{x1D700}_{j}$ is the image of $u_{i,j}:\Bbbk \rightarrow G$ defined by $u_{i,j}(s):=1+se_{i,j}$ , and the adjoint action of $u_{i,j}(s)$ on $e_{k,l}$ is given by the formula

(1) $$\begin{eqnarray}\displaystyle u_{i,j}(s)\cdot e_{k,l}=e_{k,l}+s\unicode[STIX]{x1D6FF}_{j,k}e_{i,l}-s\unicode[STIX]{x1D6FF}_{l,i}e_{k,j}-s^{2}\unicode[STIX]{x1D6FF}_{j,k}\unicode[STIX]{x1D6FF}_{l,i}e_{i,j}. & & \displaystyle\end{eqnarray}$$

Where it is convenient we allow ourselves to view a character $\unicode[STIX]{x1D6FC}\in X^{\ast }(T)$ as an element of $\mathfrak{t}^{\ast }$ by writing $\unicode[STIX]{x1D6FC}$ for $d\unicode[STIX]{x1D6FC}:\mathfrak{t}\rightarrow \Bbbk$ ; this is a slight abuse of notation, because $d\unicode[STIX]{x1D6FC}=0$ for any $\unicode[STIX]{x1D6FC}\in pX^{\ast }(T)$ .

There is a natural restricted structure on $\mathfrak{g}$ , where the $p$ -power map $x\mapsto x^{[p]}$ is given by taking the $p$ th power of $x$ as a matrix. In particular, we note that $e_{i,j}^{[p]}=\unicode[STIX]{x1D6FF}_{i,j}e_{i,j}$ for $1\leqslant i,j\leqslant N$ . The $p$ -centre of $U(\mathfrak{g})$ is the subalgebra of the centre of $U(\mathfrak{g})$ generated by $\{e_{i,j}^{p}-e_{i,j}^{[p]}\mid 1\leqslant i,j\leqslant N\}$ . It follows from the PBW theorem that $U(\mathfrak{g})$ is a free $Z_{p}(\mathfrak{g})$ -module of rank $p^{\dim \mathfrak{g}}$ . Further, there is a natural identification $Z_{p}(\mathfrak{g})\cong \Bbbk [(\mathfrak{g}^{\ast })^{(1)}]$ , where $(\mathfrak{g}^{\ast })^{(1)}$ denotes the Frobenius twist of $\mathfrak{g}^{\ast }$ . Given $\unicode[STIX]{x1D712}\in \mathfrak{g}^{\ast }$ , we define $J_{\unicode[STIX]{x1D712}}$ to be the ideal of $U(\mathfrak{g})$ generated by $\{x^{p}-x^{[p]}-\unicode[STIX]{x1D712}(x)^{p}\mid x\in \mathfrak{g}\}$ , and the reduced enveloping algebra corresponding to $\unicode[STIX]{x1D712}$ to be $U_{\unicode[STIX]{x1D712}}(\mathfrak{g}):=U(\mathfrak{g})/J_{\unicode[STIX]{x1D712}}$ .

As stated in the introduction, the Kac–Weisfeiler conjecture, which is a theorem of Premet, states that $p^{d_{\unicode[STIX]{x1D712}}}$ is a factor of the dimension of any $U_{\unicode[STIX]{x1D712}}(\mathfrak{g})$ -module. We refer to $U_{\unicode[STIX]{x1D712}}(\mathfrak{g})$ -modules of dimension $p^{d_{\unicode[STIX]{x1D712}}}$ as minimal-dimensional modules, and note that such modules are clearly irreducible.

Let $\unicode[STIX]{x1D712}\in \mathfrak{g}^{\ast }$ . There is unique $x\in \mathfrak{g}$ such that $\unicode[STIX]{x1D712}=(x,\cdot )$ . We have a Jordan decomposition $x=x_{\text{s}}+x_{\text{n}}$ of $x$ , and thus a corresponding decomposition $\unicode[STIX]{x1D712}=\unicode[STIX]{x1D712}_{\text{s}}+\unicode[STIX]{x1D712}_{\text{n}}$ . We say that $\unicode[STIX]{x1D712}$ is nilpotent if $\unicode[STIX]{x1D712}=\unicode[STIX]{x1D712}_{\text{n}}$ . Next we recall the ‘reduction’ to the case $\unicode[STIX]{x1D712}$ nilpotent in the representation theory of $U_{\unicode[STIX]{x1D712}}(\mathfrak{g})$ from [Reference Friedlander and ParshallFP88, §3]; as is noted in [Reference Friedlander and ParshallFP88, §8], this reduction can also be deduced from [Reference Veisfeiler and KatsVK71, Theorem 2]. Let $\mathfrak{l}=\mathfrak{g}^{x_{\text{s}}}$ , let $\mathfrak{q}$ be a parabolic subalgebra of $\mathfrak{g}$ with Levi factor $\mathfrak{l}$ and let $\mathfrak{u}$ denote the nilradical of $\mathfrak{q}$ . We can parabolically induce a $U_{\unicode[STIX]{x1D712}}(\mathfrak{l})$ -module $M$ , to obtain the $U_{\unicode[STIX]{x1D712}}(\mathfrak{g})$ -module $U_{\unicode[STIX]{x1D712}}(\mathfrak{g})\otimes _{U_{\unicode[STIX]{x1D712}}(\mathfrak{q})}M$ , where $M$ is the $U_{\unicode[STIX]{x1D712}}(\mathfrak{q})$ -module on which $\mathfrak{u}$ acts trivially. This gives a functor $U_{\unicode[STIX]{x1D712}}(\mathfrak{l})\operatorname{ - mod}\rightarrow U_{\unicode[STIX]{x1D712}}(\mathfrak{g})\operatorname{ - mod}$ and it is proved in [Reference Friedlander and ParshallFP88, Theorem 3.2] that this is an equivalence of categories; in turn there is an equivalence $U_{\unicode[STIX]{x1D712}}(\mathfrak{l})\operatorname{ - mod}\cong U_{\unicode[STIX]{x1D712}_{\text{n}}}(\mathfrak{l})\operatorname{ - mod}$ as follows from [Reference Friedlander and ParshallFP88, Corollary 3.3]. Further, the theory of Jordan normal forms implies that $\dim (G\cdot \unicode[STIX]{x1D712})=\dim (L\cdot \unicode[STIX]{x1D712}_{\text{n}})+2\dim \mathfrak{u}$ . Therefore, through the equivalence of categories $U_{\unicode[STIX]{x1D712}_{\text{n}}}(\mathfrak{l})\operatorname{ - mod}\cong U_{\unicode[STIX]{x1D712}}(\mathfrak{g})\operatorname{ - mod}$ , minimal-dimensional modules for $U_{\unicode[STIX]{x1D712}_{\text{n}}}(\mathfrak{l})$ correspond to minimal-dimensional modules for $U_{\unicode[STIX]{x1D712}}(\mathfrak{g})\operatorname{ - mod}$ . This justifies our restriction to nilpotent $\unicode[STIX]{x1D712}$ in the statements of Theorem 1.1 and Corollary 1.2.

2.2 Pyramids

We require the combinatorics of pyramids to set up some notation. For more details on this we refer to [Reference Brundan and KleshchevBK06, §7].

We fix a partition $\boldsymbol{p}=(p_{1},\ldots ,p_{n})$ on $N$ with $p_{1}\leqslant \cdots \leqslant p_{n}$ . A pyramid $\unicode[STIX]{x1D70B}$ associated to $\boldsymbol{p}$ is a diagram with $p_{n}$ boxes in the bottom row, $p_{n-1}$ boxes in the row above it, and so forth, stacked in such a way that every box which is not in the bottom row lies directly above a box in the row beneath it, and boxes occur consecutively in each row. The boxes in the pyramid are numbered along rows from left to right and from top to bottom. For example, the pyramids associated to the partition $\boldsymbol{p}=(2,5)$ are

(2)

Let $l=p_{n}$ . The columns of $\unicode[STIX]{x1D70B}$ are labelled $1,2,\ldots ,l$ from left to right and the rows are labelled $1,2,\ldots ,n$ from top to bottom. We denote the heights of the columns in $\unicode[STIX]{x1D70B}$ by $q_{1},q_{2},\ldots ,q_{l}$ . The box in $\unicode[STIX]{x1D70B}$ containing $i$ is referred to as the $i$ th box, and we write $\operatorname{row}(i)$ and $\operatorname{col}(i)$ for the row and column of the $i$ th box, respectively.

We fix a pyramid $\unicode[STIX]{x1D70B}$ corresponding to $\boldsymbol{p}$ for the rest of this paper. From $\unicode[STIX]{x1D70B}$ , we define the shift matrix $\unicode[STIX]{x1D70E}=(s_{i,j})$ as follows. For $1\leqslant i<j\leqslant n$ we let $s_{j,i}$ be the left indentation of the $i$ th row of $\unicode[STIX]{x1D70B}$ relative to the $j$ th row, and we let $s_{i,j}$ be the right indentation of the $i$ th row of $\unicode[STIX]{x1D70B}$ relative to the $j$ th row; also we set $s_{i,i}=0$ . For example, the shift matrices associated to the pyramids in (2) are

$$\begin{eqnarray}\displaystyle \left(\begin{array}{@{}cc@{}}0 & 3\\ 0 & 0\end{array}\right),\left(\begin{array}{@{}cc@{}}0 & 2\\ 1 & 0\end{array}\right),\left(\begin{array}{@{}cc@{}}0 & 1\\ 2 & 0\end{array}\right)\text{ and }\left(\begin{array}{@{}cc@{}}0 & 0\\ 3 & 0\end{array}\right). & & \displaystyle \nonumber\end{eqnarray}$$

2.3 The nilpotent element and subalgebras

We define the nilpotent element

(3) $$\begin{eqnarray}\displaystyle e:=\mathop{\sum }_{\substack{ \operatorname{row}(i)=\operatorname{row}(j) \\ \operatorname{col}(i)=\operatorname{col}(j)-1}}e_{i,j}\in \mathfrak{g}. & & \displaystyle\end{eqnarray}$$

For example, for each of the pyramids in (2), we have $e=e_{1,2}+e_{3,4}+e_{4,5}+e_{5,6}+e_{6,7}$ . Observe that $e$ has Jordan blocks of size $p_{1},p_{2},\ldots ,p_{n}$ . We define $\unicode[STIX]{x1D712}:=(e,\cdot )\in \mathfrak{g}^{\ast }$ . We also note that $\unicode[STIX]{x1D712}$ is in standard Levi form (in the sense of [Reference Friedlander and ParshallFP90, Definition 3.1]) with respect to the simple roots corresponding to the Borel subalgebra $\mathfrak{b}$ .

The first part of the following lemma gives a basis of $\mathfrak{g}^{e}$ , and can be verified by observing that the proof of [Reference Brundan and KleshchevBK06, Lemma 7.3] is also valid in positive characteristic. The second part of the lemma is verified by direct calculation.

Lemma 2.1. Let

$$\begin{eqnarray}c_{i,j}^{(r)}:=\mathop{\sum }_{\substack{ 1\leqslant h,k,\leqslant N \\ \operatorname{row}(h)=i,\operatorname{row}(k)=j \\ \operatorname{col}(k)-\operatorname{col}(h)+1=r}}e_{h,k}\end{eqnarray}$$

for $0\leqslant i,j\leqslant n$ and $r>s_{i,j}$ .

1. (a) The centralizer $\mathfrak{g}^{e}$ of $e$ in $\mathfrak{g}$ has basis

   $$\begin{eqnarray}\{c_{i,j}^{(r)}\mid 0\leqslant i,j\leqslant n,s_{i,j}<r\leqslant s_{i,j}+p_{\min (i,j)}\}.\end{eqnarray}$$

2. (b) We have

   $$\begin{eqnarray}[c_{i,j}^{(r)},c_{k,l}^{(s)}]=\unicode[STIX]{x1D6FF}_{j,k}c_{i,l}^{(r+s-1)}-\unicode[STIX]{x1D6FF}_{i,l}c_{k,j}^{(r+s-1)}.\end{eqnarray}$$

Consider the cocharacter $\unicode[STIX]{x1D707}:\Bbbk ^{\times }\rightarrow T\subseteq G$ defined by $\unicode[STIX]{x1D707}(t)=\operatorname{diag}(t^{\operatorname{col}(1)},\ldots ,t^{\operatorname{col}(n)})$ . Using $\unicode[STIX]{x1D707}$ , we define the $\mathbb{Z}$ -grading

(4) $$\begin{eqnarray}\displaystyle \mathfrak{g}=\bigoplus _{k\in \mathbb{Z}}\mathfrak{g}(k)\quad \text{where }\mathfrak{g}(k):=\{x\in \mathfrak{g}\mid \unicode[STIX]{x1D707}(t)x=t^{k}x\text{ for all }t\in \Bbbk ^{\times }\}. & & \displaystyle\end{eqnarray}$$

Since the adjoint action of $\unicode[STIX]{x1D707}(t)$ on a matrix unit is given by $\unicode[STIX]{x1D707}(t)\cdot e_{i,j}=t^{\operatorname{col}(j)-\operatorname{col}(i)}e_{i,j}$ , we have $\mathfrak{g}(k)=\operatorname{span}\{e_{i,j}\mid \operatorname{col}(j)-\operatorname{col}(i)=k\}$ . From the classification of good gradings in [Reference Elashvili, Kac and VinbergEK05, §4], we see that the grading in (4) is a good grading for $e$ . In fact to get a good grading we should scale the grading by a factor of 2, as we have $e\in \mathfrak{g}(1)$ . We refer also to [Reference Goodwin and TopleyGT18, §3] where good gradings are considered in positive characteristic, and it is shown that the ‘same classification’ of good gradings holds. Since the grading in (4) is good we have that $\mathfrak{g}^{e}\subseteq \bigoplus _{k\geqslant 0}\mathfrak{g}(k)$ , which can also be seen directly from Lemma 2.1, Now it follows from [Reference Elashvili, Kac and VinbergEK05, Theorem 1.4] that $\dim \mathfrak{g}^{e}=\dim \mathfrak{g}(0)$ ; this can also be verified directly from the basis given in Lemma 2.1.

We define the following subalgebras of $\mathfrak{g}$ :

(5) $$\begin{eqnarray}\displaystyle \mathfrak{p}:=\bigoplus _{k\geqslant 0}\mathfrak{g}(k),\quad \mathfrak{h}:=\mathfrak{g}(0)\quad \text{and}\quad \mathfrak{m}:=\bigoplus _{k<0}\mathfrak{g}(k). & & \displaystyle\end{eqnarray}$$

Then $\mathfrak{p}$ is a parabolic subalgebra of $\mathfrak{g}$ , and $\mathfrak{h}$ is the Levi factor of $\mathfrak{p}$ containing $\mathfrak{t}$ . Further, $\mathfrak{m}$ is the nilradical of the opposite parabolic to $\mathfrak{p}$ . We recall that the heights of the columns in $\unicode[STIX]{x1D70B}$ are $q_{1},q_{2},\ldots ,q_{l}$ , and we see that $\mathfrak{h}$ is isomorphic to $\mathfrak{g}\mathfrak{l}_{q_{1}}(\Bbbk )\oplus \mathfrak{g}\mathfrak{l}_{q_{2}}(\Bbbk )\oplus \cdots \oplus \mathfrak{g}\mathfrak{l}_{q_{l}}(\Bbbk )$ . Also $\mathfrak{m}$ is the Lie algebra of the closed subgroup $M$ of $G$ generated by the root subgroups $u_{i,j}(\Bbbk )$ with $\operatorname{col}(j)<\operatorname{col}(i)$ .

We recall that $d_{\unicode[STIX]{x1D712}}$ denotes half the dimension of the coadjoint $G$ -orbit of $\unicode[STIX]{x1D712}$ . So we also have that $d_{\unicode[STIX]{x1D712}}$ is half the dimension of the adjoint $G$ -orbit of $e$ , and thus we see that $d_{\unicode[STIX]{x1D712}}:=\dim \mathfrak{m}$ , because $\dim \mathfrak{g}^{e}=\dim \mathfrak{g}(0)$ .

2.4 Tableaux and weights

We require various weights in $\mathfrak{t}^{\ast }$ , which are used as shifts and to label certain modules. These weights can be encoded by fillings of $\unicode[STIX]{x1D70B}$ as we explain below. We then move on to give the weights we need.

A $\unicode[STIX]{x1D70B}$ -tableau is a diagram obtained by filling the boxes of $\unicode[STIX]{x1D70B}$ with elements of $\Bbbk$ . The set of all tableaux of shape $\unicode[STIX]{x1D70B}$ is denoted $\operatorname{Tab}_{\Bbbk }(\unicode[STIX]{x1D70B})$ , and we write $\operatorname{Tab}_{\mathbb{F}_{p}}(\unicode[STIX]{x1D70B})\subseteq \operatorname{Tab}_{\Bbbk }(\unicode[STIX]{x1D70B})$ for those tableaux with entries in $\mathbb{F}_{p}$ . For $A\in \operatorname{Tab}_{\Bbbk }(\unicode[STIX]{x1D70B})$ , we write $a_{i}$ for the entry in the $i$ th box of $A$ . Two tableaux are called row equivalent if one can be obtained from the other by permuting the entries in the rows. A tableau $A\in \operatorname{Tab}_{\Bbbk }(\unicode[STIX]{x1D70B})$ is column connected if whenever the $j$ th box of $\unicode[STIX]{x1D70B}$ is directly below the $i$ th box we have $a_{i}=a_{j}+1$ .

For $A\in \operatorname{Tab}_{\Bbbk }(\unicode[STIX]{x1D70B})$ we define a weight $\unicode[STIX]{x1D706}_{A}\in \mathfrak{t}^{\ast }$ by

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706}_{A}:=\mathop{\sum }_{i=1}^{N}a_{i}\unicode[STIX]{x1D700}_{i}. & & \displaystyle \nonumber\end{eqnarray}$$

To understand the required weights it helps for us to give a decomposition of $\unicode[STIX]{x1D6F7}$ . We define

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F7}_{+} & := & \displaystyle \{\unicode[STIX]{x1D700}_{i}-\unicode[STIX]{x1D700}_{j}\in \unicode[STIX]{x1D6F7}\mid \operatorname{row}(i)<\operatorname{row}(j)\},\nonumber\\ \displaystyle \unicode[STIX]{x1D6F7}_{0} & := & \displaystyle \{\unicode[STIX]{x1D700}_{i}-\unicode[STIX]{x1D700}_{j}\in \unicode[STIX]{x1D6F7}\mid \operatorname{row}(i)=\operatorname{row}(j)\},\nonumber\\ \displaystyle \unicode[STIX]{x1D6F7}_{-} & := & \displaystyle \{\unicode[STIX]{x1D700}_{i}-\unicode[STIX]{x1D700}_{j}\in \unicode[STIX]{x1D6F7}\mid \operatorname{row}(i)>\operatorname{row}(j)\}.\nonumber\end{eqnarray}$$

Also we define

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F7}(+) & := & \displaystyle \{\unicode[STIX]{x1D700}_{i}-\unicode[STIX]{x1D700}_{j}\in \unicode[STIX]{x1D6F7}\mid \operatorname{col}(i)<\operatorname{col}(j)\},\nonumber\\ \displaystyle \unicode[STIX]{x1D6F7}(0) & := & \displaystyle \{\unicode[STIX]{x1D700}_{i}-\unicode[STIX]{x1D700}_{j}\in \unicode[STIX]{x1D6F7}\mid \operatorname{col}(i)=\operatorname{col}(j)\},\nonumber\\ \displaystyle \unicode[STIX]{x1D6F7}(-) & := & \displaystyle \{\unicode[STIX]{x1D700}_{i}-\unicode[STIX]{x1D700}_{j}\in \unicode[STIX]{x1D6F7}\mid \operatorname{col}(i)>\operatorname{col}(j)\}.\nonumber\end{eqnarray}$$

Then for $\unicode[STIX]{x1D702},\unicode[STIX]{x1D709}\in \{-,0,+\}$ , we define

$$\begin{eqnarray}\unicode[STIX]{x1D6F7}(\unicode[STIX]{x1D702})_{\unicode[STIX]{x1D709}}=\unicode[STIX]{x1D6F7}(\unicode[STIX]{x1D702})\cap \unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D709}}.\end{eqnarray}$$

We note that $\unicode[STIX]{x1D6F7}(0)_{0}=\varnothing$ and that $\unicode[STIX]{x1D6F7}_{+}\cup \unicode[STIX]{x1D6F7}(+)_{0}$ is the system of positive roots corresponding to $\mathfrak{b}$ . Further, $\unicode[STIX]{x1D6F7}(+)\cup \unicode[STIX]{x1D6F7}(0)_{+}$ is the system of positive roots corresponding to a Borel subalgebra contained in $\mathfrak{p}$ , and $\unicode[STIX]{x1D6F7}(-)\cup \unicode[STIX]{x1D6F7}(0)_{+}$ is another system of positive roots.

Having set up this notation, we are in a position to give the weights that we require. First we define

(6) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70C}:=-\mathop{\sum }_{i=1}^{N}i\unicode[STIX]{x1D700}_{i}; & & \displaystyle\end{eqnarray}$$

this is a shifted half sum of positive roots for $\mathfrak{b}$ , and is given by

$$\begin{eqnarray}\unicode[STIX]{x1D70C}=\frac{1}{2}\biggl(\mathop{\sum }_{\unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D6F7}_{+}\cup \unicode[STIX]{x1D6F7}(+)_{0}}\unicode[STIX]{x1D6FC}\biggr)-\unicode[STIX]{x1D6FF},\end{eqnarray}$$

where

$$\begin{eqnarray}\unicode[STIX]{x1D6FF}=\frac{N+1}{2}\mathop{\sum }_{i=1}^{N}\unicode[STIX]{x1D700}_{i}.\end{eqnarray}$$

We note that we should be careful in the above formulas when $p=2$ , though as the final value of $\unicode[STIX]{x1D70C}$ only involves integer coefficients this is not a problem.

We also require a ‘choice of $\unicode[STIX]{x1D70C}$ ’ corresponding to the system of positive roots $\unicode[STIX]{x1D6F7}(+)\cup \unicode[STIX]{x1D6F7}(0)_{+}$ , and we define

(7) $$\begin{eqnarray}\displaystyle \overline{\unicode[STIX]{x1D70C}}:=\frac{1}{2}\biggl(\mathop{\sum }_{\unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D6F7}(+)\cup \unicode[STIX]{x1D6F7}(0)_{+}}\unicode[STIX]{x1D6FC}\biggr)-\unicode[STIX]{x1D6FF}. & & \displaystyle\end{eqnarray}$$

More explicitly, we have

$$\begin{eqnarray}\overline{\unicode[STIX]{x1D70C}}=-\mathop{\sum }_{i=1}^{N}((q_{1}+\cdots +q_{\operatorname{col}(i)-1})+\operatorname{row}(i)-(n-q_{\operatorname{col}(i)}))\unicode[STIX]{x1D700}_{i}.\end{eqnarray}$$

The weight

$$\begin{eqnarray}\unicode[STIX]{x1D6FE}:=\mathop{\sum }_{\unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D6F7}(-)_{+}}\unicode[STIX]{x1D6FC}\end{eqnarray}$$

is important for Theorem 2.2, because

(8) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70C}=\overline{\unicode[STIX]{x1D70C}}+\unicode[STIX]{x1D6FE}. & & \displaystyle\end{eqnarray}$$

We define

(9) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D702}:=\mathop{\sum }_{i=1}^{N}(n-q_{\operatorname{col}(i)}-\cdots -q_{l})\unicode[STIX]{x1D700}_{i} & & \displaystyle\end{eqnarray}$$

and

$$\begin{eqnarray}\unicode[STIX]{x1D70C}_{\mathfrak{h}}:=-\mathop{\sum }_{i=1}^{N}\operatorname{row}(i)\unicode[STIX]{x1D700}_{i},\end{eqnarray}$$

which is a shifted choice $\unicode[STIX]{x1D70C}$ for the Borel subalgebra $\mathfrak{b}\cap \mathfrak{h}$ of $\mathfrak{h}$ . Further, we define

$$\begin{eqnarray}\unicode[STIX]{x1D6FD}:=\mathop{\sum }_{i=1}^{N}((q_{1}+\cdots +q_{\operatorname{col}(i)-1})-(q_{\operatorname{col}(i)+1}+\cdots +q_{l}))\unicode[STIX]{x1D700}_{i}=\mathop{\sum }_{\unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D6F7}(-)}\unicode[STIX]{x1D6FC}\end{eqnarray}$$

and

$$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D70C}}:=\overline{\unicode[STIX]{x1D70C}}+\unicode[STIX]{x1D6FD}.\end{eqnarray}$$

We note that $\widetilde{\unicode[STIX]{x1D70C}}$ is a shifted choice of $\unicode[STIX]{x1D70C}$ for the system of positive roots $\unicode[STIX]{x1D6F7}(-)\cup \unicode[STIX]{x1D6F7}(0)_{+}$ . An important identity for us is

(10) $$\begin{eqnarray}\displaystyle \widetilde{\unicode[STIX]{x1D70C}}=\overline{\unicode[STIX]{x1D70C}}+\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D702}+\unicode[STIX]{x1D70C}_{\mathfrak{h}}=\mathop{\sum }_{\unicode[STIX]{x1D6FC}\in \unicode[STIX]{x1D6F7}(-)\cup \unicode[STIX]{x1D6F7}(0)_{+}}\unicode[STIX]{x1D6FC}. & & \displaystyle\end{eqnarray}$$

2.5 Some modules for $U_{\unicode[STIX]{x1D712}}(\mathfrak{g})$

The weights introduced in the previous subsection are required to define some modules for $\mathfrak{h}$ and for $\mathfrak{g}$ . In what follows it is helpful to note that $\unicode[STIX]{x1D712}|_{\mathfrak{p}}=0$ , so that we can view $U_{0}(\mathfrak{h})\subseteq U_{0}(\mathfrak{p})\subseteq U_{\unicode[STIX]{x1D712}}(\mathfrak{g})$ .

We note that $\unicode[STIX]{x1D706}\in \mathfrak{t}^{\ast }$ is the weight of a one-dimensional $U(\mathfrak{h})$ -module if and only if $\unicode[STIX]{x1D706}(e_{i,i})=\unicode[STIX]{x1D706}(e_{j,j})$ whenever $\operatorname{col}(i)=\operatorname{col}(j)$ , and also that $\widetilde{\unicode[STIX]{x1D70C}}(e_{i,i})=\widetilde{\unicode[STIX]{x1D70C}}(e_{j,j})-1$ , when the $i$ th box in $\unicode[STIX]{x1D70B}$ is directly above the $j$ th box. Thus, for $A\in \operatorname{Tab}_{\Bbbk }(\unicode[STIX]{x1D70B})$ , we deduce that $\unicode[STIX]{x1D706}_{A}-\widetilde{\unicode[STIX]{x1D70C}}$ is the weight of a one-dimensional $U(\mathfrak{h})$ -module if and only if $A$ is column connected. For column connected $A$ we denote this one-dimensional $U(\mathfrak{h})$ -module by $\widetilde{\Bbbk }_{A}$ .

Similarly, given $A\in \operatorname{Tab}_{\Bbbk }(\unicode[STIX]{x1D70B})$ , we have that $\unicode[STIX]{x1D706}_{A}-\overline{\unicode[STIX]{x1D70C}}$ is the weight of a one-dimensional $U(\mathfrak{h})$ -module if and only if $A$ is column connected. In this case we denote the one-dimensional $U(\mathfrak{h})$ -module by $\overline{\Bbbk }_{A}$ , and note that it factors to a module for $U_{0}(\mathfrak{h})$ if and only if $A\in \operatorname{Tab}_{\mathbb{F}_{p}}(\unicode[STIX]{x1D70B})$ . For $A\in \operatorname{Tab}_{\mathbb{F}_{p}}(\unicode[STIX]{x1D70B})$ , we can inflate $\overline{\Bbbk }_{A}$ to a $U_{0}(\mathfrak{p})$ -module and consider the induced module $N_{\unicode[STIX]{x1D712}}(A):=U_{\unicode[STIX]{x1D712}}(\mathfrak{g})\otimes _{U_{0}(\mathfrak{p})}\overline{\Bbbk }_{A}$ . We have that $N_{\unicode[STIX]{x1D712}}(A)\cong U_{\unicode[STIX]{x1D712}}(\mathfrak{m})$ as a $U_{\unicode[STIX]{x1D712}}(\mathfrak{m})$ -module, so that $\dim N_{\unicode[STIX]{x1D712}}(A)=p^{\dim \mathfrak{m}}=p^{d_{\unicode[STIX]{x1D712}}}$ and $N_{\unicode[STIX]{x1D712}}(A)$ is a minimal-dimensional $U_{\unicode[STIX]{x1D712}}(\mathfrak{g})$ -module.

Let $A\in \operatorname{Tab}_{\mathbb{F}_{p}}(\unicode[STIX]{x1D70B})$ . We define $\Bbbk _{A}$ to be the one-dimensional $U_{0}(\mathfrak{b})$ -module where $\mathfrak{t}$ acts by $\unicode[STIX]{x1D706}_{A}-\unicode[STIX]{x1D70C}$ , and the nilradical of $\mathfrak{b}$ acts trivially. The baby Verma module $Z_{\unicode[STIX]{x1D712}}(A)$ is defined to be $Z_{\unicode[STIX]{x1D712}}(A):=U_{\unicode[STIX]{x1D712}}(\mathfrak{g})\otimes _{U_{0}(\mathfrak{b})}\Bbbk _{A}$ . Since $\unicode[STIX]{x1D712}$ is in standard Levi form for the Levi subalgebra $\mathfrak{g}_{0}$ with basis $\{e_{i,j}\mid \operatorname{row}(i)=\operatorname{row}(j)\}$ , $Z_{\unicode[STIX]{x1D712}}(A)$ has a simple head, which we denote by $L_{\unicode[STIX]{x1D712}}(A)$ ; this essentially follows from the results in [Reference Friedlander and ParshallFP90, §3] (see also [Reference Jantzen and BroerJan98, Proposition 10.7]). Moreover, we have that $L_{\unicode[STIX]{x1D712}}(A)\cong L_{\unicode[STIX]{x1D712}}(A^{\prime })$ if and only if $A$ is row equivalent to $A^{\prime }$ ; see [Reference Friedlander and ParshallFP90, Corollary 3.5] or [Reference Jantzen and BroerJan98, Proposition 10.8]. To see this we note that the shift by $\unicode[STIX]{x1D70C}$ in our labelling of the simple modules transforms the dot action of the $W_{0}$ on $\mathfrak{t}^{\ast }$ in [Reference Friedlander and ParshallFP90] to the standard action, where $W_{0}$ is the Weyl group of $\mathfrak{g}_{0}$ with respect to $T$ ; and then this action corresponds to permutations of the entries of a row of a tableau. Given a $U_{\unicode[STIX]{x1D712}}(\mathfrak{g})$ -module $M$ , we say $v\in M$ is a highest weight vector (for $\mathfrak{b}$ ) of weight $\unicode[STIX]{x1D706}\in \mathfrak{t}^{\ast }$ if $\mathfrak{b}v\subseteq \Bbbk v$ and $tv=\unicode[STIX]{x1D706}(t)v$ for all $t\in \mathfrak{t}$ ; so if $v\in M$ is a highest weight vector of weight $\unicode[STIX]{x1D706}_{A}-\unicode[STIX]{x1D70C}$ , then there is a homomorphism $Z_{\unicode[STIX]{x1D712}}(A)\rightarrow M$ sending $1\otimes 1_{A}$ to $v$ , where $1_{A}$ denotes the generator of $\Bbbk _{A}$ .

The following theorem is key to this paper and gives a compatibility between the modules $L_{\unicode[STIX]{x1D712}}(A)$ and $N_{\unicode[STIX]{x1D712}}(A)$ .

Proof. From the discussion above, we know that $N_{\unicode[STIX]{x1D712}}(A)$ has dimension $p^{d_{\unicode[STIX]{x1D712}}}$ , so it is a minimal module for $U_{\unicode[STIX]{x1D712}}(\mathfrak{g})$ and thus simple. It follows that if we can find a highest weight vector $v\in N_{\unicode[STIX]{x1D712}}(A)$ for $\mathfrak{b}$ of weight $\unicode[STIX]{x1D706}_{A}-\unicode[STIX]{x1D70C}$ , then $L_{\unicode[STIX]{x1D712}}(A)\cong N_{\unicode[STIX]{x1D712}}(A)$ as required. This can be seen by noting that the homomorphism $Z_{\unicode[STIX]{x1D712}}(A)\rightarrow N_{\unicode[STIX]{x1D712}}(A)$ will factor to give this isomorphism. The claim regarding row equivalence was justified in the remarks preceding the statement of the theorem.

We observe that the root vectors corresponding to roots in $\unicode[STIX]{x1D6F7}(-)_{+}$ span a $p$ -nilpotent subalgebra $\mathfrak{a}$ of $\mathfrak{g}$ . We let

$$\begin{eqnarray}I=\{(i,j)\mid \unicode[STIX]{x1D700}_{i}-\unicode[STIX]{x1D700}_{j}\in \unicode[STIX]{x1D6F7}(-)_{+}\}=\{(i,j)\mid \operatorname{col}(i)>\operatorname{col}(j),\operatorname{row}(i)<\operatorname{row}(j)\}\subseteq \{1,\ldots ,N\}^{2},\end{eqnarray}$$

so that $\mathfrak{a}$ has basis $\{e_{i,j}\mid (i,j)\in I\}$ . Since all of the elements of this basis have non-zero $\mathfrak{t}^{e}$ weight, we see that the restriction of $\unicode[STIX]{x1D712}$ to $\mathfrak{a}$ is zero. Hence, the restricted enveloping algebra $U_{0}(\mathfrak{a})$ embeds in $U_{\unicode[STIX]{x1D712}}(\mathfrak{g})$ , and consequently $e_{i,j}^{p}=0$ in $U_{0}(\mathfrak{a})\subseteq U_{\unicode[STIX]{x1D712}}(\mathfrak{g})$ for $(i,j)\in I$ .

There is an action of $T$ on $U_{0}(\mathfrak{a})$ , and

(11) $$\begin{eqnarray}\displaystyle u=\mathop{\prod }_{(i,j)\in I}e_{i,j}^{p-1} & & \displaystyle\end{eqnarray}$$

is in the unique weight space of maximal weight (with respect to the positive roots for $\mathfrak{b}$ ). Further, this weight space is one-dimensional, which implies that the product in (11) can be taken in any order (up to rescaling).

Let $\overline{1}_{A}$ denote the generator of $\overline{\Bbbk }_{A}$ . Observe that under the adjoint action $\mathfrak{t}$ acts on $u$ with weight $(p-1)\unicode[STIX]{x1D6FE}=-\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D70C}-\overline{\unicode[STIX]{x1D70C}}$ by (8). Therefore, $v:=u\otimes \overline{1}_{A}$ is a weight vector for $\mathfrak{t}$ with weight $\unicode[STIX]{x1D706}_{A}-\overline{\unicode[STIX]{x1D70C}}+(\overline{\unicode[STIX]{x1D70C}}-\unicode[STIX]{x1D70C})=\unicode[STIX]{x1D706}_{A}-\unicode[STIX]{x1D70C}$ . In order to complete the proof we must show $v$ is a highest weight vector for the action of $\mathfrak{b}$ , which requires us to show that $e_{i,i+1}v=0$ for $i=1,\ldots ,N-1$ .

We first deal with the case where $\operatorname{row}(i)=\operatorname{row}(i+1)$ and we let $r:=\operatorname{row}(i)$ . We begin by decomposing $I$ into four subsets:

$$\begin{eqnarray}\displaystyle I_{1} & := & \displaystyle \{(j,k)\in I\mid \operatorname{row}(j)=r\};\nonumber\\ \displaystyle I_{2} & := & \displaystyle \{(j,k)\in I\mid \operatorname{row}(k)=r\};\nonumber\\ \displaystyle I_{3} & := & \displaystyle \{(j,k)\in I\mid \operatorname{row}(j)<r,\operatorname{row}(k)>r\};\nonumber\\ \displaystyle I_{4} & := & \displaystyle \{(j,k)\in I\mid (j,k)\notin I_{1}\cup I_{2}\cup I_{3}\}.\nonumber\end{eqnarray}$$

We record three facts about commuting elements which are straightforward to verify directly.

Fact 1. $e_{i,i+1}$ commutes with $e_{j,k}$ for $(j,k)\in I_{3}\cup I_{4}$ .

Fact 2. The elements $\{e_{j,k}\mid (j,k)\in I_{1}\cup I_{3}\}$ pairwise commute.

Fact 3. The elements $\{e_{j,k}\mid (j,k)\in I_{2}\cup I_{3}\}$ pairwise commute.

For $s=1,2,3$ , we see that $\{e_{j,k}\mid (j,k)\in I_{s}\}$ is the basis of an abelian subalgebra of $\mathfrak{a}$ . Therefore, the element $u_{s}:=\prod _{(j,k)\in I_{s}}e_{j,k}^{p-1}$ does not depend on the order of the product. We choose an arbitrary ordering of $I_{4}$ and let $u_{4}:=\prod _{(j,k)\in I_{4}}e_{j,k}^{p-1}$ .

We proceed with three claims, which we use to show that $e_{i,i+1}v=0$ .

Claim 1. $(\operatorname{ad}(e_{i,i+1})u_{1})\otimes \overline{1}_{A}=0$ .

Observe that $\operatorname{ad}(e_{i,i+1})u_{1}$ is a sum of expressions of the form

(12) $$\begin{eqnarray}\displaystyle u_{1}^{(i+1,l)}:=e_{i,l}(e_{i+1,l}^{p-2})\prod \mathop{e}_{j,k}^{p-1}, & & \displaystyle\end{eqnarray}$$

where $(i+1,l)\in I_{1}$ , and the product is taken over all $(j,k)\neq (i+1,l)\in I_{1}$ . Since all matrix units occurring in (12) are of the form $e_{a,b}$ with $\operatorname{row}(a)=r$ and $\operatorname{row}(b)>r$ , all of these factors commute so can be reordered.

We consider two cases to complete the proof of Claim 1. The first case is when $(i,l)\in I_{1}$ . Then $u_{1}^{(i+1,l)}$ contains a factor of $e_{i,l}^{p}$ , so that $u_{1}^{(i+1,l)}=0$ . The second case is when $\operatorname{col}(i)=\operatorname{col}(l)$ and so $e_{i,l}\in [\mathfrak{h},\mathfrak{h}]$ . In this case $e_{i,l}\overline{1}_{A}=0$ and so $u_{1}^{(i+1,l)}\otimes \overline{1}_{A}=0$ .

Claim 2. $u_{3}e_{j,k}u_{1}\otimes \overline{1}_{A}=0$ whenever $\operatorname{col}(j)=\operatorname{col}(k)$ and $\operatorname{row}(j)<\operatorname{row}(k)=r$ .

We have $e_{j,k}\in [\mathfrak{h},\mathfrak{h}]$ , so $e_{j,k}\overline{1}_{A}=0$ . Thus it suffices to show that $u_{3}(\operatorname{ad}(e_{j,k})u_{1})=0$ . Observe that $\operatorname{ad}(e_{j,k})u_{1}$ is a sum of monomials of the form

(13) $$\begin{eqnarray}\displaystyle e_{j,l}(e_{k,l}^{p-2})\prod \mathop{e}_{k^{\prime },l^{\prime }}^{p-1}, & & \displaystyle\end{eqnarray}$$

where $(k,l)\in I_{1}$ and the product is taken over $(k^{\prime },l^{\prime })\neq (k,l)\in I_{1}$ . Similar to the comments following (12), the matrix units occurring in (13) all commute and so can be reordered. Since $\operatorname{row}(j)<r$ and $\operatorname{row}(l)>\operatorname{row}(k)=r$ , we have $e_{j,l}\in I_{3}$ . Applying Fact 3 above, we see that $u_{3}e_{j,l}e_{k,l}^{p-2}\prod e_{k^{\prime },l^{\prime }}^{p-1}$ contains a factor of $e_{j,l}^{p}$ , hence is equal to 0. This proves Claim 2.

Claim 3. $u_{3}(\operatorname{ad}(e_{i,i+1})u_{2})u_{1}\otimes \overline{1}_{A}=0$ .

Observe that $\operatorname{ad}(e_{i,i+1})u_{2}$ is a sum of expressions of the form

(14) $$\begin{eqnarray}\displaystyle u_{2}^{(l,i)}:=-e_{l,i+1}(e_{l,i}^{p-2})\prod \mathop{e}_{j^{\prime },k}^{p-1}, & & \displaystyle\end{eqnarray}$$

where $(l,i)\in I_{2}$ and the product is taken over all $(j,k)\in I_{2}$ with $(j,k)\neq (l,i)$ . The matrix units occurring here all commute, so can be reordered. We consider two cases. The first case is when $(l,i+1)\in I_{2}$ . Then $u_{2}^{(l,i)}$ contains a factor of $e_{l,i+1}^{p}$ and $u_{2}^{(l,i)}=0$ . The second case is when $\operatorname{col}(l)=\operatorname{col}(i+1)$ . Then we can use Claim 2 along with Fact 3 to show that $u_{3}u_{2}^{(l,i)}u_{1}\otimes \overline{1}_{A}=0$ . This completes the proof of Claim 3.

We now combine these claims to prove that $e_{i,i+1}v=0$ . Since $e_{i,i+1}$ lies in the nilradical of $\mathfrak{p}$ , we have that $e_{i,i+1}\overline{1}_{A}=0$ . Thus it suffices to prove $\operatorname{ad}(e_{i,i+1})(u_{4}u_{3}u_{2}u_{1})\otimes \overline{1}_{A}=0$ . Applying Fact 1, we only need to check $u_{4}u_{3}(\operatorname{ad}(e_{i,i+1})u_{2})u_{1}\otimes \overline{1}_{A}=0$ and $u_{4}u_{3}u_{2}(\operatorname{ad}(e_{i,i+1})u_{1})\otimes \overline{1}_{A}=0$ , which are given by Claim 3 and Claim 1, respectively.

We move on to deal with the case $\operatorname{row}(i)<\operatorname{row}(i+1)$ , and show that $e_{i,i+1}v=0$ .

For this case first suppose that $\operatorname{col}(i)>\operatorname{col}(i+1)$ . Then we have $e_{i,i+1}\in \mathfrak{a}$ . By the remarks following (11), we can write $u=e_{i,i+1}^{p-1}u_{0}$ for some $u_{0}\in U_{0}(\mathfrak{a})$ and so $e_{i,i+1}u=0$ , which implies that $e_{i,i+1}v=0$ .

The case where $\operatorname{col}(i)=\operatorname{col}(i+1)$ only happens when $p_{\operatorname{row}(i)}=1$ and $s_{i+1,i}=0$ . Then we have $e_{i,i+1}\in [\mathfrak{h},\mathfrak{h}]$ and $e_{i,i+1}\overline{1}_{A}=0$ , so we are just required to show that $[e_{i,i+1},u]=0$ . This is done with commutator arguments similar to those used above, so we omit the details.◻

2.6 The $W$ -algebra $U(\mathfrak{g},e)$ and its $p$ -centre

Since $e\in \mathfrak{g}(1)$ , we have that $\unicode[STIX]{x1D712}$ vanishes on $\mathfrak{g}(k)$ for $k\neq -1$ . Therefore, $\unicode[STIX]{x1D712}$ restricts to a character of $\mathfrak{m}$ . We define $\mathfrak{m}_{\unicode[STIX]{x1D712}}:=\{x-\unicode[STIX]{x1D712}(x)\mid x\in \mathfrak{m}\}\subseteq U(\mathfrak{g})$ , which is a Lie subalgebra of $U(\mathfrak{g})$ . By the PBW theorem there is a direct sum decomposition

$$\begin{eqnarray}U(\mathfrak{g})=U(\mathfrak{g})\mathfrak{m}_{\unicode[STIX]{x1D712}}\oplus U(\mathfrak{p}).\end{eqnarray}$$

We let $\operatorname{pr}:U(\mathfrak{g})\rightarrow U(\mathfrak{p})$ be the projection onto the second factor. Also we abbreviate and write $I:=U(\mathfrak{g})\mathfrak{m}_{\unicode[STIX]{x1D712}}$ , and define $Q:=U(\mathfrak{g})/I$ .

As explained in [Reference Goodwin and TopleyGT18, §4.3], the adjoint action of $M$ on $U(\mathfrak{g})$ gives an adjoint action of $M$ on $Q$ . In [Reference Goodwin and TopleyGT18, Definition 4.3] the $W$ -algebra associated to $e$ is defined to be

$$\begin{eqnarray}\displaystyle \{u+I\in Q\mid g\cdot u+I=u+I\text{ for all }g\in M\}. & & \displaystyle \nonumber\end{eqnarray}$$

In this paper, we prefer to work with an equivalent realization of $U(\mathfrak{g},e)$ as a subalgebra of $U(\mathfrak{p})$ . For this we require the twisted adjoint action of $M$ on $U(\mathfrak{p})$ , which is defined by

$$\begin{eqnarray}\displaystyle \operatorname{tw}(g)\cdot u:=\operatorname{pr}(g\cdot u), & & \displaystyle \nonumber\end{eqnarray}$$

for $g\in M$ and $u\in U(\mathfrak{p})$ . By using $\operatorname{pr}$ to identify $U(\mathfrak{g})/I$ with $U(\mathfrak{p})$ , we can equivalently define the $W$ -algebra associated to $e$ to be the invariant subalgebra

$$\begin{eqnarray}\displaystyle U(\mathfrak{g},e):=U(\mathfrak{p})^{\operatorname{tw}(M)}=\{u\in U(\mathfrak{p})\mid \operatorname{tw}(g)\cdot u=u\text{ for all }g\in M\}. & & \displaystyle \nonumber\end{eqnarray}$$

We want to recast some of the material from [Reference Goodwin and TopleyGT18, §8] to our setting where $U(\mathfrak{g},e)=U(\mathfrak{p})^{\operatorname{tw}(M)}$ . We begin with the $p$ -centre of $U(\mathfrak{g},e)$ , and to define this we note that the $p$ -centre $Z_{p}(\mathfrak{p})$ of $U(\mathfrak{p})$ is stable under the twisted adjoint action of $M$ of $U(\mathfrak{p})$ . The $p$ -centre of $U(\mathfrak{g},e)$ is defined in [Reference Goodwin and TopleyGT18, Definition 8.1], and in our setting, it is given by

$$\begin{eqnarray}Z_{p}(\mathfrak{g},e):=Z_{p}(\mathfrak{p})^{\operatorname{tw}(M)}\subseteq U(\mathfrak{g},e).\end{eqnarray}$$

Let $\unicode[STIX]{x1D713}\in \mathfrak{p}^{\ast }\subseteq \mathfrak{g}^{\ast }$ . We write $J_{\unicode[STIX]{x1D713}}^{\mathfrak{p}}$ for the ideal of $U(\mathfrak{p})$ generated by $\{x^{p}-x^{[p]}-\unicode[STIX]{x1D713}(x)^{p}\mid x\in \mathfrak{p}\}$ . Then the reduced $W$ -algebra corresponding to $\unicode[STIX]{x1D713}$ is defined to be

$$\begin{eqnarray}\displaystyle U_{\unicode[STIX]{x1D713}}(\mathfrak{g},e):=U(\mathfrak{g},e)/(J_{\unicode[STIX]{x1D713}}^{\mathfrak{p}}\cap U(\mathfrak{g},e)). & & \displaystyle \nonumber\end{eqnarray}$$

We note that our notation here differs from that used in [Reference Goodwin and TopleyGT18, Definition 8.5] by a shift of $\unicode[STIX]{x1D712}$ , that is, $U_{\unicode[STIX]{x1D713}}(\mathfrak{g},e)$ here would be denoted $U_{\unicode[STIX]{x1D712}+\unicode[STIX]{x1D713}}(\mathfrak{g},e)$ there (to make sense of $\unicode[STIX]{x1D712}+\unicode[STIX]{x1D713}\in \mathfrak{g}^{\ast }$ we identify $\mathfrak{p}^{\ast }=\operatorname{Ann}_{\mathfrak{g}^{\ast }}(\mathfrak{m})\subseteq \mathfrak{g}^{\ast }$ ). This change in notation is partly justified by the fact that the kernel of the restriction of the projection $U(\mathfrak{p}){\twoheadrightarrow}U_{\unicode[STIX]{x1D713}}(\mathfrak{p})$ to $U(\mathfrak{g},e)$ is $J_{\unicode[STIX]{x1D713}}\cap U(\mathfrak{g},e)$ . Consequently, we can identify $U_{\unicode[STIX]{x1D713}}(\mathfrak{g},e)$ with the image of $U(\mathfrak{g},e)$ in $U_{\unicode[STIX]{x1D713}}(\mathfrak{p})$ .

It turns out that for $\unicode[STIX]{x1D713}\neq \unicode[STIX]{x1D713}^{\prime }$ we can have $J_{\unicode[STIX]{x1D713}}^{\mathfrak{p}}\cap U(\mathfrak{g},e)=J_{\unicode[STIX]{x1D713}^{\prime }}^{\mathfrak{p}}\cap U(\mathfrak{g},e)$ , so that $U_{\unicode[STIX]{x1D713}}(\mathfrak{g},e)=U_{\unicode[STIX]{x1D713}^{\prime }}(\mathfrak{g},e)$ . To explain precisely when this happens we need to translate some of the material from [Reference Goodwin and TopleyGT18, §8.2] to our setting. We write $\mathfrak{m}^{\bot }\subseteq \mathfrak{g}$ for the annihilator of $\mathfrak{m}$ with respect to $(\cdot \,,\cdot )$ , and note that we can identify $\mathfrak{p}^{\ast }\cong e+\mathfrak{m}^{\bot }$ via $(\cdot \,,\cdot )$ . There is an adjoint action of $M$ on $e+\mathfrak{m}^{\bot }$ , and this translates through the identification $\mathfrak{p}^{\ast }\cong e+\mathfrak{m}^{\bot }$ to an action of $M$ on $\mathfrak{p}^{\ast }$ , which we refer to as the twisted action of $M$ on $\mathfrak{p}^{\ast }$ . For $\unicode[STIX]{x1D719}\in \mathfrak{p}^{\ast }$ , $g\in M$ and $x\in \mathfrak{p}$ this twisted adjoint action is given by $(\operatorname{tw}(g)\cdot \unicode[STIX]{x1D719})(x)=\unicode[STIX]{x1D712}(g^{-1}\cdot x-x)+\unicode[STIX]{x1D719}(g^{-1}\cdot x)$ .

Now we state the required part of [Reference Goodwin and TopleyGT18, Lemma 8.6] in our notation.

Thanks to Quillen’s lemma, an irreducible $U(\mathfrak{g},e)$ -module $L$ factors to a module for $U_{\unicode[STIX]{x1D713}}(\mathfrak{g},e)$ for some $\unicode[STIX]{x1D713}\in \mathfrak{p}^{\ast }$ . Further, it is clear from the definitions that, for $\unicode[STIX]{x1D713},\unicode[STIX]{x1D713}^{\prime }\in \mathfrak{p}^{\ast }$ , the module $L$ factors to a module both for $U_{\unicode[STIX]{x1D713}}(\mathfrak{g},e)$ and for $U_{\unicode[STIX]{x1D713}^{\prime }}(\mathfrak{g},e)$ if and only if $J_{\unicode[STIX]{x1D713}}^{\mathfrak{p}}\cap U(\mathfrak{g},e)=J_{\unicode[STIX]{x1D713}^{\prime }}^{\mathfrak{p}}\cap U(\mathfrak{g},e)$ , which by the previous lemma occurs if and only if $\unicode[STIX]{x1D713}$ and $\unicode[STIX]{x1D713}^{\prime }$ are conjugate under the twisted $M$ -action.

We also recall Premet’s equivalence in Theorem 2.4 below. This theorem is based on [Reference PremetPre02, Theorem 2.4], and the statement here can be deduced from [Reference PremetPre10, Lemma 2.2(c)] and [Reference Goodwin and TopleyGT18, Proposition 8.7 and Lemma 8.8]; see also [Reference Goodwin and TopleyGT18, Remark 9.4]. For the statement, we view $\unicode[STIX]{x1D713}\in \mathfrak{p}^{\ast }$ as an element of $\mathfrak{g}^{\ast }$ via the identification $\mathfrak{p}^{\ast }=\operatorname{Ann}_{\mathfrak{g}^{\ast }}(\mathfrak{m})\subseteq \mathfrak{g}^{\ast }$ . Also we define $Q^{\unicode[STIX]{x1D713}}=Q/J_{\unicode[STIX]{x1D712}+\unicode[STIX]{x1D713}}Q$ and recall that, as explained in [Reference Goodwin and TopleyGT18, §8.3], $Q^{\unicode[STIX]{x1D713}}$ is a left $U_{\unicode[STIX]{x1D712}+\unicode[STIX]{x1D713}}(\mathfrak{g})$ -module and a right $U_{\unicode[STIX]{x1D713}}(\mathfrak{g},e)$ -module.

We also recall that $U(\mathfrak{g},e)$ has a PBW basis, which is described in [Reference Goodwin and TopleyGT18, Theorem 7.3]. We summarize the properties that we require in Proposition 2.5 below and adapt the statement to the case $\mathfrak{g}=\mathfrak{g}\mathfrak{l}_{N}(\Bbbk )$ . For this we first have to give some notation. We fix a basis $x_{1},\ldots ,x_{r}$ of $\mathfrak{g}^{e}$ , chosen so that $x_{i}\in \mathfrak{g}(n_{i})$ , where $n_{i}\in \mathbb{Z}_{{\geqslant}0}$ . Let $I_{\mathfrak{p}}=\{(i,j)\mid 1\leqslant i,j,\leqslant N,e_{i,j}\in \mathfrak{p}\}$ and fix an order on $I_{\mathfrak{p}}$ . For $\boldsymbol{a}=(a_{i,j})\in \mathbb{Z}_{{\geqslant}0}^{I_{\mathfrak{p}}}$ we write

(17) $$\begin{eqnarray}\displaystyle \boldsymbol{e}^{\boldsymbol{a}}:=\mathop{\prod }_{(i,j)\in I_{\mathfrak{p}}}e_{i,j}^{a_{i,j}}\in U(\mathfrak{p}), & & \displaystyle\end{eqnarray}$$

and define $|\boldsymbol{a}|=\sum _{(i,j)\in I_{\mathfrak{p}}}a_{i,j}$ and $|\boldsymbol{a}|_{e}=\sum _{(i,j)\in I_{\mathfrak{p}}}(\operatorname{col}(j)-\operatorname{col}(i)+1)a_{i,j}$ .

We can now state our proposition about the PBW basis of $U(\mathfrak{g},e)$ ; it is a consequence of [Reference Goodwin and TopleyGT18, Lemma 7.1 and Theorem 7.3]. We remind the reader that the graded degrees in this paper differ from those in [Reference Goodwin and TopleyGT18] by a factor of 2.

Proposition 2.5. $(\text{a})$ There are elements $\unicode[STIX]{x1D6E9}(x_{1}),\ldots ,\unicode[STIX]{x1D6E9}(x_{r})$ of $U(\mathfrak{g},e)$ of the form

(18) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E9}(x_{i})=x_{i}+\mathop{\sum }_{|\boldsymbol{a}|_{e}\leqslant n_{i}+1}\unicode[STIX]{x1D706}_{\boldsymbol{a},i}\boldsymbol{e}^{\boldsymbol{a}} & & \displaystyle\end{eqnarray}$$

where the $\unicode[STIX]{x1D706}_{\boldsymbol{a},i}\in \Bbbk$ satisfy $\unicode[STIX]{x1D706}_{\boldsymbol{a},i}=0$ whenever $|\boldsymbol{a}|_{e}=n_{i}+1$ and $|\boldsymbol{a}|=1$ .

$(\text{b})$ Given any elements $\unicode[STIX]{x1D6E9}(x_{1}),\ldots ,\unicode[STIX]{x1D6E9}(x_{r})\in U(\mathfrak{g},e)$ of the form in (18), the ordered monomials in $\unicode[STIX]{x1D6E9}(x_{1}),\ldots ,\unicode[STIX]{x1D6E9}(x_{r})$ form a basis of $U(\mathfrak{g},e)$ .

Let $\unicode[STIX]{x1D6E9}(x_{i}),\unicode[STIX]{x1D6E9}(x_{j})$ be elements of $U(\mathfrak{g},e)$ of the form (18). Then a commutator calculation shows that

(19) $$\begin{eqnarray}\displaystyle [\unicode[STIX]{x1D6E9}(x_{i}),\unicode[STIX]{x1D6E9}(x_{j})]=[x_{i},x_{j}]+\mathop{\sum }_{|\boldsymbol{a}|_{e}\leqslant n_{i}+n_{j}+1}\unicode[STIX]{x1D707}_{\boldsymbol{a}}\boldsymbol{e}^{\boldsymbol{a}}, & & \displaystyle\end{eqnarray}$$

where the $\unicode[STIX]{x1D707}_{\boldsymbol{a}}\in \Bbbk$ satisfy $\unicode[STIX]{x1D707}_{\boldsymbol{a}}=0$ whenever $|\boldsymbol{a}|_{e}=n_{i}+n_{j}+1$ and $|\boldsymbol{a}|=1$ . The key ingredient for this calculation is to observe that if we take the commutator $[\boldsymbol{e}^{\boldsymbol{a}},\boldsymbol{e}^{\boldsymbol{b}}]$ for $\boldsymbol{a},\boldsymbol{b}\in \mathbb{Z}_{{\geqslant}0}^{I_{\mathfrak{p}}}$ , then we get a linear combinations of terms $\boldsymbol{e}^{\boldsymbol{c}}$ with $|\boldsymbol{c}|_{e}=|\boldsymbol{a}|_{e}+|\boldsymbol{b}|_{e}-1$ and $|\boldsymbol{c}|=|\boldsymbol{a}|+|\boldsymbol{b}|-1$ , plus a linear combination of terms $\boldsymbol{e}^{\boldsymbol{d}}$ with $|\boldsymbol{d}|_{e}<|\boldsymbol{a}|_{e}+|\boldsymbol{b}|_{e}-1$ .