Adjoint Jordan blocks for simple algebraic groups of type $C_{\ell}$ in characteristic two

Mikko Korhonen SUSTech International Center for Mathematics, Southern University of Science and Technology, Shenzhen 518055, Guangdong, P. R. China [email protected] In memory of Irina Suprunenko

Abstract.

Let $G$ be a simple algebraic group over an algebraically closed field $K$ with Lie algebra $\mathfrak{g}$ . For unipotent elements $u\in G$ and nilpotent elements $e\in\mathfrak{g}$ , the Jordan block sizes of $\operatorname{Ad}(u)$ and $\operatorname{ad}(e)$ are known in most cases. In the cases that remain, the group $G$ is of classical type in bad characteristic, so $\operatorname{char}K=2$ and $G$ is of type $B_{\ell}$ , $C_{\ell}$ , or $D_{\ell}$ .

In this paper, we consider the case where $G$ is of type $C_{\ell}$ and $\operatorname{char}K=2$ . As our main result, we determine the Jordan block sizes of $\operatorname{Ad}(u)$ and $\operatorname{ad}(e)$ for all unipotent $u\in G$ and nilpotent $e\in\mathfrak{g}$ . In the case where $G$ is of adjoint type, we will also describe the Jordan block sizes on $[\mathfrak{g},\mathfrak{g}]$ .

Support by Shenzhen Science and Technology Program (Grant No. RCBS20210609104420034).

1. Introduction

Let $G$ be a simple algebraic group over an algebraically closed field $K$ , and denote the Lie algebra of $G$ by $\mathfrak{g}$ . Recall that an element $u\in G$ is unipotent, if $f(u)$ is a unipotent linear map for every rational representation $f:G\rightarrow\operatorname{GL}(W)$ . Similarly $e\in\mathfrak{g}$ is said to be nilpotent, if $\mathrm{d}f(e)$ is a nilpotent linear map for every rational representation $f:G\rightarrow\operatorname{GL}(W)$ .

We denote the adjoint representations of $G$ and $\mathfrak{g}$ by $\operatorname{Ad}:G\rightarrow\operatorname{GL}(\mathfrak{g})$ and $\operatorname{ad}:\mathfrak{g}\rightarrow\mathfrak{gl}(V)$ , respectively. In this paper, we will consider the following two problems.

Problem 1.1.

Let $u\in G$ be a unipotent element. What is the Jordan normal form of $\operatorname{Ad}(u)$ ?

Problem 1.2.

Let $e\in\mathfrak{g}$ be a nilpotent element. What is the Jordan normal form of $\operatorname{ad}(e)$ ?

In most cases, the answer to both questions is known. When $G$ is simply connected of exceptional type, the Jordan block sizes were computed in the unipotent case by Lawther [21, 22] and in the nilpotent case by Stewart [33]. In the case where $G$ is exceptional of adjoint type, both questions are easily settled using the results of Lawther and Stewart, see [20, Lemma 3.1].

When $G$ is of type $A_{\ell}$ , solutions to both questions follow from the main results of [16] and [19], see [19, Remark 1.3]. For $G$ of type $B_{\ell}$ , $C_{\ell}$ , or $D_{\ell}$ in good characteristic, the Jordan block sizes are described by well-known results on decompositions of tensor products, symmetric squares, and exterior squares — see for example [19, Remark 3.5].

Thus it remains to solve Problem 1.1 and Problem 1.2 in the case where $G$ is of type $B_{\ell}$ , $C_{\ell}$ , or $D_{\ell}$ in characteristic $\operatorname{char}K=2$ . In this paper, we consider the case where $G$ is of type $C_{\ell}$ . We make the following assumption for the rest of this paper.

Assume that $\operatorname{char}K=2$ .

Let $G$ be a simple algebraic group of type $C_{\ell}$ over $K$ . As our main result, we determine the Jordan block sizes of $\operatorname{Ad}(u)$ and $\operatorname{ad}(e)$ for all unipotent $u\in G$ and nilpotent $e\in\mathfrak{g}$ . In type $C_{\ell}$ either $G$ is simply connected or $G$ is adjoint; we deal with these two possibilities as follows.

Let $G_{sc}$ be simply connected and simple of type $C_{\ell}$ with Lie algebra $\mathfrak{g}_{sc}$ , and let $G_{ad}$ be simple of adjoint type $C_{\ell}$ with Lie algebra $\mathfrak{g}_{ad}$ . There exists an isogeny $\varphi:G_{sc}\rightarrow G_{ad}$ , which induces a bijection between the unipotent variety of $G_{sc}$ and $G_{ad}$ . Similarly the differential $\mathrm{d}\varphi:\mathfrak{g}_{sc}\rightarrow\mathfrak{g}_{ad}$ induces a bijection between the nilpotent cone of $\mathfrak{g}_{sc}$ and $\mathfrak{g}_{ad}$ .

Thus for the solution of Problem 1.1 and Problem 1.2 in type $C_{\ell}$ , it will suffice to consider the Jordan block sizes of unipotent $u\in G_{sc}$ on $\mathfrak{g}_{sc}$ and $\mathfrak{g}_{ad}$ , and the Jordan block sizes of nilpotent $e\in\mathfrak{g}_{sc}$ on $\mathfrak{g}_{sc}$ and $\mathfrak{g}_{ad}$ .

We can assume that $G_{sc}=\operatorname{Sp}(V)$ with Lie algebra $\mathfrak{g}_{sc}=\mathfrak{sp}(V)$ , where $\dim V=2\ell$ . It is well known (Lemma 8.1) that $\mathfrak{g}_{sc}\cong S^{2}(V)^{*}$ as $G_{sc}$ -modules. The Jordan block sizes of the action of unipotent $u\in\operatorname{GL}(V)$ on $S^{2}(V)$ and nilpotent $e\in\mathfrak{gl}(V)$ on $S^{2}(V)$ are described in [18]. Taking the dual does not change the Jordan block sizes, so the Jordan block sizes on $\mathfrak{g}_{sc}$ are known by previous results.

In our main results, we will describe the Jordan block sizes on $\mathfrak{g}_{ad}$ in terms of the Jordan block sizes on $\mathfrak{g}_{sc}$ . To state the result in the unipotent case, we first need describe the classification of unipotent conjugacy classes in $\operatorname{Sp}(V)$ , which in characteristic two is due to Hesselink [8]. We discuss this in some more detail in Section 5.

Let $C_{q}$ be a cyclic group of order $q$ , where $q=2^{\alpha}$ for some $\alpha\geq 0$ . Then there are a total of $q$ indecomposable $K[C_{q}]$ -modules $V_{1}$ , $\ldots$ , $V_{q}$ up to isomorphism, where $\dim V_{i}=i$ and a generator of $C_{q}$ acts on $V_{i}$ with a single $i\times i$ unipotent Jordan block. For convenience we will denote $V_{0}=0$ . If $W$ is a vector space over $K$ , we denote $W^{0}=0$ , and $W^{d}=W\oplus\cdots\oplus W$ ( $d$ summands) for an integer $d>0$ .

Suppose then that $u\in\operatorname{GL}(V)$ is unipotent, so $u$ has order $q$ for some $q=2^{\alpha}$ . We denote the group algebra of $\langle u\rangle$ by $K[u]$ . Then $V\downarrow K[u]\cong V_{d_{1}}^{n_{1}}\oplus\cdots\oplus V_{d_{t}}^{n_{t}}$ for some integers $0<d_{1}<\cdots<d_{t}$ , where $n_{i}>0$ for all $1\leq i\leq t$ . Equivalently, the Jordan normal form of $u$ has Jordan blocks of sizes $d_{1}$ , $\ldots$ , $d_{t}$ , and a block of size $d_{i}$ has multiplicity $n_{i}$ .

For unipotent $u\in\operatorname{Sp}(V)$ , we have a decomposition $V\downarrow K[u]=U_{1}\perp\cdots\perp U_{t}$ , where $U_{i}$ are orthogonally indecomposable $K[u]$ -modules. Here orthogonally indecomposable means that if $U_{i}=U^{\prime}\perp U^{\prime\prime}$ as $K[u]$ -modules, then $U^{\prime}=0$ or $U^{\prime\prime}=0$ . The orthogonally indecomposable $K[u]$ -modules fall into two types: one denoted by $V(m)$ (for $m$ even) and another denoted by $W(m)$ (for $m\geq 1$ ). We define these in Section 5, for now we only mention that $V(m)\cong V_{m}$ and $W(m)\cong V_{m}\oplus V_{m}$ as $K[u]$ -modules.

Our first main result is the following. (Below we denote by $\nu_{2}$ the $2$ -adic valuation on the integers, so $\nu_{2}(a)$ is the largest integer $k\geq 0$ such that $2^{k}$ divides $a$ .)

Theorem 1.3.

Let $u\in\operatorname{Sp}(V)$ be unipotent, with orthogonal decomposition

V\downarrow K[u]=\sum_{1\leq i\leq t}W(m_{i})\perp\sum_{1\leq j\leq s}V(2k_{j}).

Denote by $\alpha\geq 0$ the largest integer such that $2^{\alpha}\mid m_{i},k_{j}$ for all $i$ and $j$ . Then the following hold:

(i)

Suppose that $s=0$ . Then:

(a)

If $\alpha=0$ , then $\mathfrak{g}_{sc}\cong\mathfrak{g}_{ad}$ as $K[u]$ -modules.

(b)

If $\alpha>0$ , then

	$\displaystyle\mathfrak{g}_{sc}$	$\displaystyle\cong V_{2^{\alpha}}\oplus V^{\prime}$
	$\displaystyle\mathfrak{g}_{ad}$	$\displaystyle\cong V_{1}\oplus V_{2^{\alpha}-1}\oplus V^{\prime}$

for some $K[u]$ -module $V^{\prime}$ .

(ii)

Suppose that $s>0$ , and let $\beta=\max_{1\leq j\leq s}\nu_{2}(k_{j})$ . Then:

(a)

If $\beta=\nu_{2}(k_{j})$ for all $1\leq j\leq s$ and $\nu_{2}(m_{i})>\beta$ for all $1\leq i\leq t$ , then $\mathfrak{g}_{sc}\cong\mathfrak{g}_{ad}$ as $K[u]$ -modules.

(b)

If $\beta>\nu_{2}(k_{j})$ for some $1\leq j\leq s$ , or $\nu_{2}(m_{i})\leq\beta$ for some $1\leq i\leq t$ , then

	$\displaystyle\mathfrak{g}_{sc}$	$\displaystyle\cong V_{2^{\alpha}}\oplus V_{2^{\beta}}\oplus V^{\prime}$
	$\displaystyle\mathfrak{g}_{ad}$	$\displaystyle\cong V_{2^{\alpha}-1}\oplus V_{2^{\beta}+1}\oplus V^{\prime}$

for some $K[u]$ -module $V^{\prime}$ .

To describe our results in the nilpotent case, we recall the classification of nilpotent orbits in $\mathfrak{sp}(V)$ , due to Hesselink [8]. For more details we refer to Section 6.

Suppose that $q=2^{\alpha}$ for some $\alpha\geq 0$ . Let $\mathfrak{w}_{q}$ be the abelian $2$ -Lie algebra over $K$ generated by a single nilpotent element $e\in\mathfrak{w}_{q}$ such that $e^{[2^{\alpha}]}=0$ and $e^{[2^{\alpha-1}]}\neq 0$ . There are a total of $q$ indecomposable $\mathfrak{w}_{q}$ -modules $W_{1}$ , $\ldots$ , $W_{q}$ up to isomorphism, where $\dim W_{i}=i$ and $e$ acts on $W_{i}$ with a single $i\times i$ nilpotent Jordan block. Throughout we will denote $W_{0}=0$ .

Consider then a nilpotent linear map $e\in\mathfrak{gl}(V)$ , and denote the $2$ -Lie subalgebra generated by $e$ with $K[e]$ . Then $K[e]\cong\mathfrak{w}_{q}$ , where $q$ is the smallest power of two such that $e^{q}=0$ . We have $V\downarrow K[e]\cong W_{d_{1}}^{n_{1}}\oplus\cdots\oplus W_{d_{t}}^{n_{t}}$ for some integers $0<d_{1}<\cdots<d_{t}$ , where $n_{i}>0$ for all $1\leq i\leq t$ . As in the unipotent case, this amounts to the statement that the Jordan normal form of $e$ has Jordan blocks of sizes $d_{1}$ , $\ldots$ , $d_{t}$ , and a block of size $d_{i}$ has multiplicity $n_{i}$ .

For nilpotent $e\in\mathfrak{sp}(V)$ , we have an orthogonal decomposition $V\downarrow K[e]=U_{1}\perp\cdots\perp U_{t}$ , where the $U_{i}$ are orthogonally indecomposable $K[e]$ -modules. Here orthogonally indecomposable is defined as in the unipotent case. For the orthogonally indecomposable $K[e]$ -modules, there are several different types: $V(m)$ (for $m$ even), $W_{k}(m)$ (for $0<k<m/2$ and $m>2$ ), and $W(m)$ (for $m\geq 1$ ). We give the definitions and more details in Section 6, for now we just note that $V(m)\cong W_{m}$ , $W_{k}(m)\cong W_{m}\oplus W_{m}$ , and $W(m)\cong W_{m}\oplus W_{m}$ as $K[e]$ -modules.

Our main result in the nilpotent case is the following.

Theorem 1.4.

Let $e\in\mathfrak{sp}(V)$ be nilpotent, with orthogonal decomposition

V\downarrow K[e]=\sum_{1\leq i\leq t}W(m_{i})\perp\sum_{1\leq j\leq t^{\prime}}W_{k_{j}}(\ell_{j})\perp\sum_{1\leq r\leq t^{\prime\prime}}V(2d_{r}).

Denote by $\alpha\geq 0$ the largest integer such that $2^{\alpha}\mid m_{i},\ell_{j},2d_{r}$ for all $i$ , $j$ , and $r$ . Then the following statements hold:

(i)

Suppose that $V\downarrow K[e]=\sum_{1\leq i\leq t}W(m_{i})$ . Then:

(a)

If $\alpha=0$ , then $\mathfrak{g}_{sc}\cong\mathfrak{g}_{ad}$ as $K[e]$ -modules.

(b)

If $\alpha>0$ , then

	$\displaystyle\mathfrak{g}_{sc}$	$\displaystyle\cong W_{2^{\alpha}}\oplus V^{\prime}$
	$\displaystyle\mathfrak{g}_{ad}$	$\displaystyle\cong W_{1}\oplus W_{2^{\alpha}-1}\oplus V^{\prime}$

for some $K[e]$ -module $V^{\prime}$ .

(ii)

Suppose that $V\downarrow K[e]$ is not of the form $\sum_{1\leq i\leq t}W(m_{i})$ . Then:

(a)

If $\alpha=1$ , then $\mathfrak{g}_{sc}\cong\mathfrak{g}_{ad}$ as $K[e]$ -modules.

(b)

If $\alpha\neq 1$ , then

	$\displaystyle\mathfrak{g}_{sc}$	$\displaystyle\cong W_{1}\oplus W_{2^{\alpha}}\oplus V^{\prime}$
	$\displaystyle\mathfrak{g}_{ad}$	$\displaystyle\cong W_{2}\oplus W_{2^{\alpha}-1}\oplus V^{\prime}$

for some $K[e]$ -module $V^{\prime}$ .

Example 1.5.

In Table 1 and Table 2, we illustrate Theorem 1.3 and Theorem 1.4 in the case where $G$ is of type $C_{\ell}$ for $2\leq\ell\leq 4$ . In the tables we have also included the Jordan block sizes on $[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ , which we prove as an intermediate result in Proposition 9.5 and Proposition 10.5. Furthermore, the values of $\alpha$ and $\beta$ that appear in Theorem 1.3 and Theorem 1.4 are included.

In the tables, we use the notation

d_{1}^{n_{1}},\ldots,d_{t}^{n_{t}}

for $0<d_{1}<\cdots<d_{t}$ and $n_{i}>0$ to denote that the Jordan block sizes are $d_{1}$ , $\ldots$ , $d_{t}$ , and a block of size $d_{i}$ has multiplicity $n_{i}$ . For the orthogonal decompositions in the first column, notation such as $V(m)^{k}$ denotes an orthogonal direct sum $V(m)\perp\cdots\perp V(m)$ ( $k$ summands).

In particular, from Theorem 1.3 and Theorem 1.4 we get the number of Jordan blocks of $\operatorname{Ad}(u)$ and $\operatorname{ad}(e)$ , which is equal to the dimension of the Lie algebra centralizer of $u$ and $e$ , respectively.

Corollary 1.6.

Let $u\in\operatorname{Sp}(V)$ be unipotent and denote $\alpha$ , $s$ as in Theorem 1.3. Then the following hold:

(i)

Suppose that $s=0$ . Then

\dim\mathfrak{g}_{ad}^{u}=\begin{cases}\dim\mathfrak{g}_{sc}^{u},&\text{ if }\alpha=0.\\ \dim\mathfrak{g}_{sc}^{u}+1,&\text{ if }\alpha>0.\end{cases}

(ii)

Suppose that $s>0$ . Then

\dim\mathfrak{g}_{ad}^{u}=\begin{cases}\dim\mathfrak{g}_{sc}^{u}-1,&\text{ if }\alpha=0\text{ and }\nu_{2}(k_{j})>0\text{ for some }j.\\ \dim\mathfrak{g}_{sc}^{u}-1,&\text{ if }\alpha=0\text{ and }\nu_{2}(m_{i})=0\text{ for some }i.\\ \dim\mathfrak{g}_{sc}^{u},&\text{ if }\alpha=0\text{ and }\nu_{2}(k_{j})=0,\nu_{2}(m_{i})>0\text{ for all }i\text{ and }j.\\ \dim\mathfrak{g}_{sc}^{u},&\text{ if }\alpha>0.\end{cases}

Corollary 1.7.

Let $e\in\mathfrak{sp}(V)$ be nilpotent and denote $\alpha$ as in Theorem 1.3. Then the following hold:

(i)

Suppose that $V\downarrow K[e]=\sum_{1\leq i\leq t}W(m_{i})$ . Then

\dim\mathfrak{g}_{ad}^{e}=\begin{cases}\dim\mathfrak{g}_{sc}^{e},&\text{ if }\alpha=0.\\ \dim\mathfrak{g}_{sc}^{e}+1,&\text{ if }\alpha>0.\end{cases}

(ii)

Suppose that $V\downarrow K[e]$ is not of the form $\sum_{1\leq i\leq t}W(m_{i})$ . Then

\dim\mathfrak{g}_{ad}^{e}=\begin{cases}\dim\mathfrak{g}_{sc}^{e}-1,&\text{ if }\alpha=0.\\ \dim\mathfrak{g}_{sc}^{e},&\text{ if }\alpha>0.\end{cases}

In the case of regular elements, this amounts to the following. (An element $x\in G$ or $x\in\mathfrak{g}$ is regular, if $\dim C_{G}(x)=\operatorname{rank}G$ . In the case of $G=\operatorname{Sp}(V)$ , a regular unipotent element is characterized by $V\downarrow K[u]=V(2\ell)$ , and a regular nilpotent element is characterized by $V\downarrow K[e]=V(2\ell)$ .)

Corollary 1.8.

Suppose that $G$ is of type $C_{\ell}$ in characteristic two (adjoint or simply connected). Let $u\in G$ be a regular unipotent element and $e\in\mathfrak{g}$ a regular nilpotent element. Then $\dim\mathfrak{g}^{u}=\ell+1$ and $\dim\mathfrak{g}^{e}=2\ell$ .

For the proofs of our main results, our basic approach is as follows. We will first describe the Jordan block sizes of unipotent $u\in G_{sc}$ and nilpotent $e\in\mathfrak{g}_{sc}$ on $\mathfrak{g}_{sc}/Z(\mathfrak{g}_{sc})$ , in terms of the Jordan block sizes on $\mathfrak{g}_{sc}$ . This is attained in Section 9 and Section 10.

It is known that $\mathfrak{g}_{sc}/Z(\mathfrak{g}_{sc})\cong[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ as $G_{sc}$ -modules (Lemma 8.2), so we then have the Jordan block sizes on $[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ as well. In Section 11 and Section 12 we will describe the Jordan block sizes of unipotent $u\in G_{sc}$ and nilpotent $e\in\mathfrak{g}_{sc}$ on $\mathfrak{g}_{ad}$ , in terms of the Jordan block sizes on $[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ . Combining these results, we get the Jordan block sizes of $u$ and $e$ on $\mathfrak{g}_{ad}$ , in terms of the Jordan block sizes on $\mathfrak{g}_{sc}$ (Theorem 1.3, Theorem 1.4).

The other sections of this paper are organized as follows. The notation, terminology, and some preliminary results are stated in Section 2 and Section 3. In Section 4, we state results on Jordan block sizes of unipotent and nilpotent elements on tensor products, symmetric squares, and exterior squares. The classification of unipotent classes in $\operatorname{Sp}(V)$ and nilpotent orbits in $\mathfrak{sp}(V)$ is described in Section 5 and Section 6, respectively.

In Section 7, we discuss the Chevalley construction of simple algebraic groups, and in particular the action of $G_{sc}=\operatorname{Sp}(V)$ on $\mathfrak{g}_{ad}$ . We then make some observations about the structure of $\mathfrak{g}_{sc}$ and $\mathfrak{g}_{ad}$ as a $G_{sc}$ -module in Section 8.

As mentioned earlier, in Section 9 and Section 10 we describe the Jordan block sizes of unipotent and nilpotent elements on $\mathfrak{g}_{sc}/Z(\mathfrak{g}_{sc})$ , in terms of the Jordan block sizes on $\mathfrak{g}_{sc}$ . In Section 11 and Section 12 we similarly describe the Jordan block sizes on $[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ , in terms of Jordan block sizes on $\mathfrak{g}_{ad}$ . These results allow us to prove our main results, and the proofs of the results stated in this introduction are given in Section 13.

Table 1. For

G_{sc}=\operatorname{Sp}(V)

simply connected of type

C_{\ell}

with

2\leq\ell\leq 4

and

\operatorname{char}K=2

, Jordan block sizes of unipotent

u\in G_{sc}

\mathfrak{g}_{sc}

[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]

, and

\mathfrak{g}_{ad}

. See Example 1.5.

$\ell$	$V\downarrow K[u]$	$\mathfrak{g}_{sc}\downarrow K[u]$	$[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]\downarrow K[u]$	$\mathfrak{g}_{ad}\downarrow K[u]$	$\alpha$	$\beta$
$2$	$V(4)$	$2,4^{2}$	$1,4^{2}$	$2,4^{2}$	$1$	$1$
	$V(2)^{2}$	$1^{2},2^{4}$	$1,2^{4}$	$1^{2},2^{4}$	$0$	$0$
	$W(1)\perp V(2)$	$1^{4},2^{3}$	$1^{3},2^{3}$	$1^{2},2^{4}$	$0$	$0$
	$W(2)$	$1^{2},2^{4}$	$1^{3},2^{3}$	$1^{4},2^{3}$	$1$	$-$
	$W(1)^{2}$	$1^{10}$	$1^{9}$	$1^{10}$	$0$	$-$
$3$	$V(6)$	$1,4,8^{2}$	$4,8^{2}$	$1,4,8^{2}$	$0$	$0$
	$V(2)\perp V(4)$	$1,2^{2},4^{4}$	$2^{2},4^{4}$	$2,3,4^{4}$	$0$	$1$
	$V(2)^{3}$	$1^{3},2^{9}$	$1^{2},2^{9}$	$1^{3},2^{9}$	$0$	$0$
	$W(1)^{2}\perp V(2)$	$1^{11},2^{5}$	$1^{10},2^{5}$	$1^{9},2^{6}$	$0$	$0$
	$W(1)\perp V(4)$	$1^{3},2,4^{4}$	$1^{2},2,4^{4}$	$1^{2},3,4^{4}$	$0$	$1$
	$W(1)\perp V(2)^{2}$	$1^{5},2^{8}$	$1^{4},2^{8}$	$1^{3},2^{9}$	$0$	$0$
	$W(3)$	$1,2^{2},4^{4}$	$2^{2},4^{4}$	$1,2^{2},4^{4}$	$0$	$-$
	$W(1)\perp W(2)$	$1^{5},2^{8}$	$1^{4},2^{8}$	$1^{5},2^{8}$	$0$	$-$
	$W(1)^{3}$	$1^{21}$	$1^{20}$	$1^{21}$	$0$	$-$
$4$	$V(8)$	$4,8^{4}$	$3,8^{4}$	$4,8^{4}$	$2$	$2$
	$V(2)\perp V(6)$	$1^{2},2,4,6^{2},8^{2}$	$1,2,4,6^{2},8^{2}$	$1^{2},2,4,6^{2},8^{2}$	$0$	$0$
	$V(4)^{2}$	$2^{2},4^{8}$	$1,2,4^{8}$	$2^{2},4^{8}$	$1$	$1$
	$V(2)^{2}\perp V(4)$	$1^{2},2^{5},4^{6}$	$1,2^{5},4^{6}$	$1,2^{4},3,4^{6}$	$0$	$1$
	$V(2)\perp W(3)$	$1^{2},2^{5},4^{6}$	$1,2^{5},4^{6}$	$2^{6},4^{6}$	$0$	$0$
	$W(1)\perp V(2)^{3}$	$1^{6},2^{15}$	$1^{5},2^{15}$	$1^{4},2^{16}$	$0$	$0$
	$W(1)^{3}\perp V(2)$	$1^{22},2^{7}$	$1^{21},2^{7}$	$1^{20},2^{8}$	$0$	$0$
	$W(2)\perp V(4)$	$1^{2},2^{5},4^{6}$	$1^{3},2^{4},4^{6}$	$1^{3},2^{3},3,4^{6}$	$1$	$1$
	$V(2)^{4}$	$1^{4},2^{16}$	$1^{3},2^{16}$	$1^{4},2^{16}$	$0$	$0$
	$W(1)^{2}\perp V(4)$	$1^{10},2,4^{6}$	$1^{9},2,4^{6}$	$1^{9},3,4^{6}$	$0$	$1$
	$W(1)^{2}\perp V(2)^{2}$	$1^{12},2^{12}$	$1^{11},2^{12}$	$1^{10},2^{13}$	$0$	$0$
	$W(1)\perp V(6)$	$1^{4},4,6^{2},8^{2}$	$1^{3},4,6^{2},8^{2}$	$1^{2},2,4,6^{2},8^{2}$	$0$	$0$
	$W(1)\perp V(2)\perp V(4)$	$1^{4},2^{4},4^{6}$	$1^{3},2^{4},4^{6}$	$1^{3},2^{3},3,4^{6}$	$0$	$1$
	$W(4)$	$2^{2},4^{8}$	$2^{2},3,4^{7}$	$1,2^{2},3,4^{7}$	$2$	$-$
	$W(1)\perp W(3)$	$1^{4},2^{2},3^{4},4^{4}$	$1^{3},2^{2},3^{4},4^{4}$	$1^{4},2^{2},3^{4},4^{4}$	$0$	$-$
	$W(2)^{2}$	$1^{4},2^{16}$	$1^{5},2^{15}$	$1^{6},2^{15}$	$1$	$-$
	$W(1)^{2}\perp W(2)$	$1^{12},2^{12}$	$1^{11},2^{12}$	$1^{12},2^{12}$	$0$	$-$
	$W(1)^{4}$	$1^{36}$	$1^{35}$	$1^{36}$	$0$	$-$

Table 2. For

G_{sc}=\operatorname{Sp}(V)

simply connected of type

C_{\ell}

with

2\leq\ell\leq 4

and

\operatorname{char}K=2

, Jordan block sizes of nilpotent

e\in\mathfrak{g}_{sc}

\mathfrak{g}_{sc}

[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]

, and

\mathfrak{g}_{ad}

. See Example 1.5.

$\ell$	$V\downarrow K[e]$	$\mathfrak{g}_{sc}\downarrow K[e]$	$[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]\downarrow K[e]$	$\mathfrak{g}_{ad}\downarrow K[e]$	$\alpha$
$2$	$V(4)$	$1^{2},4^{2}$	$1^{2},3,4$	$1,2,3,4$	$2$
	$V(2)^{2}$	$1^{2},2^{4}$	$1^{3},2^{3}$	$1^{2},2^{4}$	$1$
	$W(1)\perp V(2)$	$1^{4},2^{3}$	$1^{3},2^{3}$	$1^{2},2^{4}$	$0$
	$W(2)$	$1^{2},2^{4}$	$1^{3},2^{3}$	$1^{4},2^{3}$	$1$
	$W(1)^{2}$	$1^{10}$	$1^{9}$	$1^{10}$	$0$
$3$	$V(6)$	$1^{3},2,8^{2}$	$1^{4},8^{2}$	$1^{3},2,8^{2}$	$1$
	$W_{1}(3)$	$1^{5},4^{4}$	$1^{4},4^{4}$	$1^{3},2,4^{4}$	$0$
	$V(2)\perp V(4)$	$1^{3},2,4^{4}$	$1^{4},4^{4}$	$1^{3},2,4^{4}$	$1$
	$W(1)^{2}\perp V(2)$	$1^{11},2^{5}$	$1^{10},2^{5}$	$1^{9},2^{6}$	$0$
	$W(1)\perp V(2)^{2}$	$1^{5},2^{8}$	$1^{4},2^{8}$	$1^{3},2^{9}$	$0$
	$W(1)\perp V(4)$	$1^{5},4^{4}$	$1^{4},4^{4}$	$1^{3},2,4^{4}$	$0$
	$V(2)^{3}$	$1^{3},2^{9}$	$1^{4},2^{8}$	$1^{3},2^{9}$	$1$
	$W(1)\perp W(2)$	$1^{5},2^{8}$	$1^{4},2^{8}$	$1^{5},2^{8}$	$0$
	$W(3)$	$1^{5},4^{4}$	$1^{4},4^{4}$	$1^{5},4^{4}$	$0$
	$W(1)^{3}$	$1^{21}$	$1^{20}$	$1^{21}$	$0$
$4$	$V(8)$	$1^{4},8^{4}$	$1^{4},7,8^{3}$	$1^{3},2,7,8^{3}$	$3$
	$W_{1}(4)$	$1^{4},4^{8}$	$1^{4},3,4^{7}$	$1^{3},2,3,4^{7}$	$2$
	$W(1)\perp W_{1}(3)$	$1^{6},2^{3},4^{6}$	$1^{5},2^{3},4^{6}$	$1^{4},2^{4},4^{6}$	$0$
	$V(2)\perp V(6)$	$1^{4},2^{2},6^{2},8^{2}$	$1^{5},2,6^{2},8^{2}$	$1^{4},2^{2},6^{2},8^{2}$	$1$
	$V(4)^{2}$	$1^{4},4^{8}$	$1^{4},3,4^{7}$	$1^{3},2,3,4^{7}$	$2$
	$V(2)^{2}\perp V(4)$	$1^{4},2^{4},4^{6}$	$1^{5},2^{3},4^{6}$	$1^{4},2^{4},4^{6}$	$1$
	$W(1)\perp V(6)$	$1^{6},2,6^{2},8^{2}$	$1^{5},2,6^{2},8^{2}$	$1^{4},2^{2},6^{2},8^{2}$	$0$
	$W(1)\perp V(2)\perp V(4)$	$1^{6},2^{3},4^{6}$	$1^{5},2^{3},4^{6}$	$1^{4},2^{4},4^{6}$	$0$
	$W(1)^{2}\perp V(4)$	$1^{12},4^{6}$	$1^{11},4^{6}$	$1^{10},2,4^{6}$	$0$
	$W(1)^{2}\perp V(2)^{2}$	$1^{12},2^{12}$	$1^{11},2^{12}$	$1^{10},2^{13}$	$0$
	$W(1)^{3}\perp V(2)$	$1^{22},2^{7}$	$1^{21},2^{7}$	$1^{20},2^{8}$	$0$
	$W(2)\perp V(4)$	$1^{4},2^{4},4^{6}$	$1^{5},2^{3},4^{6}$	$1^{4},2^{4},4^{6}$	$1$
	$V(2)^{4}$	$1^{4},2^{16}$	$1^{5},2^{15}$	$1^{4},2^{16}$	$1$
	$W(1)\perp V(2)^{3}$	$1^{6},2^{15}$	$1^{5},2^{15}$	$1^{4},2^{16}$	$0$
	$V(2)\perp W(3)$	$1^{6},2^{3},4^{6}$	$1^{5},2^{3},4^{6}$	$1^{4},2^{4},4^{6}$	$0$
	$W(1)^{2}\perp W(2)$	$1^{12},2^{12}$	$1^{11},2^{12}$	$1^{12},2^{12}$	$0$
	$W(2)^{2}$	$1^{4},2^{16}$	$1^{5},2^{15}$	$1^{6},2^{15}$	$1$
	$W(1)\perp W(3)$	$1^{8},3^{4},4^{4}$	$1^{7},3^{4},4^{4}$	$1^{8},3^{4},4^{4}$	$0$
	$W(4)$	$1^{4},4^{8}$	$1^{4},3,4^{7}$	$1^{5},3,4^{7}$	$2$
	$W(1)^{4}$	$1^{36}$	$1^{35}$	$1^{36}$	$0$

2. Notation and terminology

We will use the following notation and terminology throughout the paper.

2.1. Generalities

We will always assume that $K$ is an algebraically closed field of characteristic two.

For a $K$ -vector space $V$ and integer $n>0$ , we denote $V^{n}=V\oplus\cdots\oplus V$ , where $V$ occurs $n$ times. Furthermore, we denote $V^{0}=0$ .

We will always use $G$ to denote a group, and all the $K[G]$ -modules that we consider will be finite-dimensional. Suppose that $V=U_{1}\supset U_{2}\supset\cdots\supset U_{t}\supset U_{t+1}=0$ is the socle filtration of a $K[G]$ -module $V$ , so $Z_{i}:=U_{i}/U_{i+1}=\operatorname{soc}(V/U_{i+1})$ for all $1\leq i\leq t$ . Then we will denote this by $V=Z_{1}|Z_{2}|\cdots|Z_{t}$ , and $V$ is said to be uniserial if $Z_{i}$ is irreducible for all $1\leq i\leq t$ .

We denote by $\nu_{2}$ the $2$ -adic valuation on the integers, so $\nu_{2}(a)$ is the largest integer $k\geq 0$ such that $2^{k}$ divides $a$ .

2.2. Algebraic groups

Suppose that $G$ is a simple linear algebraic group over $K$ . In the context of algebraic groups, the notation that we use will be as in [14]. We will denote the Lie algebra of $G$ by $\mathfrak{g}$ .

When $G$ is an algebraic group, by a $G$ -module we will always mean a finite-dimensional rational $K[G]$ -module. Fix a maximal torus $T$ of $G$ with character group $X(T)$ , and a base $\Delta=\{\alpha_{1},\ldots,\alpha_{\ell}\}$ for the root system of $G$ , where $\ell=\operatorname{rank}G$ . We will always use the standard Bourbaki labeling of the simple roots $\alpha_{i}$ , as given in [11, 11.4, p. 58]. We denote the dominant weights with respect to $\Delta$ by $X(T)^{+}$ , and the fundamental dominant weight corresponding to $\alpha_{i}$ is denoted by $\varpi_{i}$ . For a dominant weight $\lambda\in X(T)^{+}$ , we denote the rational irreducible $K[G]$ -module with highest weight $\lambda$ by $L_{G}(\lambda)$ .

An element $u\in G$ is unipotent, if $f(u)$ is a unipotent linear map for every rational representation $f:G\rightarrow\operatorname{GL}(W)$ . Since $\operatorname{char}K=2$ , equivalently $u$ is unipotent if and only if $u$ has order $2^{\alpha}$ for some $\alpha\geq 0$ .

Similarly $e\in\mathfrak{g}$ is said to be nilpotent, if $\mathrm{d}f(e)$ is a nilpotent linear map for every rational representation $f:G\rightarrow\operatorname{GL}(W)$ . Equivalently $e$ is nilpotent if and only if $e^{[2^{\alpha}]}=0$ for some $\alpha>0$ , where $x\mapsto x^{[2]}$ is the canonical $p$ -mapping on $\mathfrak{g}$ .

We will mostly be concerned with the case where $G=\operatorname{Sp}(V)$ , in which case $G$ is of type $C_{\ell}$ , where $\dim V=2\ell$ . In this case $u\in\operatorname{Sp}(V)$ is unipotent if and only if $u$ is a unipotent linear map on $V$ . Furthermore $\mathfrak{g}=\mathfrak{sp}(V)$ , and $e\in\mathfrak{g}$ is nilpotent if and only if $e$ is a nilpotent linear map on $V$ .

2.3. Actions of unipotent elements

Let $q=2^{\alpha}$ for some $\alpha\geq 0$ . Then a cyclic group $C_{q}=\langle u\rangle$ has $q$ indecomposable $K[C_{q}]$ -modules, up to isomorphism. We denote them by $V_{1}$ , $\ldots$ , $V_{q}$ , where $\dim V_{i}=i$ and $u$ acts on $V_{i}$ as a single $i\times i$ unipotent Jordan block. For convenience we denote $V_{0}=0$ .

Let $u$ be an element of a group $G$ and let $V$ be a finite-dimensional $K[G]$ -module on which $u$ acts as a unipotent linear map. We denote $V^{u}:=\{v\in V:uv=v\}$ , which is the fixed point space of $u$ on $V$ . Note that $\dim V^{u}$ is the number of Jordan blocks of $u$ on $V$ .

If $u\in\operatorname{GL}(V)$ is unipotent, we denote by $K[u]$ the group algebra of $\langle u\rangle$ . Then $V\downarrow K[u]=V_{d_{1}}^{n_{1}}\oplus\cdots\oplus V_{d_{t}}^{n_{t}}$ , where $0<d_{1}<\cdots<d_{t}$ are the Jordan block sizes of $u$ on $V$ , and $n_{i}$ is the multiplicity of a Jordan block of size $d_{i}$ .

2.4. Actions of nilpotent elements

Let $q=2^{\alpha}$ for some $\alpha\geq 0$ . We denote by $\mathfrak{w}_{q}$ the abelian $2$ -Lie algebra over $K$ generated by a nilpotent element $e$ with $e^{[2^{\alpha}]}=0$ and $e^{[2^{\alpha-1}]}\neq 0$ . Then as a $K$ -vector space

\mathfrak{w}_{q}=\bigoplus_{0\leq i<\alpha}\langle e^{[2^{i}]}\rangle.

There are a total of $q$ indecomposable $\mathfrak{w}_{q}$ -modules, up to isomorphism. We denote them by $W_{1}$ , $\ldots$ , $W_{q}$ , where $\dim W_{i}=i$ and $e$ acts on $W_{i}$ as a single $i\times i$ nilpotent Jordan block. For convenience we denote $W_{0}=0$ .

If $V$ is a finite-dimensional module for a Lie algebra $\mathfrak{g}$ and $e\in\mathfrak{g}$ acts on $V$ as a nilpotent element, we denote $V^{e}:=\{v\in V:ev=0\}$ . Then $\dim V^{e}$ is the number of Jordan blocks of $e$ on $V$ .

If $e\in\mathfrak{gl}(V)$ is nilpotent, we denote by $K[e]$ the $2$ -Lie algebra generated by $e$ . Then $K[e]\cong\mathfrak{w}_{q}$ , where $q$ is the smallest power of two such that $e^{q}=0$ . Furthermore $V\downarrow K[e]=W_{d_{1}}^{n_{1}}\oplus\cdots\oplus W_{d_{t}}^{n_{t}}$ , where $0<d_{1}<\cdots<d_{t}$ are the Jordan block sizes of $e$ on $V$ , and $n_{i}$ is the multiplicity of a Jordan block of size $d_{i}$ .

3. Jordan normal forms on subspaces and quotients

In the proofs of our main results, the basic tool that we use to describe Jordan block sizes are the following two lemmas.

Lemma 3.1 ([16, Lemma 3.3]).

Let $e\in\mathfrak{gl}(V)$ be a nilpotent linear map. Suppose that $W\subseteq V$ is a subspace invariant under $e$ such that $\dim V/W=1$ . Let $m\geq 1$ be such that $\operatorname{Ker}e^{m-1}\subseteq W$ and $\operatorname{Ker}e^{m}\not\subseteq W$ . Then

	$\displaystyle V$	$\displaystyle\cong W_{m}\oplus V^{\prime}$
	$\displaystyle W$	$\displaystyle\cong W_{m-1}\oplus V^{\prime}$

for some $K[e]$ -module $V^{\prime}$ .

Lemma 3.2 ([17, Lemma 3.4]).

Let $e\in\mathfrak{gl}(V)$ be a nilpotent linear map. Suppose that $W\subseteq V$ is a subspace invariant under $e$ such that $\dim W=1$ . Let $m\geq 1$ be such that $\operatorname{Im}e^{m-1}\supseteq W$ and $\operatorname{Im}e^{m}\not\supseteq W$ . Then

	$\displaystyle V$	$\displaystyle\cong W_{m}\oplus V^{\prime}$
	$\displaystyle V/W$	$\displaystyle\cong W_{m-1}\oplus V^{\prime}$

for some $K[e]$ -module $V^{\prime}$ .

Remark 3.3.

Recall that we define $W_{0}=0$ . Thus if $m=1$ in Lemma 3.1, we have

V\cong W_{1}\oplus W,

and so the Jordan normal form of $e$ on $W$ is given by removing a Jordan block of size $1$ from the Jordan normal form of $e$ on $V$ . (Similar remarks apply to Lemma 3.2.)

4. Decomposition of tensor products and symmetric squares

Let $q=2^{\alpha}$ for some $\alpha\geq 0$ . In this section, we state some basic results about the decomposition of tensor products, exterior squares, and symmetric squares of $K[C_{q}]$ -modules and $\mathfrak{w}_{q}$ -modules.

4.1. Unipotent case

Let $u$ be a generator for the cyclic group $C_{q}$ . To describe the indecomposable summands of tensor products $V\otimes W$ of $K[C_{q}]$ -modules, it is clear that it suffices to do so in the case where $V$ and $W$ are indecomposable.

The decomposition of $V_{m}\otimes V_{n}$ into indecomposable summands has been studied extensively in all characteristics, see for example (in chronological order) [31], [28], [24], [30], [26], [10], [27], [12], and [1]. Although there is no closed formula in general, there are various recursive formulae which can be used to determine the indecomposable summands efficiently. We are assuming $\operatorname{char}K=2$ , in which case we have the following result.

Theorem 4.1 ([7, (2.5a)], [6, Lemma 1, Corollary 3]).

Let $0<m\leq n\leq q$ and suppose that $q/2<n\leq q$ . Then the following statements hold:

(i)

If $n=q$ , then $V_{m}\otimes V_{n}\cong V_{q}^{m}$ as $K[C_{q}]$ -modules.
(ii)

If $m+n>q$ , then $V_{m}\otimes V_{n}\cong V_{q}^{n+m-q}\oplus(V_{q-n}\otimes V_{q-m})$ as $K[C_{q}]$ -modules.
(iii)

If $m+n\leq q$ , then $V_{m}\otimes V_{n}\cong V_{q-d_{t}}\oplus\cdots\oplus V_{q-d_{1}}$ as $K[C_{q}]$ -modules, where $V_{m}\otimes V_{q-n}\cong V_{d_{1}}\oplus\cdots\oplus V_{d_{t}}$ .

With Theorem 4.1, every tensor product $V_{m}\otimes V_{n}$ is either described explicitly (case (i)), or in terms of another tensor product $V_{m^{\prime}}\otimes V_{n^{\prime}}$ with $0<m^{\prime}\leq n^{\prime}<n$ . Thus by repeated applications of Theorem 4.1, we can rapidly find the indecomposable summands of $V_{m}\otimes V_{n}$ for any given $m$ and $n$ .

For exterior squares and symmetric squares, similarly it suffices to consider the indecomposable case. This follows from the fact that for all $K[C_{q}]$ -modules $V$ and $W$ , we have isomorphisms

	$\displaystyle\wedge^{2}(V\oplus W)$	$\displaystyle\cong\wedge^{2}(V)\oplus\wedge^{2}(W)\oplus V\otimes W$
	$\displaystyle S^{2}(V\oplus W)$	$\displaystyle\cong S^{2}(V)\oplus S^{2}(W)\oplus V\otimes W$

as $K[C_{q}]$ -modules. For the decomposition of $\wedge^{2}(V_{n})$ and $S^{2}(V_{n})$ , we have the following results.

Theorem 4.2 ([6, Theorem 2]).

Suppose that $q/2<n\leq q$ . Then we have

\wedge^{2}(V_{n})\cong\wedge^{2}(V_{q-n})\oplus V_{q}^{n-q/2-1}\oplus V_{3q/2-n}

as $K[C_{q}]$ -modules.

Theorem 4.3 ([18, Theorem 1.3]).

Suppose that $q/2<n\leq q$ . Then we have

S^{2}(V_{n})\cong\wedge^{2}(V_{q-n})\oplus V_{q}^{n-q/2}\oplus V_{q/2}

as $K[C_{q}]$ -modules.

Similarly to Theorem 4.1, with Theorem 4.2 and Theorem 4.3 we can quickly decompose $\wedge^{2}(V_{n})$ and $S^{2}(V_{n})$ for any given $n$ .

Lemma 4.4.

Let $n>0$ be an integer. Then the following hold:

(i)

$\dim\wedge^{2}(V_{n})^{u}=\lfloor\frac{n}{2}\rfloor$ .
(ii)

$\dim S^{2}(V_{n})^{u}=\lfloor\frac{n}{2}\rfloor+1$ .

Proof.

We first consider (i). If $n=1$ , then $\wedge^{2}(V_{1})=0$ has dimension $0$ and thus (i) holds. Suppose then that $n>1$ and proceed by induction on $n$ . Let $q$ be a power of two such that $q/2<n\leq q$ . If $n=q$ , it follows from Theorem 4.2 that $\wedge^{2}(V_{n})\cong V_{q}^{n-q/2-1}\oplus V_{q/2}$ , so $\dim\wedge^{2}(V_{n})^{u}=n-q/2=n/2$ . If $q/2<n<q$ , we have

\wedge^{2}(V_{n})\cong\wedge^{2}(V_{q-n})\oplus V_{q}^{n-q/2-1}\oplus V_{3q/2-n}

by Theorem 4.2. Then by induction

\dim\wedge^{2}(V_{n})^{u}=\left\lfloor\frac{q-n}{2}\right\rfloor+n-q/2=\left\lfloor\frac{n}{2}\right\rfloor,

as claimed by (i).

Next we will prove (ii). If $n=1$ , then $S^{2}(V_{1})=V_{1}$ so (ii) holds. Suppose then that $n>1$ , and let $q$ be a power of two such that $q/2<n\leq q$ . If $n=q$ , then $S^{2}(V_{n})\cong V_{q}^{n-q/2}\oplus V_{q/2}$ , and thus $\dim S^{2}(V_{n})^{u}=n-q/2+1=n/2+1$ . Suppose then that $q/2<n<q$ . It follows from Theorem 4.3 and (i) that

\dim S^{2}(V_{n})^{u}=\left\lfloor\frac{q-n}{2}\right\rfloor+n-q/2+1=\left\lfloor\frac{n}{2}\right\rfloor+1,

as claimed by (ii).∎

Lemma 4.5.

Let $\ell>0$ and define $\alpha=\nu_{2}(\ell)$ . Then the smallest Jordan block size in $S^{2}(V_{2\ell})$ is $2^{\alpha}$ , occurring with multiplicity one.

Proof.

(cf. [17, Lemma 4.12]) If $\ell=2^{\alpha}$ , it follows from Theorem 4.3 that $S^{2}(V_{2\ell})=V_{2^{\alpha+1}}^{2^{\alpha}}\oplus V_{2^{\alpha}}$ , so the claim holds. If $\ell\neq 2^{\alpha}$ , we have $q/2<2\ell<q$ for some $q=2^{\beta}$ . Then

(4.1)

S^{2}(V_{2\ell})=\wedge^{2}(V_{q-2\ell})\oplus V_{q}^{2\ell-q/2}\oplus V_{q/2}

by Theorem 4.3.

Now $\nu_{2}(q-2\ell)=2^{\alpha+1}$ since $q>2^{\alpha+1}$ , so by [17, Lemma 4.12] the smallest Jordan block size in $\wedge^{2}(V_{q-2\ell})$ is $2^{\alpha}$ , occurring with multiplicity one. Furthermore $2^{\alpha}<q/2$ because $q/2<2\ell<q$ , so the lemma follows from (4.1).∎

4.2. Nilpotent case

As in the previous section, to determine the indecomposable summands of tensor products, exterior squares, and symmetric squares of $\mathfrak{w}_{q}$ -modules, it suffices to do so in the indecomposable case.

For the indecomposable summands of $W_{m}\otimes W_{n}$ , it turns out that we get the same decomposition as for $V_{m}\otimes V_{n}$ in the unipotent case.

Proposition 4.6 ([4, Section III], [25, Corollary 5 (a)]).

Let $0<n,m\leq q$ and suppose that we have $V_{m}\otimes V_{n}\cong V_{r_{1}}\oplus\cdots\oplus V_{r_{t}}$ as $K[C_{q}]$ -modules for some $r_{1},\ldots,r_{t}>0$ . Then $W_{m}\otimes W_{n}\cong W_{r_{1}}\oplus\cdots\oplus W_{r_{t}}$ as $\mathfrak{w}_{q}$ -modules.

Thus we can apply Theorem 4.1 to find the decomposition of $W_{m}\otimes W_{n}$ into indecomposable summands.

Following [5, p. 231], we call the consecutive-ones binary expansion of an integer $n>0$ the alternating sum $n=\sum_{1\leq i\leq r}(-1)^{i+1}2^{\beta_{i}}$ such that $\beta_{1}>\cdots>\beta_{r}\geq 0$ and $r$ is minimal. (Here $\beta_{r-1}>\beta_{r}+1$ if $r>1$ .)

The decomposition of $V_{n}\otimes V_{n}$ into indecomposable summands can be given explicitly in terms of the consecutive-ones binary expansion of $n$ [5, Theorem 15]. Such descriptions can also be given for $\wedge^{2}(V_{n})$ and $S^{2}(V_{n})$ , by using Theorem 4.2 and Theorem 4.3.

For the decomposition of $W_{n}\otimes W_{n}$ , $\wedge^{2}(W_{n})$ , and $S^{2}(W_{n})$ we have the following result.

Theorem 4.7 ([18, Theorem 1.6, Theorem 1.7, Theorem 3.7]).

Let $n>0$ be an integer, with consecutive-ones binary expansion $n=\sum_{1\leq i\leq r}(-1)^{i+1}2^{\beta_{i}}$ , where $\beta_{1}>\cdots>\beta_{r}\geq 0$ . For $1\leq k\leq r$ , define $d_{k}:=2^{\beta_{k}}+\sum_{k<i\leq r}(-1)^{k+i}2^{\beta_{i}+1}$ . Then

	$\displaystyle W_{n}\otimes W_{n}$	$\displaystyle\cong\bigoplus_{1\leq k\leq r}W_{2^{\beta_{k}}}^{d_{k}}$
	$\displaystyle\wedge^{2}(W_{n})$	$\displaystyle\cong\bigoplus_{\begin{subarray}{c}1\leq k\leq r\\ \beta_{k}>0\end{subarray}}W_{2^{\beta_{k}}-1}^{d_{k}/2}$
	$\displaystyle S^{2}(W_{n})$	$\displaystyle\cong W_{1}^{\lceil n/2\rceil}\oplus\bigoplus_{\begin{subarray}{c}1\leq k\leq r\\ \beta_{k}>0\end{subarray}}W_{2^{\beta_{k}}}^{d_{k}/2}$

as $\mathfrak{w}_{q}$ -modules.

5. Unipotent elements in $\operatorname{Sp}(V)$

Consider $\operatorname{Sp}(V)$ with $\dim V=2\ell$ . In this section, we will recall the description of unipotent conjugacy classes in $\operatorname{Sp}(V)$ , due to Hesselink [8]. For more details, see for example [8], [23, Chapter 4, Chapter 6], or [17, Section 6].

For a group $G$ , a bilinear $K[G]$ -module $(W,\beta)$ is a finite-dimensional $K[G]$ -module $W$ equipped with a $G$ -invariant bilinear form $\beta$ . Two bilinear $K[G]$ -modules $(W,\beta)$ and $(W^{\prime},\beta^{\prime})$ are said to be isomorphic, if there exists an isomorphism $W\rightarrow W^{\prime}$ of $K[G]$ -modules which is also an isometry.

Let $u,u^{\prime}\in\operatorname{Sp}(V)$ be unipotent. Let $q$ be a power of two such that $u^{q}=(u^{\prime})^{q}=1$ , so that $V\downarrow K[u]$ and $V\downarrow K[u^{\prime}]$ are $K[C_{q}]$ -modules. Then $u$ and $u^{\prime}$ are conjugate in $\operatorname{Sp}(V)$ if and only if $V\downarrow K[u]\cong V\downarrow K[u^{\prime}]$ as bilinear $K[C_{q}]$ -modules.

For $u\in\operatorname{Sp}(V)$ unipotent, it is clear that we can write $V\downarrow K[u]=U_{1}\perp\cdots\perp U_{t}$ , where $U_{i}$ are orthogonally indecomposable $K[u]$ -modules. Here orthogonally indecomposable means that if $U_{i}=U^{\prime}\perp U^{\prime\prime}$ as $K[u]$ -modules, then $U^{\prime}=0$ or $U^{\prime\prime}=0$ . There are two basic types of orthogonally indecomposable $K[u]$ -modules, which we can define as follows. (Similar definitions are given in [23, Section 6.1].)

Definition 5.1.

For $\ell\geq 1$ , we define the module $V(2\ell)$ as follows. Let $n=2\ell$ , and suppose that $V$ has basis $v_{1}$ , $\ldots$ , $v_{n}$ with $b(v_{i},v_{j})=1$ if $i+j=n+1$ and $0$ otherwise. Define $u:V\rightarrow V$ by

	$\displaystyle uv_{1}$	$\displaystyle=v_{1}$
	$\displaystyle uv_{i}$	$\displaystyle=v_{i}+v_{i-1}+\cdots+v_{1}\text{ for all }1<i\leq\ell+1$
	$\displaystyle uv_{i}$	$\displaystyle=v_{i}+v_{i-1}\text{ for all }\ell+1<i\leq n.$

Then $u\in\operatorname{Sp}(V)$ , and we define $V(2\ell)$ as the bilinear $K[u]$ -module $V\downarrow K[u]$ .

Definition 5.2.

For $\ell\geq 1$ , we define the module $W(\ell)$ as follows. Let $n=2\ell$ , and suppose that $V$ has basis $v_{1}$ , $\ldots$ , $v_{n}$ with $b(v_{i},v_{j})=1$ if $i+j=n+1$ and $0$ otherwise. Define $u:V\rightarrow V$ by

	$\displaystyle uv_{1}$	$\displaystyle=v_{1}$
	$\displaystyle uv_{i}$	$\displaystyle=v_{i}+v_{i-1}+\cdots+v_{1}\text{ for all }1<i\leq\ell$
	$\displaystyle uv_{\ell+1}$	$\displaystyle=v_{\ell+1}$
	$\displaystyle uv_{i}$	$\displaystyle=v_{i}+v_{i-1}\text{ for all }\ell+1<i\leq n.$

Then $u\in\operatorname{Sp}(V)$ , and we define $W(\ell)$ as the bilinear $K[u]$ -module $V\downarrow K[u]$ .

The fact that the modules are isomorphic to those described by Hesselink [8, Proposition 3.5] is seen as follows.

•

For the module $V(2\ell)$ in Definition 5.1 we have $V(2\ell)\downarrow K[u]=V_{2\ell}$ , so this agrees with [8, Proposition 3.5].
•

In Definition 5.2, we have totally singular decomposition $V=W\oplus Z$ , where $W=\langle v_{1},\ldots,v_{\ell}\rangle$ and $Z=\langle v_{\ell+1},\ldots,v_{n}\rangle$ are $K[u]$ -modules with $W\cong Z\cong V_{\ell}$ . From this it follows that $V$ is isomorphic to the module $W(\ell)$ defined by Hesselink, see for example [17, Lemma 6.12].

The classification of unipotent conjugacy classes in $\operatorname{Sp}(V)$ is based on the following result.

Theorem 5.3 ([8, Proposition 3.5]).

Let $u\in\operatorname{Sp}(V)$ be unipotent such that $V\downarrow K[u]$ is orthogonally indecomposable. Then $V\downarrow K[u]$ is isomorphic to $V(2\ell)$ or $W(\ell)$ , where $\dim V=2\ell$ .

By Theorem 5.3, for every unipotent $u\in\operatorname{Sp}(V)$ we have an orthogonal decomposition

V\downarrow K[u]=U_{1}\perp\cdots\perp U_{t},

where for all $1\leq i\leq t$ we have $U_{i}\cong V(2\ell_{i})$ or $U_{i}\cong W(\ell_{i})$ for some integer $\ell_{i}\geq 1$ .

In general there can be several different ways to decompose $V\downarrow K[u]$ into orthogonally indecomposable summands, and even the number of summands is not uniquely determined. This is due to the fact that for even $m$ , we have an isomorphism

W(m)\perp V(m)\cong V(m)\perp V(m)\perp V(m)

of bilinear $K[u]$ -modules. However, there are normal forms which are uniquely determined, such as the Hesselink normal form [8, 3.7] [17, Theorem 6.4] or the distinguished normal form defined by Liebeck and Seitz in [23, p. 61].

6. Nilpotent elements in $\mathfrak{sp}(V)$

We consider $G=\operatorname{Sp}(V)$ with Lie algebra $\mathfrak{sp}(V)$ , where $\dim V=2\ell$ . We recall the classification of nilpotent orbits in $\mathfrak{sp}(V)$ due to Hesselink [8]. For more details, we refer to [8] and [23, Chapter 4, Chapter 5].

For a nilpotent element $e\in\mathfrak{sp}(V)$ , define the index function $\chi_{V}:\mathbb{Z}_{\geq 0}\rightarrow\mathbb{Z}_{\geq 0}$ corresponding to $e$ by

\chi_{V}(m):=\operatorname{min}\{n\geq 0:b(e^{n+1}v,e^{n}v)=0\text{ for all }v\in\operatorname{Ker}e^{m}\}.

Let $0<d_{1}<\cdots<d_{t}$ be the Jordan block sizes of $e$ , and let $n_{i}$ be the multiplicity of Jordan block size $d_{i}$ for $e$ . By a result of Hesselink [8, Theorem 3.8], the nilpotent orbit of $e$ is determined by the integers $d_{i}$ , $n_{i}$ , and the function $\chi_{V}$ .

Hesselink also proved that it suffices to only consider the values of $\chi_{V}$ on the Jordan block sizes $d_{1}$ , $\ldots$ , $d_{t}$ .

Theorem 6.1 ([8, 3.9]).

Let $e\in\mathfrak{sp}(V)$ be nilpotent with index function $\chi=\chi_{V}$ on $V$ . Let $0<d_{1}<\cdots<d_{t}$ be the Jordan block sizes of $e$ , and let $n_{i}$ be the multiplicity of a Jordan block of size $d_{i}$ for $e$ . Then the following statements hold:

(i)

The nilpotent orbit of $e$ is determined by the symbol $({d_{1}}_{\chi(d_{1})}^{n_{1}},\ldots,{d_{t}}_{\chi(d_{t})}^{n_{t}})$ .
(ii)

$\chi(d_{1})\leq\cdots\leq\chi(d_{t})$ and $d_{1}-\chi(d_{1})\leq\cdots\leq d_{t}-\chi(d_{t})$ .
(iii)

$0\leq\chi(d_{i})\leq d_{i}/2$ for all $1\leq i\leq t$ .
(iv)

$\chi(d_{i})=d_{i}/2$ if $n_{i}$ is odd.

Remark 6.2.

Conversely, consider integers $d_{i}$ , $n_{i}$ , $\chi(d_{i})$ with $0<d_{1}<\cdots<d_{t}$ and $\sum_{i=1}^{t}n_{i}d_{i}=\dim V$ , such that conditions (ii) – (iv) of Theorem 6.1 hold. Then there exists a nilpotent element $e\in\mathfrak{sp}(V)$ with corresponding symbol $({d_{1}}_{\chi(d_{1})}^{n_{1}},\ldots,{d_{t}}_{\chi(d_{t})}^{n_{t}})$ [8, 3.9].

Similarly to the unipotent case, we can phrase the classification in terms of bilinear modules. For a Lie algebra $\mathfrak{w}$ , a bilinear $\mathfrak{w}$ -module $(W,\beta)$ is a finite-dimensional $\mathfrak{w}$ -module $W$ equipped with a $\mathfrak{w}$ -invariant bilinear form $\beta$ , so $\beta(Xv,w)+\beta(v,Xw)=0$ for all $X\in\mathfrak{w}$ and $v,w\in W$ . Two bilinear $\mathfrak{w}$ -modules $(W,\beta)$ and $(W^{\prime},\beta^{\prime})$ are said to be isomorphic, if there exists an isomorphism $W\rightarrow W^{\prime}$ of $\mathfrak{w}$ -modules which is also an isometry.

Let $e,e^{\prime}\in\mathfrak{sp}(V)$ be nilpotent. Choose a power of two $q$ such that $e^{q}=(e^{\prime})^{q}=0$ , so that $V\downarrow K[e]$ and $V\downarrow K[e^{\prime}]$ are $\mathfrak{w}_{q}$ -modules. Then $e$ and $e^{\prime}$ are conjugate under the action of $\operatorname{Sp}(V)$ if and only if $V\downarrow K[e]\cong V\downarrow K[e^{\prime}]$ as bilinear $\mathfrak{w}_{q}$ -modules.

For nilpotent $e\in\mathfrak{sp}(V)$ , it is clear that we can write $V\downarrow K[e]=V_{1}\perp\cdots\perp V_{t}$ , where $V_{i}$ are orthogonally indecomposable $K[e]$ -modules. (Here orthogonally indecomposable is defined similarly to the group case.) By the next lemma, the index function $\chi_{V}$ is determined by its restriction to the orthogonally indecomposable summands of $V\downarrow K[e]$ .

Lemma 6.3 ([23, Lemma 5.2]).

Let $e\in\mathfrak{sp}(V)$ be nilpotent and assume $V\downarrow K[e]=W_{1}\perp W_{2}$ as $K[e]$ -modules. Then $\chi_{V}(m)=\max\{\chi_{W_{1}}(m),\chi_{W_{2}}(m)\}$ for all $m\geq 0$ .

The orthogonally indecomposable modules were classified by Hesselink. In the case of $\mathfrak{sp}(V)$ , there are three types of orthogonally indecomposable modules, defined as follows. (Similar definitions are given in [23, Section 5.1].)

Definition 6.4.

	$\displaystyle ev_{1}$	$\displaystyle=0$
	$\displaystyle ev_{i}$	$\displaystyle=v_{i-1}\text{ for all }1<i\leq n.$

Then $e\in\mathfrak{sp}(V)$ , and we define $V(2\ell)$ as the bilinear $K[e]$ -module $V\downarrow K[e]$ .

Definition 6.5.

	$\displaystyle ev_{1}$	$\displaystyle=0,$	$\displaystyle ev_{i}$	$\displaystyle=v_{i-1}\text{ for all }1<i\leq\ell.$
	$\displaystyle ev_{\ell+1}$	$\displaystyle=0,$	$\displaystyle ev_{i}$	$\displaystyle=v_{i-1}\text{ for all }\ell+1<i\leq n.$

Then $e\in\mathfrak{sp}(V)$ , and we define $W(\ell)$ as the bilinear $K[e]$ -module $V\downarrow K[e]$ .

Definition 6.6.

For $\ell\geq 1$ and $0<k<\ell/2$ we define the module $W_{k}(\ell)$ as follows. Let $n=2\ell$ , and suppose that $V$ has basis $v_{1}$ , $\ldots$ , $v_{n}$ with $b(v_{i},v_{j})=1$ if $i+j=n+1$ and $0$ otherwise. Define $e:V\rightarrow V$ by

	$\displaystyle ev_{1}$	$\displaystyle=0,$	$\displaystyle ev_{\ell+1}$	$\displaystyle=0$
	$\displaystyle ev_{n-k+1}$	$\displaystyle=v_{n-k}+v_{k}$	$\displaystyle ev_{i}$	$\displaystyle=v_{i-1}\text{ for all }i\not\in\{1,\ell+1,n-k+1\}$

Then $e\in\mathfrak{sp}(V)$ , and we define $W_{k}(\ell)$ as the bilinear $K[e]$ -module $V\downarrow K[e]$ .

The fact that the modules in Definition 6.4 – 6.6 agree with those described by Hesselink in [8, Proposition 3.5] is seen as follows.

•

In Definition 6.4, this is clear from the fact that $V\downarrow K[e]=W_{2\ell}$ , so $V\downarrow K[e]=V(2\ell)$ as defined by Hesselink.
•

In Definition 6.5, we have $V\downarrow K[e]=W_{\ell}\oplus W_{\ell}$ . It is easy to see that $b(ev,v)=0$ for all $v\in V$ , so $\chi_{V}(m)=0$ for all $m\geq 0$ . It follows from [8, Proposition 3.5, Theorem 3.8] that $V\downarrow K[e]$ is isomorphic to the module $W(\ell)$ defined by Hesselink.
•

In Definition 6.6 we have used the representative given in [20], which gives $W_{k}(\ell)$ by [20, Lemma 3.4].

Theorem 6.7 ([8, Proposition 3.5]).

Let $e\in\mathfrak{sp}(V)$ be nilpotent such that $V\downarrow K[e]$ is orthogonally indecomposable. Then $V\downarrow K[e]$ is isomorphic to $V(2\ell)$ , $W(\ell)$ , or $W_{k}(\ell)$ for some $0<k<\ell/2$ . These modules are characterized by the following properties:

$V\downarrow K[e]$	Jordan normal form on $V$	$\chi_{V}$
$V(2\ell)$	$W_{2\ell}$	$\chi_{V}(2\ell)=\ell$
$W(\ell)$	$W_{\ell}\oplus W_{\ell}$	$\chi_{V}(\ell)=0$
$W_{k}(\ell)$ ( $0<k<\ell/2$ )	$W_{\ell}\oplus W_{\ell}$	$\chi_{V}(\ell)=k$

By Theorem 6.7, for every nilpotent $e\in\mathfrak{sp}(V)$ we have an orthogonal decomposition

V\downarrow K[e]=U_{1}\perp\cdots\perp U_{t},

where for all $1\leq i\leq t$ we have $U_{i}\cong V(2\ell_{i})$ , $U_{i}\cong W(\ell_{i})$ , or $U_{i}\cong W_{k_{i}}(\ell_{i})$ ( $0<k_{i}<\ell_{i}/2$ ) for some integer $\ell_{i}\geq 1$ .

As in the unipotent case, the orthogonally indecomposable summands and their number is not uniquely determined. For example, by Lemma 6.3 and Theorem 6.1 (i), we have isomorphisms

	$\displaystyle W(d)\perp V(d^{\prime})$	$\displaystyle\cong W_{d^{\prime}/2}(d)\perp V(d^{\prime})$	$\displaystyle\text{ for }d>d^{\prime}>0\text{ even};$
	$\displaystyle W(d)\perp V(d)$	$\displaystyle\cong V(d)\perp V(d)\perp V(d)$	$\displaystyle\text{ for }d>0\text{ even};$

of bilinear $K[e]$ -modules.

We end this section with two observations about orthogonally indecomposable modules of the form $\sum_{1\leq i\leq t}W(m_{i})$ .

Lemma 6.8.

Let $e\in\mathfrak{sp}(V)$ be nilpotent. Then the following statements are equivalent:

(i)

There is an orthogonal decomposition $V\downarrow K[e]=\sum_{1\leq i\leq t}W(m_{i})$ for some integers $m_{1}$ , $\ldots$ , $m_{t}$ .
(ii)

There is a totally singular decomposition $V=W\oplus Z$ , where $W$ and $Z$ are $K[e]$ -submodules of $V$ .
(iii)

$b(ev,v)=0$ for all $v\in V$ .

Proof.

(cf. [17, Lemma 6.12]) We prove that (i) $\Rightarrow$ (ii) $\Rightarrow$ (iii) $\Rightarrow$ (i).

(i) $\Rightarrow$ (ii): It is clear from Definition 6.5 that each $W(m_{i})$ has a decomposition $W(m_{i})=W_{i}\oplus Z_{i}$ , where $W_{i}$ and $Z_{i}$ are totally singular $K[e]$ -submodules with $W_{i}\cong W_{m_{i}}\cong Z_{i}$ . Therefore (i) implies (ii).

(ii) $\Rightarrow$ (iii): We have $b(ew,w)=b(ez,z)=0$ for all $w\in W$ and $z\in Z$ since $W$ and $Z$ are $e$ -invariant and totally singular. Thus $b(e(w+z),w+z)=b(ew,z)+b(ez,w)=0$ for all $w\in W$ and $z\in Z$ , since $e\in\mathfrak{sp}(V)$ .

(iii) $\Rightarrow$ (i): Suppose that $b(ev,v)=0$ for all $v\in V$ . Then for the index function of $e$ we have $\chi_{V}(m)=0$ for all $m\geq 1$ . Therefore every orthogonally indecomposable summand of $V\downarrow K[e]$ must be of the form $W(m)$ for some $m\geq 1$ (Theorem 6.7 and Lemma 6.3), so (i) holds.∎

Lemma 6.9.

Let $e\in\mathfrak{sp}(V)$ be nilpotent. Then $V\downarrow K[e^{2}]=\sum_{1\leq i\leq t}W(m_{i})$ for some integers $m_{1}$ , $\ldots$ , $m_{t}$ .

Proof.

We have $b(e^{2}v,v)+b(ev,ev)=0$ for all $v\in V$ since $e\in\mathfrak{sp}(V)$ , so $b(e^{2}v,v)=0$ for all $v\in V$ . Now the claim follows from Lemma 6.8.∎

7. Chevalley construction

In this section, we will recall basics of the Chevalley construction of Lie algebras and simple algebraic groups, and in particular how it applies for groups of type $C_{\ell}$ in characteristic two. For more details, see for example [32] or [11, Chapter VII]. We will also make some preliminary observations about the actions of unipotent and nilpotent elements on the adjoint Lie algebra $\mathfrak{g}_{ad}$ of type $C_{\ell}$ .

7.1. Chevalley construction

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$ . Fix a Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$ , and let $\Phi$ be the corresponding root system, so

\mathfrak{g}=\mathfrak{h}\oplus\bigoplus_{\alpha\in\Phi}\mathfrak{g}_{\alpha},

where $\mathfrak{g}_{\alpha}=\{X\in\mathfrak{g}:[H,X]=\alpha(H)X\text{ for all }H\in\mathfrak{h}.\}.$ The Killing form $\kappa$ is non-degenerate on $\mathfrak{h}$ , so for all $\alpha\in\Phi$ there exists $H_{\alpha}^{\prime}\in\mathfrak{h}$ such that $\kappa(H,H_{\alpha}^{\prime})=\alpha(H)$ for all $H\in\mathfrak{h}$ . We define $H_{\alpha}:=\frac{2}{\kappa(H_{\alpha}^{\prime},H_{\alpha}^{\prime})}H_{\alpha}^{\prime}$ for all $\alpha\in\Phi$ .

It was shown by Chevalley [2, Théorème 1] that one can choose $X_{\alpha}\in\mathfrak{g}_{\alpha}$ such that the following properties hold:

(a)

$[X_{\alpha},X_{-\alpha}]=H_{\alpha}$ for all $\alpha\in\Phi$ ,
(b)

If $\alpha,\beta\in\Phi$ and $\alpha+\beta\in\Phi$ , then $[X_{\alpha},X_{\beta}]=\pm(r+1)X_{\alpha+\beta}$ , where $r\geq 0$ is the largest integer such that $\beta-r\alpha$ is a root.
(c)

If $\alpha,\beta\in\Phi$ and $\alpha+\beta\not\in\Phi$ , then $[X_{\alpha},X_{\beta}]=0$ .

For a choice of root vectors $X_{\alpha}$ satisfying (a) – (c) above, we define $\mathfrak{g}_{\mathbb{Z}}$ to be the $\mathbb{Z}$ -span of all $X_{\alpha}$ and $H_{\alpha}$ for $\alpha\in\Phi$ . Let $\Delta$ be a base for $\Phi$ and let $\Phi^{+}$ be the corresponding system of positive roots. Then $\{X_{\alpha}:\alpha\in\Phi\}\cup\{H_{\alpha}:\alpha\in\Delta\}$ is a $\mathbb{Z}$ -basis of $\mathfrak{g}_{\mathbb{Z}}$ , called a Chevalley basis for $\mathfrak{g}$ .

Fix a Chevalley basis for $\mathfrak{g}$ . Let $\mathscr{U}_{\mathbb{Z}}$ be the corresponding Kostant $\mathbb{Z}$ -form, which is the subring of the universal enveloping algebra generated by $1$ and $\frac{X_{\alpha}^{k}}{k!}$ for all $\alpha\in\Phi$ and $k\geq 1$ .

Let $V$ be a faithful finite-dimensional $\mathfrak{g}$ -module over $\mathbb{C}$ . We denote the set of weights of $\mathfrak{h}$ in $V$ by $\Lambda(V)$ . Then $\mathbb{Z}\Phi\subseteq\mathbb{Z}\Lambda(V)\subseteq\Lambda$ , where $\Lambda$ is the weight lattice.

A lattice in $V$ is the $\mathbb{Z}$ -span of a basis of $V$ . We say that a lattice $L\subseteq V$ is admissible, if $L$ is $\mathscr{U}_{\mathbb{Z}}$ -invariant.

Let $F$ be an algebraically closed field of characteristic $p>0$ , and $L$ an admissible lattice in $V$ . Set $V_{F}:=F\otimes_{\mathbb{Z}}L$ . Then for all $X\in\mathscr{U}_{\mathbb{Z}}$ , the reduction modulo $p$ of $X$ is the $F$ -linear map defined by $1\otimes X^{\prime}:V_{F}\rightarrow V_{F}$ , where $X^{\prime}:L\rightarrow L$ is the action of $X$ on $L$ .

In particular, for all $\alpha\in\Phi$ and $k\geq 0$ we have a linear map $X_{\alpha,k}:V_{F}\rightarrow V_{F}$ which is the reduction modulo $p$ of $X_{\alpha}^{k}/k!$ on $V_{F}$ . Since $X_{\alpha}$ acts nilpotently on $V$ , we can define for all $t\in F$ the root element $x_{\alpha}(t)$ as the exponential $x_{\alpha}(t):=\sum_{k\geq 0}t^{k}X_{\alpha,k}$ . We have $x_{\alpha}(t)\in\operatorname{GL}(V_{F})$ , and the Chevalley group (over $F$ ) corresponding to $V$ and $L$ is defined as

G(V,L):=\langle x_{\alpha}(t):\alpha\in\Phi,t\in F\rangle.

Then $G(V,L)$ is a simple algebraic group over $F$ with root system $\Phi$ [32, Theorem 6]. Furthermore, we have a maximal torus

T=\langle h_{\alpha}(t):\alpha\in\Delta,t\in K^{\times}\rangle,

where $h_{\alpha}(t)$ is defined as in [32, Lemma 19, p. 22]. Then the weights of $T$ on $V$ can be identified with $\Lambda(V)$ , and the character group $X(T)$ can be identified with $\mathbb{Z}\Lambda(V)$ [32, p. 39].

We say that $G(V,L)$ is simply connected if $\mathbb{Z}\Lambda(V)=\Lambda$ , and adjoint if $\mathbb{Z}\Lambda(V)=\mathbb{Z}\Phi$ .

Lemma 7.1 ([32, Corollary 1, p. 41]).

Suppose that $V^{\prime}$ is another faithful finite-dimensional $\mathfrak{g}$ -module with admissible lattice $L^{\prime}$ , and denote the corresponding root elements in $G(V^{\prime},L^{\prime})$ by $x_{\alpha}^{\prime}(t)$ . Then:

(i)

If $G(V^{\prime},L^{\prime})$ is simply connected, there exists a morphism $\varphi:G(V^{\prime},L^{\prime})\rightarrow G(V,L)$ of algebraic groups with $x_{\alpha}^{\prime}(t)\mapsto x_{\alpha}(t)$ for all $\alpha\in\Phi$ and $t\in F$ .
(ii)

If $G(V^{\prime},L^{\prime})$ is adjoint, there exists a morphism $\varphi:G(V,L)\rightarrow G(V^{\prime},L^{\prime})$ of algebraic groups with $x_{\alpha}(t)\mapsto x_{\alpha}^{\prime}(t)$ for all $\alpha\in\Phi$ and $t\in F$ .

The Lie algebra of $G(V,L)$ is identified as follows. The stabilizer of $L$ in $\mathfrak{g}$ and $\mathfrak{h}$ is given by [32, Corollary 2, p. 16]

	$\displaystyle\mathfrak{h}^{L}$	$\displaystyle=\{H\in\mathfrak{h}:\mu(H)\in\mathbb{Z}\text{ for all }\mu\in\Lambda(V)\},$
	$\displaystyle\mathfrak{g}^{L}$	$\displaystyle=\mathfrak{h}^{L}\oplus\bigoplus_{\alpha\in\Phi}\mathbb{Z}X_{\alpha}.$

In particular $\mathfrak{g}^{L}$ and $\mathfrak{h}^{L}$ only depend on the weights in $V$ , not the choice of the admissible lattice $L$ . Furthermore, under the adjoint action $\mathfrak{g}^{L}$ is an admissible lattice in $\mathfrak{g}$ [11, Proposition 27.2].

Then $\mathfrak{g}^{L}$ is a $\mathbb{Z}$ -Lie algebra which acts on $L$ . The Chevalley algebra

\mathfrak{g}(V,L):=F\otimes_{\mathbb{Z}}\mathfrak{g}^{L}

is a Lie algebra over $F$ which acts faithfully on $V_{F}$ , and $\mathfrak{g}(V,L)\subseteq\mathfrak{gl}(V_{F})$ is precisely the Lie algebra of $G(V,L)$ . The adjoint action of $G(V,L)$ can be realized by applying the Chevalley construction with $V^{\prime}=\mathfrak{g}$ and $L^{\prime}=\mathfrak{g}^{L}$ , and then taking the morphism $G(V,L)\rightarrow G(V^{\prime},L^{\prime})$ of Lemma 7.1 (ii).

Note that then the Lie algebra $\mathfrak{g}(V,L)$ is a $G_{sc}$ -module for a simply connected Chevalley group $G_{sc}$ with root system $\Phi$ (Lemma 7.1 (i)). In all cases, the structure of $\mathfrak{g}(V,L)$ as a Lie algebra and as a $G_{sc}$ -module is described by Hogeweij in [9]. Here the composition factors of $\mathfrak{g}(V,L)$ are completely determined by $V$ , but the submodule structure depends on the choice of the admissible lattice $L$ .

In the simply connected case we have $\mathfrak{g}^{L}=\mathfrak{g}_{\mathbb{Z}}$ . In the adjoint case we denote $\mathfrak{g}_{\mathbb{Z}}^{ad}:=\mathfrak{g}^{L}$ , so $\mathfrak{g}_{\mathbb{Z}}^{ad}$ is the $\mathbb{Z}$ -span of $\mathfrak{g}_{\mathbb{Z}}$ and $\mathfrak{h}^{ad}:=\{H\in\mathfrak{h}:\alpha(H)\in\mathbb{Z}\text{ for all }\alpha\in\Phi\}$ .

7.2. Chevalley construction for type $C_{\ell}$

We will now setup the Chevalley construction for groups of type $C_{\ell}$ over $K$ . (Recall that by $K$ we always denote an algebraically closed field of characteristic two.)

Let $V_{\mathbb{C}}$ be a $\mathbb{C}$ -vector space of dimension $n=2\ell$ , with basis $v_{1}$ , $\ldots$ , $v_{n}$ . We define a non-degenerate alternating bilinear form $(-,-)$ on $V_{\mathbb{C}}$ by $(v_{i},v_{n-i+1})=1=-(v_{n-i+1},v_{i})$ for all $1\leq i\leq\ell$ , and $(v_{i},v_{j})=0$ if $i+j\neq n+1$ .

Let $\mathfrak{sp}(V_{\mathbb{C}})=\{X\in\mathfrak{gl}(V_{\mathbb{C}}):(Xv,w)+(v,Xw)=0\text{ for all }v,w\in V_{\mathbb{C}}\}$ , so $\mathfrak{sp}(V_{\mathbb{C}})$ is a simple Lie algebra of type $C_{\ell}$ . Let $\mathfrak{h}_{\mathbb{C}}$ be the Cartan subalgebra formed by the diagonal matrices in $\mathfrak{sp}(V_{\mathbb{C}})$ . Then $\mathfrak{h}_{\mathbb{C}}=\{\operatorname{diag}(h_{1},\ldots,h_{\ell},-h_{\ell},\ldots,-h_{1}):h_{i}\in\mathbb{C}\}$ .

For $1\leq i\leq\ell$ , define linear maps $\varepsilon_{i}:\mathfrak{h}_{\mathbb{C}}\rightarrow\mathbb{C}$ by $\varepsilon_{i}(h)=h_{i}$ where $h$ is a diagonal matrix with diagonal entries $(h_{1},\ldots,h_{\ell},-h_{\ell},\ldots,-h_{1})$ . Then

\Phi=\{\pm(\varepsilon_{i}\pm\varepsilon_{j}):1\leq i<j\leq\ell\}\cup\{\pm 2\varepsilon_{i}:1\leq i\leq\ell\}

is the root system for $\mathfrak{sp}(V_{\mathbb{C}})$ , and

\Phi^{+}=\{\varepsilon_{i}\pm\varepsilon_{j}:1\leq i<j\leq\ell\}\cup\{2\varepsilon_{i}:1\leq i\leq\ell\}

is a system of positive roots. The set of simple roots corresponding to $\Phi^{+}$ is $\Delta=\{\alpha_{1},\ldots,\alpha_{\ell}\}$ , where $\alpha_{i}=\varepsilon_{i}-\varepsilon_{i+1}$ for $1\leq i<{\ell}$ and $\alpha_{\ell}=2\varepsilon_{\ell}$ .

For all $i,j$ let $E_{i,j}$ be the linear endomorphism on $V_{\mathbb{C}}$ such that $E_{i,j}(v_{j})=v_{i}$ and $E_{i,j}(v_{k})=0$ for $k\neq j$ . Throughout we will use the following Chevalley basis of $\mathfrak{sp}(V_{\mathbb{C}})$ , which is taken from [13, Section 11, p. 38].

$\displaystyle X_{\varepsilon_{i}-\varepsilon_{j}}$	$\displaystyle=E_{i,j}-E_{n-j+1,n-i+1}$	$\displaystyle\text{ for all }i\neq j,$
$\displaystyle X_{\varepsilon_{i}+\varepsilon_{j}}$	$\displaystyle=E_{j,n-i+1}+E_{i,n-j+1}$	$\displaystyle\text{ for all }i\neq j,$
$\displaystyle X_{-(\varepsilon_{i}+\varepsilon_{j})}$	$\displaystyle=E_{n-j+1,i}+E_{n-i+1,j}$	$\displaystyle\text{ for all }i\neq j,$
$\displaystyle X_{2\varepsilon_{i}}$	$\displaystyle=E_{i,n-i+1}$	$\displaystyle\text{ for all }i,$
$\displaystyle X_{-2\varepsilon_{i}}$	$\displaystyle=E_{n-i+1,i}$	$\displaystyle\text{ for all }i.$
$\displaystyle H_{\alpha_{i}}$	$\displaystyle=[X_{\alpha_{i}},X_{-\alpha_{i}}]$	$\displaystyle\text{ for all }1\leq i\leq\ell.$

Let $\Lambda\subset\mathfrak{h}_{\mathbb{C}}^{*}$ be the weight lattice. The maps $\varepsilon_{1}$ , $\ldots$ , $\varepsilon_{\ell}$ form $\mathbb{Z}$ -basis for $\Lambda$ , and for $1\leq i\leq\ell$ , the $i$ th fundamental highest weight is equal to $\varpi_{i}:=\varepsilon_{1}+\cdots+\varepsilon_{i}$ .

As in the previous subsection, we denote by $\mathfrak{g}_{\mathbb{Z}}$ be the $\mathbb{Z}$ -span of the Chevalley basis above. Then $\mathfrak{g}_{\mathbb{Z}}^{ad}$ is the $\mathbb{Z}$ -span of $\mathfrak{g}_{\mathbb{Z}}$ and $H$ , where $H\in\mathfrak{h}_{\mathbb{C}}$ is the diagonal matrix $H=\operatorname{diag}(1/2,\ldots,1/2,-1/2,\ldots,-1/2)$ . In terms of the Chevalley basis, we have

H=\frac{1}{2}\left(H_{\alpha_{1}}+2H_{\alpha_{2}}+\cdots+\ell H_{\alpha_{\ell}}\right).

7.3. Simply connected groups of type $C_{\ell}$

Let $V_{\mathbb{Z}}$ be the $\mathbb{Z}$ -span of the basis $v_{1}$ , $\ldots$ , $v_{n}$ of $V_{\mathbb{C}}$ . It is clear that $V_{\mathbb{Z}}$ is an admissible lattice. Denote $V:=K\otimes_{\mathbb{Z}}V_{\mathbb{Z}}$ . By abuse of notation we denote $v_{i}:=1\otimes v_{i}$ for all $1\leq i\leq n$ , so $v_{1}$ , $\ldots$ , $v_{n}$ is a basis of $V$ . The alternating bilinear form $(-,-)$ induces an alternating bilinear form $b:V\times V\rightarrow K$ on $V$ , with $b(v_{i},v_{j})=1$ if $i+j=n+1$ and $b(v_{i},v_{j})=0$ otherwise.

Then the Chevalley group corresponding to $V_{\mathbb{C}}$ and $V_{\mathbb{Z}}$ is simply connected of type $C_{\ell}$ , and it is precisely the symplectic group $\operatorname{Sp}(V)$ corresponding to $b$ [29, 5]. We denote $G_{sc}=\operatorname{Sp}(V)$ . Then the Lie algebra of $G_{sc}$ is

\mathfrak{g}_{sc}=\mathfrak{sp}(V)=\{X\in\mathfrak{gl}(V):b(Xv,w)+b(v,Xw)=0\text{ for all }v,w\in V\}.

We denote by $\mathfrak{g}_{ad}:=K\otimes_{\mathbb{Z}}\mathfrak{g}_{\mathbb{Z}}^{\operatorname{ad}}$ the adjoint Lie algebra of type $C_{\ell}$ . For any $\mathfrak{sp}(V_{\mathbb{C}})$ -module $W$ with admissible lattice $L$ , we will consider $K\otimes_{\mathbb{Z}}L$ as a $G_{sc}$ -module via the action provided by Lemma 7.1 (i).

7.4. Lie algebra of adjoint type $C_{\ell}$

To compute with the action of $G_{sc}=\operatorname{Sp}(V)$ on $\mathfrak{g}_{ad}$ , it will be convenient to use the following well-known identification of $\mathfrak{sp}(V_{\mathbb{C}})$ with the symmetric square $S^{2}(V_{\mathbb{C}})$ .

Lemma 7.2.

For $x,y\in V_{\mathbb{C}}$ , define a linear map $\psi_{x,y}:V_{\mathbb{C}}\rightarrow V_{\mathbb{C}}$ by $v\mapsto(y,v)x+(x,v)y$ . Then we have an isomorphism $S^{2}(V_{\mathbb{C}})\rightarrow\mathfrak{sp}(V_{\mathbb{C}})$ of $\mathfrak{sp}(V_{\mathbb{C}})$ -modules, defined by $xy\mapsto\psi_{x,y}$ for all $x,y\in V_{\mathbb{C}}$ .

Proof.

We have an isomorphism $\tau:V_{\mathbb{C}}\otimes V_{\mathbb{C}}^{*}\rightarrow\mathfrak{gl}(V_{\mathbb{C}})$ of $\mathfrak{gl}(V_{\mathbb{C}})$ -modules, where $\tau(v\otimes f)$ is the linear map $w\mapsto f(w)v$ for all $v\in V_{\mathbb{C}}$ and $f\in V_{\mathbb{C}}^{*}$ . Moreover, we have an isomorphism $\tau^{\prime}:V_{\mathbb{C}}\otimes V_{\mathbb{C}}\rightarrow V_{\mathbb{C}}\otimes V_{\mathbb{C}}^{*}$ of $\mathfrak{sp}(V_{\mathbb{C}})$ -modules defined by $x\otimes y\mapsto x\otimes f_{y}$ , where $f_{y}(v)=(y,v)$ for all $v\in V$ .

Now identifying $S^{2}(V_{\mathbb{C}})\subseteq V_{\mathbb{C}}\otimes V_{\mathbb{C}}$ via $xy\mapsto x\otimes y+y\otimes x$ , the restriction of $\tau\tau^{\prime}$ to $S^{2}(V_{\mathbb{C}})$ is a map $\psi:S^{2}(V_{\mathbb{C}})\rightarrow\mathfrak{gl}(V_{\mathbb{C}})$ defined by $xy\mapsto\psi_{x,y}$ . Then $\psi$ is an injective morphism of $\mathfrak{sp}(V_{\mathbb{C}})$ -modules, and it is clear that $\psi_{x,y}\in\mathfrak{sp}(V_{\mathbb{C}})$ for all $x,y\in V_{\mathbb{C}}$ . Since $S^{2}(V_{\mathbb{C}})$ and $\mathfrak{sp}(V_{\mathbb{C}})$ have the same dimension, we conclude that $\psi$ defines an isomorphism $S^{2}(V_{\mathbb{C}})\rightarrow\mathfrak{sp}(V_{\mathbb{C}})$ of $\mathfrak{sp}(V_{\mathbb{C}})$ -modules.∎

We denote by $\psi:S^{2}(V_{\mathbb{C}})\rightarrow\mathfrak{sp}(V_{\mathbb{C}})$ the isomorphism $\psi(xy)=\psi_{x,y}$ as in Lemma 7.2. Let $1\leq i,j\leq\ell$ with $i\neq j$ . Then under the map $\psi$ , the root vectors $X_{\pm\varepsilon_{i}\pm\varepsilon_{j}}$ in the Chevalley basis correspond to elements of $S^{2}(V_{\mathbb{C}})$ as follows.

	$\displaystyle\psi\left(v_{i}v_{n-j+1}\right)$	$\displaystyle=-X_{\varepsilon_{i}-\varepsilon_{j}}$
	$\displaystyle\psi\left(v_{i}v_{j}\right)$	$\displaystyle=X_{\varepsilon_{i}+\varepsilon_{j}}$
	$\displaystyle\psi\left(v_{n-i+1}v_{n-j+1}\right)$	$\displaystyle=-X_{-(\varepsilon_{i}+\varepsilon_{j})}$
	$\displaystyle\psi\left(\frac{1}{2}v_{i}^{2}\right)$	$\displaystyle=X_{2\varepsilon_{i}}$
	$\displaystyle\psi\left(\frac{1}{2}v_{n-i+1}^{2}\right)$	$\displaystyle=-X_{-2\varepsilon_{i}}$

Furthermore, define

\delta:=\frac{1}{2}\sum_{i=1}^{\ell}v_{i}v_{n-i+1}.

Then $\psi(\delta)=-H$ .

We define

	$\displaystyle L_{sc}$	$\displaystyle:=\mathbb{Z}\text{-span of }v_{i}v_{j}\text{ and }\frac{1}{2}v_{i}^{2}\text{ for all }1\leq i,j\leq n;$
	$\displaystyle L_{ad}$	$\displaystyle:=\mathbb{Z}\text{-span of }L_{sc}\text{ and }\delta;$

so that $\psi(L_{sc})=\mathfrak{g}_{\mathbb{Z}}$ and $\psi(L_{ad})=\mathfrak{g}_{\mathbb{Z}}^{ad}$ .

Then $L_{sc}\subset L_{ad}$ are admissible lattices in $S^{2}(V_{\mathbb{C}})$ , since $\mathfrak{g}_{\mathbb{Z}}$ and $\mathfrak{g}_{\mathbb{Z}}^{ad}$ are admissible lattices in $\mathfrak{sp}(V_{\mathbb{C}})$ .

From the Chevalley construction $K\otimes_{\mathbb{Z}}\mathfrak{g}_{\mathbb{Z}}=\mathfrak{g}_{sc}$ and $K\otimes_{\mathbb{Z}}\mathfrak{g}_{\mathbb{Z}}^{ad}=\mathfrak{g}_{ad}$ as $G_{sc}$ -modules, so $\psi$ induces isomorphisms

(7.1)			$\displaystyle\psi^{\prime}:\mathfrak{g}_{sc}\rightarrow K\otimes_{\mathbb{Z}}L_{sc}$
(7.1)			$\displaystyle\psi^{\prime\prime}:\mathfrak{g}_{ad}\rightarrow K\otimes_{\mathbb{Z}}L_{ad}$

of $G_{sc}$ -modules.

By [9, Lemma 2.2] we have $\dim Z(\mathfrak{g}_{sc})=1$ and $Z(\mathfrak{g}_{sc})$ is spanned by $1\otimes 2H$ . Similarly by [9, Table 1], we have $\dim\mathfrak{g}_{ad}/[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]=1$ and $[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ is spanned by $1\otimes v$ with $v\in\mathfrak{g}_{\mathbb{Z}}$ . Thus under the isomorphisms in (7.1), we get

(7.2)		$\displaystyle\psi^{\prime}(Z(\mathfrak{g}_{sc}))$	$\displaystyle=\langle 1\otimes 2\delta\rangle,$
(7.2)		$\displaystyle\psi^{\prime\prime}([\mathfrak{g}_{ad},\mathfrak{g}_{ad}])$	$\displaystyle=\langle 1\otimes v:v\in L_{sc}\rangle.$

Note that

(7.3)

L_{ad}/L_{sc}=\{L_{sc},\delta+L_{sc}\}\cong\mathbb{Z}/2\mathbb{Z},

so for the linear map $\pi:K\otimes_{\mathbb{Z}}L_{sc}\rightarrow K\otimes_{\mathbb{Z}}L_{ad}$ induced by the inclusion $L_{sc}\subset L_{ad}$ , we have $\operatorname{Ker}\pi=\langle 1\otimes(2\delta)\rangle$ and $\operatorname{Im}\pi=\langle 1\otimes v:v\in L_{sc}\rangle$ .

7.5. Action of unipotent elements on $\mathfrak{g}_{ad}$ , orthogonally indecomposable case

Let $G_{sc}=\operatorname{Sp}(V)$ as in the previous sections, and let $u\in\operatorname{Sp}(V)$ be a unipotent element.

We will describe how to construct a representative for the conjugacy class of $u$ in terms of root elements, and how to compute the action of $u$ on $\mathfrak{g}_{ad}\cong K\otimes_{\mathbb{Z}}L_{ad}$ . We first do this in the case where $V\downarrow K[u]$ is orthogonally indecomposable, so $V\downarrow K[u]=V(2\ell)$ or $V\downarrow K[u]=W(\ell)$ (Theorem 5.3). In this case it is well known that by replacing $u$ with a conjugate, we can take

u=\begin{cases}x_{\alpha_{1}}(1)\cdots x_{\alpha_{\ell}}(1),&\text{ if }V\downarrow K[u]=V(2\ell).\\ x_{\alpha_{1}}(1)\cdots x_{\alpha_{\ell-1}}(1),&\text{ if }V\downarrow K[u]=W(\ell).\end{cases}

For $\alpha\in\Phi$ , let

x_{\mathbb{Z}}^{(\alpha)}:=1+X_{\alpha}+\frac{X_{\alpha}^{2}}{2!}\in\mathscr{U}_{\mathbb{Z}}.

Then $x_{\alpha}(1)$ is the reduction modulo $p$ of the action of $x_{\mathbb{Z}}^{(\alpha)}$ on $V_{\mathbb{Z}}$ . Furthermore, since $X_{\alpha}^{3}\cdot S^{2}(V_{\mathbb{C}})=0$ , the action of $x_{\alpha}(1)$ on $\mathfrak{g}_{ad}\cong K\otimes_{\mathbb{Z}}L_{ad}$ is the reduction modulo $p$ of the action of $x_{\mathbb{Z}}^{(\alpha)}$ on $L_{ad}$ .

Therefore we define

u_{\mathbb{Z}}:=\begin{cases}x_{\mathbb{Z}}^{(\alpha_{1})}\cdots x_{\mathbb{Z}}^{(\alpha_{\ell})},&\text{ if }V\downarrow K[u]=V(2\ell).\\ x_{\mathbb{Z}}^{(\alpha_{1})}\cdots x_{\mathbb{Z}}^{(\alpha_{\ell-1})},&\text{ if }V\downarrow K[u]=W(\ell).\end{cases}

If $V\downarrow K[u]=V(2\ell)$ , we get

	$\displaystyle u_{\mathbb{Z}}\cdot v_{1}$	$\displaystyle=v_{1},$
	$\displaystyle u_{\mathbb{Z}}\cdot v_{i}$	$\displaystyle=v_{i}+v_{i-1}+\cdots+v_{1}\text{ if }1<i\leq\ell+1,$
	$\displaystyle u_{\mathbb{Z}}\cdot v_{i}$	$\displaystyle=v_{i}-v_{i-1}\text{ if }\ell+1<i\leq 2\ell.$

Similarly for $V\downarrow K[u]=W(\ell)$ , we get

	$\displaystyle u_{\mathbb{Z}}\cdot v_{1}$	$\displaystyle=v_{1},$
	$\displaystyle u_{\mathbb{Z}}\cdot v_{i}$	$\displaystyle=v_{i}+v_{i-1}+\cdots+v_{1}\text{ if }1<i\leq\ell,$
	$\displaystyle u_{\mathbb{Z}}\cdot v_{\ell+1}$	$\displaystyle=v_{\ell+1}$
	$\displaystyle u_{\mathbb{Z}}\cdot v_{i}$	$\displaystyle=v_{i}-v_{i-1}\text{ if }\ell+1<i\leq 2\ell.$

Thus the action of $u$ on $V$ is exactly as described in Definition 5.1 or Definition 5.2.

Since $X_{\alpha_{i}}^{3}\cdot S^{2}(V_{\mathbb{C}})=0$ , for the action on $S^{2}(V_{\mathbb{C}})$ we have

x_{\mathbb{Z}}^{(\alpha)}\cdot(vw)=(x_{\mathbb{Z}}^{(\alpha)}v)(x_{\mathbb{Z}}^{(\alpha)}w)

for all $v,w\in V_{\mathbb{C}}$ . Thus

u_{\mathbb{Z}}\cdot(vw)=(u_{\mathbb{Z}}v)(u_{\mathbb{Z}}w)

for all $v,w\in V_{\mathbb{C}}$ .

7.6. Action of unipotent elements on $\mathfrak{g}_{ad}$ , general case

For all unipotent elements $u\in G_{sc}$ , in general we can proceed as follows. We have an orthogonal decomposition $V\downarrow K[u]=U_{1}\perp\cdots\perp U_{t}$ , where $\dim U_{i}>0$ and $U_{i}$ is an orthogonally indecomposable $K[u]$ -module for all $1\leq i\leq t$ . Write $\dim U_{i}=n_{i}=2\ell_{i}$ for all $1\leq i\leq t$ .

Then $u=u_{1}\cdots u_{t}$ , where $u_{i}\in\operatorname{Sp}(U_{i})$ is the action of $u$ on $U_{i}$ . For all $1\leq i\leq t$ , we have $U_{i}\downarrow K[u_{i}]\cong W(\ell_{i})$ or $U_{i}\downarrow K[u_{i}]\cong V(2\ell_{i})$ (Theorem 5.3).

Relabel the basis $v_{1}$ , $\ldots$ , $v_{n}$ of $V_{\mathbb{C}}$ as

v_{1}^{(1)},\ldots,v_{n_{1}}^{(1)},\ldots,v_{1}^{(t)},\ldots,v_{n_{t}}^{(t)},

where $(v_{i}^{(j)},v_{i^{\prime}}^{(j^{\prime})})=1$ if $j=j^{\prime}$ and $i+i^{\prime}=n_{j}$ , and $0$ otherwise. Then the Cartan subalgebra $\mathfrak{h}$ consists of diagonal matrices of the form

h=\operatorname{diag}(a_{1}^{(1)},\ldots,a_{\ell_{1}}^{(1)},-a_{\ell_{1}}^{(1)},\ldots,-a_{1}^{(1)},\ \ldots,\ a_{1}^{(t)},\ldots,a_{\ell_{t}}^{(t)},-a_{\ell_{t}}^{(t)},\ldots,-a_{1}^{(t)}).

We define then for $1\leq i\leq t$ and $1\leq k\leq\ell_{i}$ the linear map $\varepsilon_{k}^{(i)}:\mathfrak{h}\rightarrow\mathbb{C}$ by $h\mapsto a_{k}^{(i)}$ .

Denote by $V_{\mathbb{Z}}^{(i)}$ (resp. $V_{\mathbb{C}}^{(i)}$ ) the $\mathbb{Z}$ -span (resp. $\mathbb{C}$ -span) of $v_{1}^{(i)},\ldots,v_{n_{i}}^{(i)}$ , so we have

	$\displaystyle V_{\mathbb{Z}}$	$\displaystyle=V_{\mathbb{Z}}^{(1)}\oplus\cdots\oplus V_{\mathbb{Z}}^{(t)},$
	$\displaystyle V_{\mathbb{C}}$	$\displaystyle=V_{\mathbb{C}}^{(1)}\perp\cdots\perp V_{\mathbb{C}}^{(t)}.$

Since $V=K\otimes_{\mathbb{Z}}V_{\mathbb{Z}}$ , we can assume that $U_{i}=K\otimes_{\mathbb{Z}}V_{\mathbb{Z}}^{(i)}$ for all $1\leq i\leq t$ .

We have $\mathfrak{sp}(V_{\mathbb{C}}^{(i)})\subseteq\mathfrak{sp}(V_{\mathbb{C}})$ , and $\mathfrak{sp}(V_{\mathbb{C}}^{(i)})$ has a root system $\Phi^{(i)}$ of type $C_{\ell_{i}}$ with simple roots $\Delta^{(i)}=\{\beta_{1},\ldots,\beta_{\ell_{i}}\}$ , where

\beta_{j}=\begin{cases}\varepsilon_{j}^{(i)}-\varepsilon_{j+1}^{(i)},&\text{ if }1\leq j<\ell_{i},\\ 2\varepsilon_{\ell_{i}}^{(i)},&\text{ if }j=\ell_{i}.\end{cases}

Note that $\{X_{\alpha}:\alpha\in\Phi^{(i)}\}\cup\{H_{\alpha}:\alpha\in\Delta^{(i)}\}$ is a Chevalley basis of $\mathfrak{sp}(V_{\mathbb{C}}^{(i)})$ , and the corresponding Kostant $\mathbb{Z}$ -form $\mathscr{U}_{\mathbb{Z}}^{(i)}$ is a subring of $\mathscr{U}_{\mathbb{Z}}$ . Then $V_{\mathbb{Z}}^{(i)}$ is an admissible lattice for $\mathfrak{sp}(V_{\mathbb{C}}^{(i)})$ , and applying the Chevalley construction we get $\operatorname{Sp}(U_{i})$ .

As in the orthogonally indecomposable case, we define

u_{\mathbb{Z},i}:=\begin{cases}x_{\mathbb{Z}}^{(\beta_{1})}\cdots x_{\mathbb{Z}}^{(\beta_{\ell_{i}})},&\text{ if }U_{i}\downarrow K[u_{i}]=V(2\ell_{i}).\\ x_{\mathbb{Z}}^{(\beta_{1})}\cdots x_{\mathbb{Z}}^{(\beta_{\ell_{i}-1})},&\text{ if }U_{i}\downarrow K[u_{i}]=W(\ell_{i}).\end{cases}

Then $u_{\mathbb{Z},i}\in\mathscr{U}_{\mathbb{Z}}^{(i)}$ , and the reduction modulo $p$ of the action of $u_{\mathbb{Z},i}$ on $V_{\mathbb{Z}}^{(i)}$ is precisely $u_{i}\in\operatorname{Sp}(U_{i})$ .

Thus we can define

u_{\mathbb{Z}}:=u_{\mathbb{Z},1}\cdots u_{\mathbb{Z},t}\in\mathscr{U}_{\mathbb{Z}},

so that $u$ is the reduction modulo $p$ of the action of $u_{\mathbb{Z}}$ on $V_{\mathbb{Z}}$ . Furthermore, the action of $u$ on $\mathfrak{g}_{ad}$ is the reduction modulo $p$ of the action of $u_{\mathbb{Z}}$ on $L_{ad}$ .

7.7. Action of nilpotent elements on $\mathfrak{g}_{ad}$

Let $e\in\mathfrak{sp}(V)$ be nilpotent. Similarly to the unipotent case, we can find $e_{\mathbb{Z}}\in\mathfrak{g}_{\mathbb{Z}}$ such that $e$ is the reduction modulo $p$ of the action of $e_{\mathbb{Z}}$ on $V_{\mathbb{Z}}$ .

We first consider the orthogonally indecomposable case, so $V\downarrow K[e]=V(2\ell)$ , $V\downarrow K[e]=W(\ell)$ , or $V\downarrow K[e]=W_{k}(\ell)$ for some $0<k<\ell/2$ (Theorem 6.7). In this case we define

e_{\mathbb{Z}}:=\begin{cases}X_{\alpha_{1}}+\cdots+X_{\alpha_{\ell}},&\text{ if }V\downarrow K[e]=V(2\ell).\\ X_{\alpha_{1}}+\cdots+X_{\alpha_{\ell-1}},&\text{ if }V\downarrow K[e]=W(\ell).\\ X_{\alpha_{1}}+\cdots+X_{\alpha_{\ell-1}}+X_{2\varepsilon_{k}},&\text{ if }V\downarrow K[e]=W_{k}(\ell).\end{cases}

Then for the reduction modulo $p$ of $e_{\mathbb{Z}}$ , the action of $V=K\otimes_{\mathbb{Z}}V_{\mathbb{Z}}$ is exactly as in Definition 6.4 – 6.6. (Here the expression for $W_{k}(\ell)$ is taken from [20, Lemma 3.4].)

In the general case, we have an orthogonal decomposition $V\downarrow K[e]=U_{1}\perp\cdots\perp U_{t}$ , where $U_{i}$ is an orthogonally indecomposable $K[e]$ -module, with $\dim U_{i}=2\ell_{i}$ . We have $e=e_{1}+\cdots+e_{t}$ , where $e_{i}\in\mathfrak{sp}(U_{i})$ is the action of $e$ on $U_{i}$ . In the notation of the previous section, we take $U_{i}=K\otimes_{\mathbb{Z}}V_{\mathbb{Z}}^{(i)}$ . We can then define

e_{\mathbb{Z}}:=e_{\mathbb{Z},1}+\cdots+e_{\mathbb{Z},t},

where $e_{\mathbb{Z},i}\in\mathfrak{g}_{\mathbb{Z}}\cap\mathfrak{sp}(V_{\mathbb{C}}^{(i)})$ is defined as in the orthogonally indecomposable case, such that $e_{i}$ is the reduction modulo $p$ of $e_{\mathbb{Z},i}$ .

Then $e$ is the reduction modulo $p$ of $e_{\mathbb{Z}}$ . Moreover, given any irreducible $\mathfrak{sp}(V_{\mathbb{C}})$ -module $W_{\mathbb{C}}$ and admissible lattice $L\subset W_{\mathbb{C}}$ , the action of $e$ on $K\otimes_{\mathbb{Z}}L$ is precisely the reduction modulo $p$ of the action of $e_{\mathbb{Z}}$ on $L$ .

7.8. Elements of $\mathscr{U}_{\mathbb{Z}}$ acting nilpotently

In this section, we consider $X\in\mathscr{U}_{\mathbb{Z}}$ which act nilpotently on $S^{2}(V_{\mathbb{C}})$ . Then by reduction modulo $p$ , we get an action $\widehat{X}$ on $\mathfrak{g}_{sc}$ , and $\widetilde{X}$ on $\mathfrak{g}_{ad}$ . We will describe when $\operatorname{Im}(\widehat{X})\supseteq Z(\mathfrak{g}_{sc})$ and when $\operatorname{Ker}(\widetilde{X})\subseteq[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ . For our purposes this is mostly relevant for $X=(u_{\mathbb{Z}}-1)^{m}$ in the unipotent case, and $X=e_{\mathbb{Z}}^{m}$ in the nilpotent case, for integer $m\geq 1$ .

Lemma 7.3.

Let $X\in\mathscr{U}_{\mathbb{Z}}$ be such that $X$ acts nilpotently on $S^{2}(V_{\mathbb{C}})$ , and denote the action of $X$ on $\mathfrak{g}_{sc}$ by $\widehat{X}$ . Then $\operatorname{Im}(\widehat{X})\supseteq Z(\mathfrak{g}_{sc})$ if and only if there exists $v\in L_{sc}$ such that $X\cdot v\equiv 2\delta\mod 2L_{sc}$ .

Proof.

Identifying $\mathfrak{g}_{sc}=K\otimes_{\mathbb{Z}}L_{sc}$ via (7.1), we have $Z(\mathfrak{g}_{sc})=\langle 1\otimes 2\delta\rangle$ by (7.2). Thus if $v\in L_{sc}$ is such that $X\cdot v\equiv 2\delta\mod 2L_{sc}$ , we have $\widehat{X}\cdot(1\otimes v)=1\otimes 2\delta$ .

Conversely, suppose that $\operatorname{Im}(\widehat{X})\supseteq Z(\mathfrak{g}_{sc})$ . Then there exists $w\in\mathfrak{g}_{sc}$ with $\widehat{X}\cdot w=1\otimes 2\delta$ . Since $\widehat{X}=1\otimes X^{\prime}$ where $X^{\prime}$ is the action of $X$ on $L_{sc}$ , we can assume that $w=1\otimes v$ with $v\in L_{sc}$ . Then

1\otimes 2\delta=\widehat{X}\cdot w=1\otimes(X\cdot v)

implies that $X\cdot v\equiv 2\delta\mod 2L_{sc}$ .∎

For actions of unipotent elements, Lemma 7.3 gives the following.

Lemma 7.4.

Let $u\in G_{sc}$ be unipotent, and suppose that $u$ is the reduction modulo $p$ of $u_{\mathbb{Z}}\in\mathscr{U}_{\mathbb{Z}}$ . Denote the action of $u$ on $\mathfrak{g}_{sc}$ by $\widehat{u}$ . Let $m\geq 1$ be an integer. Then $\operatorname{Im}(\widehat{u}-1)^{m}\supseteq Z(\mathfrak{g}_{sc})$ if and only if there exists $v\in L_{sc}$ such that $(u_{\mathbb{Z}}-1)^{m}\cdot(v)\equiv 2\delta\mod 2L_{sc}$ .

Proof.

Follows from Lemma 7.3, with $X=(u_{\mathbb{Z}}-1)^{m}$ . ∎

Lemma 7.5.

Let $X\in\mathscr{U}_{\mathbb{Z}}$ be such that $X$ acts nilpotently on $S^{2}(V_{\mathbb{C}})$ , and denote the action of $X$ on $\mathfrak{g}_{ad}$ by $\widetilde{X}$ . Then the following statements are equivalent:

(i)

$\operatorname{Ker}(\widetilde{X})\not\subseteq[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ .
(ii)

There exists $v\in L_{sc}$ such that $X\cdot(\delta+v)\in 2L_{ad}$ .
(iii)
There exists $v\in L_{sc}$ such that one of the following holds:
1. (a)
  
  $X\cdot(\delta+v)\equiv 0\mod{2L_{sc}}$ .
2. (b)
  
  $X\cdot(\delta+v)\equiv 2\delta\mod{2L_{sc}}$ .

Proof.

We first prove that (i) and (ii) are equivalent. We identify $\mathfrak{g}_{ad}=K\otimes_{\mathbb{Z}}L_{ad}$ via (7.1), in which case $\mathfrak{g}_{ad}=\langle 1\otimes\delta,[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]\rangle$ as $K$ -vector spaces. Here $[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ is the subspace spanned by $1\otimes v$ with $v\in L_{sc}$ , as seen in (7.2).

If there exists $v\in L_{sc}$ such that $X\cdot(\delta+v)\in 2L_{ad}$ , we have

\widetilde{X}\cdot(1\otimes\delta+1\otimes v)=1\otimes(X\cdot(\delta+v))=0,

so $\operatorname{Ker}(\widetilde{X})\not\subseteq[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ .

Conversely, suppose that $\operatorname{Ker}(\widetilde{X})\not\subseteq[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ . Then there exists $w\in[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ such that $\widetilde{X}\cdot(1\otimes\delta+w)=0$ . We have $\widetilde{X}=1\otimes X^{\prime}$ where $X^{\prime}$ is the action of $X$ on $L_{ad}$ , so we can assume that $w=1\otimes v$ for some $v\in L_{sc}$ . Then

0=\widetilde{X}\cdot(1\otimes\delta+1\otimes v)=1\otimes(X\cdot(\delta+v))

implies that $X\cdot(\delta+v)\in 2L_{ad}$ .

Next we show that (ii) and (iii) are equivalent. It follows from (7.3) that $2L_{ad}/2L_{sc}\cong L_{ad}/L_{sc}\cong\mathbb{Z}/2\mathbb{Z}$ contains only two elements, $2L_{sc}$ and $2\delta+2L_{sc}$ . Therefore $X\cdot(\delta+v)\in 2L_{ad}$ if and only if (iii)(a) or (iii)(b) holds. ∎

Next we make some observations that will allow us to reduce the proofs of our main results to the orthogonally indecomposable case. Continue with the notation as in Section 7.6, so $V=U_{1}\perp\cdots\perp U_{t}$ with $U_{i}=K\otimes_{\mathbb{Z}}V_{\mathbb{Z}}^{(i)}$ , and $\mathscr{U}_{\mathbb{Z}}^{(i)}$ is the Kostant $\mathbb{Z}$ -form for $\mathfrak{sp}(V_{\mathbb{C}}^{(i)})$ . Define

\delta_{i}=\frac{1}{2}\sum_{1\leq j\leq\ell_{i}}v_{j}^{(i)}v_{2\ell_{i}-j+1}^{(i)}

for all $1\leq i\leq t$ . (Note that $\delta=\delta_{1}+\cdots+\delta_{t}$ .) Furthermore, for $1\leq i\leq t$ we define

	$\displaystyle L_{sc}^{(i)}$	$\displaystyle:=\mathbb{Z}\text{-span of }v_{j}^{(i)}v_{k}^{(i)}\text{ and }\frac{1}{2}\left(v_{j}^{(i)}\right)^{2}\text{ for all }1\leq j,k\leq n_{i}.$
	$\displaystyle L_{ad}^{(i)}$	$\displaystyle:=\mathbb{Z}\text{-span of }L_{sc}^{(i)}\text{ and }\delta_{i}.$

Then

(7.4)

L_{sc}=L_{sc}^{(1)}\oplus\cdots\oplus L_{sc}^{(t)}\oplus\bigoplus_{i\neq i^{\prime}}V_{\mathbb{Z}}^{(i)}V_{\mathbb{Z}}^{(i^{\prime})}

as $\mathbb{Z}$ -modules.

Moreover $L_{ad}^{(i)}\subset S^{2}(V_{\mathbb{C}}^{(i)})$ is an admissible lattice for the action of $\mathfrak{sp}(V_{\mathbb{C}}^{(i)})$ , and

\mathfrak{g}_{ad}^{(i)}:=K\otimes_{\mathbb{Z}}L_{ad}^{(i)}

is an adjoint Lie algebra of type $C_{\ell_{i}}$ . Here $\operatorname{Sp}(U_{i})$ acts on $\mathfrak{g}_{ad}^{(i)}$ via the Chevalley construction.

Lemma 7.6.

Let $X=X_{1}+\cdots+X_{t}$ , where $X_{i}\in\mathscr{U}_{\mathbb{Z}}^{(i)}$ acts nilpotently on $S^{2}(V_{\mathbb{C}})$ for all $1\leq i\leq t$ . Denote by $\widetilde{X}$ the linear map acting on $\mathfrak{g}_{ad}$ , given by reducing the action of $X$ on $L_{ad}$ modulo $p$ . For $1\leq i\leq t$ , let $\widetilde{X_{i}}$ be the linear map acting on $\mathfrak{g}_{ad}^{(i)}$ , given by reducing the action of $X_{i}$ on $L_{ad}^{(i)}$ modulo $p$ .

Then the following statements hold.

(i)
$\operatorname{Ker}(\widetilde{X})\not\subseteq[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ if and only if there exist $w_{1}$ , $\ldots$ , $w_{t}$ with $w_{i}\in L_{sc}^{(i)}$ such that one of the following holds:
1. (a)
  
  $X_{i}\cdot(\delta_{i}+w_{i})\equiv 0\mod{2L_{sc}^{(i)}}$ for all $1\leq i\leq t$ .
2. (b)
  
  $X_{i}\cdot(\delta_{i}+w_{i})\equiv 2\delta_{i}\mod{2L_{sc}^{(i)}}$ for all $1\leq i\leq t$ .
(ii)

If $\operatorname{Ker}(\widetilde{X})\not\subseteq[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ , then $\operatorname{Ker}(\widetilde{X_{i}})\not\subseteq[\mathfrak{g}_{ad}^{(i)},\mathfrak{g}_{ad}^{(i)}]$ for all $1\leq i\leq t$ .

Proof.

For (i), suppose first that there exist $w_{1}$ , $\ldots$ , $w_{t}$ with $w_{i}\in L_{sc}^{(i)}$ such that (i)(a) or (i)(b) holds. For $w=w_{1}+\cdots+w_{t}$ , we have

X\cdot(\delta+w)=X_{1}\cdot(\delta_{1}+w_{1})+\cdots+X_{t}\cdot(\delta_{t}+w_{t}).

Thus $X\cdot(\delta+w)\equiv 0\mod{2L_{sc}}$ if (a) holds, and $X\cdot(\delta+w)\equiv 2\delta\mod{2L_{sc}}$ if (b) holds. Therefore $\operatorname{Ker}(\widetilde{X})\not\subseteq[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ by Lemma 7.5.

Conversely, suppose that $\operatorname{Ker}(\widetilde{X})\not\subseteq[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ . Then by Lemma 7.5, there exists $w\in L_{sc}$ such that $X\cdot(\delta+w)\equiv 0\mod{2L_{sc}}$ or $X\cdot(\delta+w)\equiv 2\delta\mod{2L_{sc}}$ . By (7.4) we can write $w=w_{1}+\cdots+w_{t}+w^{\prime}$ , where $w_{i}\in L_{sc}^{(i)}$ for all $1\leq i\leq t$ and $w^{\prime}\in\bigoplus_{i\neq i^{\prime}}V_{\mathbb{Z}}^{(i)}V_{\mathbb{Z}}^{(i^{\prime})}$ .

Note that $L_{sc}^{(1)}\oplus\cdots\oplus L_{sc}^{(t)}$ and $\bigoplus_{i\neq i^{\prime}}V_{\mathbb{Z}}^{(i)}V_{\mathbb{Z}}^{(i^{\prime})}$ are $\mathscr{U}_{\mathbb{Z}}^{(i)}$ -invariant. Thus

X\cdot(\delta+w)=X_{1}\cdot(\delta_{1}+w_{1})+\cdots+X_{t}\cdot(\delta_{t}+w_{t})+X\cdot w^{\prime}

with $X_{i}\cdot(\delta_{i}+w_{i})\in L_{ad}^{(i)}$ for all $1\leq i\leq t$ , and $X\cdot w^{\prime}\in\bigoplus_{i\neq i^{\prime}}V_{\mathbb{Z}}^{(i)}V_{\mathbb{Z}}^{(i^{\prime})}$ . Since $2\delta\in L_{sc}^{(1)}\oplus\cdots\oplus L_{sc}^{(t)}$ , it follows that $X\cdot w^{\prime}=0$ . Moreover we have assumed that $X\cdot(\delta+w)\equiv 0\mod{2L_{sc}}$ or $X\cdot(\delta+w)\equiv 2\delta\mod{2L_{sc}}$ , so either (i)(a) or (i)(b) holds.

Claim (ii) follows from (i) and Lemma 7.5. ∎

For the action of unipotent elements, we have the following result.

Lemma 7.7.

Let $u=u_{1}\cdots u_{t}\in\operatorname{Sp}(V)$ be unipotent with $u_{\mathbb{Z}}=u_{\mathbb{Z},1}\cdots u_{\mathbb{Z},t}\in\mathscr{U}_{\mathbb{Z}}$ as in Section 7.6. Denote the action of $u$ on $\mathfrak{g}_{ad}$ by $\widetilde{u}$ , and denote the action of $u_{i}$ on $\mathfrak{g}_{ad}^{(i)}$ by $\widetilde{u_{i}}$ .

Let $m\geq 1$ be an integer. Then the following statements hold.

(i)
$\operatorname{Ker}(\widetilde{u}-1)^{m}\not\subseteq[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ if and only if there exists $w_{1}$ , $\ldots$ , $w_{t}$ with $w_{i}\in L_{sc}^{(i)}$ such that one of the following holds:
1. (a)
  
  $(u_{\mathbb{Z},i}-1)^{m}\cdot(\delta_{i}+w_{i})\equiv 0\mod{2L_{sc}^{(i)}}$ for all $1\leq i\leq t$ .
2. (b)
  
  $(u_{\mathbb{Z},i}-1)^{m}\cdot(\delta_{i}+w_{i})\equiv 2\delta_{i}\mod{2L_{sc}^{(i)}}$ for all $1\leq i\leq t$ .
(ii)

If $\operatorname{Ker}(\widetilde{u}-1)^{m}\not\subseteq[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ , then $\operatorname{Ker}(\widetilde{u_{i}}-1)^{m}\not\subseteq[\mathfrak{g}_{ad}^{(i)},\mathfrak{g}_{ad}^{(i)}]$ for all $1\leq i\leq t$ .

Proof.

It follows from Lemma 7.5 that $\operatorname{Ker}(\widetilde{u}-1)^{m}\not\subseteq[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ if and only if there exists $w\in L_{sc}$ such that $(u_{\mathbb{Z}}-1)\cdot(\delta+w)\equiv 0\mod{2L_{ad}}$ .

By (7.4) we can write every $w\in L_{sc}$ in the form $w=w_{1}+\cdots+w_{t}+w^{\prime}$ , where $w_{i}\in L_{sc}^{(i)}$ for all $1\leq i\leq t$ and $w^{\prime}\in\bigoplus_{i\neq i^{\prime}}V_{\mathbb{Z}}^{(i)}V_{\mathbb{Z}}^{(i^{\prime})}$ . Since $u_{\mathbb{Z},i}$ acts trivially on $V_{\mathbb{Z}}^{(i^{\prime})}$ for $i\neq i^{\prime}$ , we have

(u_{\mathbb{Z}}-1)^{m}\cdot(\delta+w)=(u_{\mathbb{Z},1}-1)\cdot(\delta_{1}+w_{1})+\cdots+(u_{\mathbb{Z},t}-1)\cdot(\delta_{t}+w_{t})+(u_{\mathbb{Z}}-1)\cdot w^{\prime}.

Here $(u_{\mathbb{Z},i}-1)\cdot(\delta_{i}+w_{i})\in L_{ad}^{(i)}$ for all $1\leq i\leq t$ , and $(u_{\mathbb{Z}}-1)\cdot w^{\prime}\in\bigoplus_{i\neq i^{\prime}}V_{\mathbb{Z}}^{(i)}V_{\mathbb{Z}}^{(i^{\prime})}$ . Thus the result follows similarly to the proof of Lemma 7.6.∎

8. Lie algebras of type $C_{\ell}$ as $G_{sc}$ -modules

In the notation of Section 7.3, let $G_{sc}=\operatorname{Sp}(V)$ be a simply connected simple algebraic group of type $C_{\ell}$ , with Lie algebra $\mathfrak{g}_{sc}=\mathfrak{sp}(V)$ . In this section, we will make some initial observations about the structure of $\mathfrak{g}_{sc}$ and $\mathfrak{g}_{ad}$ as $G_{sc}$ -modules.

Lemma 8.1.

Let $G_{sc}=\operatorname{Sp}(V)$ with Lie algebra $\mathfrak{g}_{sc}=\mathfrak{sp}(V)$ . Then $\mathfrak{g}_{sc}\cong S^{2}(V)^{*}$ as $G_{sc}$ -modules.

Proof.

A proof is given in [3, 5.4], alternatively this follows from the isomorphism $K\otimes_{\mathbb{Z}}L_{sc}\cong\mathfrak{g}_{sc}$ given in Section 7.4.∎

Lemma 8.2.

We have $\mathfrak{g}_{sc}/Z(\mathfrak{g}_{sc})\cong[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ as $G_{sc}$ -modules.

Proof.

Let $G_{ad}$ be a group of adjoint type $C_{\ell}$ , and let $\psi:G_{sc}\rightarrow G_{ad}$ be an isogeny as in Lemma 7.1. Then the map $\mathrm{d}\psi:\mathfrak{g}_{sc}\rightarrow\mathfrak{g}_{ad}$ is a morphism of $G_{sc}$ -modules, with $\operatorname{Ker}\mathrm{d}\psi=Z(\mathfrak{g}_{sc})$ [9, Lemma 2.2]. The image of $\mathrm{d}\psi$ contains all root elements of $\mathfrak{g}_{ad}$ , so by [9, Table 1] it is equal to $[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ . Therefore $\mathfrak{g}_{sc}/\operatorname{Ker}\mathrm{d}\psi=\mathfrak{g}_{sc}/Z(\mathfrak{g}_{sc})\cong[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ as $G_{sc}$ -modules.∎

Let $\varphi$ the linear map $\varphi:S^{2}(V)\rightarrow K$ defined by $\varphi(xy)=b(x,y)$ for all $x,y\in V$ . Since $b$ is $G_{sc}$ -invariant, it follows that $\varphi$ is a surjective morphism of $G_{sc}$ -modules.

Lemma 8.3.

Let $G_{sc}=\operatorname{Sp}(V)$ with Lie algebra $\mathfrak{g}_{sc}=\mathfrak{sp}(V)$ . Then $\operatorname{Ker}\varphi\cong\left(\mathfrak{g}_{sc}/Z(\mathfrak{g}_{sc})\right)^{*}$ as $G_{sc}$ -modules.

Proof.

It is clear that $\varphi$ is surjective, so we have a short exact sequence

0\rightarrow\operatorname{Ker}\varphi\rightarrow S^{2}(V)\rightarrow K\rightarrow 0

of $G_{sc}$ -modules. This induces a short exact sequence

0\rightarrow K\rightarrow S^{2}(V)^{*}\rightarrow\left(\operatorname{Ker}\varphi\right)^{*}\rightarrow 0

of $G_{sc}$ -modules. Now $S^{2}(V)^{*}\cong\mathfrak{g}_{sc}$ as $G_{sc}$ -modules (Lemma 8.1), and $\mathfrak{g}_{sc}$ has a unique trivial submodule since $Z(\mathfrak{g}_{sc})\cong K$ [9, Table 1]. Thus we conclude that $\left(\operatorname{Ker}\varphi\right)^{*}\cong\mathfrak{g}_{sc}/Z(\mathfrak{g}_{sc})$ as $G_{sc}$ -modules, from which the lemma follows. ∎

Lemma 8.4.

Let $G=\operatorname{Sp}(V)$ be simply connected and simple of type $C_{\ell}$ . Then:

(i)

There exists a uniserial $G$ -module $W$ with $W=L_{G}(0)|L_{G}(2\varpi_{1})|L_{G}(0)$ .
(ii)

A $G$ -module $W$ as in (i) is unique up to isomorphism.

(iii)

Let $u\in G$ be a unipotent element. Then

\dim W^{u}=\begin{cases}\dim V^{u}+1,&\text{ if }\dim V^{u}\text{ is odd.}\\ \dim V^{u}+2,&\text{ if }\dim V^{u}\text{ is even.}\end{cases}

Proof.

Let $G^{\prime}=\operatorname{SO}(V^{\prime})$ with $\dim V^{\prime}=2\ell+1$ , so $G^{\prime}$ is a simple algebraic group of adjoint type $B_{\ell}$ . Let $\tau:G\rightarrow G^{\prime}$ be an exceptional isogeny as in [32, Theorem 28]. We can embed $G^{\prime}$ into a simple algebraic group $\operatorname{SO}(W)$ of type $D_{\ell+1}$ as the stabilizer of a nonsingular vector, see for example [23, Section 6.8]. Here $\dim W=2\ell+2$ , and as in [23, Section 6.8], we identify $V^{\prime}=\langle v\rangle^{\perp}\subset W$ , where $v\in W$ is a nonsingular vector.

It is straightforward to see that $W$ is a uniserial $G^{\prime}$ -module with

W=L_{G^{\prime}}(0)|L_{G^{\prime}}(\varpi_{1}^{\prime})|L_{G^{\prime}}(0),

where $\varpi_{1}^{\prime}$ is the first fundamental highest weight for $G^{\prime}$ . Then the twist of $W$ by $\tau$ is a uniserial $G$ -module $W^{\tau}$ as in (i). For a unipotent element $u\in G$ , the fixed point space of $u$ on the Frobenius twist $L_{G}(2\varpi_{1})\cong L_{G}(\varpi_{1})^{[1]}$ is the same as on $L_{G}(\varpi_{1})\cong V$ , because the Frobenius endomorphism preserves unipotent conjugacy classes. Therefore (iii) holds for $W$ by [15, Lemma 3.8].

It remains to check that $W$ is unique. To this end, note first that

\operatorname{Ext}_{G}^{1}(K,L_{G}(2\varpi_{1}))\cong\operatorname{Hom}_{G}(\wedge^{2}(V)^{*},K)

[14, II.2.14] and [3, 5.4]. Here $\operatorname{Hom}_{G}(\wedge^{2}(V)^{*},K)\cong\wedge^{2}(V)^{G}\cong K$ , since $V$ has a unique $G$ -invariant alternating bilinear form up to a scalar. Thus there exists a unique nonsplit extension

0\rightarrow K\rightarrow Z\rightarrow L_{G}(2\varpi_{1})\rightarrow 0,

up to isomorphism of $G$ -modules.

Since $\operatorname{Ext}_{G}^{1}(K,K)=\operatorname{Ext}_{G}^{2}(K,K)=0$ [14, II.4.11] and $\operatorname{Ext}_{G}^{1}(K,L_{G}(2\varpi_{1}))\cong K$ , we have $\operatorname{Ext}_{G}^{1}(K,Z)\cong K$ . Hence there exists a unique nonsplit extension

0\rightarrow Z\rightarrow W\rightarrow K\rightarrow 0,

up to isomorphism of $G$ -modules. Every $W$ as in (i) is such an extension, so we conclude that $W$ is unique up to isomorphism.∎

Lemma 8.5.

Let $G_{sc}=\operatorname{Sp}(V)$ , so $G_{sc}$ is simply connected and simple of type $C_{\ell}$ . Assume that $\ell$ is even. Then there is a short exact sequence

0\rightarrow L_{G}(\varpi_{2})\rightarrow\mathfrak{g}_{ad}\rightarrow W\rightarrow 0

of $G_{sc}$ -modules, where $W$ is as in Lemma 8.4 (i).

Proof.

As observed in [3, 5.6], in this case as a $G_{sc}$ -module $\mathfrak{g}_{ad}$ is uniserial with $\mathfrak{g}_{ad}=L_{G}(\varpi_{2})|L_{G}(0)|L_{G}(2\varpi_{1})|L_{G}(0)$ . Thus the result follows from Lemma 8.4 (ii). ∎

Lemma 8.6.

Let $G=\operatorname{Sp}(V)$ be simply connected and simple of type $C_{\ell}$ . Assume that $\ell\equiv 2\mod{4}$ and let $u\in G$ be unipotent with $V\downarrow K[u]=V(2\ell)$ . Then $\dim\mathfrak{g}_{ad}^{u}\leq\dim\mathfrak{g}_{sc}^{u}$ .

Proof.

It follows from Lemma 4.4 that $\dim\wedge^{2}(V)^{u}=\ell$ . Because $\ell\equiv 2\mod{4}$ , the smallest Jordan block size of $u$ on $\wedge^{2}(V)$ is equal to $2$ [17, Lemma 4.12], and by [17, Theorem B] we have $\dim L_{G}(\varpi_{2})^{u}=\ell-1$ .

For $W$ as in Lemma 8.5 (i), we have $\dim W^{u}=2$ by Lemma 8.5 (iii). Thus it follows from Lemma 8.5 that $\dim\mathfrak{g}_{ad}^{u}\leq\dim L_{G}(\varpi_{2})^{u}+\dim W^{u}=\ell+1$ . By Lemma 8.1 and Lemma 4.4 (ii) we have $\dim\mathfrak{g}_{sc}^{u}=\ell+1$ , so the result follows.∎

Remark 8.7.

As a corollary of our results, we will see later that equality $\dim\mathfrak{g}_{ad}^{u}=\dim\mathfrak{g}_{sc}^{u}$ holds in Lemma 8.6, without assumptions on $\ell$ (Corollary 1.8).

9. Jordan block sizes of unipotent elements on $\mathfrak{g}_{sc}/Z(\mathfrak{g}_{sc})$

Let $G_{sc}=\operatorname{Sp}(V)$ be simply connected of type $C_{\ell}$ , and let $b$ be the $G_{sc}$ -invariant alternating bilinear form defining $G_{sc}$ . We have $\mathfrak{g}_{sc}\cong S^{2}(V)^{*}$ by Lemma 8.1, so for every unipotent element $u\in G$ the Jordan block sizes of $u$ on $\mathfrak{g}_{sc}$ are known by the results described in Section 4. In this section, we will describe the Jordan block sizes of unipotent elements $u\in G$ on $\mathfrak{g}_{sc}/Z(\mathfrak{g}_{sc})$ , in terms of Jordan block sizes of $u$ on $\mathfrak{g}_{sc}$ .

Throughout this section, we will denote by $\varphi$ the linear map $\varphi:S^{2}(V)\rightarrow K$ defined by $\varphi(xy)=b(x,y)$ for all $x,y\in V$ . Since $b$ is $G_{sc}$ -invariant, it follows that $\varphi$ is a surjective morphism of $G_{sc}$ -modules. By Lemma 8.3, the Jordan normal form of a unipotent element $u\in G$ on $\mathfrak{g}_{sc}/Z(\mathfrak{g}_{sc})$ is the same as on $\operatorname{Ker}\varphi$ .

We will then describe the Jordan block sizes of unipotent elements $u\in G$ on $\operatorname{Ker}\varphi$ . By Lemma 3.1, this amounts to finding the largest integer $m\geq 0$ such that $\operatorname{Ker}(\widetilde{u}-1)^{m}\subseteq\operatorname{Ker}\varphi$ , where $\widetilde{u}$ is the action of $u$ on $S^{2}(V)$ .

Lemma 9.1.

Let $u\in\operatorname{Sp}(V)$ be unipotent. Suppose that we have a orthogonal decomposition $V=U_{1}\perp\cdots\perp U_{t}$ as $K[u]$ -modules. Denote the action of $u$ on $S^{2}(V)$ by $\widetilde{u}$ , and the action of $u$ on $S^{2}(U_{i})$ by $\widetilde{u_{i}}$ . Let $m\geq 0$ be an integer.

Then $\operatorname{Ker}(\widetilde{u}-1)^{m}\subseteq\operatorname{Ker}\varphi$ if and only if $\operatorname{Ker}(\widetilde{u_{i}}-1)^{m}\subseteq\operatorname{Ker}\varphi$ for all $1\leq i\leq t$ .

Proof.

The orthogonal decomposition $V=U_{1}\perp\cdots\perp U_{t}$ into $K[u]$ -modules induces a decomposition

S^{2}(V)=\bigoplus_{1\leq i\leq t}S^{2}(U_{i})\perp\bigoplus_{1\leq i<j\leq t}U_{i}U_{j}

into $K[u]$ -modules, where $U_{i}U_{j}=\{xy:x\in U_{i},y\in U_{j}\}$ . We have $U_{i}U_{j}\subseteq\operatorname{Ker}\varphi$ for all $1\leq i<j\leq t$ , from which the lemma follows.∎

With Lemma 9.1, we reduce to the case where $V\downarrow K[u]$ is orthogonally indecomposable. We first consider the case where $V\downarrow K[u]=V(2\ell)$ , so $n=2\ell$ . Let $v_{1}$ , $\ldots$ , $v_{n}$ be a basis as in the definition of $V(2\ell)$ (Definition 5.1). Then $b(v_{i},v_{j})=1$ if $i+j=n+1$ , and $0$ otherwise. Furthermore, the action of $u$ is defined by

	$\displaystyle uv_{1}$	$\displaystyle=v_{1},$
	$\displaystyle uv_{i}$	$\displaystyle=v_{i}+v_{i-1}+\cdots+v_{1}\text{ for all }2\leq i\leq\ell+1,$
	$\displaystyle uv_{i}$	$\displaystyle=v_{i}+v_{i-1}\text{ for all }\ell+1<i\leq n.$

We will denote $v_{j}=0$ for all $j\leq 0$ and $j>n$ .

Define $\gamma=\sum_{1\leq i\leq\ell}v_{i}v_{n+1-i}$ . We have a short exact sequence

0\rightarrow V^{[2]}\rightarrow S^{2}(V)\rightarrow\wedge^{2}(V)\rightarrow 0,

where $V^{[2]}$ is the subspace generated by $v^{2}$ for all $v\in V$ . As noted in [17, Section 9], the image of $\gamma$ in $\wedge^{2}(V)$ is fixed by the action of $\operatorname{Sp}(V)$ . Thus $u\cdot\gamma=\gamma+x$ for some $x\in V^{[2]}$ , and more precisely we have the following.

Lemma 9.2.

Let $u$ and $\gamma$ be as above. Then $u\cdot\gamma=\gamma+\sum_{1\leq i\leq\ell}v_{i}^{2}$ , and $\gamma+v_{\ell+1}^{2}$ is fixed by the action of $u$ .

Proof.

Denote $s_{i}:=\sum_{j\geq 0}v_{i-j}v_{n-i}$ . First we note that

	$\displaystyle u\cdot\sum_{1\leq i<\ell}v_{i}v_{n-i+1}$	$\displaystyle=\sum_{1\leq i<\ell}\left(\sum_{j\geq 0}v_{i-j}\right)\left(v_{n-i+1}+v_{n-i}\right)$
		$\displaystyle=\sum_{1\leq i<\ell}\left(s_{i}+s_{i-1}+v_{i}v_{n-i+1}\right)$
(9.1)			$\displaystyle=s_{\ell-1}+\sum_{1\leq i<\ell}v_{i}v_{n-i+1}.$

By another calculation, we get

(9.2)

u\cdot v_{\ell}v_{\ell+1}=\left(\sum_{1\leq j\leq\ell}v_{j}\right)\left(\sum_{1\leq j\leq\ell+1}v_{j}\right)=s_{\ell-1}+v_{\ell}v_{\ell+1}+\sum_{1\leq j\leq\ell}v_{j}^{2}.

Adding (9.1) and (9.2) together, we conclude that $u\cdot\gamma=\gamma+\sum_{1\leq i\leq\ell}v_{i}^{2}$ .

Since $u\cdot v_{\ell+1}^{2}=\left(\sum_{1\leq i\leq\ell+1}v_{i}\right)^{2}=\sum_{1\leq i\leq\ell+1}v_{i}^{2}$ , it follows that $\gamma+v_{\ell+1}^{2}$ is fixed by the action of $u$ . ∎

Lemma 9.3.

Let $u\in\operatorname{Sp}(V)$ be unipotent such that $V\downarrow K[u]=V(2\ell)$ . Let $\widetilde{u}$ be the action of $u$ on $S^{2}(V)$ , and denote $\alpha=\nu_{2}(\ell)$ . Then the following hold:

(i)

$\operatorname{Ker}(\widetilde{u}-1)^{2^{\alpha}-1}\subseteq\operatorname{Ker}\varphi$ .
(ii)

$\operatorname{Ker}(\widetilde{u}-1)^{2^{\alpha}}\not\subseteq\operatorname{Ker}\varphi$ .

Proof.

By Lemma 4.5 the smallest Jordan block size of $\widetilde{u}$ is $2^{\alpha}$ , so (i) follows from Lemma 3.1.

For (ii), we first consider the case where $\alpha=0$ , so $\ell$ is odd. In the notation used before the lemma, by Lemma 9.2 the vector $\gamma+v_{\ell+1}^{2}$ is fixed by the action of $u$ . Furthermore $\varphi(\gamma+v_{\ell+1}^{2})=\ell\neq 0$ , so $\operatorname{Ker}(\widetilde{u}-1)\not\subseteq\operatorname{Ker}\varphi$ , as claimed.

Suppose then that $\alpha>0$ . We have $V\downarrow K[u^{2^{\alpha}}]=V(\ell/2^{\alpha-1})^{2^{\alpha}}$ by [17, Lemma 6.13]. Combining this with Lemma 9.1 and the fact that $(\widetilde{u}-1)^{2^{\alpha}}=\widetilde{u}^{2^{\alpha}}-1$ , the claim follows from the $\alpha=0$ case.∎

Lemma 9.4.

Let $u\in\operatorname{Sp}(V)$ be unipotent such that $V\downarrow K[u]=W(\ell)$ . Let $\widetilde{u}$ be the action of $u$ on $S^{2}(V)$ , and denote $\alpha=\nu_{2}(\ell)$ . Then the following hold:

(i)

$\operatorname{Ker}(\widetilde{u}-1)^{2^{\alpha}-1}\subseteq\operatorname{Ker}\varphi$ .
(ii)

$\operatorname{Ker}(\widetilde{u}-1)^{2^{\alpha}}\not\subseteq\operatorname{Ker}\varphi$ .

Proof.

We have $V=W\oplus W^{*}$ as $K[u]$ -modules, where $W$ and $W^{*}$ are totally isotropic subspaces. This induces a decomposition

S^{2}(V)=S^{2}(W)\oplus S^{2}(W^{*})\oplus WW^{*}

of $K[u]$ -modules, where $WW^{*}\cong W\otimes W^{*}$ is the subspace $\{xy:x\in W,y\in W^{*}\}$ . We have $S^{2}(W),S^{2}(W^{*})\subseteq\operatorname{Ker}\varphi$ and the smallest Jordan block size in $W\otimes W^{*}$ is $2^{\alpha}$ [16, Lemma 4.2], so (i) follows from Lemma 3.1.

For (ii), we first consider the case where $\alpha=0$ , so $\ell$ is odd. Choose a basis $v_{1}$ , $\ldots$ , $v_{\ell}$ of $W$ , and let $w_{1}$ , $\ldots$ , $w_{\ell}$ be the corresponding dual basis in $W^{*}$ , so $b(v_{i},w_{j})=\delta_{i,j}$ for all $1\leq i,j\leq\ell$ . Then $\gamma^{\prime}=\sum_{1\leq i\leq\ell}v_{i}w_{i}$ is fixed by the action of $u$ , see for example [16, Lemma 3.7]. We have $\varphi(\gamma^{\prime})=\ell\neq 0$ , so $\operatorname{Ker}(\widetilde{u}-1)\not\subseteq\operatorname{Ker}\varphi$ , as claimed.

Suppose then that $\alpha>0$ . We have $(\widetilde{u}-1)^{2^{\alpha}}=\widetilde{u}^{2^{\alpha}}-1$ , and furthermore $V\downarrow K[u^{2^{\alpha}}]=W(\ell/2^{\alpha})^{2^{\alpha}}$ by [17, Lemma 6.12]. Thus as in the proof of Lemma 9.3, the claim follows from Lemma 9.1 and the $\alpha=0$ case.∎

Proposition 9.5.

Let $u\in\operatorname{Sp}(V)$ be unipotent, with orthogonal decomposition

V\downarrow K[u]=\sum_{1\leq i\leq t}W(m_{i})\perp\sum_{1\leq j\leq s}V(2k_{j}).

Denote by $\alpha\geq 0$ the largest integer such that $2^{\alpha}\mid m_{i},k_{j}$ for all $i$ and $j$ . Then

	$\displaystyle\mathfrak{g}_{sc}$	$\displaystyle\cong V_{2^{\alpha}}\oplus V^{\prime}$
	$\displaystyle\mathfrak{g}_{sc}/Z(\mathfrak{g}_{sc})$	$\displaystyle\cong V_{2^{\alpha}-1}\oplus V^{\prime}$

for some $K[u]$ -module $V^{\prime}$ .

Proof.

Let $\widetilde{u}$ be the action of $u$ on $S^{2}(V)$ . By Lemma 8.1, Lemma 8.3, and Lemma 3.1, the proposition is equivalent to the statement that $\operatorname{Ker}(\widetilde{u}-1)^{2^{\alpha}-1}\subseteq\operatorname{Ker}\varphi$ and $\operatorname{Ker}(\widetilde{u}-1)^{2^{\alpha}}\not\subseteq\operatorname{Ker}\varphi$ .

By Lemma 9.1, Lemma 9.3, and Lemma 9.4, we have $\operatorname{Ker}(\widetilde{u}-1)^{2^{\alpha}-1}\subseteq\operatorname{Ker}\varphi$ . Next note that either $\alpha=\nu_{2}(m_{i})$ for some $i$ , or $\alpha=\nu_{2}(k_{j})$ for some $j$ . It follows then from Lemma 9.1, together with Lemma 9.4 (if $\alpha=\nu_{2}(m_{i})$ ) or Lemma 9.3 (if $\alpha=\nu_{2}(k_{j})$ ) that $\operatorname{Ker}(\widetilde{u}-1)^{2^{\alpha}}\not\subseteq\operatorname{Ker}\varphi$ .∎

10. Jordan block sizes of nilpotent elements on $\mathfrak{g}_{sc}/Z(\mathfrak{g}_{sc})$

We continue with the setup of the previous section. Let $G_{sc}=\operatorname{Sp}(V)$ be simply connected of type $C_{\ell}$ with Lie algebra $\mathfrak{g}_{sc}=\mathfrak{sp}(V)$ . In this section, we will describe the Jordan block sizes of nilpotent elements $e\in\mathfrak{sp}(V)$ on $\mathfrak{g}_{sc}/Z(\mathfrak{g}_{sc})\cong(\operatorname{Ker}\varphi)^{*}$ (Lemma 8.3). As in the previous section, we describe the Jordan block sizes of $e$ on $\operatorname{Ker}\varphi$ in terms of Jordan block sizes of $e$ on $S^{2}(V)$ . Note that the Jordan normal form of $e$ on $S^{2}(V)$ is known by the results described in Section 4.

We begin by reducing the calculation to the orthogonally indecomposable case. After this we consider the different orthogonally indecomposable $K[e]$ -modules in turn, and by combining the results we obtain Proposition 10.5, which is analogous to Proposition 9.5.

Lemma 10.1.

Let $e\in\mathfrak{sp}(V)$ be unipotent. Suppose that we have a orthogonal decomposition $V=U_{1}\perp\cdots\perp U_{t}$ as $K[e]$ -modules. Denote the action of $e$ on $S^{2}(V)$ by $\widetilde{e}$ , and the action of $e$ on $S^{2}(U_{i})$ by $\widetilde{e_{i}}$ . Let $m\geq 0$ be an integer.

Then $\operatorname{Ker}(\widetilde{e})^{m}\subseteq\operatorname{Ker}\varphi$ if and only if $\operatorname{Ker}(\widetilde{e_{i}})^{m}\subseteq\operatorname{Ker}\varphi$ for all $1\leq i\leq t$ .

Proof.

Follows with the same proof as Lemma 9.1.∎

Lemma 10.2.

Let $e\in\mathfrak{sp}(V)$ be nilpotent such that $V\downarrow K[e]=W(\ell)$ . Let $\widetilde{e}$ be the action of $e$ on $S^{2}(V)$ , and denote $\alpha=\nu_{2}(\ell)$ . Then the following hold:

(i)

$\operatorname{Ker}(\widetilde{e})^{2^{\alpha}-1}\subseteq\operatorname{Ker}\varphi$ .
(ii)

$\operatorname{Ker}(\widetilde{e})^{2^{\alpha}}\not\subseteq\operatorname{Ker}\varphi$ .

Proof.

We proceed similarly to the proof of Lemma 9.4. By definition of $W(\ell)$ , we have a totally singular decomposition $V=W\oplus W^{*}$ , where $W$ and $W^{*}$ are $K[e]$ -modules on which $e$ acts with a single Jordan block of size $\ell$ . This induces a decomposition

S^{2}(V)=S^{2}(W)\oplus S^{2}(W^{*})\oplus WW^{*}

of $K[e]$ -modules, where $WW^{*}\cong W\otimes W^{*}$ . Thus

\operatorname{Ker}(\widetilde{e})=\operatorname{Ker}(\widetilde{e}_{S^{2}(W)})\oplus\operatorname{Ker}(\widetilde{e}_{S^{2}(W)})\oplus\operatorname{Ker}(\widetilde{e}_{WW^{*}}).

We have $S^{2}(W),S^{2}(W^{*})\subseteq\operatorname{Ker}\varphi$ since $W$ and $W^{*}$ are totally singular. Furthermore, the smallest Jordan block size in $W\otimes W^{*}$ is $2^{\alpha}$ by Proposition 4.6 and [16, Lemma 4.2], so (i) follows from Lemma 3.1.

Next we consider (ii). We have $V\downarrow K[e^{2^{\alpha}}]=W(\ell/2^{\alpha})^{2^{\alpha}}$ by Lemma 6.9. Therefore by Lemma 10.1, it suffices to consider the case where $\alpha=0$ , in which case $\ell$ is odd. In this case, choose a basis $v_{1}$ , $\ldots$ , $v_{\ell}$ of $W$ , and let $w_{1}$ , $\ldots$ , $w_{\ell}$ be the corresponding dual basis in $W^{*}$ , so $b(v_{i},w_{j})=\delta_{i,j}$ for all $1\leq i,j\leq\ell$ . Then $\gamma=\sum_{1\leq i\leq\ell}v_{i}w_{i}$ is annihilated by $e$ and $\varphi(\gamma)=\ell\neq 0$ , so $\operatorname{Ker}(\widetilde{e})\not\subseteq\operatorname{Ker}\varphi$ , as required.∎

Lemma 10.3.

Let $e\in\mathfrak{sp}(V)$ be nilpotent such that $V\downarrow K[e]=W_{k}(\ell)$ , where $0<k<\ell/2$ . Let $\widetilde{e}$ be the action of $e$ on $S^{2}(V)$ , and denote $\alpha=\nu_{2}(\ell)$ . Then the following hold:

(i)

$\operatorname{Ker}(\widetilde{e})^{2^{\alpha}-1}\subseteq\operatorname{Ker}\varphi$ .
(ii)

$\operatorname{Ker}(\widetilde{e})^{2^{\alpha}}\not\subseteq\operatorname{Ker}\varphi$ .

Proof.

Let $v_{1}$ , $\ldots$ , $v_{n}$ be the basis used in the definition of $W_{k}(\ell)$ . Then $b(v_{i},v_{j})=1$ if $i+j=n+1$ and $0$ otherwise. Furthermore, the action of $e$ is defined by

	$\displaystyle ev_{1}$	$\displaystyle=0,$
	$\displaystyle ev_{\ell+1}$	$\displaystyle=0$
	$\displaystyle ev_{i}$	$\displaystyle=v_{i-1}\text{ for all }i\not\in\{1,\ell+1,n-k+1\}$
	$\displaystyle ev_{n-k+1}$	$\displaystyle=v_{n-k}+v_{k}.$

We will denote $v_{j}=0$ for all $j\leq 0$ and $j>n$ .

Let $W=\langle v_{i}:1\leq i\leq\ell\rangle$ and $Z=\langle e^{i}v_{n}:0\leq i<\ell\rangle$ . Then $V=W\oplus Z$ , where $W$ and $Z$ are $K[e]$ -modules on which $e$ acts as a single Jordan block of size $\ell$ . We take $w_{1}$ , $\ldots$ , $w_{\ell}$ as a basis of $Z$ , where $w_{i}:=e^{i-1}v_{n}$ for all $1\leq i\leq\ell$ .

For the claims, we will first consider the case where $\alpha=0$ , so $\ell$ is odd. In this case (i) is trivial. For (ii), note that the vector $\gamma=\sum_{1\leq j\leq\ell}v_{j}w_{j}$ is annihilated by the action of $e$ . For all $1\leq j\leq\ell$ , we have $w_{j}=v_{n-j+1}$ or $w_{j}=v_{n-j+1}+v_{r}$ for some $1\leq r\leq k$ . Therefore $\varphi(\gamma)=\ell\neq 0$ , so $\operatorname{Ker}(\widetilde{e})\not\subseteq\operatorname{Ker}\varphi$ , as claimed.

Suppose then that $\alpha>0$ . For (i), we will first prove that $\operatorname{Ker}(\widetilde{e})\subseteq\operatorname{Ker}\varphi$ . To this end, note that the decomposition $V=W\oplus Z$ induces a decomposition

S^{2}(V)=S^{2}(W)\oplus S^{2}(Z)\oplus WZ

of $K[e]$ -modules, where $WZ=\{wz:w\in W,z\in Z\}$ is isomorphic to $W\otimes Z$ . It follows from [18, Proof of Theorem 1.6] that $S^{2}(W)^{e}=W^{[2]}$ and $S^{2}(Z)^{e}=Z^{[2]}$ , so

\operatorname{Ker}(\widetilde{e})=W^{[2]}\oplus Z^{[2]}\oplus\operatorname{Ker}(\widetilde{e}_{WZ}).

It is clear that $W^{[2]},Z^{[2]}\subseteq\operatorname{Ker}\varphi$ , and $\operatorname{Ker}(\widetilde{e}_{WZ})\subseteq\operatorname{Ker}\varphi$ since the smallest Jordan block size in $W\otimes Z$ is $2^{\alpha}$ (Proposition 4.6 and [16, Lemma 4.2]).

Thus $\operatorname{Ker}(\widetilde{e})\subseteq\operatorname{Ker}\varphi$ . This also proves (i) in the case where $\alpha=1$ , so suppose next that $\alpha>1$ . We have $V\downarrow K[e^{2^{\alpha-1}}]=W(\ell/2^{\alpha-1})^{2^{\alpha-1}}$ by Lemma 6.9. Since $\ell/2^{\alpha-1}$ is even, it follows from Lemma 10.2 and Lemma 10.1 that

(10.1)

\operatorname{Ker}(\widetilde{e})^{2^{\alpha-1}}\subseteq\operatorname{Ker}\varphi.

In $S^{2}(W)$ , $S^{2}(Z)$ , and $W\otimes Z$ all Jordan block sizes of $e$ are powers of two (Theorem 4.7). In particular $\widetilde{e}$ has no Jordan blocks of size $2^{\alpha-1}<d<2^{\alpha}$ , so by (10.1) and Lemma 3.1 we conclude that $\operatorname{Ker}(\widetilde{e})^{2^{\alpha}-1}\subseteq\operatorname{Ker}\varphi$ , as required.

It remains to prove (ii) in the case where $\alpha>0$ . To this end, note first that we have $V\downarrow K[e^{2^{\alpha}}]=W(\ell/2^{\alpha})^{2^{\alpha}}$ by Lemma 6.9. It follows then from Lemma 10.2 and Lemma 10.1 that $\operatorname{Ker}(\widetilde{e})^{2^{\alpha}}\not\subseteq\operatorname{Ker}\varphi$ .∎

Lemma 10.4.

Let $e\in\mathfrak{sp}(V)$ be nilpotent such that $V\downarrow K[e]=V(2\ell)$ . Let $\widetilde{e}$ be the action of $e$ on $S^{2}(V)$ , and denote $\alpha=\nu_{2}(2\ell)$ . Then the following hold:

(i)

$\operatorname{Ker}(\widetilde{e})^{2^{\alpha}-1}\subseteq\operatorname{Ker}\varphi$ .
(ii)

$\operatorname{Ker}(\widetilde{e})^{2^{\alpha}}\not\subseteq\operatorname{Ker}\varphi$ .

Proof.

We have $\operatorname{Ker}(\widetilde{e})=V^{[2]}$ by [18, Proof of Theorem 1.6], so $\operatorname{Ker}(\widetilde{e})\subseteq\operatorname{Ker}\varphi$ . Since $e$ has no Jordan blocks of size $1<d<2^{\alpha}$ on $S^{2}(V)$ (Theorem 4.7), it follows from Lemma 3.1 that $\operatorname{Ker}(\widetilde{e})^{2^{\alpha}-1}\subseteq\operatorname{Ker}\varphi$ .

Next we consider claim (ii). Since $\alpha=\nu_{2}(2\ell)>0$ , we have $V\downarrow K[e^{2^{\alpha}}]=W(\ell/2^{\alpha-1})^{2^{\alpha-1}}$ by Lemma 6.9. Because $\ell/2^{\alpha-1}$ is odd, we conclude from Lemma 10.2 and Lemma 10.1 that $\operatorname{Ker}(\widetilde{e})^{2^{\alpha}}\not\subseteq\operatorname{Ker}\varphi$ .∎

Proposition 10.5.

Let $e\in\mathfrak{sp}(V)$ be nilpotent, with orthogonal decomposition

V\downarrow K[e]=\sum_{1\leq i\leq t}W(m_{i})\perp\sum_{1\leq j\leq t^{\prime}}W_{k_{j}}(\ell_{j})\perp\sum_{1\leq r\leq t^{\prime\prime}}V(2d_{r}).

Denote by $\alpha\geq 0$ the largest integer such that $2^{\alpha}\mid m_{i},\ell_{j},2d_{r}$ for all $i$ , $j$ , and $r$ .

Then

	$\displaystyle\mathfrak{g}_{sc}$	$\displaystyle\cong W_{2^{\alpha}}\oplus V^{\prime}$
	$\displaystyle\mathfrak{g}_{sc}/Z(\mathfrak{g}_{sc})$	$\displaystyle\cong W_{2^{\alpha}-1}\oplus V^{\prime}$

for some $K[e]$ -module $V^{\prime}$ .

Proof.

Let $\widetilde{e}$ be the action of $e$ on $S^{2}(V)$ . By Lemma 8.1, Lemma 8.3, and Lemma 3.1, the proposition is equivalent to the statement that $\operatorname{Ker}(\widetilde{e})^{2^{\alpha}-1}\subseteq\operatorname{Ker}\varphi$ and $\operatorname{Ker}(\widetilde{e})^{2^{\alpha}}\not\subseteq\operatorname{Ker}\varphi$ .

Thus similarly to the proof of Proposition 9.5, the result follows from Lemma 10.1, together with Lemma 10.2, Lemma 10.3, and Lemma 10.4.∎

11. Jordan block sizes of unipotent elements on $\mathfrak{g}_{ad}$

Let $G_{sc}=\operatorname{Sp}(V)$ be simply connected and simple of type $C_{\ell}$ . In this section, we will describe the Jordan block sizes of unipotent elements $u\in G_{sc}$ acting on $\mathfrak{g}_{ad}$ . Our approach is to describe the Jordan block sizes of $u$ on $\mathfrak{g}_{ad}$ in terms of the Jordan block sizes on $[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ , using Lemma 3.1. Since the Jordan block sizes of $u$ on $[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]\cong\mathfrak{g}_{sc}/Z(\mathfrak{g}_{sc})$ (Lemma 8.2) are known by the results of Section 9, we then get an explicit description of the Jordan block sizes of $u$ on $\mathfrak{g}_{ad}$ .

Throughout this section we will consider $\mathfrak{g}_{ad}$ as constructed in Section 7.2 – 7.4, and we use the notation established there.

Lemma 11.1.

Let $u\in\operatorname{Sp}(V)$ be the reduction modulo $p$ of $u_{\mathbb{Z}}=x_{\mathbb{Z}}^{(\alpha_{1})}\cdots x_{\mathbb{Z}}^{(\alpha_{\ell-1})}$ as in Section 7.5, so that $V\downarrow K[u]=W(\ell)$ . Then the following hold:

(i)

$(u_{\mathbb{Z}}-1)\cdot\delta=0$ .
(ii)

There exists $v\in L_{sc}$ such that $(u_{\mathbb{Z}}-1)\cdot(\delta+v)\equiv 2\delta\mod{2L_{sc}}$ if and only if $\ell$ is even.

Proof.

For claim (i), consider $X_{\alpha_{i}}=E_{i,i+1}-E_{n-i,n-i+1}$ for $1\leq i\leq\ell-1$ from the Chevalley basis of $\mathfrak{sp}(V_{\mathbb{C}})$ . It is clear that $X_{\alpha_{i}}\cdot\delta=0$ , so $x_{\mathbb{Z}}^{(\alpha_{i})}\cdot\delta=\delta$ for all $1\leq i\leq\ell-1$ , from which (i) follows.

For (ii), we have $(u_{\mathbb{Z}}-1)\cdot(\delta+v)=(u_{\mathbb{Z}}-1)\cdot v$ for all $v\in L_{sc}$ by (i). By Lemma 7.4, there exists $v\in L_{sc}$ such that $(u_{\mathbb{Z}}-1)\cdot(v)\equiv 2\delta\mod{2L_{sc}}$ if and only if $\operatorname{Im}(\widehat{u}-1)\supseteq Z(\mathfrak{g}_{sc})$ , where $\widehat{u}$ is the action of $u$ on $\mathfrak{g}_{sc}$ . It follows from Proposition 9.5 and Lemma 3.2 that $\operatorname{Im}(\widehat{u}-1)\supseteq Z(\mathfrak{g}_{sc})$ if and only if $\ell$ is even, so we conclude that (ii) holds.∎

Lemma 11.2.

Assume that $\ell$ is odd. Let $u\in\operatorname{Sp}(V)$ be the reduction modulo $p$ of $u_{\mathbb{Z}}=x_{\mathbb{Z}}^{(\alpha_{1})}\cdots x_{\mathbb{Z}}^{(\alpha_{\ell})}$ as in Section 7.5, so that $V\downarrow K[u]=V(2\ell)$ . Then the following hold:

(i)

There exists $v\in L_{sc}$ such that $(u_{\mathbb{Z}}-1)\cdot(\delta+v)\equiv 2\delta\mod{2L_{sc}}$ .
(ii)

There does not exist $v\in L_{sc}$ such that $(u_{\mathbb{Z}}-1)\cdot(\delta+v)\equiv 0\mod{2L_{sc}}$ .

Proof.

For (i), first we calculate

$\displaystyle u_{\mathbb{Z}}\cdot\delta$	$\displaystyle=x_{\mathbb{Z}}^{(\alpha_{1})}\cdots x_{\mathbb{Z}}^{(\alpha_{\ell-1})}\cdot(\delta+\frac{1}{2}v_{\ell}^{2})$	$\displaystyle(\text{by }x_{\mathbb{Z}}^{(\alpha_{\ell})}\cdot\delta=\delta+\frac{1}{2}v_{\ell}^{2})$
	$\displaystyle=\delta+\frac{1}{2}\left(v_{1}+\cdots+v_{\ell}\right)^{2}$	$\displaystyle(\text{by Lemma \ref{lemma:deltaWLunip} (i)})$
	$\displaystyle=\delta+\sum_{1\leq i\leq\ell}\frac{1}{2}v_{i}^{2}+\sum_{1\leq i<j\leq\ell}v_{i}v_{j}.$

Furthermore

u_{\mathbb{Z}}\cdot\frac{1}{2}v_{\ell+1}^{2}=\frac{1}{2}\left(v_{1}+\cdots+v_{\ell+1}\right)^{2}=\sum_{1\leq i\leq\ell+1}\frac{1}{2}v_{i}^{2}+\sum_{1\leq i<j\leq\ell+1}v_{i}v_{j},

(11.1)

u_{\mathbb{Z}}\cdot\left(\delta+\frac{1}{2}v_{\ell+1}^{2}\right)\equiv\delta+\frac{1}{2}v_{\ell+1}^{2}+\sum_{1\leq i\leq\ell}v_{i}v_{\ell+1}\mod{2L_{sc}}.

Next define

\gamma:=\sum_{1\leq j\leq(\ell-1)/2}v_{2j}\left(v_{n-2j+1}+v_{n-2j+2}\right).

For $1\leq j\leq(\ell-1)/2$ , we denote $s_{j}:=\sum_{1\leq t\leq 2j}v_{t}v_{n-2j}$ . Then modulo $2L_{sc}$ , we have

	$\displaystyle u_{\mathbb{Z}}\cdot\gamma$	$\displaystyle\equiv\sum_{1\leq j\leq(\ell-1)/2}\left(\sum_{1\leq t\leq 2j}v_{t}\right)\left(v_{n-2j+2}+v_{n-2j}\right)$
		$\displaystyle\equiv\sum_{1\leq j\leq(\ell-1)/2}\left(s_{j}+s_{j-1}+v_{2j-1}v_{n-2j+2}+v_{2j}v_{n-2j+2}\right)$
		$\displaystyle\equiv s_{(\ell-1)/2}+\sum_{1\leq j\leq(\ell-1)/2}v_{2j-1}v_{n-2j+2}+\sum_{1\leq j\leq(\ell-1)/2}v_{2j}v_{n-2j+2}$
		$\displaystyle\equiv s_{(\ell-1)/2}+\sum_{1\leq j\leq(\ell-1)/2}v_{2j-1}v_{n-2j+2}+\left(\gamma+\sum_{1\leq j\leq(\ell-1)/2}v_{2j}v_{n-2j+1}\right)$
		$\displaystyle\equiv\gamma+s_{(\ell-1)/2}+\sum_{1\leq j\leq\ell-1}v_{j}v_{n-j+1}$
		$\displaystyle\equiv\gamma+s_{(\ell-1)/2}+\left(2\delta+v_{\ell}v_{\ell+1}\right)$
		$\displaystyle\equiv\gamma+2\delta+\sum_{1\leq j\leq\ell}v_{j}v_{\ell+1}.$

Combining this with (11.1), we get

u_{\mathbb{Z}}\cdot\left(\delta+\frac{1}{2}v_{\ell+1}^{2}+\gamma\right)\equiv\delta+\frac{1}{2}v_{\ell+1}^{2}+\gamma+2\delta\mod{2L_{sc}}.

We conclude then that (i) holds, with $v=\frac{1}{2}v_{\ell+1}^{2}+\gamma$ .

For claim (ii), suppose for the sake of contradiction that there exists $v\in L_{sc}$ such that $(u_{\mathbb{Z}}-1)\cdot(\delta+v)\equiv 0\mod{2L_{sc}}$ . Then it follows from (i) that there exists $w\in L_{sc}$ such that $(u_{\mathbb{Z}}-1)\cdot w\equiv 2\delta\mod{2L_{sc}}$ . By Lemma 7.4, we have $\operatorname{Im}(\widehat{u}-1)\supseteq Z(\mathfrak{g}_{sc})$ , where $\widehat{u}$ is the action of $u$ on $\mathfrak{g}_{sc}$ . On the other hand $\operatorname{Im}(\widehat{u}-1)\not\supseteq Z(\mathfrak{g}_{sc})$ by Proposition 9.5 and Lemma 3.2, so we have a contradiction.∎

Lemma 11.3.

Let $u\in\operatorname{Sp}(V)$ be unipotent such that $V\downarrow K[u]=\sum_{1\leq i\leq t}W(\ell_{i})$ , where $\ell_{i}\geq 1$ for all $1\leq i\leq t$ . Then $\operatorname{Ker}(\widetilde{u}-1)\not\subseteq[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ .

Proof.

Follows from Lemma 7.7 (i) and Lemma 11.1 (i). ∎

Lemma 11.4.

Let $u\in\operatorname{Sp}(V)$ be unipotent such that $V\downarrow K[u]=\sum_{1\leq i\leq t}W(m_{i})\perp\sum_{1\leq j\leq s}V(2k_{j})$ , where $k_{j}\geq 1$ is odd for all $1\leq j\leq s$ . Assume that $s>0$ .

Denote the action of $u$ on $\mathfrak{g}_{ad}$ by $\widetilde{u}$ . Then $\operatorname{Ker}(\widetilde{u}-1)\not\subseteq[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ if and only if $m_{i}$ is even for all $1\leq i\leq t$ .

Proof.

In view of Lemma 11.2 (ii) and Lemma 7.7 (i), we have $\operatorname{Ker}(\widetilde{u}-1)\not\subseteq[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ if and only if Lemma 7.7 (i)(b) holds. Then by Lemma 11.1 (ii), we conclude that $\operatorname{Ker}(\widetilde{u}-1)\not\subseteq[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ if and only if $m_{i}$ is even for all $1\leq i\leq t$ .∎

Lemma 11.5.

Let $u\in\operatorname{Sp}(V)$ be unipotent such that $V\downarrow K[u]=V(2\ell)$ . Let $\widetilde{u}$ be the action of $u$ on $\mathfrak{g}_{ad}$ , and denote $\alpha=\nu_{2}(\ell)$ . Then the following hold:

(i)

$\operatorname{Ker}(\widetilde{u}-1)^{2^{\alpha}-1}\subseteq[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ .
(ii)

$\operatorname{Ker}(\widetilde{u}-1)^{2^{\alpha}}\not\subseteq[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ .

Proof.

We begin with the proof of (i), which is trivially true when $\alpha=0$ , so suppose that $\alpha>0$ .

We first consider $\alpha=1$ , in which case $\ell\equiv 2\mod{4}$ . Then

$\displaystyle\dim\mathfrak{g}_{ad}^{u}$	$\displaystyle\leq\dim\mathfrak{g}_{sc}^{u}$	by Lemma 8.6
	$\displaystyle=\dim[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]^{u}$	by Lemma 8.2 and Proposition 9.5
	$\displaystyle\leq\dim\mathfrak{g}_{ad}^{u}.$

Thus we conclude that $\dim\mathfrak{g}_{ad}^{u}=\dim\mathfrak{g}_{sc}^{u}=\dim[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]^{u}.$ It follows then from Lemma 3.1 that $\operatorname{Ker}(\widetilde{u}-1)\subseteq[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ , as claimed by (i).

Suppose then that $\alpha>1$ . We have $V\downarrow K[u^{2^{\alpha-1}}]=V(\ell/2^{\alpha-2})^{2^{\alpha-1}}$ by [17, Lemma 6.13]. It follows then from Lemma 7.7 (ii) and the case $\alpha=1$ that

\operatorname{Ker}(\widetilde{u}-1)^{2^{\alpha-1}}=\operatorname{Ker}(\widetilde{u}^{2^{\alpha-1}}-1)\subseteq[\mathfrak{g}_{ad},\mathfrak{g}_{ad}].

In particular, we have

(11.2)

\operatorname{Ker}(\widetilde{u}-1)\subseteq[\mathfrak{g}_{ad},\mathfrak{g}_{ad}].

Next we note that it follows from Lemma 8.2, Proposition 9.5, and Lemma 4.5 that the smallest Jordan block size of $u$ on $[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ is equal to $2^{\alpha}-1$ . In particular, there are no Jordan blocks of size $1\leq d<2^{\alpha}-1$ for $u$ on $[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ , which combined with Lemma 3.1 and (11.2) implies $\operatorname{Ker}(\widetilde{u}-1)^{2^{\alpha}-1}\subseteq[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ .

For (ii), note that $V\downarrow K[u^{2^{\alpha}}]=V(\ell/2^{\alpha-1})^{2^{\alpha}}$ by [17, Lemma 6.13]. Thus it follows from Lemma 11.4 that $\operatorname{Ker}(\widetilde{u}-1)^{2^{\alpha}}=\operatorname{Ker}(\widetilde{u}^{2^{\alpha}}-1)\not\subseteq[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ .∎

Proposition 11.6.

Let $u\in\operatorname{Sp}(V)$ be unipotent, with orthogonal decomposition $V\downarrow K[u]=\sum_{1\leq i\leq t}W(m_{i})\perp\sum_{1\leq j\leq s}V(2k_{j})$ . Assume that $s>0$ , and let $\alpha=\max_{1\leq j\leq s}\nu_{2}(k_{j})$ . Let $\widetilde{u}$ be the action of $u$ on $\mathfrak{g}_{ad}$ . Then the following statements hold:

(i)

$\operatorname{Ker}(\widetilde{u}-1)^{2^{\alpha}-1}\subseteq[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ .
(ii)

$\operatorname{Ker}(\widetilde{u}-1)^{2^{\alpha}}\not\subseteq[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ if and only if $\alpha=\nu_{2}(k_{j})$ for all $1\leq j\leq s$ , and $\nu_{2}(m_{i})>\alpha$ for all $1\leq i\leq t$ .
(iii)

$\operatorname{Ker}(\widetilde{u}-1)^{2^{\alpha}+1}\not\subseteq[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ .

Proof.

Claim (i) follows from Lemma 7.7 and Lemma 11.5.

Next we will prove (ii). In the case where $\alpha=0$ we have $k_{j}$ odd for all $1\leq j\leq s$ . Then the claim of (ii) is that $\operatorname{Ker}(\widetilde{u}-1)\not\subseteq[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ if and only if $m_{i}$ is even for all $1\leq i\leq t$ , which is precisely Lemma 11.4.

Suppose then that $\alpha>0$ . Write $m_{i}=m_{i}^{\prime}2^{\alpha}+m_{i}^{\prime\prime}$ with $0\leq m_{i}^{\prime\prime}<2^{\alpha}$ and $k_{j}=k_{j}^{\prime}2^{\alpha-1}+k_{j}^{\prime\prime}$ with $0\leq k_{j}^{\prime\prime}<2^{\alpha-1}$ . Then for the restrictions of the summands in $V\downarrow K[u^{2^{\alpha}}]$ , we have

	$\displaystyle W(m_{i})\downarrow K[u^{2^{\alpha}}]$	$\displaystyle=W(m_{i}^{\prime})^{2^{\alpha}-m_{i}^{\prime\prime}}\perp W(m_{i}^{\prime}+1)^{m_{i}^{\prime\prime}}$
	$\displaystyle V(2k_{j})\downarrow K[u^{2^{\alpha}}]$	$\displaystyle=\begin{cases}V(k_{j}/2^{\alpha-1})^{2^{\alpha}},&\text{ if }\nu_{2}(k_{j})=\alpha.\\ W(k_{j}^{\prime})^{2^{\alpha-1}-k_{j}^{\prime\prime}}\perp W(k_{j}^{\prime}+1)^{k_{j}^{\prime\prime}},&\text{ if }\nu_{2}(k_{j})<\alpha.\end{cases}$

for all $1\leq i\leq t$ and $1\leq j\leq s$ , by [17, Lemma 3.1, Lemma 6.12, Lemma 6.13]. In the above we denote $W(0)=0$ .

We have $\alpha=\nu_{2}(k_{j})$ for some $j$ , so $V(k_{j}/2^{\alpha-1})$ appears as a summand of $V\downarrow K[u^{2^{\alpha}}]$ . Thus by Lemma 11.4, we have $\operatorname{Ker}(\widetilde{u}-1)^{2^{\alpha}}=\operatorname{Ker}(\widetilde{u}^{2^{\alpha}}-1)\not\subseteq[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ if and only if all the summands of the form $W(m)$ in $V\downarrow K[u^{2^{\alpha}}]$ have $m$ even.

Claim 1: Assume that $\nu_{2}(m_{i})\leq\alpha$ for some $i$ . Then $\operatorname{Ker}(\widetilde{u}-1)^{2^{\alpha}}\subseteq[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ .
It suffices to show that $W(m_{i})\downarrow K[u^{2^{\alpha}}]$ has a summand $W(m)$ with $m$ odd. If $0<m_{i}^{\prime\prime}<2^{\alpha}$ , then $W(m_{i})\downarrow K[u^{2^{\alpha}}]$ has $W(m_{i}^{\prime})$ and $W(m_{i}^{\prime}+1)$ as a summand. If $m_{i}^{\prime\prime}=0$ , then $W(m_{i})\downarrow K[u^{2^{\alpha}}]=W(m_{i}^{\prime})^{2^{\alpha}}$ with $m_{i}^{\prime}$ odd, since $\nu_{2}(m_{i})\leq\alpha$ .

Claim 2: Assume that $\nu_{2}(k_{j})<\alpha$ for some $j$ . Then $\operatorname{Ker}(\widetilde{u}-1)^{2^{\alpha}}\subseteq[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ .
It suffices to show that $V(2k_{j})\downarrow K[u^{2^{\alpha}}]$ has a summand $W(m)$ with $m$ odd. If $0<k_{j}^{\prime\prime}<2^{\alpha-1}$ , then $V(2k_{j})\downarrow K[u^{2^{\alpha}}]$ has $W(k_{j}^{\prime})$ and $W(k_{j}^{\prime}+1)$ as a summand. Suppose then that $k_{j}^{\prime\prime}=0$ . Then $V(2k_{j})\downarrow K[u^{2^{\alpha}}]=W(k_{j}^{\prime})^{2^{\alpha-1}}$ , with $k_{j}^{\prime}$ is odd since $\nu_{2}(k_{j})<\alpha$ .

The “only if” part of (ii) follows from Claim 1 and Claim 2. Conversely, if $\alpha=\nu_{2}(k_{j})$ for all $1\leq j\leq s$ and $\nu_{2}(m_{i})>\alpha$ for all $1\leq i\leq t$ , we have

V\downarrow K[u^{2^{\alpha}}]=\sum_{1\leq i\leq t}W(m_{i}/2^{\alpha})^{2^{\alpha}}\perp\sum_{1\leq j\leq s}V(k_{j}/2^{\alpha-1}).

Thus $\operatorname{Ker}(\widetilde{u}-1)^{2^{\alpha}}=\operatorname{Ker}(\widetilde{u}^{2^{\alpha}}-1)\not\subseteq[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ by Lemma 11.4.

For (iii), let $u_{\mathbb{Z}}=u_{\mathbb{Z},1}\cdots u_{\mathbb{Z},t+s}\in\mathscr{U}_{\mathbb{Z}}$ be as in Section 7.5, so that $u$ is the reduction modulo $p$ of the action of $u_{\mathbb{Z}}$ on $V_{\mathbb{Z}}$ . Denote $\delta_{1}$ , $\ldots$ , $\delta_{t+s}$ as in Section 7.5. It follows from Lemma 7.7 (i), Lemma 11.1, and Lemma 11.5 that there exist $w_{i}\in L_{sc}^{(i)}$ such that

	$\displaystyle(u_{\mathbb{Z},i}-1)^{2^{\alpha}}\cdot(\delta_{i}+w_{i})$	$\displaystyle\equiv 0\mod{2L_{sc}^{(i)}}$	$\displaystyle\text{ if }1\leq i\leq t,$
	$\displaystyle(u_{\mathbb{Z},i}-1)^{2^{\alpha}}\cdot(\delta_{i}+w_{i})$	$\displaystyle\equiv 2\delta_{i}\mod{2L_{sc}^{(i)}}$	$\displaystyle\text{ if }t+1\leq i\leq t+s.$

Then

(u_{\mathbb{Z}}-1)^{2^{\alpha}}\cdot(\delta+w_{1}+\cdots+w_{t+s})\equiv 2\delta_{t+1}+\cdots+2\delta_{t+s}\mod{2L_{sc}}.

On the other hand, we have $u_{\mathbb{Z},i}\cdot\delta_{i}=\delta_{i}+z_{i}$ for some $z_{i}\in L_{sc}^{(i)}$ , as seen in the beginning of the proof of Lemma 11.2. Thus $(u_{\mathbb{Z}}-1)\cdot 2\delta_{i}\equiv 0\mod{2L_{sc}^{(i)}}$ for all $t+1\leq i\leq t+s$ , and consequently

(u_{\mathbb{Z}}-1)^{2^{\alpha}+1}\cdot(\delta+w_{1}+\cdots+w_{t+s})\equiv 0\mod{2L_{sc}}.

It follows then from Lemma 7.7 that $\operatorname{Ker}(\widetilde{u}-1)^{2^{\alpha}+1}\not\subseteq[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ , as claimed by (iii).∎

Proposition 11.7.

Let $u\in\operatorname{Sp}(V)$ be unipotent, with orthogonal decomposition $V\downarrow K[u]=\sum_{1\leq i\leq t}W(m_{i})\perp\sum_{1\leq j\leq s}V(2k_{j})$ . Then the following statements hold:

(i)

Suppose that $s=0$ . Then

$\mathfrak{g}_{ad}\cong V_{1}\oplus[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$

as $K[u]$ -modules.

(ii)

Suppose that $s>0$ , and let $\alpha=\max_{1\leq j\leq s}\nu_{2}(k_{j})$ . Then:

(a)

If $\alpha=\nu_{2}(k_{j})$ for all $1\leq j\leq s$ and $\nu_{2}(m_{i})>\alpha$ for all $1\leq i\leq t$ , then

	$\displaystyle\mathfrak{g}_{ad}$	$\displaystyle\cong V_{2^{\alpha}}\oplus V^{\prime}$
	$\displaystyle[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$	$\displaystyle\cong V_{2^{\alpha}-1}\oplus V^{\prime}$

for some $K[u]$ -module $V^{\prime}$ .

(b)

If $\alpha>\nu_{2}(k_{j})$ for some $1\leq j\leq s$ , or $\nu_{2}(m_{i})\leq\alpha$ for some $1\leq i\leq t$ , then

	$\displaystyle\mathfrak{g}_{ad}$	$\displaystyle\cong V_{2^{\alpha}+1}\oplus V^{\prime}$
	$\displaystyle[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$	$\displaystyle\cong V_{2^{\alpha}}\oplus V^{\prime}$

for some $K[u]$ -module $V^{\prime}$ .

Proof.

Claim (i) follows from Lemma 11.3 and Lemma 3.1. Similarly (ii) follows from Proposition 11.6 and Lemma 3.1. ∎

12. Jordan block sizes of nilpotent elements on $\mathfrak{g}_{ad}$

Continuing with the setup of the previous section, in this section we describe the Jordan block sizes of nilpotent $e\in\mathfrak{g}_{sc}$ on $\mathfrak{g}_{ad}$ , in terms of the Jordan block sizes on $[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ . The basic approach is similar to the previous section, but the proofs will be more simple due to the fact that $V\downarrow K[e^{2}]$ is always has an orthogonal decomposition of the form $\sum_{1\leq i\leq t}W(m_{i})$ (Lemma 6.9).

Lemma 12.1.

Let $e\in\mathfrak{g}_{sc}$ be nilpotent such that $V\downarrow K[e]=\sum_{1\leq i\leq t}W(m_{i})$ , and let $\widetilde{e}$ be the action of $e$ on $\mathfrak{g}_{ad}$ . Then $\operatorname{Ker}\widetilde{e}\not\subseteq[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ .

Proof.

Let $e_{\mathbb{Z}}=e_{\mathbb{Z},1}+\cdots+e_{\mathbb{Z},t}\in\mathfrak{g}_{\mathbb{Z}}$ be as in Section 7.7, so that $e$ is the reduction modulo $p$ of $e_{\mathbb{Z}}$ . Then

e_{\mathbb{Z},i}=X_{\alpha_{t_{i}+1}}+\cdots+X_{\alpha_{t_{i}+m_{i}-1}},

where $t_{1}=0$ and $t_{i}=m_{1}+\cdots+m_{i-1}$ for all $1<i\leq t$ .

As noted earlier (proof of Lemma 11.1), we have $X_{\alpha_{i}}\cdot\delta=0$ for all $1\leq i<\ell$ . Therefore $e_{\mathbb{Z}}\cdot\delta=0$ , from which it follows that $\operatorname{Ker}\widetilde{e}\not\subseteq[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ (Lemma 7.5). ∎

Lemma 12.2.

Let $e\in\mathfrak{g}_{sc}$ be nilpotent with $V\downarrow K[e]=V(2\ell)$ or $V\downarrow K[e]=W_{k}(\ell)$ for some $0<k<\ell/2$ . Let $\widetilde{e}$ be the action of $e$ on $\mathfrak{g}_{ad}$ . Then $\operatorname{Ker}\widetilde{e}\subseteq[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ .

Proof.

Let $e_{\mathbb{Z}}\in\mathfrak{g}_{\mathbb{Z}}$ be as in Section 7.7, so that $e$ is the reduction modulo $p$ of $e_{\mathbb{Z}}$ . Then

e_{\mathbb{Z}}=X_{\alpha_{1}}+\cdots+X_{\alpha_{\ell-1}}+X_{2\varepsilon_{r}},

where $r=\ell$ if $V\downarrow K[e]=V(2\ell)$ , and $r=k$ if $V\downarrow K[e]=W_{k}(\ell)$ .

Suppose that $\operatorname{Ker}\widetilde{e}\not\subseteq[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ . Then it follows from Lemma 7.5 that there exists $v\in L_{sc}$ such that $e_{\mathbb{Z}}\cdot(\delta+v)\in 2L_{ad}$ . We have $e_{\mathbb{Z}}\cdot\delta=\frac{1}{2}v_{r}^{2},$ so

(12.1)

\frac{1}{2}v_{r}^{2}=-e_{\mathbb{Z}}\cdot v+2w

for some $w\in L_{ad}$ .

Note that

e_{\mathbb{Z}}\cdot\frac{1}{2}v_{i}^{2}=\pm v_{i}v_{i-1}

for all $1\leq i\leq n$ . Therefore $e_{\mathbb{Z}}L_{sc}\subseteq S^{2}(V_{\mathbb{Z}})$ . Since also $2L_{ad}\subseteq S^{2}(V_{\mathbb{Z}})$ , it follows from (12.1) that $\frac{1}{2}v_{r}^{2}\in S^{2}(V_{\mathbb{Z}})$ , which is impossible. We have a contradiction, so we conclude that $\operatorname{Ker}\widetilde{e}\subseteq[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ .∎

Lemma 12.3.

Let $e\in\mathfrak{g}_{sc}$ be nilpotent, and let $\widetilde{e}$ be the action of $e$ on $\mathfrak{g}_{ad}$ . Then $\operatorname{Ker}\widetilde{e}\subseteq[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ , except possibly when $V\downarrow K[e]=\sum_{1\leq i\leq t}W(\ell_{i})$ for some $\ell_{1}$ , $\ldots$ , $\ell_{t}$ .

Proof.

Write $V\downarrow K[e]=U_{1}\perp\cdots\perp U_{t}$ , where $U_{i}$ is orthogonally indecomposable for all $1\leq i\leq t$ , and $\dim U_{i}=2\ell_{i}$ with $\ell_{i}>0$ . Let $e_{\mathbb{Z}}=e_{\mathbb{Z},1}+\cdots+e_{\mathbb{Z},t}\in\mathfrak{g}_{\mathbb{Z}}$ be as in Section 7.7, so that $e$ is the reduction modulo $p$ of $e_{\mathbb{Z}}$ .

The result follows from Lemma 7.6 (ii) (with $X_{i}=e_{\mathbb{Z},i}$ ) and Lemma 12.2.∎

Lemma 12.4.

Let $e\in\mathfrak{g}_{sc}$ be nilpotent, and let $\widetilde{e}$ be the action of $e$ on $\mathfrak{g}_{ad}$ . Then $\operatorname{Ker}(\widetilde{e})^{2}\not\subseteq[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$ .

Proof.

We have $V\downarrow K[e^{2}]=\sum_{1\leq i\leq t}W(m_{i})$ by Lemma 6.9, so the result follows from Lemma 12.1. ∎

Proposition 12.5.

Let $e\in\mathfrak{g}_{sc}$ be nilpotent. Then the following hold:

(i)

If $V\downarrow K[e]=\sum_{1\leq i\leq t}W(m_{i})$ , then

$\mathfrak{g}_{ad}\cong W_{1}\oplus[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$

as $K[e]$ -modules.

(ii)

If $V\downarrow K[e]$ is not of the form $\sum_{1\leq i\leq t}W(m_{i})$ , then

	$\displaystyle\mathfrak{g}_{ad}$	$\displaystyle\cong W_{2}\oplus V^{\prime}$
	$\displaystyle[\mathfrak{g}_{ad},\mathfrak{g}_{ad}]$	$\displaystyle\cong W_{1}\oplus V^{\prime}$

for some $K[e]$ -module $V^{\prime}$ .

Proof.

In case (i), the claim follows from Lemma 12.1 and Lemma 3.1. Similarly (ii) follows from Lemma 12.3, Lemma 12.4, and Lemma 3.1.∎

13. Proofs of main results

We can now prove the main results stated in the introduction; all of them are straightforward consequences of the results from previous sections.

Proof of Theorem 1.3.

By Proposition 9.5, we have

(13.1)		$\displaystyle\mathfrak{g}_{sc}$	$\displaystyle\cong V_{2^{\alpha}}\oplus V^{\prime}$
(13.1)		$\displaystyle\mathfrak{g}_{sc}/Z(\mathfrak{g}_{sc})$	$\displaystyle\cong V_{2^{\alpha}-1}\oplus V^{\prime}$

for some $K[u]$ -module $V^{\prime}$ .

If $s=0$ , then by Proposition 11.7 and Lemma 8.2 we have $\mathfrak{g}_{ad}\cong V_{1}\oplus\mathfrak{g}_{sc}/Z(\mathfrak{g}_{sc})\cong V_{1}\oplus V_{2^{\alpha}-1}\oplus V^{\prime}$ as $K[u]$ -modules. Thus (i) holds.

For (ii), suppose that $s>0$ and let $\beta=\max_{1\leq j\leq s}\nu_{2}(k_{j})$ . Consider first the case where (ii)(a) holds, so $\nu_{2}(k_{j})=\beta$ for all $1\leq j\leq s$ and $\nu_{2}(m_{i})>\beta$ for all $1\leq i\leq t$ . Then $\alpha=\beta$ , so by Proposition 11.7 and Lemma 8.2 we have

	$\displaystyle\mathfrak{g}_{ad}$	$\displaystyle\cong V_{2^{\alpha}}\oplus V^{\prime\prime}$
	$\displaystyle\mathfrak{g}_{sc}/Z(\mathfrak{g}_{sc})$	$\displaystyle\cong V_{2^{\alpha}-1}\oplus V^{\prime\prime}$

for some $K[u]$ -module $V^{\prime\prime}$ . We have $V^{\prime}\cong V^{\prime\prime}$ by (13.1), so $\mathfrak{g}_{ad}\cong\mathfrak{g}_{sc}$ as $K[u]$ -modules.

We consider then the case where (ii)(b) holds, so either $\nu_{2}(k_{j})<\beta$ for some $1\leq j\leq s$ , or $\nu_{2}(m_{i})\leq\beta$ for some $1\leq i\leq t$ . Then by Proposition 11.7 and Lemma 8.2, we have

	$\displaystyle\mathfrak{g}_{ad}$	$\displaystyle\cong V_{2^{\beta}+1}\oplus V^{\prime\prime}$
	$\displaystyle\mathfrak{g}_{sc}/Z(\mathfrak{g}_{sc})$	$\displaystyle\cong V_{2^{\beta}}\oplus V^{\prime\prime}$

for some $K[u]$ -module $V^{\prime\prime}$ . By (13.1) we have $V^{\prime\prime}\cong V_{2^{\alpha}-1}\oplus V^{\prime\prime\prime}$ and $V^{\prime}\cong V_{2^{\beta}}\oplus V^{\prime\prime\prime}$ for some $K[u]$ -module $V^{\prime\prime\prime}$ . Thus

	$\displaystyle\mathfrak{g}_{sc}$	$\displaystyle\cong V_{2^{\alpha}}\oplus V_{2^{\beta}}\oplus V^{\prime\prime\prime},$
	$\displaystyle\mathfrak{g}_{ad}$	$\displaystyle\cong V_{2^{\alpha}-1}\oplus V_{2^{\beta}+1}\oplus V^{\prime\prime\prime}$

so claim (ii)(b) holds.∎

Proof of Theorem 1.4.

Similarly to Theorem 1.3, the result follows from Proposition 10.5, Lemma 8.2, and Proposition 12.5. ∎

Proof of Corollary 1.6 and Corollary 1.7.

Immediate from Theorem 1.3 and Theorem 1.4. ∎

Proof of Corollary 1.8.

For a regular unipotent $u\in\operatorname{Sp}(V)$ we have $V\downarrow K[u]=V(2\ell)$ [23, p. 61]. Then $\dim\mathfrak{g}_{sc}^{u}=\ell+1$ by Lemma 8.1 and Lemma 4.4. It follows then from Corollary 1.6 that $\dim\mathfrak{g}_{ad}^{u}=\dim\mathfrak{g}_{sc}^{u}=\ell+1$ .

For regular nilpotent $e\in\mathfrak{sp}(V)$ we have similarly $V\downarrow K[e]=V(2\ell)$ [23, p. 60]. It follows from Lemma 8.1 and [18, Corollary 4.2] that $\dim\mathfrak{g}_{sc}^{e}=2\ell$ . Then $\dim\mathfrak{g}_{ad}^{e}=\dim\mathfrak{g}_{sc}^{e}=2\ell$ by Corollary 1.7, as claimed. ∎

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Adjoint Jordan blocks for simple algebraic groups of type CℓC_{\ell} in characteristic two

Abstract.

1. Introduction

Problem 1.1.

Problem 1.2.

Theorem 1.3.

Theorem 1.4.

Example 1.5.

Corollary 1.6.

Corollary 1.7.

Corollary 1.8.

2. Notation and terminology

2.1. Generalities

2.2. Algebraic groups

2.3. Actions of unipotent elements

2.4. Actions of nilpotent elements

3. Jordan normal forms on subspaces and quotients

Lemma 3.1 ([16, Lemma 3.3]).

Lemma 3.2 ([17, Lemma 3.4]).

Remark 3.3.

4. Decomposition of tensor products and symmetric squares

4.1. Unipotent case

Theorem 4.1 ([7, (2.5a)], [6, Lemma 1, Corollary 3]).

Theorem 4.2 ([6, Theorem 2]).

Theorem 4.3 ([18, Theorem 1.3]).

Lemma 4.4.

Proof.

Lemma 4.5.

Proof.

4.2. Nilpotent case

Proposition 4.6 ([4, Section III], [25, Corollary 5 (a)]).

Theorem 4.7 ([18, Theorem 1.6, Theorem 1.7, Theorem 3.7]).

5. Unipotent elements in Sp⁡(V)\operatorname{Sp}(V)

Definition 5.1.

Definition 5.2.

Theorem 5.3 ([8, Proposition 3.5]).

6. Nilpotent elements in 𝔰​𝔭​(V)\mathfrak{sp}(V)

Theorem 6.1 ([8, 3.9]).

Remark 6.2.

Lemma 6.3 ([23, Lemma 5.2]).

Definition 6.4.

Definition 6.5.

Definition 6.6.

Theorem 6.7 ([8, Proposition 3.5]).

Lemma 6.8.

Proof.

Lemma 6.9.

Proof.

7. Chevalley construction

7.1. Chevalley construction

Lemma 7.1 ([32, Corollary 1, p. 41]).

7.2. Chevalley construction for type CℓC_{\ell}

7.3. Simply connected groups of type CℓC_{\ell}

7.4. Lie algebra of adjoint type CℓC_{\ell}

Lemma 7.2.

Proof.

7.5. Action of unipotent elements on 𝔤a​d\mathfrak{g}_{ad}, orthogonally indecomposable case

7.6. Action of unipotent elements on 𝔤a​d\mathfrak{g}_{ad}, general case

7.7. Action of nilpotent elements on 𝔤a​d\mathfrak{g}_{ad}

7.8. Elements of 𝒰ℤ\mathscr{U}_{\mathbb{Z}} acting nilpotently

Lemma 7.3.

Proof.

Lemma 7.4.

Proof.

Lemma 7.5.

Proof.

Lemma 7.6.

Proof.

Lemma 7.7.

Proof.

8. Lie algebras of type CℓC_{\ell} as Gs​cG_{sc}-modules

Lemma 8.1.

Proof.

Lemma 8.2.

Proof.

Lemma 8.3.

Proof.

Lemma 8.4.

Proof.

Lemma 8.5.

Adjoint Jordan blocks for simple algebraic groups of type $C_{\ell}$ in characteristic two

5. Unipotent elements in $\operatorname{Sp}(V)$

6. Nilpotent elements in $\mathfrak{sp}(V)$

7.2. Chevalley construction for type $C_{\ell}$

7.3. Simply connected groups of type $C_{\ell}$

7.4. Lie algebra of adjoint type $C_{\ell}$

7.5. Action of unipotent elements on $\mathfrak{g}_{ad}$ , orthogonally indecomposable case

7.6. Action of unipotent elements on $\mathfrak{g}_{ad}$ , general case

7.7. Action of nilpotent elements on $\mathfrak{g}_{ad}$

7.8. Elements of $\mathscr{U}_{\mathbb{Z}}$ acting nilpotently

8. Lie algebras of type $C_{\ell}$ as $G_{sc}$ -modules

9. Jordan block sizes of unipotent elements on $\mathfrak{g}_{sc}/Z(\mathfrak{g}_{sc})$

10. Jordan block sizes of nilpotent elements on $\mathfrak{g}_{sc}/Z(\mathfrak{g}_{sc})$

11. Jordan block sizes of unipotent elements on $\mathfrak{g}_{ad}$

12. Jordan block sizes of nilpotent elements on $\mathfrak{g}_{ad}$