3.1. Complex reducible groups

Symplectic irreducibility does not imply complex irreducibility since $V$ might decompose into nonsymplectic $G$ -submodules. However, this can only happen if $G$ preserves a Lagrangian subspace $\mathfrak{h}\subset V$ . In this case, $G$ acts on $\mathfrak{h}$ by complex reflections; see [Reference Bellamy and Schedler6, Section 4.1].

Assume now that the action of $G$ on $V$ is induced by a complex reflection group $W\leq \mathrm{GL}(\mathfrak h)$ , where $V\cong \mathfrak h\oplus \mathfrak h^\ast$ . For clarity, we will write $H^\vee$ , if we mean the induced action of a subgroup $H\leq W$ on $\mathfrak h\oplus \mathfrak h^\ast$ .

This particular case of Theorem 1.1 was already proved as part of [Reference Brown and Gordon3, Proposition 7.7]. Since our claim is weaker than the statement of loc. cit., a shorter argument suffices. We give it here for the sake of completeness.

Proof. Let $G$ be defined by a complex reflection group $W$ as above. Let $v\in V$ , so there are $v_1\in \mathfrak h$ and $v_2^\ast \in \mathfrak h^\ast$ with $v = v_1 + v_2^\ast$ and we have

\begin{equation*}\mathrm {Stab}_G(v) = \mathrm {Stab}_W(v_1)^\vee \cap \mathrm {Stab}_W(v_2^\ast )^\vee \;.\end{equation*}

Since $\mathfrak h^\ast$ is the dual of the representation $\mathfrak h$ , there exists $v_2\in \mathfrak h$ with $\mathrm{Stab}_W(v_2) = \mathrm{Stab}_W(v_2^\ast )$ .

By Steinberg’s Theorem [Reference Steinberg26, Theorem 1.5], the group $\mathrm{Stab}_W(v_1)$ is generated by complex reflections. We have

\begin{equation*}\mathrm {Stab}_W(v_1) \cap \mathrm {Stab}_W(v_2) = \mathrm {Stab}_{\mathrm {Stab}_W(v_1)}(v_2)\;,\end{equation*}

and a second application of Steinberg’s Theorem implies that this intersection is generated by complex reflections. Hence, $\mathrm{Stab}_G(v)$ is generated by symplectic reflections as claimed.

3.2. Symplectically imprimitive groups

We assume from now on that $G$ is complex irreducible.

In this section, we assume, moreover, that $G$ is symplectically imprimitive. Let $V = V_1\oplus \cdots \oplus V_n$ , with $n \geq 2$ , be a system of imprimitivity. It is explained in the proof of [Reference Cohen9, Theorem 2.9] that we have $\dim V_i = 2$ in this case. Note that Cohen works over the quaternions and shows that $\dim V_i = 1$ as a quaternionic vector space; this is the symplectic analogue of [Reference Cohen8, Proposition 2.2].

By Lemma 2.2, we may assume $\dim V \gt 4$ . Then $G$ is constructed as follows. Let $K\leq \mathrm{SL}_2(\mathbb{C})$ be a finite group and let $H\leq K$ be a subgroup containing $[K,K]$ . Let $G_n(K, H)$ be the subgroup of $K\wr S_n$ consisting of all pairs $(\sigma, k)$ , where $k = (k_1,\dots, k_n)\in K^n$ and $\sigma \in S_n$ , satisfying $k_1\cdots k_n\in H$ ; see [Reference Cohen9, Notation 2.8]. Then [Reference Cohen9, Theorem 2.9] says that $G$ is conjugate to some $G_n(K, H)$ , where $\dim V = 2n$ .

Notice that the transpositions in $S_n$ act as symplectic reflections on $V$ ; they simply swap two summands in the system of imprimitivity.

Proof. We may assume $ G = G_n(K, H)$ as described above.

Let $v = (v_1,\dots, v_n)\in V = V_1\oplus \dots \oplus V_n$ . Now let $\sigma \in S_n$ such that for the permutation $v'$ of $v$ given by $v'_j := v_{\sigma (i)}$ we have a ‘block structure’

\begin{equation*}(v'_1,\dots, v'_{n_0},v'_{n_0 + 1},\dots, v'_{n_0 + n_1},\dots, v'_{n_0+\cdots +n_{r - 1} + 1},\dots, v'_{n_0+\cdots +n_r})\;,\end{equation*}

given by the condition that $Kv_i' = Kv_j'$ if and only if there exists $0\leq s\leq r$ with $ (\sum _{t = -1}^{s - 1} n_t ) + 1 \leq i, j\leq \sum _{t = 0}^{s}n_t$ , where we set $n_{-1} := 0$ . That is, we permute the entries of $v$ so that elements in the same $K$ -orbit lie next to each other and the number of elements lying in the same orbit is given by the $n_i$ . Without loss of generality, we may assume $v'_1 = \cdots = v'_{n_0} = 0$ . After fixing representatives $w_0,\dots, w_r$ for the occurring orbits, we can find an element $k\in K^n$ such that

\begin{equation*}k.v' = w = (w_0,\dots, w_0,w_1,\dots, w_1,\dots, w_r, \dots, w_r)\;,\end{equation*}

where $(Kw_i)\cap (Kw_j) = \emptyset$ for $i\neq j$ and $w_0 = 0$ . Combining $\sigma$ and $k$ hence gives an element $g\in K\wr S_n$ with $g.v = w$ .

If an element $\tau h\in G_n(K, H)$ stabilizes the vector $w$ , then $\tau \in S_{n_0}\times S_{n_1}\times \cdots \times S_{n_r}$ . Furthermore, we must have $h = (h_1,\dots,h_{n_0}, 1,\dots, 1)$ where $h_1,\dots,h_{n_0}\in K$ with $h_1\cdots h_{n_0}\in H$ .

Hence, $\mathrm{Stab}_{G_n(K, H)}(v)$ is $(K\wr S_n)$ -conjugate to $G_{n_0}(K, H)\times S_{n_1}\times \cdots \times S_{n_r}$ , which is a (in general, reducible) symplectic reflection group. Notice that we may have $n_i = 1$ for some of the blocks, resulting in trivial factors in the above product. The claim now follows as symplectic reflections are preserved under conjugation.

3.3. Symplectically primitive groups

The only remaining case is where $G$ is complex irreducible and symplectically primitive. Then $G$ is conjugate to one of the groups classified in [Reference Cohen9, Theorem 3.6] and [Reference Cohen9, Theorem 4.2]. Once again, we may assume $\dim V \gt 4$ by Lemma 2.2. This leaves only seven groups to consider. These are given explicitly via the root systems $Q$ to $U$ in [Reference Cohen9, Table II] and one can check with the help of a computer that all stabilizer subgroups are indeed generated by symplectic reflections. A list of the groups occurring in this way can be found in Section 7.

This finishes the proof of Theorem 1.1. Finally, we note how Theorem 1.1 implies the statement in Remark 1.2. The proof is the same easy induction as in [Reference Steinberg26, Section 7] (see also [Reference Lehrer and Taylor20, Corollary 9.51]); we repeat it for the reader’s convenience.

Proof. As the action of $G$ is linear, we may replace $U$ be the linear span $\langle U\rangle$ and assume in the following that $U$ is a subspace of $V$ .

Let $u_1,\dots, u_k$ be a basis of $U$ . Recalling that $\mathrm{Stab}_G(U)$ fixes $U$ pointwise in this discussion, we have

\begin{equation*}\mathrm {Stab}_G(U) = \mathrm {Stab}_G(u_1)\cap \mathrm {Stab}_G(\langle u_2,\dots, u_k\rangle )\;.\end{equation*}

Now

\begin{equation*}\mathrm {Stab}_G(u_1)\cap \mathrm {Stab}_G(\langle u_2,\dots, u_k\rangle ) = \mathrm {Stab}_{\mathrm {Stab}_G(u_1)}(\langle u_2,\dots, u_k\rangle )\end{equation*}

and $\mathrm{Stab}_G(u_1)$ is a symplectic reflection group by Theorem 1.1. Hence, the claim follows by induction on $k$ .

4.1. Minimal and maximal parabolic subgroups

We note the following results on the rank of minimal and maximal parabolic subgroups where minimality and maximality are to be understood with respect to inclusion.

Proof. By Corollary 3.6, the parabolic subgroup $H$ must contain a symplectic reflection $s$ . Set $K := \mathrm{Stab}_G(V^s)$ . Then

\begin{equation*}\mathrm {Stab}_G(V^H + V^K) = \mathrm {Stab}_H(V^K) = H\cap K\;,\end{equation*}

so $H = K$ by minimality of $H$ . Hence $\dim V^H = \dim V^s = \dim V - 2$ .

The analogous result for maximal parabolic subgroups is easier and does not require Theorem 1.1.

Proof. Let $S\subset G$ be the set of symplectic reflections. Since $G = \langle S \rangle$ , there must exist some $s \in S$ that is not in $H$ . Then $\dim V^s = \dim V - 2$ and we know

\begin{equation*}\dim V^s + \dim V^H - \dim V^s \cap V^H = \dim V\end{equation*}

as $V^s + V^H = V$ . So if $\dim V^H \gt 2$ then $V^s \cap V^H \neq \{0\}$ . If this is the case, then let $K := \mathrm{Stab}_G(V^s\cap V^H)$ . Since $V^K \neq \{0\}$ , we have $K \neq G$ . But $\langle s,H\rangle \leq K$ so $H$ is a proper subgroup of $K$ . This is a contradiction. Thus, $\dim V^H = 2$ .

4.2. Codimension of symplectic quotient singularities

As another application, we have the following result on the singular locus of the symplectic quotient $V/G$ .

Proof. Let $Z \subset V/G$ be an irreducible component of the singular locus. Choose $p \in Z$ generic and $q \in V$ a point mapping to $p$ under the quotient map $V \to V/G$ . Let $H$ be the stabilizer of $q$ in $G$ . Then Luna’s slice theorem [Reference Luna21] says that the map $V/H \to V/G$ induced by the map $V \to V$ , $v \mapsto v + q$ , is étale at $0 \in V/H$ . In particular, the codimension at $p$ of the singular locus in $V/G$ equals the codimension at $0$ of the singular locus in $V/H$ .

Decompose $V = W \oplus V^H$ as a $H$ -module. The fact that $p$ is generic in $Z$ means that $0$ is an isolated singularity in $W/H$ . But Theorem 1.1 says that $H$ acts on $W$ as a symplectic reflection group. This implies that the singular locus of $W/H$ has at least one irreducible component of codimension two (for instance, along the points stabilized by a symplectic reflection). We deduce that $\dim W = 2$ and the irreducible component $Z$ of the singular locus has codimension 2 in $V/G$ .

7.1. An example: the group $W(S_1)$

As an illustration, we show that there is a parabolic subgroup $H$ of $W(S_1)$ which is isomorphic (as a symplectic reflection group) to the complex reducible symplectic group coming from the complex reflection group $G(3, 3, 3)$ .

The necessary computer calculations were carried out and cross-checked using the software package Hecke [Reference Fieker, Hart, Hofmann and Johansson16] and the computer algebra systems GAP [12] and Magma [Reference Bosma, Cannon and Playoust1].

7.1.1. The group

The group $W(S_1)$ is a subgroup of $\mathrm{Sp}_8(\mathbb{C})$ of order $2^8\cdot 3^3 ={6912}$ . Like all complex primitive groups, it is given by a root system [Reference Cohen9, Table II]. Cohen lists 36 root lines for the group. However, four are enough to generate a group of the correct order. A choice of root lines are

\begin{align*} &( 1, i, 0, 0, 0, 0, 1, -i), &&( 1 - i, 1 - i, 0, 0, 0, 0, 0, 0),\\ \\[-9pt] &( 1 - i, 0, 1 - i, 0, 0, 0, 0, 0), &&( 1 - i, 0, 0, 1 - i, 0, 0, 0, 0). \end{align*}

Note that these are the ‘complexified’ versions of the vectors over the quaternions given in [Reference Cohen9]. Thus, the group $W(S_1)\leq \mathrm{Sp}_8(\mathbb{C})$ is generated by the symplectic reflection matrices

\begin{align*} M_1 := \frac{1}{2}&\left (\begin{array}{cccccccc} 1 & i & & & & & -1 & -i \\ -i & 1 & & & & & -i & 1 \\ & & 1 & i & 1 & i & & \\ & & -i & 1 & i & -1 & & \\ & & 1 & -i & 1 & -i & & \\ & & -i & -1 & i & 1 & & \\ -1 & i & & & & & 1 & -i \\ i & 1 & & & & & i & 1 \end{array}\right ), &&M_2 := \left (\begin{array}{cccccccc} & -1 & & & & & & \\ -1 & & & & & & & \\ & & 1 & & & & & \\ & & & 1 & & & & \\ & & & & & -1 & & \\ & & & & -1 & & & \\ & & & & & & 1 & \\ & & & & & & & 1 \end{array}\right ), \end{align*}

\begin{align*} M_3 := &\left (\begin{array}{cccccccc} & & -1 & & & & & \\ & 1 & & & & & & \\ -1 & & & & & & & \\ & & & 1 & & & & \\ & & & & & & -1 & \\ & & & & & 1 & & \\ & & & & -1 & & & \\ & & & & & & & 1 \end{array}\right ), &&M_4 := \left (\begin{array}{cccccccc} & & & -1 & & & & \\ & 1 & & & & & & \\ & & 1 & & & & & \\ -1 & & & & & & & \\ & & & & & & & -1 \\ & & & & & 1 & & \\ & & & & & & 1 & \\ & & & & -1 & & & \end{array}\right ), \end{align*}

through these root lines.

7.1.2. The parabolic subgroup

Let $v := (0, 0, 0, 1, -\alpha, \alpha, -\alpha, \alpha + 1)^\top \in \mathbb{C}^8$ , where $\alpha := \frac{1}{2}(i - 1)$ . Let $H\leq W(S_1)$ be the stabilizer of $v$ . Using the command Stabilizer in either GAP or Magma one can compute this group:

\begin{equation*}H = \langle M_2, M_3M_1M_3, M_4M_1M_4\rangle \;.\end{equation*}

The space $V^H \subset \mathbb{C}^8$ of vectors fixed by $H$ is generated by $v$ and $(1, -1, 1, 0, \alpha + 1, -\alpha - 1, \alpha + 1, 3\alpha )^\top$ . Its $H$ -invariant complement $W$ has a basis given by the columns $w_1,\dots, w_6\in \mathbb{C}^8$ of the matrix

{\begin{equation*} \left (\begin{array}{cccccc} \zeta ^2 + \zeta + 1 & \zeta ^3 + \zeta ^2 - \zeta - 2 & \zeta & -\zeta ^3 + \zeta ^2 + \zeta - 2 & \zeta ^2 - \zeta + 1 & -\zeta ^3 + \zeta \\ -\zeta ^3 - \zeta ^2 + \zeta + 2 & -\zeta ^2 - \zeta - 1 & -\zeta & -\zeta ^2 + \zeta - 1 & \zeta ^3 - \zeta ^2 - \zeta + 2 & \zeta ^3 - \zeta \\ -\zeta ^3 - 2\zeta ^2 + 1 & -\zeta ^3 - 2\zeta ^2 + 1 & \zeta & \zeta ^3 - 2\zeta ^2 + 1 & \zeta ^3 - 2\zeta ^2 + 1 & -\zeta ^3 + \zeta \\ 0 & 0 & -2\zeta ^3 + \zeta & 0 & 0 & \zeta ^3 + \zeta \\ -\zeta ^2 - \zeta + 1 & -\zeta ^3 + \zeta ^2 + \zeta & \zeta ^3 + \zeta ^2 & \zeta ^3 - \zeta ^2 - \zeta & \zeta ^2 + \zeta - 1 & -\zeta ^3 + \zeta ^2 - 1 \\ \zeta ^3 - \zeta ^2 - \zeta & \zeta ^2 + \zeta - 1 & -\zeta ^3 - \zeta ^2 & -\zeta ^2 - \zeta + 1 & -\zeta ^3 + \zeta ^2 + \zeta & \zeta ^3 - \zeta ^2 + 1 \\ \zeta ^3 - 1 & \zeta ^3 - 1 & \zeta ^3 + \zeta ^2 & -\zeta ^3 + 1 & -\zeta ^3 + 1 & -\zeta ^3 + \zeta ^2 - 1 \\ 0 & 0 & -\zeta ^3 - \zeta ^2 + 2\zeta + 2 & 0 & 0 & -\zeta ^3 - \zeta ^2 + 2\zeta - 1 \\ \end{array}\right ) \end{equation*}}

where $\zeta \in \mathbb{C}$ is a primitive 12-th root of unity such that $\zeta ^3 = i$ .

By changing the basis from $\mathbb{C}^8$ to $W \oplus V^H$ and restricting to $W$ we may identify $H$ with the subgroup $H_W$ of $\mathrm{Sp}(W)$ generated by the matrices

\begin{equation*}\left (\begin{array}{cccccc} & 1 & & & & \\ 1 & & & & & \\ & & 1 & & & \\ & & & & 1 & \\ & & & 1 & & \\ & & & & & 1 \end{array}\right ), \left (\begin{array}{cccccc} & & 1 & & & \\ & 1 & & & & \\ 1 & & & & & \\ & & & & & 1 \\ & & & & 1 & \\ & & & 1 & & \end{array}\right ), \left (\begin {array}{cccccc} & & \zeta ^2 - 1 & & & \\ & 1 & & & & \\ -\zeta ^2 & & & & & \\ & & & & & -\zeta ^2 \\ & & & & 1 & \\ & & & \zeta ^2 - 1 & & \end{array}\right ). \end{equation*}

The basis of $W$ was chosen so that the symplectic form on $W$ is, up to a constant, given by the matrix

\begin{equation*} \begin {pmatrix} & I_3 \\ -I_3 & \end {pmatrix}. \end{equation*}

One can see directly that $H_W$ leaves the subspace $\langle w_1, w_2, w_3\rangle$ invariant and that this subspace is Lagrangian. Hence, $H_W$ is a complex reducible, but symplectically irreducible, group coming from a complex reflection group in $\mathrm{GL}(\langle w_1, w_2, w_3\rangle )$ . Since the complex reflection group has rank 3 and order 54, it must be conjugate to $G(3, 3, 3)$ in the classification [Reference Shephard and Todd25].

7.2. Maximal parabolic subgroups

In this section, we list the maximal parabolic subgroups up to conjugation.

7.2.1. $W(Q)$

The group $W(Q)$ is a subgroup of $\mathrm{Sp}_6(\mathbb{C})$ of order $2^6\cdot 3^3\cdot 7 ={12,096}$ . It is generated by the symplectic reflections corresponding to the root lines

\begin{align*} &(2, 0, 0, 0, 0, 0)\;, && \frac{1}{2}(2i, 2i, -i + 1, 0, 0, i\sqrt{5} - 1)\;, \\[3pt] &\frac{1}{2}(2i, 2, i + 1, 0, 0, i + \sqrt{5})\;, && \frac{1}{2}(2, 2i, i + 1, 0, 0, i + \sqrt{5}). \end{align*}

The maximal parabolic subgroups are each conjugate to $H_1 := G(3, 3, 2)$ or $H_2 := G(4, 2, 2)$ . They stabilize the following vectors:

where $\alpha := \frac{1}{6}(i\sqrt{5} + i - \sqrt{5} + 1)$ and $\beta := \frac{1}{3}(-i + \sqrt{5})$ .

7.2.2. $W(R)$

The group $W(R)$ is a subgroup of $\mathrm{Sp}_6(\mathbb{C})$ of order $2^8\cdot 3^3\cdot 5^2\cdot 7 ={12,09,600}$ . It is generated by the symplectic reflections corresponding to the root lines

\begin{align*} &(2, 0, 0, 0, 0, 0)\;, && \frac{1}{2}(i + 1, i - 1, 0, -i\sqrt{5} + 1, -i - \sqrt{5}, 0)\;, \\[3pt] &\frac{1}{2}(0, 0, i + 1, -i, 1, i + \sqrt{5})\;, && \frac{1}{2}(0, 2i, i + \sqrt{5}, 2, 0, -i - 1)\;. \end{align*}

The maximal parabolic subgroups are conjugate to $H_1 := G(3, 3, 2)$ , $H_2 := G(5, 5, 2)$ , or $H_3 := G(\textsf{D}_2, \textsf{C}_2, 1)$ . They stabilize the following vectors:

7.2.3. $W(S_1)$

The group $W(S_1)$ is a subgroup of $\mathrm{Sp}_8(\mathbb{C})$ of order $2^8\cdot 3^3 ={6912}$ . It is generated by the symplectic reflections corresponding to the root lines

\begin{align*} & (1, i, 0, 0, 0, 0, 1, -i)\;, && (-i + 1, -i + 1, 0, 0, 0, 0, 0, 0)\;,\\[3pt] & (-i + 1, 0, -i + 1, 0, 0, 0, 0, 0)\;, && (-i + 1, 0, 0, -i + 1, 0, 0, 0, 0)\;. \end{align*}

The maximal parabolic subgroups are conjugate to $H_1 := \textsf{C}_2\times \textsf{C}_2 \times \textsf{C}_2$ , $H_2 := G(2, 2, 3)$ , or $H_3 := G(3, 3, 3)$ . They stabilize the following vectors:

Note that multiple occurrences of $H_2$ in the above table means that there are distinct maximal parabolic subgroups which are conjugate in $\mathrm{GL}_8(\mathbb{C})$ , but not in $W(S_1)$ .

7.2.4. $W(S_2)$

The group $W(S_2)$ is a subgroup of $\mathrm{Sp}_8(\mathbb{C})$ of order $2^{10}\cdot 3^4 ={82,944}$ . It is generated by the symplectic reflections corresponding to the root lines

\begin{align*} & (1, i, 0, 0, 0, 0, 1, -i)\;, && (-i + 1, -i + 1, 0, 0, 0, 0, 0, 0)\;, \\[3pt] & (-i + 1, 0, -i + 1, 0, 0, 0, 0, 0)\;, && (2, 0, 0, 0, 0, 0, 0, 0)\;. \end{align*}

The maximal parabolic subgroups are conjugate to $H_1 := \textsf{C}_2 \times G(3, 3, 2)$ , $H_2 := G(2, 2, 3)$ , $H_3 := G(2, 1, 3)$ , $H_4 := G(3, 3, 3)$ , and $H_5 := G(4, 4, 3)$ . They stabilize the following vectors:

7.2.5. $W(S_3)$

The group $W(S_3)$ is a subgroup of $\mathrm{Sp}_8(\mathbb{C})$ of order $2^{13}\cdot 3^4\cdot 5 ={3,317,760}$ . It is generated by the symplectic reflections corresponding to the root lines

\begin{align*} & (1, i, 0, 0, 0, 0, 1, -i)\;, && (-i + 1, -i + 1, 0, 0, 0, 0, 0, 0)\;,\\[3pt] & (-i + 1, 0, -i + 1, 0, 0, 0, 0, 0)\;, && (2, 0, 0, 0, 0, 0, 0, 0)\;,\\[3pt] & (-i + 1, 0, 0, 0, 0, -i + 1, 0, 0)\;. \end{align*}

The maximal parabolic subgroups are conjugate to $H_1 := \textsf{C}_2 \times G(3, 3, 2)$ , $H_2 := G(2, 2, 3)$ , $H_3 := G(3, 3, 3)$ , and $H_4 := G_3(\textsf{D}_2, \textsf{C}_2)$ . They stabilize the following vectors:

7.2.6. $W(T)$

The group $W(T)$ is a subgroup of $\mathrm{Sp}_8(\mathbb{C})$ of order $2^8\cdot 3^4\cdot 5^3 ={2,592,000}$ . It is generated by the symplectic reflections corresponding to the root lines

\begin{align*} & (-\zeta ^3 + \zeta ^2 + 1, \zeta ^3 - \zeta ^2, -1, 0, 0, 0, 0, 0)\;, && (1, 0, 0, 0, 0, 0, 0, 0)\;,\\[3pt] & (1, 1, 1, 1, 0, 0, 0, 0)\;, && (1, i, 0, 0, 0, 0, -1, i)\;, \end{align*}

where $\zeta$ is a primitive 10-th root of unity.

The maximal parabolic subgroups are conjugate to $H_1 := \textsf{C}_2 \times G(3, 3, 2)$ , $H_2 := \textsf{C}_2 \times G(5, 5, 2)$ , $H_3 := G(2, 2, 3)$ , $H_4 := G(3, 3, 3)$ , $H_5 := G_{23}$ , and $H_6 := G(5, 5, 3)$ . They stabilize the following vectors:

Note that there are two distinct maximal parabolic subgroups which are conjugate in $\mathrm{GL}_8(\mathbb{C})$ , but not in $W(T)$ .

7.2.7. $W(U)$

The group $W(U)$ is a subgroup of $\mathrm{Sp}_{10}(\mathbb{C})$ of order $2^{11}\cdot 3^5\cdot 5\cdot 11 = 12,096$ . It is generated by the symplectic reflections corresponding to the root lines

\begin{align*} &(2, 0, 0, 0, 0, 0, 0, 0, 0, 0)\;,\\ \\[-9pt] &(0, 2, i - 1, i - 1, 2, 0, 0, -i + 1, -i + 1, 0)\;,\\ \\[-9pt] &(0, 2, i - 1, -i - 1, 2i, 0, 0, i - 1, -i - 1, 0)\;,\\ \\[-9pt] &(0, 2, -i - 1, i - 1, 0, 0, 0, i + 1, i - 1, 2)\;,\\ \\[-9pt] &(2, i - 1, i - 1, 2, 0, 0, -i + 1, -i + 1, 0, 0)\;. \end{align*}

The maximal parabolic subgroups are conjugate to $H_1 := \textsf{C}_2 \times G(2, 2, 3)$ , $H_2 := \textsf{C}_2 \times G(3, 3, 3)$ , $H_3 := \mathfrak S_5$ , $H_4 := G(3, 3, 4)$ , and $H_5 := W(S_1)$ . They stabilize the following vectors: