Globally maximal arithmetic groups.

Benedict Gross

Dept. of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 01238, USA

gross@math.harvard.edu

Gabriele Nebe ^{∗}

Abteilung Reine Mathematik, Universität Ulm, 89069 Ulm, Germany nebe@mathematik.uni-ulm.de

November 13, 2002

1 Introduction.

Let G be a linear algebraic group defined over ℚ, and assume that G(ℝ) is compact and meets every connected component of G(ℂ). Let ℚ̂ := ℤ̂ ⊗ ℚ be the ring of finite adèles. Every arithmetic subgroup Γ of G(ℚ) is finite, and is obtained by choosing an open, compact subgroup K of G(ℚ̂) and defining Γ = K ∩ G(ℚ) in G(ℚ̂). We note that G(ℚ) is discrete and co-compact in G(ℚ̂).

In this paper, we consider the cases where the arithmetic subgroup Γ is contained in a unique maximal compact subgroup K_{p} of G(ℚ_{p}), for all primes p. We call such Γ globally maximal; examples are provided by finite groups Γ with globally irreducible representations V over ℚ where G is a classical group O(V), SU_{K}(V), or SU_{D}(V), according to whether the commuting algebra of V is ℚ, an imaginary quadratic field K or a definite quaternion algebra D. Other, in general not globally irreducible, examples are provided by the finite absolutely irreducible rational matrix groups that are “lattice sparse” of even type (see [5]). These are finite subgroups Γ ≤ GL_{n}(ℚ) for which the natural representation is absolutely irreducible such that all Γ-invariant lattices can be obtained from any Γ-invariant lattice L, by successively taking the dual lattice, scalar multiples, intersections and sums of lattices that are already constructed (there are many such groups, e.g. for n = 24 there are 34 such maximal finite groups). Here the algebraic group G is G = O(V) and the maximal compact subgroup G(ℚ_{p}) containing Γ is O(L ⊗ ℤ_{p}) for any Γ-invariant lattice L.

Another simple example of a globally maximal Γ is the group Γ = S_{4} = 2^{2} ⋊ SL_{2}(2), which has a unique irreducible representation V of dimension 3 and determinant 1. This representation is orthogonal, and Γ is an arithmetic subgroup of G = SO(V). The unique maximal compact K_{p} containing Γ is hyperspecial, for p ≠ 2, and K_{2} = G(ℚ_{2}). In this paper, we will consider similar examples, when G is the unique anisotropic form of G_{2}, F_{4}, and E_{8} over ℚ. In these cases, G is split over ℚ_{p} for all primes p.

∗Most of this work was carried out during G. Nebe’s visit to Harvard University in Spring and Summer of 2002

In [4] the first author has already given some examples of globally maximal Γ, where K_{p} is hyperspecial for all primes p. These are groups over ℤ, such as Γ = G_{2}(2) in G of type G_{2}, and Γ = ^{3}D_{4}(2).3 in G of type F_{4}. Here we will consider the more exotic cases of Jordan subgroups Γ, where K_{p} is not hyperspecial at a single prime p. To identify the maximal parahoric subgroup K_{p} containing Γ at this prime, we will determine the discriminant of its Lie algebra with respect to a multiple of the Killing form.

We begin with a review of the structure of simple, simply-connected complex Lie groups G = G(ℂ) and their Lie algebras 𝔤_{ℂ}. We describe the Chevalley lattice 𝔤 and the associated split group G over ℤ. This gives us a hyperspecial maximal compact subgroup G(ℤ_{p}) in G(ℚ_{p}) and we describe the other maximal parahoric subgroups K_{p} and their Lie algebras starting from G(ℤ_{p}). We then consider the Killing form on 𝔤 and show that it is divisible by 2h^{∨}, where h^{∨} is the dual Coxeter number. The same holds for the Lie algebras of the other maximal parahorics. We compute the discriminants of the resulting scaled forms. Finally we consider the Jordan subgroups Γ = 2^{3}.SL_{3}(2) ≤ G_{2}, 3^{3} ⋊ SL_{3}(3) ≤ F_{4}, and 2^{5}.SL_{5}(2) ≤ 2^{5}.2^{10}.SL_{5}(2) ≤ E_{8} and determine the Γ-invariant lattices in 𝔤_{ℚ}. The Γ-invariant Lie brackets on 𝔤 are unique up to scalar multiples, except for Γ = 3^{3} ⋊ SL_{3}(3) ≤ F_{4}, where there are two possible Lie brackets (which are interchanged by an outer automorphism). We show that these Jordan subgroups are globally maximal and determine their maximal compact overgroups K_{p} ≤ G(ℚ_{p}). The last section treats the Jordan subgroups of the classical groups.

2 Simple Lie groups.

(cf. [7])

Let G be a simple, simply-connected, complex Lie group. Let

T ⊂ B ⊂ G

be a maximal torus contained in a Borel subgroup of G. Let X^{•} denote the character group of T. This is a free abelian group, containing the finite set Φ of roots - the non-zero characters of T which occur on 𝔤 = Lie(G). Let Φ_{+} ⊂ Φ be the positive roots, which occur on Lie(B), and let

Δ ⊂ Φ_{+}

be the root basis determined by B. Every root β in Φ_{+} can be written uniquely as

β = ∑_{α∈Δ} n_{α}(β)α n_{α} ≥ 0.

Since G is simple, there is a highest root β_{0} with the property that

n_{α}(β_{0}) ≥ 1 for all α ∈ Δ
n_{α}(β_{0}) ≥ n_{α}(β) for all β ∈ Φ_{+}, α ∈ Δ

The sum

h := 1 + ∑_{α∈Δ} n_{α}(β_{0})

is the Coxeter number of G. Let 𝔱 := Lie(T). Then, as a representation of T,

𝔤 = 𝔱 + ⨁_{β∈Φ} 𝔤^{β}

where each root space 𝔤^{β} has dimension 1. The space 𝔱^{β} := [𝔤^{β}, 𝔤^{−β}] has dimension 1 and is contained in 𝔱. It has a unique basis H_{β} which satisfies β(H_{β}) = 2, here we have identified Hom(𝔱, ℂ) with X^{•} ⊗ ℂ. If X_{β} is a basis for 𝔤^{β}, there is a unique basis vector Y_{β} for 𝔤^{−β} with

[X_{β}, Y_{β}] = H_{β}.

Furthermore we have [H_{β}, X_{β}] = 2X_{β}, [H_{β}, Y_{β}] = −2Y_{β}. Hence

𝔤_{β} := ⟨H_{β}, X_{β}, Y_{β}⟩

is a sub-algebra of 𝔤 isomorphic to 𝔰l_{2}.

By Lie’s theorem, the homomorphism 𝔰l_{2} → 𝔤 given by the root β lifts to a homomorphism of complex Lie groups SL_{2} → G. The unipotent subgroup

𝔾_{a} ≅ ( \begin{matrix}1 & ⋆ \\ 0 & 1 \end{matrix} ) of SL_{2} maps to the root group U_{β} of G, with Lie algebra 𝔤_{β}. The

map of the tori 𝔾_{m} ≅ ( \begin{matrix}t & 0 \\ 0 & t^{−1} \end{matrix} ) → T is the co-root β^{∨} in X_{•} = Hom(X^{•}, ℤ).

Under the identification X_{•} ⊗ ℂ = Lie(T), β^{∨} maps to the vector H_{β} in 𝔱^{β}. Since G is simply-connected, the co-roots span X_{•}, and the simple co-roots α^{∨}, α ∈ Δ give a ℤ-basis.

The Weyl group W = N_{G}(T)/T acts on X_{•} and X^{•}, the pairing X_{•} ⊗X^{•} → ℤ is W -invariant. Since G is simple, the action of W on X_{•} ⊗ ℚ is (absolutely) irreducible. Hence there is a unique W -invariant pairing

⟨ , ⟩ : X_{•} ⊗ X_{•} → ℤ

which is even, indivisible, and positive definite. We have

⟨α^{∨}, α^{∨}⟩ = 2

if α is a long root, and

⟨α^{∨}, α^{∨}⟩ = 2c

if α is a short root (so α^{∨} is a long co-root), with c = 2 or 3. The dual Coxeter number h^{∨} is defined by

h^{∨} = 1 + ∑_{α long} n_{α}(β_{0}) + \frac{1}{c} ∑_{α short} n_{α}(β_{0})

Here is a table:

\begin{matrix}G & h & c & h^{∨} \\ A_{n} & n + 1 & 1 & n + 1 \\ B_{n} & 2n & 2 & 2n − 1 \\ C_{n} & 2n & 2 & n + 1 \\ D_{n} & 2n − 2 & 1 & 2n − 2 \\ G_{2} & 6 & 3 & 4 \\ F_{4} & 12 & 2 & 9 \\ E_{6} & 12 & 1 & 12 \\ E_{7} & 18 & 1 & 18 \\ E_{8} & 30 & 1 & 30 \end{matrix}

3 Integral Theory

We now modify our notation slightly: the complex Lie groups and Lie algebras of the previous section will now be denoted by G_{ℂ} and 𝔤_{ℂ} to preserve G and 𝔤 for the integral forms.

Chevalley proved that one can choose the basis elements X_{β} of the root eigenspaces 𝔤^{β} ⊂ 𝔤_{ℂ} so that

[X_{β}, X_{−β}] = H_{β}
[X_{β}, X_{α}] = 0 if α + β ≠ 0 is not a root
[X_{β}, X_{α}] = ±(m + 1)X_{α+β} if α + β is a root.

Here m ≥ 0 is the largest integer such that β − mα is a root; an examination of the root systems of rank 2 shows that m = 0, 1, or 2. The abelian subgroup 𝔤 of 𝔤_{ℂ} spanned by the H_{β} and X_{β}, β ∈ Φ is a Lie order with ℤ-basis (H_{α}, X_{β} | α ∈ Δ, β ∈ Φ). This is the Lie algebra of the split, simply-connected group G over ℤ with complex points G(ℂ) = G_{ℂ}. The group G is generated by the integral Torus T = X_{•} ⊗ 𝔾_{m}, and the root subgroups U_{β} ≅ 𝔾_{a} with Lie algebras ℤX_{β}.

The group G(ℤ_{p}) gives a hyperspecial maximal compact subgroup of G(ℚ_{p}) for every prime p. This contains the Iwahori subgroup I_{p}, with reduction to the Borel B mod p. We have

Lie(I_{p}) = ⨁_{α∈Δ} ℤ_{p}H_{α} ⊕ ⨁_{β<0} ℤ_{p}X_{β} ⊕ ⨁_{β>0} pℤ_{p}X_{β}.

We want to describe the maximal parahoric subgroups of G(ℚ_{p}) which contain I_{p}. Besides G(ℤ_{p}), they are indexed by the simple roots α in Δ, and the groups

{G(ℤ_{p}), G_{α}(ℤ_{p}) | α ∈ Δ}

represent the (l + 1) distinct conjugacy classes of maximal compact subgroups of G(ℚ_{p}).

To each simple root α ∈ Δ we can associate a maximal parabolic subgroup P_{α} of G(𝔽_{p}), which contains B. Its inverse image J_{α} in G(ℤ_{p}) has Lie algebra

Lie(J_{α}) = ⨁_{γ∈Δ} ℤ_{p}H_{γ} ⊕ ⨁_{n_{α}(β)≤0} ℤ_{p}X_{β} ⊕ ⨁_{n_{α}(β)>0} pℤ_{p}X_{β}.

J_{α} is a non-maximal parahoric subgroup, and we will see that

J_{α} = G(ℤ_{p}) ∩ G_{α}(ℤ_{p}).

The next theorem follows from Bruhat-Tits theory.

Theorem 1 Let α ∈ Δ be a simple root. Then there is a maximal compact subgroup

G_{α} := G_{α}(ℤ_{p}) ≤ G(ℚ_{p})

with Lie-algebra

Lie(G_{α}) = 𝔤_{α} := ⨁_{γ∈Δ} ℤ_{p}H_{γ} ⊕ ⨁_{n_{α}(β)=−n} \frac{1}{p}ℤ_{p}X_{β} ⊕ ⨁_{−n<n_{α}(β)≤0} ℤ_{p}X_{β} ⊕ ⨁_{n_{α}(β)>0} pℤ_{p}X_{β}

where n := n_{α}(β_{0}) is the multiplicity of α in the highest root β_{0}. The group \overline{G_{α}} := G_{α}(𝔽_{p}) is a semidirect product

\overline{G_{α}} = G_{α}^{red} ⋉ R(\overline{G_{α}})

where R(\overline{G_{α}}) is the unipotent radical, and G_{α}^{red} is semi-simple, with root system

Φ_{α} = {β ∈ Φ | n_{α}(β) ≡ 0 (mod n)}.

This root system has simple roots Δ − {α} ∪ {−β_{0}} with respect to the Borel subgroup reducing to I_{p}. The unipotent radical R := R(\overline{G_{α}}) is filtered as a G_{α}^{red}- module, with n − 1 abelian subquotients U_{i}

R = R_{1} ⊃ R_{2} ⊃ … ⊃ R_{n} = {0}

U_{i} = R_{i}/R_{i+1} ≅ ⨁_{n_{α}(β)≡i (mod n)} 𝔽_{p}X_{β}.

\underline{Proof.} The ℤ_{p}-lattice 𝔤_{α} is a sub-Lie order of 𝔤 ⊗ ℚ_{p}. This will be the Lie-algebra of G_{α}. Indeed, we may define G_{α} by adjoining to J_{α} the elements e_{β}(1/p) in the root groups U_{β} ⊗ ℚ_{p}, where β is a root with n_{α}(β) = −n = n_{α}(−β_{0}), and e_{β} : 𝔾_{a} → U_{β} is the isomorphism over ℤ_{p}. This gives a compact subgroup with desired Lie algebra, by the Chevalley relations. The theory of Bruhat and Tits shows that G_{α} defines a smooth group scheme over ℤ_{p}, and describes its special fiber. The filtration of R is obtained by looking at the orbits of the Weyl group of G_{α}^{red} on Φ. □

Remark 2 When G is simply-laced, each U_{i} is a minuscule, irreducible representation of G_{α}^{red}. In general, there are at most two orbits of the Weyl group of G_{α}^{red} on the weights in U_{i}, corresponding to the roots β of different lengths with n_{α}(β) ≡ i (mod n). In this case, U_{i} need not be irreducible, if p = c.

Remark 3 The semi-direct product structure of \overline{G_{α}} gives a Lie-ideal M with

L = Lie(G_{α}) ⊃ M ⊃ pL

The quotient L/M is isomorphic to the Lie algebra of G_{α}^{red} and M/pL has order p^{dim(R(\overline{G_{α}}))}.

4 An example - the maximal parahorics in E_{8}.

We illustrate the theory of the previous section with a discussion of the 9 conjugacy classes of maximal parahoric subgroups of E_{8} over ℚ_{p}. For each, we determine G_{α}^{red}, as well as the minuscule representations in the filtration of R(\overline{G_{α}}). The representations U_{i} are explicitely identified using the description of their roots in Theorem 1 with the help of the system LIE [12].

The distinct conjugacy classes of maximal parahoric subgroups of E_{8}(ℚ_{p}) correspond bijectively to the nodes of the extended Dynkin diagram:

[figure: 1; 2; 3; 4; 5; 6; 4; 2; 3]

We have labelled the nodes with the multiplicity n_{α}(β_{0}) of the corresponding simple root α in the highest root β_{0}. The extended vertex, with label n = 1, corresponds to the longest root −β_{0}.

We discuss the parahorics from left to right. μ_{a} ≤ G_{m} denotes the group of a-th roots of unity. By Δ we understand a diagonal embedding. The unique vertex labelled 1 corresponds to the hyperspecial compact G(ℤ_{p}). This has

G^{red} = E_{8}, R(\overline{G}) = 0, dim(R(\overline{G})) = 0.

The adjacent vertex, labelled 2, has

G_{α}^{red}  ≅ (SL_{2} × E_{7})/Δμ_{2}
R(\overline{G_{α}}) = U_{1} =  2 ⊗ 56
dim(R(\overline{G_{α}})) =  112

where we have indicated a minuscule representation of a factor by its dimension. The adjacent vertex, labelled 3, has

G_{α}^{red}  ≅ (SL_{3} × E_{6})/Δμ_{3}
U_{1}  =  3 ⊗ 27
U_{2}  =  3′ ⊗ 27′
dim(R(\overline{G_{α}})) =  162

Here 3′ is the contragredient representation of the natural representation 3 of SL_{3}, and the representations 27 and 27′ of E_{6} are also dual. The adjacent vertex, labelled 4, has

G_{α}^{red}  ≅ (SL_{4} × Spin_{10})/Δμ_{4}
U_{1}  =  4 ⊗ 16
U_{2}  =  6 ⊗ 10
U_{3}  =  4′ ⊗ 16′
dim(R(\overline{G_{α}})) =  188

Here 4 is the natural representation of SL_{4}, 6 = Λ^{2}(4), and 4′ = Λ^{3}(4) is the dual of 4. The representations 16 and 16′ are the half spin representations of Spin(10). The adjacent vertex, labelled 5, has

G_{α}^{red}  ≅ (SL_{5} × SL_{5})/Δμ_{5}
U_{1}  =  5 ⊗ 10
U_{2}  =  10 ⊗ 5′
U_{3}  =  10′ ⊗ 5
U_{4}  =  5′ ⊗ 10′
dim(R(\overline{G_{α}})) =  200

Here 5 is the natural representation of SL_{5}, 10 = Λ^{2}5, 10′ = Λ^{3}5, and 5′ = Λ^{4}5. The adjacent vertex, labelled 6, has

G_{α}^{red}  ≅ (SL_{2} × SL_{3} × SL_{6})/Δμ_{6}
U_{1}  =  2 ⊗ 3 ⊗ 6
U_{2}  =  1 ⊗ 3′ ⊗ 15
U_{3}  =  2 ⊗ 1 ⊗ 20
U_{4}  =  1 ⊗ 3 ⊗ 15′
U_{5}  =  2 ⊗ 3′ ⊗ 6′
dim(R(\overline{G_{α}})) =  202

Here 6 is the natural representation of SL_{6}, 15 = Λ^{2}6, 20 = Λ^{3}6, 15′ = Λ^{4}6, and 6′ = Λ^{5}(6). The bottom vertex, adjacent with 6 and labelled 3, has

G_{α}^{red}  ≅ SL_{9}/μ_{3}
U_{1}  = 84 = Λ^{3}9
U_{2}  = 84′ = Λ^{6}9
dim(R(\overline{G_{α}})) =  168

where 9 is the natural representation of SL_{9}. The next vertex, adjacent to 6 and labelled 4, has

G_{α}^{red}  ≅ (SL_{2} × SL_{8})/Δμ_{2}
U_{1}  =  2 ⊗ 28
U_{2}  =  1 ⊗ 70
U_{3}  =  2 ⊗ 28′
dim(R(\overline{G_{α}})) =  182

Here 8 is the natural representation of SL_{8}, 28 = Λ^{2}8, 70 = Λ^{4}8, and 28′ = Λ^{6}8. Similarly 2 is the natural representation of SL_{2} (which is self-dual) and 1 is the trivial representation of SL_{2}. The last vertex on the right, labelled 2, has

G_{α}^{red}  ≅ Spin_{16}/μ_{2}
R(\overline{G_{α}}) = U_{1} =  128
dim(R(\overline{G_{α}})) =  128

In each case it is interesting to note that every minuscule representation of G_{α}^{red} occurs in the filtration of R(\overline{G_{α}}). This is a general phenomenon, when G is of adjoint type, as the center of G_{α}^{red} has order n = n_{α}(β_{0}).

Some other examples of maximal parahorics, which exhibit unusual symmetry, are given by the following simple roots α, indicated in the extended Dynkin diagram. G = Spin_{8} :

[figure: 2]
G_{α}^{red}  ≅ (SL_{2} × SL_{2} × SL_{2} × SL_{2})/Δμ_{2}
R(\overline{G_{α}}) = U_{1} =  2 ⊗ 2 ⊗ 2 ⊗ 2
dim(R(\overline{G_{α}})) =  16

G = E_{6} :

[figure: 3]
G_{α}^{red}  ≅ (SL_{3} × SL_{3} × SL_{3})/Δμ_{3}
U_{1}  =  3 ⊗ 3 ⊗ 3
U_{2}  =  3′ ⊗ 3′ ⊗ 3′
dim(R(\overline{G_{α}})) =  54

5 The Killing form.

We retain the notion of Section 3, so 𝔤 = Lie(G) is the Chevalley Lie algebra of the simply-connected, simple group scheme G over ℤ. The Killing form

(X, Y) := Tr(adX · adY)

is integral, symmetric, and G-invariant on 𝔤. On X_{•}(T) = Lie(T), it is integral, even, and W -invariant, so it is a multiple of the indivisible form ⟨, ⟩ with ⟨α^{∨}, α^{∨}⟩ = 2 for α a long root. Steinberg and Springer [8] show that

(H_{α}, H_{α}) = 4h^{∨}

for α a long root, with h^{∨} the dual Coxeter number. Hence

(,) = 2h^{∨} · ⟨, ⟩

as bilinear forms on Lie(T). The decomposition

Lie(G) = Lie(T) ⊕ ⨁_{β>0}(ℤX_{β} + ℤX_{−β})

is orthogonal for the Killing form. Steinberg and Springer also show that, for all roots β,

(X_{β}, X_{−β}) = \frac{1}{2}(H_{β}, H_{β}) = h^{∨}⟨H_{β}, H_{β}⟩

Hence if we define

⟨X, Y ⟩ := \frac{1}{2h^{∨}} (X, Y)

we find that

Proposition 4 The pairing

⟨ , ⟩ : 𝔤 × 𝔤 → ℤ

is even, indivisible, and is positive definite on Lie(T).

If G is simply-laced, we find that, with respect to ⟨ , ⟩

𝔤^{∗}/𝔤 ≅ Lie(T)^{∗}/Lie(T) ≅ { \atop Z(G)}

where Z(G) is the (finite) center of G. In the general case, 𝔤^{∗}/𝔤 has order #Z(G)c^{k}, where k is the number of short positive roots plus the number of short simple roots. The latter contribute to Lie(T)^{∗}/Lie(T). Here is a table:

G det⟨ , ⟩ on 𝔤 det⟨ , ⟩ on Lie(T)
A_{n}  n + 1  n + 1
B_{n}  2^{n+2}  2^{2}
C_{n}  2^{n^{2}}  2^{n}
D_{n}  2^{2}  2^{2}
G_{2}  3^{7}  3
F_{4}  2^{26}  2^{2}
E_{6}  3  3
E_{7}  2  2
E_{8}  1  1

The pairing ⟨ , ⟩ on 𝔤 ⊗ ℚ_{p} is also integral and even on the ℤ_{p}-lattices L = Lie(G_{α}), for the maximal parahorics in G(ℚ_{p}) defined in Section 3. Indeed the only change in the discriminant L^{∗}/L from that of 𝔤^{∗}/𝔤 involves the planes

ℤ_{p}X_{−β} + pℤ_{p}X_{β}

where β is a positive root with

0 < n_{α}(β) < n_{α}(β_{0}).

This contributes a factor of (ℤ/pℤ)^{2} to L^{∗}/L. Hence we find the following Proposition 5 Assume that p does not divide det(⟨ , ⟩) on 𝔤. Then

L^{∗}/L ≅ (ℤ/pℤ)^{dim R(\overline{G_{α}})}

and pL^{∗} is the Lie ideal M with L/M ≅ Lie(G_{α}^{red}).

This allows us to determine which maximal parahorics G_{α}(ℤ_{p}) can contain certain finite groups Γ ⊂ G(ℚ_{p}), once we know some information on the Γ-stable lattices in 𝔤 ⊗ ℚ_{p}.

6 The type of some Jordan subgroups of exceptional groups.

Definition 6 Let Γ be a globally maximal arithmetic subgroup of a linear algebraic group G defined over some number field K. Then the type of Γ is

T (Γ) := (T_{℘}(Γ))_{℘ prime}.

Here ℘ runs through the prime ideals of K and T_{℘}(Γ) denotes the maximal compact subgroup of G(K_{℘}) over the ℘-adic completion of K, that contains Γ.

Γ ≤ T_{℘}(Γ) ≤ G(K_{℘}).

If Γ is a globally maximal group, then the type T_{℘}(Γ) is hyperspecial for almost all primes ℘. In particular if the Γ-module Lie(G(K_{℘})) is irreducible modulo ℘ then T_{℘}(Γ) is a hyperspecial.

We now treat the different Jordan subgroups of the exceptional groups in detail. The explicit calculations are performed using the computer algebra system MAGMA [11].

6.1 2^{3}.SL_{3}(2) in G_{2}

The simple roots of G_{2} are given as follows:

[figure: >; −β_{0}; α_{1}; α_{2}]

The next theorem is already shown in [2] by calculations in the 7-dimensional representation of G_{2}.

Theorem 7 Let Γ := 2^{3}.SL_{3}(2) be the Jordan subgroup of the anisotropic form G(ℚ) of G_{2}. Then T_{p}(Γ) = G_{2} is hyperspecial for p > 2 and T_{2}(Γ) = A_{2}.

\underline{Proof.} Γ has a unique complex irreducible 14-dimensional representation V. This representation is rational. The space of Γ-invariant homomorphisms of V ⊗ V to V is one dimensional. Any generator of this space is skew symmetric and gives a Γ-invariant Lie-multiplication on V. This yields an embedding of Γ into G(ℚ). The group Γ fixes up to isomorphism 12 lattices in V of which the 2-local inclusions are given as follows:

[figure: (1); (2); (4); (3); (5); (6); 2 · (1); 2 · (2)]

Here the vertical line (e.g from (1) to (2)) and the lines parallel to (2), (4) indicate inclusions of index 2^{3} (two different 𝔽_{2}Γ-modules) and the lines parallel to (1), (3) mean inclusions of index 2^{8}. The other 6 isomorphism classes are represented by sublattices of index 3^{7} of these 6 lattices. This gives the type of Γ for all primes p > 2. The G_{α_{1}}(ℤ_{2})-composition factors of Lie(G_{α_{1}}(ℤ_{2}))/2Lie(G_{α_{1}}(ℤ_{2})) are of dimension 1,2, and 4. Hence the 2-local type of Γ is either G_{2} or A_{2}. It follows from the mass-formula (see [4], [2]) or from the calculation in [2] that T_{2}(Γ) = G_{α_{2}}(ℤ_{2}) is of typs A_{2}. □

Remark 8 The reduction map Γ → G_{α_{2}}(𝔽_{2}) is injective.

Remark 9 The possibility that T_{2}(Γ) = G_{2} cannot be ruled out looking at the Lie bracket: The maximal Γ-invariant Lie-order (which corresponds to the lattice (1) in the picture above) has discriminant 3^{7} (with respect to 1/8 times the Killing form) which is the same discriminant as the one of Lie(G(ℤ)). Indeed this Lie-order is also invariant under the maximal compact G_{α_{2}}(ℤ_{2}) of type A_{2} that contains Γ. The Lie-order Lie(G_{α_{2}}(ℤ_{2})) corresponds to the lattice number (2) in the picture above, which is contained in (1) of index 2^{3}.

6.2 3^{3} ⋊ SL_{3}(3) in F_{4}

Let Γ be the Jordan subgroup 3^{3} ⋊ SL_{3}(3) of the unique anisotropic form G(ℚ) of the algebraic group F_{4}.

The group Γ has 3 absolutely irreducible representations of degree 52. To decide which one is the action of Γ on the Lie-algebra of G(ℚ), we note that the elements of order 9 in both conjugacy classes of G(ℚ) have trace 1. This identifies the representation V = Lie(G(ℚ)) of Γ uniquely. The space H := Hom_{Γ}(Λ^{2}V, V) is 2-dimensional. The Jacobi identity gives a quadratic equation which has two solutions in H. Hence there are up to scalar multiples two Γ-invariant Lie brackets on V. They are interchanged by the outer automorphism (in 3^{3} ⋊ GL_{3}(3)) of Γ (which is not in G(ℚ)), therefore there are up to conjugacy two representations of Γ into G(ℚ) giving the same conjugacy class of groups Γ ≤ G(ℚ). We fix one of the two Γ-invariant Lie brackets.

The simple roots of F_{4} are indicated in the following diagram:

[figure: >; −β_{0}; α_{1}; α_{2}; α_{3}; α_{4}]

Theorem 10 T_{p}(Γ) = G(ℤ_{p}) is hyperspecial for p ≠ 3 and T_{3}(Γ) = G_{α_{2}}(ℤ_{3}).

\underline{Proof.} For p > 3, the theorem follows because the representation of Γ on the Lie algebra is irreducible modulo p. For p = 2, this representation has two 2-modular constituents of degree 26. This implies that T_{2}(Γ) = G(ℤ_{2}) is also hyperspecial. It remains to consider the prime 3. The unique maximal Γ-invariant Lie-order has discriminant 2^{26} · 3^{36} (with respect to 1/18 times the Killing form) which is the discriminant of the Lie-order Lie(G_{α_{2}}(ℤ_{3})). There is no other Γ-invariant Lieorder that has the discriminant of the Lie algebra of a maximal compact subgroup of G(ℚ_{3}). Therefore T_{3}(Γ) = G_{α_{2}}(ℤ_{3}). □

6.3 2^{5}.SL_{5}(2) and 2^{5}.2^{10}.SL_{5}(2) in E_{8}

Let Γ := 2^{5}.SL_{5}(2) be a Jordan subgroup of the unique anisotropic form G(ℚ) of the algebraic group E_{8} and let H := 2^{5}.2^{10}.SL_{5}(2) be the maximal finite Jordan subgroup of G(ℚ) that contains Γ.

The 248-dimensional representation V of Γ can be obtained from the 248dimensional integral representation of the Thompson group, which contains Γ as a maximal subgroup, from the matrices in the www-atlas [10]. To construct the Γ-invariant Lie-multiplication on V (which is unique up to scalar multiples) we decompose V as the direct sum of eigenspaces

V = ⊕_{χ}V_{χ}

under the maximal normal 2-subgroup T ≅ 2^{5} of Γ. All 31 nontrivial characters χ of T occur on V with multiplicity 8 and Γ permutes the V_{χ} 2-transitively. The space of Stab_{Γ}(χ_{1}, χ_{2})-invariant homomorphisms from V_{χ_{1}} ⊗ V_{χ_{2}} to V_{χ_{1}χ_{2}} is onedimensional. From this one constructs the Γ-invariant Lie-bracket on V. The maximal decomposable sublattice L_{OD} := ⊕_{χ}(V_{χ} ∩ Λ) of the Thompson-Smith lattice Λ carries a Γ-invariant integral Lie-multiplication such that the discriminant of L_{OD} (with respect to 1/60 times the Killing form) is 2^{248}.

The simple roots of G are labelled as in the following extended Dynkin diagram:

[figure: −β_{0}; α_{1}; α_{2}; α_{3}; α_{4}; α_{5}; α_{6}; α_{7}; α_{8}]

Theorem 11 Γ (and hence also H) is a globally maximal subgroup of E_{8}. The type of Γ is T (Γ) = T (H) with T_{2}(Γ) = G_{α_{4}}(ℤ_{2}) and T_{p}(Γ) = G(ℤ_{p}) for all primes p > 2.

\underline{Proof.} The 248-dimensional representation of Γ is absolutely irreducible modulo every prime p > 2. Therefore T_{p}(Γ) = T_{p}(H) = G(ℤ_{p}) for all primes p > 2. Since Γ is compact, it embeds into at least one maximal compact subgroup G_{α} of G(ℚ_{2}). Then Γ acts on Lie(G_{α}) and hence there is a Γ-invariant Lie-order in V, of the correct discriminant (with respect to 1/60 times the Killing form) 2^{dim(R(\overline{G_{α}}))} (see Section 4). With MAGMA ([11]) one calculates that Γ fixes up to isomorphism 383 lattices in V. The only lattice of one of the discriminants above, that is closed under the Lie bracket is a lattice L_{A4+A4} of discriminant 2^{200}. Hence α = α_{4}, Γ ≤ G_{α_{4}}(ℤ_{2}), and T_{2}(Γ) = G_{α_{4}}(ℤ_{2}). □

Remark 12 The representation of Γ on L_{A_{4}+A_{4}}/2L_{A_{4}+A_{4}} and hence also the reduction map of Γ to G_{α_{4}}(𝔽_{2}) is injective. From the action of Γ on Lie(G_{α_{4}}^{red}) one sees that the image is diagonally embedded in G_{α_{4}}^{red} ≅ SL_{5}(2) × SL_{5}(2)/Δμ_{5}.

Remark 13 There are two maximal Γ-invariant lattices that are closed under the Lie bracket, one of which is the orthogonal decomposition L_{OD} ≅ ^{(2)}E_{8}^{31} of discriminant 2^{248} and the other lattice is L_{A4+A4}. Therefore Γ has 2 maximal Lie orders. Since both of them are also stable under H, the same holds for H. For both Lie orders, the Lie bracket is surjective. The intersection L_{A4+A4} ∩ L_{OD} is of index 2^{5} in L_{OD} (and of index 2^{29} in L_{A4+A4}).

7 The Jordan subgroups of the classical groups.

In this section we calculate the type of the Jordan subgroups of the classical groups G. To this aim we calculate in the natural representation of G. Then the group Γ may contain a center, that acts trivially in the adjoint representation of G.

7.1 p_{+}^{1+2n}.Sp_{2n}(p) ≤ U_{p^{n}}

Let Γ = p_{+}^{1+2n} ⋊ Sp_{2n}(p) ≤ U_{p^{n}}(ℂ) if p > 2 and Γ = C_{4}Y2^{1+2n}.Sp_{2n}(2) ≤ U_{2^{n}}(ℂ) if p = 2. The minimal number field L, such that Γ is contained in U_{p^{n}}(L) is L = ℚ[ζ_{8}] for p = 2 and L = ℚ[ζ_{p}] for p > 2 and the involution is complex conjugation. Let K denote the totally real subfield of L. Then the algebraic group G is defined over K.

Theorem 14 (a) For p = 2 the type of Γ is hyperspecial for all primes ℘ of K.

(b) For p > 2 the type of Γ is hyperspecial for all primes of K not dividing p. Let q := 1/2(p^{n} − 1). Then at the prime ℘ := (1 − ζ_{p})(1 − ζ_{p}^{−1}), the type T_{℘}(Γ) is the maximal parahoric subgroup corresponding to the vertex number ⌊\frac{q}{2}⌋ + 1 of the local Dynkin diagram C − BC_{q} (with q + 1 vertices) (7th diagram on page 60 [9]):

[figure: >; >]

\underline{Proof.} (a) For p = 2 the group ⟨ζ_{8}⟩YΓ is the complex Clifford group described in [6, Section 6]. It follows from the remarks before [6, Theorem 6.5], that the natural representation of Γ is irreducible modulo all primes ℘ of K. Hence the type of Γ is hyperspecial everywhere.

(b) Since O_{p}(Γ) acts absolutely irreducible, the natural representation of Γ is clearly irreducible modulo all primes of K that do not divide p. [1] shows that the natural representation of Γ has two p-modular constituents of degree q and q+1, where p^{n} = 2q+1. Hence Γ embeds into the maximal parahoric subgroup P of U_{p^{n}}(K_{℘}) with P^{red} = O_{q}(p) × Sp_{q+1}(p) if q is odd and P^{red} = O_{q+1}(p) × Sp_{q}(p) if q is even. □

Note that also for the case p > 2, the group Γ is (up to certain scalars) the Clifford group described in [6, Section 7]. In particular Γ is (up to scalars) a maximal finite subgroup of U_{p^{n}}(ℂ) (see [6, Theorem 7.3] for p > 2 and [6, Theorem 6.5] for p = 2).

7.2 2^{2n} ⋊ S_{2n+1} ≤ O_{2n+1}

Let n ≥ 3 and Γ := 2^{2n} ⋊ S_{2n+1}. Then Γ is the determinant 1 subgroup of the full monomial subgroup M ≤ O_{2n+1}(ℚ). M is generated by −I_{2n+1} and Γ. Hence Γ fixes the same lattices as M, namely the standard lattice S := ℤ^{2n+1} with quadratic form ∑_{i=1}^{2n+1} x_{i}^{2}, its even sublattice L and the dual lattice L^{∗} (see e.g. [3]).

[figure: L^{∗}; S; L; 2L^{∗}; 2S; 2L]

The unique maximal parahoric subgroup P of O_{2n+1}(ℚ_{2}, F) that contains Γ is the orthogonal group of the lattice L ⊗ ℤ_{2}. It also stabilizes the dual lattice L^{∗} and, since L^{∗}/L ≅ C_{4}, the unique lattice S between L^{∗} and L. P corresponds to the last vertex of the local Dynkin diagram

[figure: >]

Note that \overline{P} is isomorphic to 2^{2n} ⋊ O_{2n}^{+}(2), if 2n + 1 ≡ ±3 (mod 8) i.e. the 2-adic quadratic space is split over ℚ_{2} (Dynkin diagram B_{n} on page 60 [9]) and \overline{P} ≅ 2^{2n} ⋊ O_{2n}^{−}(2), if 2n + 1 ≡ ±1 (mod 8) i.e. the 2-adic quadratic space is non split over ℚ_{2} (Dynkin diagram ^{2}B_{n} on page 63 [9]).

Theorem 15 Let Γ := 2^{2n} ⋊ S_{2n+1} ≤ O_{2n+1}(ℚ). Then Γ is globally maximal. T_{p}(Γ) is hyperspecial for all primes p > 2 and T_{2}(Γ) = P as described above.

7.3 2^{2n−1} ⋊ S_{2n} ≤ O_{2n}

Assume that n ≥ 5 and let Γ := 2^{2n−1} ⋊S_{2n} be the determinant 1 subgroup of the full monomial subgroup M of O_{2n}(ℚ). As in the last section M fixes 3 lattices, the standard lattice S := ℤ^{2n}, its even sublattice L and the dual lattice L^{∗} (see e.g. [3]). Since the dimension is even, L^{∗}/L ≅ C_{2} × C_{2} and 2L^{∗} is contained in L.

[figure: L^{∗}; S; L; 2L^{∗}]

Since Γ is a normal subgroup of index 2 in M, the only other lattices possibly fixed by Γ are the the two lattices S_{1} = ⟨L, \frac{1}{2} ∑_{i=1}^{2n} x_{i}⟩ and S_{2} = ⟨L, \frac{1}{2} ∑_{i=1}^{2n} x_{i}−x_{1}⟩. These are not fixed under Γ (the stabilizer in M of either of these two lattices is the subgroup of M generated by all permutations and all even sign changes) and hence Γ fixes the same lattices as M.

The type of Γ is clearly hyperspecial for all primes p > 2. For p = 2, the type P := T_{2}(Γ) is as follows:

Assume first that 2n ≡ 0 (mod 8), i.e. the 2-adic quadratic space is split. In this case the local Dynkin diagram is D_{n} on page 61 [9].

[figure]

The two lattices S_{1} and S_{2} are even unimodular lattices and their 2-adic stabilizers correspond to the two extremal hyperspecial vertices at one side of the diagram above. Γ interchanges the two lattices S_{1} and S_{2} and hence fixes the midpoint m of the edge joining the two hyperspecial maximal parahorics in the building. The type of Γ is the maximal compact group P = Stab(m).

Now assume that 2n ≢ 0 (mod 8). Then the dimension of the anisotropic kernel of the quadratic space is 4 and the relative local Dynkin diagram is ^{2}D′_{n} on page 65 [9].

[figure: >; <; 2; 2]

In this case, P corresponds to one of the two (special) extremal vertices labelled by 2.

Theorem 16 Let Γ := 2^{2n−1} ⋊ S_{2n} ≤ SO_{2n}(ℚ). Then Γ is globally maximal. T_{p}(Γ) is hyperspecial for p > 2. For p = 2, T_{2}(Γ) = P as described above.

7.4 2_{+}^{1+2n}.O_{2n}^{+}(2) ≤ O_{2^{n}}

Let n ≥ 3 and Γ := 2_{+}^{1+2n}.O_{2n}^{+}(2) ≤ O_{2^{n}}(ℝ). Then Γ is the full normalizer of the extraspecial group 2_{+}^{1+2n} in the orthogonal group O_{2^{n}}(ℝ). This gives an isomorphism of Γ with the real Clifford group described in [6]. Hence up to conjugacy Γ ≤ O_{2^{n}}(ℚ[\sqrt{2}]) and by [6, Lemma 5.4] Γ fixes only one lattice in ℚ[\sqrt{2}]^{2^{n}}. Hence the type of Γ is hyperspecial for all primes ℘ of ℤ[\sqrt{2}]. The reduction modulo \sqrt{2} of Γ is the natural action of O_{2n}^{+}(2) on the simple module of the Clifford algebra of the associated quadratic form.

Theorem 17 The type of Γ = 2_{+}^{1+2n}.O_{2n}^{+}(2) ≤ O_{2^{n}}(ℚ[\sqrt{2}]) is hyperspecial for all primes ℘.

Note that Γ is a maximal finite subgroup of O_{2^{n}}(ℝ) as shown in [6, Theorem 5.6].

7.5 2_{−}^{1+2n}.O_{2n}^{−}(2) ≤ Sp_{2^{n}}

The group Γ := 2_{−}^{1+2n}.O_{2n}^{−}(2) is the centralizer of one factor Q_{8} in 2_{+}^{1+2(n+1)}.O_{2n+2}^{+}(2). Let Q be the quaternion algebra with center ℚ[\sqrt{2}] ramified only at the two infinite places. Then Γ can be realized as a subgroup of U_{2^{n−1}}(Q) ≤ Sp_{2^{n}}(ℂ).

Lemma 18 The enveloping order ℤΓ of Γ in Q^{2^{n−1}×2^{n−1}} is a maximal order.

\underline{Proof.} For n ≤ 2 the lemma can be checked easily by direct computations. Assume that n ≥ 3. Then Γ contains the tensor product S̃_{4}Y2_{+}^{1+2(n−1)}.O_{2(n−1)}^{+}(2). Since

n − 1 ≥ 2, the group 2_{+}^{1+2(n−1)}.O_{2(n−1)}^{+}(2) ≤ O_{2^{n−1}}(ℚ[\sqrt{2}]) spans a maximal order ≅ ℤ[\sqrt{2}]^{2^{n−1}×2^{n−1}} by [6, Lemma 5.4]. Since S̃_{4} spans a maximal order in Q, the lemma follows by taking tensor products. □

Hence Γ fixes only one class of lattices and hence we get

Theorem 19 The type of Γ = 2_{−}^{1+2n}.O_{2n}^{−}(2) ≤ U_{2^{n−1}}(Q) ≤ Sp_{2^{n}}(ℂ) is hyperspecial for all primes ℘.

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[10] An atlas of finite group representations. Rob Wilson et al. http://www.mat.bham.ac.uk/atlas/v2.0/

[11] The  Magma  Computational  Algebra  System

http://magma.maths.usyd.edu.au/magma/

[12] LiE, A Computer algebra package for Lie group computations http://wwwmathlabo.univ-poitiers.fr/∼maavl/LiE/