Journal of Lie Theory Volume 14 (2004) 287–316 \begin{matrix}c \\ ◯ \end{matrix} 2004 Heldermann Verlag

The Structure of Parabolic Subgroups

Kenneth D. Johnson

Communicated by E. B. Vinberg

Abstract. Suppose G is a real connected simple noncompact Lie group with (using standard notation) Iwasawa decomposition G = KAN. If M = Z(A)∩K, the group B = MAN is a minimal parabolic subgroup of G. Since A is a vector group and N is a simply connected nilpotent group, the topological structure of B is determined by the structure of M. When G is a linear group the structure of M is well known. However, if G is not a linear group there is very little available information about M. Our purpose here is to give a description of the group M for any connected, simply connected, nonlinear simple group G.

1. Introduction and Notation

Let g be a real simple Lie algebra with g_{C} its complexification. Suppose G is a real connected Lie group with Lie algebra g. If G has finite center, G has an Iwasawa decomposition G = KAN. That is, K is a maximal compact subgroup of G, A is a maximal vector subgroup of G with Ad(A) consisting of semisimple elements, and N is a maximal nilpotent group normalized by A. If G does not have a finite center, G is a covering of a group with finite center; the Iwasawa decomposition still holds if K is the inverse image of the maximal compact subgroup by the covering map Ad : G ⟶ Ad(G). If M = Z(A) ∩ K, the group MAN is a minimal parabolic subgroup of G. Unlike the case when G is a linear group, the structure of M is not known when G is not a linear group.

Let k, m, a, and n be the respective Lie algebras of K, M, A, and N. When G is a linear group M = Z_{1} · M_{0} ≅ Z_{1} × M_{0} where M_{0} is the connected component of the identity of M and Z_{1} is a subgroup of (exp ia) ∩ K isomorphic to Z_{2}^{r} for some r ≤ dim a (Satake [12], Helgason [7]). If G ⊂ G_{C} where G_{C} is a connected and simply connected Lie group with Lie algebra g_{C}, it is known that r is the number of white dots in the Satake diagram of G that are neither attached to another white dot by an arrow nor adjacent to a black dot (Johnson [10]). If π : G̃ ⟶ G is a nontrivial covering of G with K̃ = π^{−1}(K), then G̃ = K̃AN where A and N are as above. If M̃ = π^{−1}(M), then M̃ = Z(A) ∩ K̃ and M̃AN is a minimal parabolic subgroup of G̃. Again M̃ = Z̃_{1} · M̃_{0} where M̃_{0} is the connected component of the identity of M̃ and Z̃_{1} is a discrete subgroup.

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag A priori, the group Z̃_{1} need not be either finite or abelian. Indeed we will show that Z̃_{1} is in general not abelian. However, it is true that the group Z̃_{1} is finite provided G/K is not a tube type domain.

Our main goal in this paper is to give a complete description of the structure of M̃ for any simply connected group G̃. To establish our footing we first prove two results that should be well known.

Proposition 1.1. The Group G is a linear group Lie group if and only if it is isomorphic to a subgroup of G_{C}.

Proof. If G ⊂ G_{C}, G is linear since G_{C} is already linear. On the other hand, if G is linear, we may assume G is a subgroup of GL(n, C) for some n. Then G is a subgroup of the analytic subgroup of GL(n, C) whose Lie algebra is g_{C}.

Theorem 1.2. Suppose G ⊂ G_{C} with G_{C} simply connected. Then no nontrivial covering group of G is linear.

Proof. Suppose G̃ is a nontrivial covering group of G. Then G̃/F ≅ G for some discrete group F with |F | > 1. Since any finite dimensional representation σ of G̃ extends to a representation of G_{C}, σ(F) = I.

In particular, note that the double cover of SL(n, R) is not a linear group. This paper begins with a brief review of the Clifford algebra and spinors

followed by an examination of the finite group D_{n} and its representations. The bulk of the paper deals with the case by case study of the exceptional groups. Moreover, since every real simple exceptional simple Lie algebra may be realized as a subalgra of a real form of e_{8}, we devote several sections to explicitly describe a real form of the Lie algebra e_{8}. Sections 2, 3, and 4 are devoted to recalling results about the Clifford algebra, the group D_{n} and spinors. In sections 5 and 6, we explicitly construct the Lie algebra e_{8,C}. In sections 7 and 8, we obtain the split real forms of e_{8,C}, e_{7,C}, e_{6,C}, f_{4,C} and the corresponding simply connected groups, and in section 9, we construct the corresponding groups M̃. The split group G̃_{2(2)} is considered in section 10. The remaining nonsplit exceptional groups are examined in sections 11 through 16, and the classical groups are dealt with in section 17. The final results describing the structure of M̃ are summarized in section 18.

This paper grew out of joint work with Adam Korányi and Martin Reimann. I am particularly indebted to Adam Korányi for comments and suggestions. I am also grateful to David Benson who acquianted me with the properties of the group D_{n}. Notation:

1. In the subsequent sections E_{6,C}, E_{7,C}, E_{8,C}, F_{4,C}, and G_{2,C} will all denote complex simply connected simple exceptional Lie groups. We will use the notation from [6] to denote specific real exceptional groups; the groups E_{6(6)}, E_{6(2)}, E_{6(−14)}, E_{6(−26)} denote connected noncompact real forms of E_{6,C}; the groups E_{7(7)}, E_{7(−5)}, E_{7(−25)} denote connected noncompact real forms of E_{7,C}; the groups E_{8(8)}, E_{8(−24)} denote connected noncompact real forms of E_{8,C}; the groups F_{4(4)}, F_{4(−20)} denote connected noncompact real forms of F_{4,C}; and, the group G_{2(2)} denotes a connected noncompact real form of G_{2,C}.

2. The corresponding simply connected covering groups will be denoted by placing tilde over the letter E, F, or G.

3. The corresponding exceptional Lie algebras will be denoted by boldface lower case letters.

4. The symbol ⟨x_{1}, …, x_{n}⟩ will denote either the complex or real linear span of x_{1}, …, x_{n} or the group generated by these terms. The precise meaning will be clear from the context.

2. The Clifford Algebra

Let (,) denote the standard inner product on R^{n}. Extend (,) to be bilinear on C^{n} × C^{n}. The respective tensor algebras of R^{n} and C^{n} will be denoted by T (R^{n}) and T (C^{n}). If I(R^{n}) is the two sided ideal of T (R^{n}) generated by all elements of the form x ⊗ x + (x, x)1 (x ∈ R^{n}) and I(C^{n}) is the two sided ideal of T (C^{n}) generated by all elements of the form z ⊗ z + (z, z)1 (z ∈ C^{n}), then C(R^{n}) = T (R^{n})/I (R^{n}) is the Clifford algebra of R^{n} and C(C^{n}) = T (C^{n})/I (C^{n}) is the Clifford algebra of C^{n}. Since T (C^{n}) = T (R^{n}) ⊗_{R} C and I(C^{n}) = I (R^{n}) ⊗_{R} C, we have C(C^{n}) = C (R^{n}) ⊗_{R} C. If a, b ∈ C(V) denote their product in C(V) by a · b. An element of an irreducible C(V)-module is called a spinor. We now use one of the many well known constructions construct irreducible C(R^{n}) and C(C^{n})-modules. The one we are using may be found in Cartan [5].

Let {e_{j} : 1 ≤ j ≤ n} be the standard basis for R^{n}. Suppose first that n = 2k. Set f_{j} = (e_{2j−1} − ie_{2j})/\sqrt{2}, and g_{j} = (e_{2j−1} + ie_{2j})/\sqrt{2} for j ≤ k. Note that (f_{j}, g_{l}) = δ_{jl}. The spaces W = ⟨f_{1}, ⋯, f_{k}⟩ and W̄ = ⟨g_{1}, ⋯, g_{k}⟩ are maximally isotropic subspaces and are dual to each other with respect to the inner product.

Set

⋀^{∗} W = ⨁_{j=0}^{k} ⋀^{j} W, ⋀^{e} W = ⨁_{j=0}^{[k/2]} ⋀^{2j} W,

and

⋀^{o} W = ⨁_{j=0}^{[(k−1)/2]} ⋀^{2j+1} W

where, as usual, ⋀^{0} W = C. Consider the map

γ : {f_{1}, ⋯, f_{k}, g_{1}, ⋯, g_{k}} → End ⋀^{∗} W

defined by setting γ(f_{j})ω = f_{j} ∧ ω and γ(g_{j})ω = −2ι(g_{j})ω. Now γ extends to a linear map of C^{n} into End ⋀^{∗} W and hence to an algebra map of T (C^{n}) into End ⋀^{∗} W. Since γ maps any z ⊗ z + (z, z)1 to zero, I(C^{n}) is in the kernel of γ. Hence γ induces an algebra map of C(C^{n}) into End ⋀^{∗} W. Since C(C^{n}) = ⟨e_{i_{1}} ⋯ e_{i_{r}} : i_{1} < ⋯ < i_{r}⟩, we have

dimC(C^{n}) ≤ 2^{n} = 2^{2k} = dimEnd ⋀^{∗} W.

Notation: Set S = {1, ⋯, k} and for I ⊂ S set f_{I} = f_{i_{1}} ∧ ⋯ ∧ f_{i_{r}} where I = {i_{1}, ⋯, i_{r}} and i_{1} < ⋯ < i_{r}.

Theorem 2.1. The space ⋀^{∗} W is an irreducible C(C^{n})-module.

Proof. If U is a C(C^{n})-submodule and 0 ≠ ω ∈ U by applying appropriate γ(g_{j})′s, we see that f_{ϕ} = 1 is in U. Similarly, by applying appropriate γ(f_{j})′s, we see that U = ⟨f_{I} : I ⊂ S⟩. Hence U = ⋀^{∗} W.

By Burnsides’ theorem (see [8]) γ(C(C^{n})) = End ⋀^{∗} W, and by dimension we have that γ is an isomorphism and dimC(C^{n}) = 2^{n}.

If n = 2k + 1, define W and W̄ as before. Also, define γ on C^{n−1} as above. We extend γ to all of C^{n} by setting γ(e_{2k+1}) = i on ⋀^{e} W and = −i on ⋀^{o} W. Thus C(C^{2k}) is a simple associative algebra.

Note that if zϵC^{2k}, γ(z) interchanges ⋀^{e} W and ⋀^{o} W, but γ(e_{2k+1}) leaves both spaces invariant.

Since C(V) is an associative algebra, it has the natural structure of a Lie algebra where [a, b] = a · b − b · a.

Let

g(n) = ⟨x · y : (x, y) = 0, x, yϵR^{n}⟩ and g(n)_{C} = ⟨u · v : (u, v) = 0, u, vϵC^{n}⟩.

Proposition 2.2. g(n) and g(n)_{C} are Lie algebras.

The proposition is well known and its proof follows from a direct calculation. Note that

[u · v, w] = 2(u, w)v − 2(v, w)u(u, v, wϵC^{n}).

The map u · v → 2(v ⊗ u^{t} − v ⊗ u^{t}) is a Lie algebra map of g(n)_{C} onto on(n, C) that restricts to a map of g(n) onto o(n). We then obtain the following result.

Proposition 2.3. g(n) ≅ o(n), g(n)_{C} ≅ on(n, C), and g(n)_{C} ≅ g(n) ⊗_{R} C.

The group G(n) will be the group of all even products of the form v_{1} ⋯ v_{2l} with v_{1}, ⋯, v_{2l} unit vectors in R^{n}.

Proposition 2.4. G(n) is an analytic group with Lie algebra g(n).

Proof. Suppose x·y ∈ g(n) with x and y both unit vectors. Since (x·y)^{2} = −1, e^{tx·y} = cos t + sin tx · y = x · (− cos tx + sin ty).

Also, if |v_{1}| = |v_{2}| = 1, v_{1} · v_{2} = v_{1} · (− cos ψv_{1} + sin ψw) where (v_{1}, w) = 0. The proof now follows.

Let G(n)_{C} be the analytic subgroup of C(C^{n}) having Lie algebra g(n)_{C}. The bracket operations on C(C^{n}) and C(R^{n}) turn C^{n} and R^{n} into respective g(n)_{C} and g(n)-modules. Let τ : g(n)_{C} → EndC^{n}, τ : g(n) → EndR^{n} denote the representations induced by the bracket operation. Then τ induces representations (also denoted by τ) of G(n)_{C} on C^{n} and of G(n) on R^{n} where τ(g)v = g·v·g^{−1}.

As τ is a Lie algebra isomorphism, we see that τ : G(n) → SO(n) is a covering homomorphism. Hence τ : G_{C} → SO(n, C) is also a covering homomorphism. As τ((e_{1}, e_{2})^{2}) = τ(−I) = 1, we see that G(n) = Spin(n) and so G(n)_{C} = Spin(n, C) for n ≥ 3

If n = 2k + 1, Z(SO(n)) is trivial and so Z(Spin(n)) = {±I}. Now

(e_{1} ⋯ e_{n}) · e_{j} = (−1)^{n+1}e_{j} · (e_{1} ⋯ e_{n}).

Hence, if n is even, e_{1} ⋯ e_{n} ∈ Z(Spin(n)). A simple calculation yields

(e_{1} ⋯ e_{2j})^{2} = (−I)^{j}.

Thus, if n is even, Z(SO(n)) = {I, −I} and Z(Spin(n)) = {±I, ±e_{1} ⋯ e_{n}}. If n ≡ 0 mod 4, (e_{1} ⋯ e_{n})^{2} = 1 and so Z(Spin(n)) = Z_{2} × Z_{2}. If n ≡ 2 mod 4, (e_{1} ⋯ e_{n})^{2} = (−1) and Z(Spin(n)) = Z_{4}.

As γ : C(C^{n}) → End ⋀^{∗} W is a Lie algebra homomorphism, it restricts to a Lie algebra homomorphism of g(n)_{C}. Hence γ induces representations of Spin(n, C) and of Spin(n) on ⋀^{∗} W. Note that for n even ⋀^{e} W and ⋀^{o} W are both Spin(n)-modules.

Proposition 2.5. If n is odd, ⋀^{∗} W is an irreducible Spin(n)-module.

If n is even, ⋀^{e} W and ⋀^{o} W are irreducible and inequivalent Spin(n)- modules.

We will not prove this result here but will prove a much stronger result in the next section.

Remark. Note that for n = 2k + 1,

p_{±} = \frac{1 ± i^{k+1}(e_{1} ⋯ e_{n})}{\sqrt{2}}

are commuting central idempotents of C(C^{n}), and so

C(C^{n}) ≅ C(C^{2k}) ⊕ C(C^{2k})

as an associative algebra.

3. The group D_{n}

Let D_{n} = {e_{i_{1}} ⋯ e_{i_{2l}} : 1 ≤ i_{1}, …, i_{2l} ≤ n}. Now D_{n} is a subgroup of Spin(n) and

D_{n}/{±I} ≡ {diag(ϵ_{1}, …, ϵ_{n}) : ϵ_{1} ⋯ ϵ_{n} = 1, ϵ_{j} = ±1}

is the centralizer in SO(n) of the diagonal matrices of SL(n, R).

The following result was told to me by David Benson.

Theorem 3.1. If n is even, the modules ⋀^{e} W and ⋀^{o} W are inequivalent irreducible D_{n}-modules. If n is odd, ⋀^{∗} W is an irreducible D_{n}-module.

Proof. If I ⊂ S, γ(e_{2j−1} · e_{2j})f_{I} = if_{I} whenever j ∈ I and γ(e_{2j−1}·e_{2j})f_{I} = −if_{I} whenever j ∉ I. Suppose n = 2k and U is a D_{n}-submodule of ⋀^{e} W. If f ∈ ⋀^{e} W, write f = ∑ a_{I}(f)f_{I} where the sum is taken over all subsets of S of even order. Set ℓ(f) = |{I : a_{I}(f) ≠ 0}|. Select f ∈ U such that ℓ(f) = inf{ℓ(F) : 0 ≠ F ∈ U}. Suppose ℓ(f) > 1 with a_{I}(f) ≠ 0 and a_{J}(f) ≠ 0 for sets I and J. Without loss of generality, we may assume there is a j ∈ I ∼ J. Then (−iγ(e_{2j−1}·e_{2j})(f) − f) = F ∈ U and ℓ(F) < ℓ(f). This is a contradiction. Hence ℓ(f) = 1. So f = f_{I} for some set I. For J ⊂ S set e_{J} = e_{2j_{1}−1} ⋯ e_{2j_{2l}−1} where J = {j_{1}, …, j_{2l}} and j_{1} < ⋯ < j_{2l}. Then for J ⊂ S, we have γ(e_{J})γ(e_{I})f_{I} = cf_{J} for some c ≠ 0. Thus we have that f_{J} ∈ U for any J ⊂ S. Therefore ⋀^{e} W is an irreducible D_{n}-module.

Since γ(e_{2j−1} · e_{2k−1})f_{k} = cf_{j} for some c ≠ 0, we see that ⋀^{o} W is also an irreducible D_{n}-module. To see that these two modules are inequivalent note that an intertwining operator maps eigenvectors of the elements γ(e_{2j−1} · e_{2j}) to eigenvectors with the same eigenvalue. Since f_{S} is the only element of ⋀^{∗} W that is an eigenvector with eigenvalue i for each γ(e_{2j−1} · e_{2j}), we see that the two modules are inequivalent.

Similarly, it is easy to see that if n is odd, ⋀^{∗} W is an irreducible D_{n}- module.

If n = 2k,

|D_{n}| = 2^{2k} = 2^{2k−1} + (2^{k−1})^{2} + (2^{k−1})^{2} =

|D_{n}/{±I}| + dim End ⋀^{e} W + dim End ⋀^{o} W.

Hence ⋀^{e} W and ⋀^{o} W are –up to equivalence– the only irreducible D_{n}-modules of dimension > 1.

Similarly, if n = 2k + 1,

|D_{n}| = 2^{2k} + 2^{2k} = |D_{n}/{±I}| + dim End ⋀^{∗} W.

Hence ⋀^{∗} W is the only irreducible D_{n}-module of dimension > 1.

4. The Lie Algebra g(n)_{C}

If n = 2k set h_{j} = −i/2e_{2j−1} · e_{2j} = −1/4(f_{j} · g_{j} − g_{j} · f_{j}) and let h = ⟨h_{j} : j ≤ k⟩. Note that [h_{j}, f_{l}] = δ_{j,l}f_{l} and [h_{j}, g_{l}] = −δ_{j,l}g_{l}. Set n = ⟨f_{j} · f_{l}, f_{j} · g_{l} : j < l ≤ k⟩ and \overline{n} = ⟨g_{j} · g_{l}, g_{j} · f_{l} : j < l ≤ k⟩. If n = 2k, n and \overline{n} are maximal nilpotent Lie subalgebras of g(n)_{C} normalized by h, and g(n)_{C} = n + h + \overline{n}. In this case we see that h is a Cartan subalgebra of g(n)_{C}. Let ϵ_{l} ∈ h^{∗} be such that for H = ∑_{j=1}^{k} z_{j}h_{j}, ϵ_{j}(H) = z_{j}. Then, if Σ(ad) denotes the root system of g(n)_{C}, we have Σ(ad) = {±ϵ_{j} ± ϵ_{l} : 1 ≤ j < l ≤ k}.

If n = 2k + 1, set ñ = n + ⟨f_{j}·e_{2k+1} : 1 ≤ j ≤ k⟩ and

\overline{ñ} = \overline{n} + ⟨g_{j}·e_{2k+1} : 1 ≤ j ≤ k⟩.

In this case, ñ and \overline{ñ} are maximal nilpotent Lie algebras normalized by h, g(n)_{C} = ñ + h + \overline{ñ} and

Σ(ad) = {±ϵ_{j} ± ϵ_{l} : 1 ≤ j < l ≤ k} ∪ {±ϵ_{j} : 1 ≤ j ≤ k}.

We now examine the representation γ of g(n)_{C}. If H = ∑_{j=1}^{k}, z_{j}h_{j} and I ⊂ S,

γ(H)f_{I} = 1/2(∑_{j∈I} z_{j} − ∑_{j∈S∼I} z_{j})f_{I}.

Hence, the weights of γ are Σ(γ) = {1/2(±ϵ_{1} ± ⋯ ± ϵ_{k})}. If n = 2k, γ = γ_{e} ⊕ γ_{o} where γ_{e} and γ_{o} are the respective representations of g(n)_{C} on ⋀^{e} W and ⋀^{o} W. Then Σ(γ_{e}) consists of those weights in Σ(γ) that have an even number of plus signs in their expansions, and Σ(γ_{o}) consists of the weights of Σ(γ) that have an odd number of plus signs in their expansions.

Proposition 4.1. As a g(2k + 1)_{C}-module, End ⋀^{∗} W ≅ ⊕_{l=0}^{k} ⋀^{l} C^{2k+1}.

Proof. Consider the linear map Γ : ⋀^{l} C^{n} → End ⋀^{∗} W such that

Γ(ψ_{1} ∧ ⋯ ∧ ψ_{l})ω = 1/l! ∑_{σ∈P_{l}}(sgnσ)γ(ψ_{σ(1)}) ⋯ γ(ψ_{σ(l)})ω.

A simple calculation shows that Γ is an intertwining operator for any l. Using the fact that as g(n)_{C}-modules, ⋀^{l} C^{n} ≅ ⋀^{n−l} C^{n}, and noting that

dim(C ⊕ ⋯ ⊕ ⋀^{k} C^{2k+1}) = 2^{2k} = End ⋀^{∗} W,

we have our result.

Since C^{2k+1} = C^{2k} ⊕ Ce_{2k+1}, we see that as g(2k)_{C}-modules,

End ⋀^{∗} W ≅ C ⊕ ⊕_{l=1}^{k}(⋀^{l} C^{2k} ⊕ ⋀^{l−1} C^{2k}).

Thus we obtain the following.

Proposition 4.2. If n = 2k, as a g(n)_{C}-module, End ⋀^{∗} W ≅ 2(⊕_{l=0}^{k}(⋀^{l} C^{2k})) ⊕ ⋀^{k} C^{2k}.

Recall that the g(2k)_{C}-module, ⋀^{l} C^{2k}, is irreducible for l ≠ k, and ⋀^{k} C^{2k} = U_{!} + U_{2} where U_{1} is the irreducible module with highest weight vector f_{1} ∧⋯∧f_{k}, and U_{2} is the irreducible module with highest weight vector f_{1} ∧⋯∧ f_{k−1} ∧ g_{k}.

Proposition 4.3. The following are g(2k)_{C}-module isomorphisms. (α) If k = 2l + 1,

End ⋀^{e} W ≅ End ⋀^{o} W ≅ ⨁_{j=0}^{l} ⋀^{2j} C^{2k},
Hom(⋀^{e} W, ⋀^{o} W) ≅ ⨁_{j=0}^{l−1} ⋀^{2j+1} C^{2k} ⊕ U_{1},

and

Hom(⋀^{o} W, ⋀^{e} W) ≅ ⨁_{j=0}^{l−1} ⋀^{2j+1} C^{2k} ⊕ U_{2}.
(β) If k = 2l,
End ⋀^{e} W ≅ ⨁_{j=0}^{k−1} ⋀^{2j} C^{2k} ⊕ U_{1}, End ⋀^{o} W ≅ ⨁_{j=0}^{k−1} ⋀^{2j} C^{2k} ⊕ U_{2},

and

Hom(⋀^{o} W, ⋀^{e} W) ≅ Hom(⋀^{e} W, ⋀^{o} W) ≅ ⨁_{j=0}^{l−1} ⋀^{2j+1} C^{2k}.

Proof. Note that Γ(f_{1} ∧ ⋯ ∧ f_{k}) = 0 on ⋀^{o} W and Γ(f_{1} ∧ ⋯ ∧ f_{k−1} ∧ g_{k}) = 0 on ⋀^{e} W. If k is even, these operators preserve the parity of a form and if k is odd they reverse the parity. Note also that Γ(f_{1}∧⋯ f_{r}) preserves parity if r is even and reverses parity if r is odd. Our proposition now follows from proposition 4.2.

We have already seen that the space of g(2k + 1)_{C}-invariants in ⋀^{∗} W ⊗ ⋀^{∗} W is one dimensional. Suppose ω = ∑_{I⊂S} a(I, S ∼ I)f_{I} ⊗ f_{S∼I} is an invariant. Then γ(f_{p}·f_{q})(ω) = 0, and if J ∩ {p, q} = Ø and K = J ∪ {p, q}, we have

a(J, S ∼ J)f_{p} ∧ f_{q} ∧ f_{J} ⊗ f_{S∼J} + a(K, S ∼ K)f_{K} ⊗ f_{p} ∧ f_{q} ∧ f_{S∼K} = 0. Without loss of generality we may assume p < q, |{1, …, p − 1} ∩ J| = l and |{1, … q − 1} ∩ J| = m. Then f_{p} ∧ f_{q} ∧ f_{J} = (−1)^{l+m}f_{K} and

f_{p} ∧ f_{q} ∧ f_{S∼K} = (−1)^{p+q+l+m+1}f_{S∼J}.

Hence we have a(K, S ∼ K) = (−1)^{p+q}a(J, S ∼ J). Similarly,
γ(f_{q}·e_{2k+1})(ω) = 0.

So, if L = J ∪ {q}, we have a(L, S ∼ L) = (−1)^{q+|S|}a(J, S ∼ J). Finally, setting a(Ø, S) = 1 and σ(I) = ∑_{j∈I} j, we have a(I, S ∼ I) = (−1)^{σ(I)+k|I|} where k = |S|. Now σ(I) ≅ σ(S ∼ I) mod 2 if and only if k ≅ 0 mod 4 or k ≅ 3 mod 4, and σ(I) ≅ −σ(S ∼ I) mod 2 if and only if k ≅ 1 mod 4 or k ≅ 2 mod 4. Thus we have a(I, S ∼ I) = a(S ∼ I, I) if k ≅ 0 mod 4 or k ≅ 1 mod 4, and a(I, S ∼ I) = −a(S ∼ I, I) if k ≅ 2 mod 4 or k ≅ 3 mod 4. Summarizing our results, we have the following.

Proposition 4.4. For any k > 0 there is a unique, up to scalar multiple, g(2k + 1)_{C}-invariant ω ∈ ⋀^{∗} W ⊗ ⋀^{∗} W with the following properties.

1. ω is nondegenerate.

2. If k is odd, ⋀^{e} W and ⋀^{o} W are maximally isotropic spaces w.r.t. ω.

3. If k is even, ⋀^{e} W and ⋀^{o} W are mutually orthogonal.

4. If k ≅ 0 mod 4 or k ≅ 1 mod 4, ω is a symmetric form.

5. If k ≅ 2 mod 4 or k ≅ 3 mod 4, ω is a skew-symmetric form.

6. The form ω is the sum of two g(2k)_{C}-invariant forms.

Corollary 4.5. 1. If k ≅ 0 mod 4 or 1 mod 4, g(2k + 1)_{C} ⊂ ⋀^{2}(⋀^{∗} W).

2. If k ≅ 2 mod 4 or 3 mod 4, g(2k + 1)_{C} ⊂ S^{2}(⋀^{∗} W).

3. If k ≅ 0 mod 4, g(2k)_{C} may be considered as a subset of either ⋀^{2}(⋀^{e} W) or of ⋀^{2}(⋀^{o} W).

4. If k > 2 and k ≅ 2 mod 4, g(2k)_{C} may be considered as a subset of either S^{2}(⋀^{e} W) or of S^{2}(⋀^{o} W).

5. g(4)_{C} is the direct sum of two ideals with one contained in S^{2}(⋀^{e} W) and the other contained in S^{2}(⋀^{o} W) We conclude this section with another result about the group D_{n}.

Proposition 4.6. If V_{1}, V_{2}, and V_{3} are irreducible D_{n}-modules, dim Hom_{D_{n}}(V_{3}, V_{1} ⊗ V_{2}) ≤ 1.

Proof. We prove this for n odd. If either V_{1} or V_{2} is one dimensional, the result is obvious. If V_{1} ≅ V_{2} ≅ ⋀^{∗} W, the space V_{1} ⊗ V_{2} is the direct sum of inequivalent one dimensional representations.

The proof for n even is similar.

5. General Results

Let k be a complex reductive Lie algebra with Lie bracket [,]_{0}. Suppose ρ : k → Endp is an irreducible representation and Φ : ⋀^{2} p → k is a k intertwining operator. Set g = k + p and define [,] : g × g → g by setting

[x + u, y + v] = [x, y]_{0} + ρ(x)v − ρ(y)u + Φ(u ∧ v) for x, y ∈ k and u, v ∈ p. Clearly, [,] is skew symmetric and bilinear. For α, β, γ ∈ g set

J(α, β, γ) = [[α, β], γ] + [[β, γ], α] + [[γ, α], β]. Now g is a Lie algebra if and only if J = 0.

(i) If α, β, γ ∈ k, then J(α, β, γ) = 0.

(ii) If |{α, β, γ} ∩ k| = 2, J(α, β, γ) = 0 as ρ is a representation of k. (iii) If |{α, β, γ} ∩ k| = 1, J(α, β, γ) = 0 as Φ intertwines k.

Hence we see that g is a Lie algebra if and only if J(α, β, γ) = 0 for any α, β, γ ∈ p. For X ∈ k and α, β, γ ∈ p we have the identity ρ(X)(J(α, β, γ)) = J(ρ(X)α, β, γ) + J(α, ρ(X)β, γ) + J(α, β, ρ(X)γ). Suppose k = n^{∗} + h + n with h a Cartan subalgebra, h + n a Borel

subalgebra, and n^{∗} + h the opposite Borel.

Lemma 5.1. J = 0 if and only if J(α_{0}, β, γ) = 0 where 0 ≠ α_{0} ∈ p^{n} and β and γ are arbitrary elements of p.

Proof. If J(α_{o}, β, γ) = 0 for any β, γ ∈ p, it follows from the equation above that J(ρ(X_{1})α_{0}, β, γ) = 0 for any X_{1} ∈ n^{∗} and any β, γ ∈ p. By induction on r, it follows that J(ρ(X_{1}) ⋯ ρ(X_{r})α_{0}, β, γ) = 0 for any X_{1}, …, X_{r} ∈ n^{∗} and any β, γ ∈ p. Since p is an irreducible k-module, it follows that J = 0. The opposite implication is trivial.

Fix 0 ≠ β_{0} ∈ p^{n^{∗}}

Theorem 5.2. J = 0 if and only if J(α_{0}, β_{0}, γ) = 0 for any γ ∈ p.

Proof. Suppose J(α_{0}, β_{0}.γ) = 0 for any γ ∈ p. For X ∈ n we have J(α_{0}, ρ(X)β_{0}, γ) = −J(α_{0}, β_{0}, ρ(X)γ) = 0, an d so, for X_{1}, …, X_{r} ∈ n we have

J(α_{0}, ρ(X_{1}) ⋯ ρ(X_{r})β_{0}, γ) = (−1)^{r}J(α_{0}, β_{0}, ρ(X_{r}) ⋯ ρ(X_{1})γ) = 0. Since ρ(U(n))β_{0} = p, it follows from lemma 1 that J = 0.

The opposite implication is obvious.

Note that we may replace Φ by cΦ for any c ≠ 0.

6. The Complex Lie Algebra e_{8,C}

In this section we will construct the complex Lie algebra e_{8,C}. Although it is apparent from Cartan’s list of symmetric spaces ( Helgason [6] ) that such a construction exists, we will need the explicit construction given here in subsequent sections. Suppose n = 2k and let g(n)_{C} and ⋀^{∗} W be as in section 3. In order to construct a Lie algebra on g(n)_{C} + ⋀^{e} W as in section 4 with k = g(n)_{C} and p = ⋀^{e} W, since k is simple whenever k > 2, we must have k ⊂ ⋀^{2} ⋀^{e} W. Hence, k = 4l for some integer l.

Since γ is an irreducible faithful representation of k not equivalent to ad, k + p must be a simple Lie algebra if it is a Lie algebra provided Φ : ⋀^{2} p → k is nontrivial. Moreover, if k + p is a Lie algebra, a Cartan subalgebra of k is also a Cartan subalgebra of k + p, since 0 ∉ Σ(γ). So, if k + p = l_{k} is a Lie algebra, it is simple of rank k and dimension k(2k − 1) + 2^{k−1}. A simple examination of dimensions and ranks shows that l_{k} can be a simple Lie algebra only for k = 4 or 8. Apriori we may have either l_{4} = B_{4} or l_{4} = C_{4}, or l_{8} = e_{8}. By construction, we see that l_{4} ≅ B_{4}. We devote the rest of this section to an examination of the case k = 8.

For the rest of this section we take S = {1, …, 8}. For I ⊂ S with I = {i_{1}, …, i_{p}} and i_{1} < ⋯ < i_{p} set f_{I} = f_{i_{1}} ∧ ⋯ ∧ f_{i_{p}}, f̃_{I} = f_{i_{1}} ⋯ f_{i_{p}}, and g̃_{I} = g_{i_{1}} ⋯ g_{i_{p}}. Set I′ = S ∼ I.

Suppose l_{8} has the structure of a Lie algebra. Let I, J ⊂ S with |I| and |J| even and [f_{I}, f_{J}] ≠ 0. Setting

μ = 1/2( ∑_{j∈I∪J}ϵ_{j} − ∑_{j∈I′∪J′}ϵ_{j})

we have μ ∈ Σ(ad) where Σ(ad) is the root system of k. Moreover, if μ ∈ Σ(ad), μ = ±ϵ_{j} ± ϵ_{r} for j ≠ r. An elementary calculation yields the following.

Lemma 6.1. Suppose I, J ⊂ S with I and J even. If [f_{I}, f_{J}] ≠ 0, one of the following holds.

1. |I ∩ J| = 2 and |I| + |J| = 10, or equivalently, |I′ ∩ J′| = 0 and |I′| + |J′| = 6.

2. |I ∩ J| = 1 and |I| + |J| = 8, or equivalently, |I′ ∩ J′| = 1 and |I′| + |J′| = 8.

3. I′ = J.

4. |I ∩ J| = 0 and |I| + |J| = 6, or equivalently, |I′ ∩ J′| = 2 and |I′| + |J′| = 10.

Lemma 6.2. Suppose I, J ⊂ S with I and J even and [f_{I}, f_{J}] ≠ 0.

1. If |I ∩ J| = 2 and |I| + |J| = 10, [f_{I}, f_{J}] = cf̃_{I∩J}.

2. If |I ∩ J| = 1 and |I| + |J| = 8, [f_{I}, f_{J}] = cf_{I∩J} · g_{I′∩J′}.

3. If J = I′, [f_{I}, f_{J}] ∈ h.

4. If |I ∩ J| = 0 and |I| + |J| = 6, [f_{I}, f_{J}] = cg̃_{I′∩J′}.

The proof of this lemma follows immediately from an examination of the root spaces of k.

Lemma 6.3. Suppose I, J, K ⊂ S with |I|, |J|, and |K| all even. If [f_{I}, f_{J}] ≠ 0 and [f_{I}, f_{K}] ≠ 0, then [f_{J}, f_{K}] = 0.

Proof. If |I| = 0, lemma 6.1 guarantees that |J| + |K| ≥ 12, and so by lemma 6.1, [f_{J}, f_{K}] = 0.

If |I| = 8, lemma 6.1 states that |J| + |K| ≤ 4, and so again by lemma 6.1, [f_{J}, f_{K}] = 0.

If |I| = 2, lemma 6.1 states that 4 ≤ |J|, |K| ≤ 8. From lemma 6.1 we need to consider only the cases (|J|, |K|) = (4, 4), (|J|, |K|) = (4, 6), and (|J|, |K|) = (6, 4). If |J| = |K| = 4, I ∩ J = I ∩ K = Ø and so |J ∩ K| ≥ 2 and hence the bracket is 0. If |J| = 4 and |K| = 6, we have I ∩J = Ø and |I ∩K| ≤ 1. Then |J ∩ K| ≥ 3 and the bracket again must be 0. The case (|J|, |K|) = (6, 4) is clear.

If |I| = 4, lemma 6.1 states that 2 ≤ |J|, |K| ≤ 6. Since [f_{J}, f_{K}] = 0 if |J| + |K| ≤ 4 or |J| + |K| ≥ 12 we need to consider the cases where (|J|, |K|) is one of the pairs (2, 4), (2, 6), (4, 2), (4, 4), (4, 6), (6, 2) or (6, 4). For (|J|, |K|) = (2, 4), |I ∩ J| = 0 and |I ∩ K| ≤ 1, and hence |J ∩ K| ≥ 1. Thus the bracket is 0in this case and also if (|J|, |K|) = (4, 2). If (|J|, |K|) = (2, 6), we now have |J ∩ K| = 2, and hence the bracket is 0 in this case and also if (|J|, |K|) = (6, 2). For (|J|, |K|) = (4, 4), we have |I ∩ J| ≤ 1 and |I ∩ K| ≤ 1. This guarantees that |J ∩ K| ≥ 2 and hence the bracket is 0. If (|J|, |K|) = (4, 6) or (6, 4), we have |J ∩ K| ≥ 3 and again the bracket is 0.

If |I| = 6, 0 ≤ |J|, |K| ≤ 4. Since the bracket is 0 if |J| + |K| ≤ 4, we need now only consider the cases where (|J|, |K|) = (2, 4), (4, 2) or (4, 4). If (|J|, |K|) = (2, 4), |I ∩ J| ≤ 1 and |I ∩ K| = 2. Thus |J ∩ K| ≥ 1 and the bracket is 0 in this case and in the case where (|J|, |K|) = (4, 2). Finally, if |J| = |K| = 4, we have |I ∩ J| = |I ∩ K| = 2 and thus |J ∩ K| ≥ 2. This guarantees that the bracket is 0 in the case.

We now proceed to define [,] : p × p → k. From the simplicity of k we know that [f_{1} ∧f_{2}, f_{S}] is a non zero multiple of f_{1} ·f_{2}. Setting [f_{1} ∧f_{2}, f_{S}] = f_{1} ·f_{2}, the intertwining condition now forces the remaining brackets.

Lemma 6.4. If |I| = 2, [f_{I}, f_{S}] = f̃_{I}.

Proof. If I = {1, 2} we already have the result. Suppose I = {1, k} with k > 2. Then we must have

−2f̃_{I} = [f_{k} · g_{2}, f_{1} · f_{2}]_{0} = [γ(f_{k} · g_{2})(f_{1} ∧ f_{2}), f_{S}] + [f_{1} ∧ f_{2}, γ(f_{k} · g_{2})f_{S}] =
[γ(f_{k} · g_{2})f_{1} ∧ f_{2}, f_{S}] + 0 = −2[f_{I}, f_{S}].

Thus we have our result in this case. The proofs for I = {2, k} (2 < k) and I = {k, l} (2 < k < l) are similar.

Recall that σ(I) = ∑_{j∈I} j.

Lemma 6.5. [f_{I}, f_{I′}] = (−1)^{σ(I)}(1/2)(∑_{j∈I} h_{j} − ∑_{j∈I′} h_{j}).

Proof. Clearly, [f_{Ø}, f_{S}] = ∑_{j=1}^{8} a_{j}h_{j} for some a_{1}, …, a_{8}. As

[f_{j} · f_{k}, [f_{Ø}, f_{S}]] = [f_{j} ∧ f_{k}, f_{S}] = −(a_{j} + a_{k})f_{j} · f_{k} = f_{j} · f_{k},

a_{j} + a_{k} = −1 for all j, k. Hence a_{j} = −1/2 for all j, and [f_{Ø}, f_{S}] = −1/2(∑_{j=1}^{8} h_{j}).

Now

[g_{j} · g_{k}, [f_{j} ∧ f_{k}, f_{S}]] = [−4f_{Ø}, f_{S}] + 4(−1)^{j+k}[f_{j} ∧ f_{k}, f_{S∼{j,k}}] =

[g_{j} · g_{k}, f_{j} · f_{k}] = −2(f_{j} · g_{j} − g_{k} · f_{k}) = 4(h_{j} + h_{k}).

Hence

[f_{j} ∧ f_{k}, f_{S∼{j,k}}] = (−1)^{j+k}(h_{j} + h_{k} − (1/2)(∑_{l=1}^{8} h_{l})),

and the result holds in this case. Moreover, since (−1)^{σ(I)} = (−1)^{σ(I′)}, the result holds if |I| is 0, 2, 6 or 8, It remains only to consider the case where |I| = 4.

Suppose I = {a, b, c, d} where a < b < c < d. Since

[f_{{c,d}}, f_{{c,d}′}] = (−1)^{c+d}(1/2)(h_{c} + h_{d} − ∑_{j∈{c,d}′} h_{j}),

we have after applying f_{a} · f_{b} that [f_{I}, f_{{c,d}′}] = (−1)^{c+d}f_{a} · f_{b}. Thus, we have

[g_{a} · g_{b}, [f_{I}, f_{{c,d}′}]] = −4[f_{{c,d}}, f_{{c,d}′}] + 4(−1)^{a+b}[f_{I}, f_{I′}] = (−1)^{c+d}4(h_{a} + h_{b}).

After rearranging terms, our result also holds in this case.

The remaining brackets will be obtained as follows. Suppose u, v ∈ p with [u, v] known and X ∈ k such that γ(X)u = 0. Then [u, γ(X)v] = [X, [u, v]].

Lemma 6.6. Suppose |I| = 2, |J| = 6, and |I ∩ J| = 1, then [f_{I}, f_{J}] = ±(1/2)f_{I∩J} · g_{I′∩J′}.

Proof. Suppose {a} = I ∩ J and {b} = I′ ∩ J′. Then γ(f_{a} · g_{b})f_{I′} = ±2f_{J}. Thus [f_{I}, f_{J}] = ±1/2[f_{a} · g_{b}, [f_{I}, f_{I′}]] and our result now follows from lemma 6.5.

Lemma 6.7. Suppose |I| = 2, |J| = 4 and I ∩ J = Ø. Then [f_{I}, f_{J}] = ±(1/4)g̃_{(I∪J)′}.

Proof. Suppose {a, b} = (I ∪ J)′. The proof follows from the identity

[f_{I}, γ(g_{a} · g_{b})f_{I′}] = [g_{a} · g_{b}, [f_{I}, f_{I′}]].

We summarize the remaining possible brackets in the following lemma; the proofs are similar to those above.

Lemma 6.8. 1. If |J| = 6, [f_{Ø}, f_{J}] = (−1)^{σ(J′)+1}(1/4)g_{J′}.

2. If |I| = 4, |J| = 6 and |I ∩ J| = 2, then [f_{I}, f_{J}] = ±f_{I∩J}.

3. If |I| = 4, |J| = 4 and |I ∩ J| = 1, then [f_{I}, f_{J}] = ±(1/2)f_{a} · g_{b} where {a} = I ∩ J and {b} = I′ ∩ J′. The remaining brackets are obtain from the skew symmetry of [,].

Theorem 6.9. The bracket operation turns k + p into a simple Lie algebra.

Proof. Since [,] : ⋀^{2} p → k is a k intertwining operator, we need only check that the Jacobi identity J(u, v, w) = 0 holds for u, v, w ∈ p. From theorem 5.2 we need only check that J(f_{S}, f_{Ø}, f_{I}) = 0 for any I ⊂ S with |I| even. For |I| = 0 or 8 the result is obvious.

For |I| = 2,

J(f_{S}, f_{Ø}, f_{I}) = [[f_{S}, f_{Ø}], f_{I}] + [[f_{I}, f_{S}], f_{Ø}] =

γ(1/2 ∑_{j=1}^{8} h_{j})f_{I} + f_{I} = 1/4(−2 + 4)f_{I} + f_{I} = 0.

For |I| = 4,

J(f_{S}, f_{Ø}, f_{I}) = [[f_{S}, f_{Ø}], f_{I}] = γ(1/2 ∑_{j=1}^{8} h_{j})f_{I} = 0.

For |I| = 6,
J(f_{S}, f_{Ø}, f_{I}) = [[f_{S}, f_{Ø}], f_{I}] + [[f_{Ø}, f_{I}]f_{S}] = f_{I} + [[f_{Ø}, f_{I}], f_{S}].

If I′ = {a, b} with a < b, [f_{Ø}, f_{I}] = (−1)^{a+b+1}1/4g_{a}·g_{b}. Hence

J(f_{S}, f_{Ø}, f_{I}) = f_{I} + (−1)^{a+b+1}(1/4)γ(g_{a} · g_{b})f_{S} = f_{I} − f_{I} = 0.

Thus we see that g = k + p has the structure of a Lie algebra. Since it is simple of rank 8 and dimension 248, it must be e_{8,C}.

7. Real Forms e_{8(8)}, e_{7(7)}, e_{6(6)}, and f_{4(4)}

Throughout this section e_{j(j)} will denote a real split Lie algebra of e_{j,C}. Fix k_{R} = g(16) = spin(16). For I ⊂ S set

X(I) = 2^{|I′|/2}f_{I} + (−1)^{σ(I)}2^{|I|/2}f_{I′}

and

Y (I) = i(2^{|I′|/2}f_{I} − (−1)^{σ(I)}2^{|I|/2}f_{I′}).

Let p_{R} = ⟨X(I), Y (I) : I ⊂ S, |I| even ⟩. For convenience set ϵ(I) = (−1)^{σ(I)}. Note that X(I) = ϵ(I)X(I′) and Y (I) = −ϵ(I)Y (I′).

Suppose I ⊂ S, l ∉ I and J = I ∪ {l}. If |I ∩ {1, …, l − 1}| = p, we have the following equations:

γ(e_{2l−1})X(I) = (−1)^{p}X(J); γ(e_{2l−1})Y (I) = (−1)^{p}Y (J);

γ(e_{2l})X(I) = (−1)^{p}Y (J); and, γ(e_{2l})Y (I) = (−1)^{p+1}X(J).
Since k_{R} = ⟨e_{j} · e_{l} : j ≠ l⟩, we have that [k_{R}, p_{R}] ⊂ p_{R}.

Theorem 7.1. g_{R} = k_{R} + p_{R} is a Lie algebra.

Proof. Since [k_{R}, p_{R}] ⊂ p_{R}, it remains only to prove [p_{R}, p_{R}] ⊂ k_{R}. Since p_{R} is an irreducible k_{R}-module, it suffices to prove [p_{R}, u] ⊂ k_{R} for a 0 ≠ u ∈ p_{R}. Take u = X(Ø). Now [X(I), X(Ø)] = 0 unless |I| = 2 or |I| = 6. Since X(I′) = ϵ(I)X(I) we need only consider the case |I| = 2. In this case,

[X(I), X(Ø)] = 2^{|I′|/2}[f_{I}, f_{S}] + ϵ(I)2^{|I|/2+4}[f_{I′}, f_{Ø}] = 2^{3}f̃_{I} + ϵ(I)2^{5}[f_{I′}, f_{Ø}].

Applying 1 of lemma 9 we see

[X(I), X(Ø)] = 2^{3}f̃_{I} + ϵ(I)2^{5}(1/4)ϵ(I)g̃_{I} = 8(f̃_{I} + g̃_{I}) ∈ k_{R}.

Similarly, for |I| = 2, we see that [Y (I), X(Ø)] = i8(f̃_{I} − g̃_{I}) ∈ k_{R}. Finally,

[Y (Ø), X(Ø)] = 2^{5}i[f_{Ø}, f_{S}] = −2^{4}i ∑_{j=1}^{8} h_{j} ∈ k_{R}.

Hence, g_{R} is a Lie algebra.

Since g_{R} ⊗ C = g, we obtain the following.

Corollary 7.2. g_{R} is a real form of e_{8,C}.

We now construct a maximal abelian subalgebra a of p_{R}. If I, J ⊂ S with |I| = |J| = 4 and |I ∩ J| = 2 observe that [X(I), X(J)] = 0. Let

C = {Ø, {1, 2, 3, 4}, {1, 2.5.6}, {1, 2, 7, 8},
{1, 3, 5, 7}, {1, 3, 6, 8}, {1, 4, 5, 8}, {1, 4, 6, 7}}.

If a = ⟨X(I) : I ∈ C ⟩, a is a maximal abelian subalgebra of p_{R}. As dim a = 8, the following is obvious.

Theorem 7.3. The Lie algebra g_{R} is a real split form of e_{8,C}. Thus g_{R} = e_{8(8)}.

Suppose T r : h → C is the map T r(∑_{j=1}^{8} z_{j}h_{j}) = ∑_{j=1}^{8} z_{j} and

h_{0} = {H : T r(H) = 0}. Setting k_{0} = h_{0} + ⟨f_{j} · g_{k} : j ≠ k⟩, we see that k_{0} ≅ sl(8, C). Now [k_{0}, ⋀^{l} W] = ⋀^{l} W and if p_{0} = ⋀^{4} W it follows from lemmas 7 and 10 that [p_{0}, p_{0}] ⊂ k_{0}. Hence g_{0} = k_{0} + p_{0} is a Lie algebra. As p_{0} is an irreducible k_{0}-module and k_{0} is simple, g_{0} is a simple Lie algebra. Moreover, since rank(g_{0}) = 7 and dim g_{0} = 133, g_{0} = e_{7,C}. Now g_{0,R} = g_{0} ∩ g_{R} is a real form of g_{0}, and g_{0,R} = k_{0,R} + p_{0,R} where k_{0,R} = k_{0} ∩ k_{R} and p_{0,R} = p_{0} ∩ p_{R}. If C_{0} = C ∼ Ø, a_{0} = ⟨X_{I} : I ∈ C_{0} ⟩ is a maximal abelian subalgebra of p_{0,R} of dimension 7. Hence we have

Theorem 7.4. The Lie algebra g_{0,R} is the split Lie algebra e_{7(7)}.

We now construct the lie algebra e_{6(6)}.

Let

Ω = f_{1} ∧ f_{2} + f_{3} ∧ f_{4} + f_{5} ∧ f_{6} + f_{7} ∧ f_{8}

and

Ω^{∗} = g_{1} ∧ g_{2} + g_{3} ∧ g_{4} + g_{5} ∧ g_{6} + g_{7} ∧ g_{8}.

Set k_{00} = {X ∈ k_{0} : X(Ω) = 0} and p_{00} = {u ∈ p_{0} : ι(Ω^{∗})u = 0}. Since Ω is a non degenerate 2-form, we obtain the following.

Johnson 301

Proposition 7.5. k_{00} is the Lie algebra sp(4, C).

It is an elementary exercise to see that

k_{00} = ⟨h_{2j−1} − h_{2j} : j ≤ 4⟩+ ⟨f_{2j−1} · g_{2j}, f_{2j} · g_{2j−1} : j ≤ 4⟩ + ⟨f_{2j−1} · g_{2k−1} − f_{2k} · g_{2j} : j ≠ k⟩+

⟨f_{2j−1} · g_{2k} + f_{2k−1} · g_{2j}, f_{2j} · g_{2k−1} + f_{2k} · g_{2j−1} : j < k⟩.

Moreover, restricting ourselves only to I ⊂ S with |I| = 4, we have p_{00} = ⟨f_{I} : |I ∩ {2j − 1, 2j}| = 1, 1 ≤ j ≤ 4⟩+

⟨f_{I} − f_{J} : |I ∩ J| = 2, |I ∩ J ∩ {2j − 1, 2j}| ≤ 1, 1 ≤ j ≤ 4, {2k − 1, 2k} ⊂ I, {2k − 1, 2k} ∩ J = Ø, {2l − 1, 2l} ⊂ J, {2l − 1, 2l} ∩ I = Ø for some k, l ≤ 4⟩+ ⟨aX_{J} +bX_{K} +cX_{L} : J = {1, 2, 3, 4}, K = {1, 2, 5, 6}, L = {1, 2, 7, 8}, a+b+c = 0⟩. Now an elementary calculation yields Proposition 7.6. g_{00} = k_{00} + p_{00} is a Lie algebra. Now g_{00} is easily seen to be simple of dimension 78 with Cartan subalgebra

⟨h_{2j−1} − h_{2j} : j ≤ 4⟩+

⟨aX_{J} +bX_{K} +cX_{L} : a+b+c = o, J = {1, 2, 3, 4}, K = {1, 2, 5, 6}, L = {1, 2, 7, 8}⟩. As rank g_{00} = 6, we see that g_{00} = e_{6,C}. If we set g_{00,R} = g_{00} ∩ g_{R}, we have g_{00,R} is a real form of e_{6,C}. Since

a_{00} = ⟨X_{I} : I = {1, 3, 5, 7}, {1, 3, 6, 8}, {1, 4, 5, 8}, {1, 4, 6, 7}⟩+

⟨aX_{J} + bX_{K} + cX_{L} : a, b, c ∈ R, a + b + c = 0,
J = {1, 2, 3, 4}, K = {1, 2, 5, 6}, L = {1, 2, 7, 8}⟩

is a maximal abelian subalgebra of p_{00,R}, and is of dimension 6, g_{00,R} is a split real form of e_{8,C}. In other words Theorem 7.7. g_{00,R} = e_{6(6)}.

Suppose Ω^{0} = f_{3} ∧ f_{4} + f_{5} ∧ f_{6} + f_{7} ∧ f_{8} and Ω^{0∗} = g_{3} ∧ g_{4} + g_{5} ∧ g_{6} + g_{7} ∧ g_{8}. Set k_{000} = {X ∈ k_{00} : XΩ^{0} = Ω^{0}} and p_{000} = {u ∈ p_{00} : ι(Ω^{0∗})u = 0}. Proposition 7.8. k_{000} ≅ sl(2, C) ⊕ sp(3, C)

Proof. A direct calculation we have:

k_{000} = ⟨h_{1} − h_{2}, f_{1} · g_{2}, f_{2} · g_{1}⟩⊕
(⟨h_{2j−1} − h_{2j} : 2 ≤ j ≤ 4⟩ + ⟨f_{2j−1} · g_{2j}, f_{2j} · g_{2j−1} : 2 ≤ j ≤ 4⟩+
⟨f_{2j−1} · f_{2k−1} − f_{2k} · g_{2j} : j, k > 1, j ≠ k⟩+
⟨f_{2j−1} · g_{2k} + f_{2k−1} · g_{2j}, f_{2j} · g_{2k−1} + f_{2k} · g_{2j−1} : 1 < j < k⟩).

Note that the first direct summand is sl(2, C) and the second is sp(3, C).

Similarly, we have

p_{000} = ⟨f_{I} : |I ∩ {2j − 1, 2j}| = 1, 1 ≤ j ≤ 4⟩+

⟨f_{I} − f_{J} : |I ∩ J| = 2, |I ∩ {1, 2}| = 1, I ∩ {1, 2} = J ∩ {1, 2}, {2j − 1, 2j} ⊂ I,

{2j − 1, 2j} ∩ J = Ø, {2k − 1, 2k} ⊂ J, {2k − 1, 2k} ∩ I = Ø for some j, k >

Now g_{000} = k_{000} + p_{000} is a simple Lie algebra of rank 4 and dimension 52. Setting g_{000,R} = g_{000} ∩ g_{00,R} we have

a_{000} = ⟨X_{I} : I = {1, 3, 5, 7}, {1, 3, 6, 8}, {1, 4, 5, 8}, {1, 4, 6, 7}⟩

is a maximal abelian subalgebra of, p_{000,R}. Thus we have

Theorem 7.9. g_{000} = f_{4,C}, and g_{000,R} = f_{4(4)}.

8. The Groups Ẽ_{6(6)}, Ẽ_{7(7)}, Ẽ_{8(8)}, and F̃_{4(4)}

From Bourbaki [4], we know that Z(E_{6,C}) = Z_{3}, Z(E_{7,C}) = Z_{2}, and Z(E_{8,C}) = {I}. Since Z(Ẽ_{8(8)}) = {1, e_{1} ⋯ e_{16}} ≅ Z_{2}, Ẽ_{8(8)} is not a linear group. However, Ẽ_{8(8)}/Z(Ẽ_{8(8)}) = E_{8(8)} is a linear group and hence so are all of its subgroups. Also, note from [4] that Z(F_{4,C}) = {1}.

Proposition 8.1. The analytic subgroup of Ẽ_{8(8)} with Lie algebra e_{7(7)} is Ẽ_{7(7)}.

Proof. Let G_{0} be the analytic subgroup of Ẽ_{8(8)} with Lie algebra e_{7(7)}. Now the maximal compact subgroup of Ẽ_{7(7)} is SU(8), and the center of Ẽ_{7(7)} is Z_{4}. If u_{j} = (1 + e_{2j−1} · e_{2j})/\sqrt{2} and z = u_{1} ⋯ u_{8}, we have z·f_{j}·z^{−1} = if_{j} and z·g_{j}·z^{−1} = −ig_{j} and so z centralizes e_{7(7)}. As z^{2} = e_{1}⋯e_{16}, it suffices to show z ∈ G_{0}. Since exp(πi/2)h_{j} = u_{j}, u_{j}^{4} = 1, and h_{0} = {H ∈ h : T r(H) = 0} is a Cartan sualgebra of e_{7(7)}, it follows that u_{1}^{−1}u_{j} ∈ G_{0} for any j ≤ 8. Hence (u_{1}^{−1}·u_{2}) ⋯ (u_{1}^{−1}u_{8}) = z ∈ G_{0} and so G_{0} = Ẽ_{7(7)}.

Note that since Z(Ẽ_{7(7)}) = Z_{4} and Z(E_{7,C}) = Z_{2}, Ẽ_{7(7)} is not a linear group.

Proposition 8.2. The analytic subgroup of Ẽ_{7(7)} with Lie algebra e_{6(6)} is Ẽ_{6(6)}.

Proof. Let G_{00} be the analytic subgroup of Ẽ_{7(7)} with Lie algebra e_{6(6)}. Since the maximal compact subgroup of Ẽ_{6(6)} is Sp(4), Z(Ẽ_{6(6)}) = Z_{2}. It thus suffices to show that z^{2} ∈ G_{00}. Since ⟨h_{2j−1} − h_{2j} : j ≤ 4⟩ ⊂ k_{00,R},

exp(πi) ∑_{j=1}^{4}(h_{2j−1} − h_{2j}) = z^{2} ∈ G_{00} and hence G_{00} = Ẽ_{6(6)}.

Again since Z(E_{6,C}) = Z_{3}, Ẽ_{6(6)} is not a linear group.

Remark. We have Z(Ẽ_{8(8)}) ⊂ Ẽ_{6(6)} ⊂ Ẽ_{7(7)} ⊂ Ẽ_{8(8)} with no Ẽ_{j(j)} linear. However, all Ẽ_{j(j)}/Z(Ẽ_{8(8)}) are linear. As Z(F̃_{4(4)}) ≅ Z_{2}, F̃_{4(4)} is not a linear group. Since Z(F̃_{4(4)}) = Z(Ẽ_{8(8)}), F̃_{4(4)}/Z(Ẽ_{8(8)}) = F_{4(4)} is a linear group.

Since ∑_{j=1}^{4} i(h_{2j−1} − h_{2j}) ∈ f_{4(4)} and exp π ∑_{j=1}^{4} i(h_{2j−1} − h_{2j}) = e_{1} ⋯ e_{16}, the analytic subgroup of Ẽ_{6(6)} having Lie algebra f_{4(4)}, is F̃_{4(4)}. We state the following.

Proposition 8.3. The analytic subgroup of Ẽ_{6(6)} having Lie algebra f_{4(4)} is F̃_{4(4)}.

9. The Group M̃_{j} for the Groups Ẽ_{j(j)} and F̃_{4(4)}

Recall a, a_{0},a_{00}, and a_{000} from section 7. For convenience set a_{8} = a, a_{7} = a_{0}, a_{6} = a_{00} and a_{4} = a_{000}. Let K̃_{j} be the maximal compact subgroup of Ẽ_{j(j)} for j = 6, 7 or 8 or of F̃_{4(4)} for j = 4; set M̃_{j} = K̃_{j} ∩ Z(a_{j}). We now give an explicit description of the groups M̃_{j}. Recall from [10] that M̃_{j}/Z(Ẽ_{8(8)}) ≅ (Z_{2})^{j} and hence |M̃_{j}| = 2^{j+1}.

Consider the element ∗ = e_{1}·e_{3}⋯e_{15} of K̃_{8} = Spin(16) and note that ∗^{2} = 1. A simple calculation yields the following.

Proposition 9.1. If I ⊂ S, γ(∗)f_{I} = (−1)^{σ(I)+|I|}2^{(|I|−|I′|)/2}f_{I′}

Corollary 9.2. ∗ ∈ M̃_{8}.

Recall that

C = {Ø, {1, 2, 3, 4}, {1, 2.5.6}, {1, 2, 7, 8},
{1, 3, 5, 7}, {1, 3, 6, 8}, {1, 4, 5, 8}, {1, 4, 6, 7}}

and a = ⟨X_{I} : I ∈ C ⟩. Note that for I ∈ C , σ(I) = 1 and so X_{I} = 4(f_{I} + f_{I′}). We first give an explicit description of M̃_{7}. If P_{8} denotes the permutation group on 8 elements P_{8} ⊂ U(8) and the elements of even order are in SU(8).Suppose

P = {(12)(34)(56)(78), (13)(24)(57)(68), (14)(23)(58)(67), (15)(26)(37)(48), (16)(25)(38)(47), (17)(28)(35)(46), (18)(27)(36)(45), I} ⊂ P_{8}.

It is easy to see that P is an abelian group of order 8 with every element other than the identity of order 2.

Now let Q be the group generated by the following elements:

ω_{1} = iI, ω_{2} = diag(−1, −1, 1, 1, 1, 1, −1, −1),

ω_{3} = diag(−1, −1, 1, 1, −1, −1, 1, 1), and ω_{4} = diag(−1, 1, −1, 1, −1, 1, −1, 1).

Proposition 9.3. M̃_{7} = P Q and [M̃_{7}, M̃_{7}] = Z(Ẽ_{8(8)}). It has precisely two inequivalent irreducible representations of dimension 8, and all other irreducible representations are one dimensional.

Proof. If ω_{1}^{k_{1}}ω_{2}^{k_{2}}ω_{3}^{k_{3}}ω_{4}^{k_{4}} = 1, the (4,4)-entry is 1 and so it follows that ω_{1}^{k_{1}} = 1. Now, since the (3,3)-entry is 1, ω_{4}^{k_{4}} = 1. Finally, since the (8,8)-entry is 1, ω_{2}^{k_{2}} = 1 and hence ω_{3}^{k_{3}} = 1. Thus Q is a an abelian group of order 32. Now ω_{1}^{2} = e_{1} ⋯ e_{16} = −I, and observe that for p ∈ P and q ∈ Q, pq = ±qp. Now P ⊂ M̃_{7} and Q ⊂ M̃_{7} and hence P Q = {pq : p ∈ P, q ∈ Q} is a subgroup of M̃_{7}. Since |P Q| = |M̃_{7}| = 2^{8}, P Q = |M̃_{7}|. Now M̃_{7} is a subgroup of SU(8) and both W ≅ C^{8} and ⋀^{7} W ≅ (C^{8})^{∗} are irreducible inequivalent M̃_{7}-modules. As |χ(M̃_{7})| = 2^{7} and |M̃_{7}| = 8^{2} + 8^{2} + 2^{7} = 2^{8}, these are the only irreducible modules of M̃_{7} of dimension > 1.

We now describe M̃_{8}.

Proposition 9.4. M̃_{8} = M̃_{7} ∪ ∗M̃_{7} and [M̃_{8}, M̃_{8}] = Z(Ẽ_{8(8)}). It has –up to equivalence– precisely one irreducible representation of dimension 16, and all other irreducible representations are one dimensional.

Proof. Now M̃_{7} ⊂ M̃_{8} and ∗ ∈ M̃_{8}. Since |M̃_{8}| = 2^{9}, we have M̃_{8} = M̃_{7}∪∗M̃_{7}. Moreover, since M̃_{8}/Z(Ẽ_{8(8)}) ≅ (Z_{2})^{8} and M̃_{8} is not abelian, we have [M̃_{8}, M̃_{8}] = {1, e_{1}⋯e_{16}}. Since ∗ : ⋀^{r} W → ⋀^{8−r} W and W and ⋀^{7} W are inequivalent irreducible M̃_{7}-modules, W + ⋀^{7} W is an irreducible 16-dimensional M̃_{8}-module. As |χ(M̃_{8})| = 2^{8} and |M̃_{8}| = 2^{9} = 2^{8} + (16)^{2} we see that this –up to equivalence– is the only irreducible M̃_{8}-module of dimension > 1. Observe that for p ∈ P with p(1) = j that

γ(∗p)(f_{1}) = γ(∗)(f_{j}) = (−1)^{j+1}2^{−3}f_{{j}′}

and

γ(p∗)(f_{1}) = 2^{−3}γ(p)f_{{1}′} = 2^{−3}(−1)^{j+1}f_{{j}′}.

Hence [∗, p] = 1 for any p ∈ P. Similarly, we obtain [∗, ω_{j}] = 1 for 2 ≤ j ≤ 4, and [∗, ω_{1}] = −I = e_{1}⋯e_{16}. The remaining commutation relations in M̃_{8} are easy to compute.

Proposition 9.5. M̃_{6} = {m ∈ M̃_{7} : ∧^{2}γ(m)Ω = Ω}, and – up to equivalence – M̃_{6} has exactly one irreducible representation of dimension > 1.

Proof. Let F = {m ∈ M̃_{7} : ∧^{2}γ(m)Ω = Ω}. Clearly, F is a subgroup of M̃_{6}. Now

P′ = F ∩ P = {(13)(24)(57)(68), (15)(26)(37)(48), (17)(28)(35)(46), 1}

a subgroup of order 4, and Q′ = F ∩ Q a subgroup of Q generated by ω_{1}^{2}, ω_{2}, ω_{3} and ω_{1}ω_{4} and the order of Q′ is 16. So P′Q′ is a subgroup of F of order 2^{6}.

Setting τ = (12)(34)(56)(78), we see that τω_{1} ∈ F. Thus P′Q′ ∪ τω_{1}P′Q′ is a subgroup of F of order 2^{7}. Hence we must have P′Q′ ∪ τω_{1}P′Q′ = F = M̃_{6}. Moreover, up to equivalence, W is the only irreducible M̃_{6}-module of dimension > 1.

We now construct the group M̃_{4}. Let L = {m ∈ M̃_{6} : ∧^{2}γ(m)Ω^{∗} = Ω^{∗}}. Clearly, L ⊂ M_{4} and so to prove L = M̃_{4}, it suffices to prove |L| = 2^{5}. Now τω_{1}, τω_{4}, ω_{2}, ω_{3} ∈ L with (τω_{1})^{2} = (τω_{4})^{2} = −1 and ω_{2}^{2} = ω_{3}^{2} = 1. It is easy to see that the group ⟨τω_{1}, ω_{2}, ω_{3}⟩ is a subgroup of L of order 16. Since τω_{4} is not in this group we have that |L| = 32.

Proposition 9.6. M̃_{4} = L, [M̃_{4}, M̃_{4}] = {±I} and M̃_{4} has – up to equivalence – 4 irreducible inequivalent representations of degree 2. The remaining representations are one dimensional.

Proof. The fact that M̃_{4} = L and [M̃_{4}, M̃_{4}] = {±1} are clear. It remains only to exhibit 4 inequivalent irreducible modules of dimension 2. The spaces ⟨f_{1}, f_{2}⟩, ⟨f_{3}, f_{4}⟩,⟨f_{5}, f_{6}⟩, and ⟨f_{7}, f_{8}⟩ are all M̃_{4}-modules. Note that Z(M̃_{4}) = {±1, ±ω_{2}, ±ω_{3}, ±ω_{2}ω_{3}} acts on each of these spaces with a different character. Therefore these modules are inequivalent M̃_{4}-modules.

Remark. It is easy to see that M̃_{j} ≠ D_{j+1} for j = 4, 6, 7, or 8.

10. The Construction of g_{2(2)} and G̃_{2(2)}

Let

k′ = ⨁_{j=1}^{4}⟨f_{2j−1} · g_{2j}, f_{2j} · g_{2j−1}, h_{2j−1} − h_{2j}⟩

and

p′ = ⟨f_{I} : |I| = 4, |I ∩ {2j − 1, 2j}| = 1, 1 ≤ j ≤ 4⟩.

Now k′ ⊂ k_{000} and p′ ⊂ p_{000}. It is easy to see that g′ = k′ + p′ is a simple Lie subalgebra of f_{4,C} and, in fact, g′ ≅ so(8, C). Consider the linear map σ : g′ → g′ defined as follows:

f_{1} · g_{2} → f_{1} · g_{2},f_{2} · g_{1} → f_{2} · g_{1}, h_{1} − h_{2} → h_{1} − h_{2},

f_{3} · g_{4} → f_{5} · g_{6} → f_{7} · g_{8} → f_{3} · g_{4}, f_{4} · g_{3} → f_{6} · g_{5} → f_{8} · g_{7} → f_{4} · g_{3},
h_{3} − h_{4} → h_{5} − h_{6} → h_{7} − h_{8} → h_{3} − h_{4},
and for 1 ≤ j ≤ 2
f_{{j,3,5,7}} → f_{{j,3,5,7}}, f_{{j,4,6,8}} → f_{{j,4,6,8}},
f_{{j,4,5,7}} → −f_{{j,3,6,7}} → f_{{j,3,5,8}} → f_{{j,4,5,7}},
f_{{j,3,6,8}} → −f_{{j,4,5,8}} → f_{{j,4,6,7}} → f_{{j,3,6,8}}.

Proposition 10.1. The map σ : g′ → g′ is a Lie algebra automorphism.

Proof. Clearly, σ : k′ → k′ is a Lie algebra homomorphism, and it is also easy to see that σ : p′ → p′ intertwines the action of k′. It suffices to show that σ([f_{I}, f_{J}]) = [σ(f_{I}), σ(f_{J})] for any f_{I}, f_{J} ∈ p′. We recall the following identities:

[f_{{1,3,5,7}}, f_{{2,4,6,8}}] = 1/2 ∑_{j=1}^{4}(h_{2j−1} − h_{2j}),

[f_{{1,3,5,7}}, f_{{1,4,6,8}}] = 1/2f_{1} · g_{2},
[f_{{1,3,5,7}}, f_{{2,3,6,8}}] = −1/2f_{3} · g_{4},
[f_{{1,3,5,7}}, f_{{2,4,5,8}}] = 1/2f_{5} · g_{6},
[f_{{1,3,5,7}}, f_{{2,4,5,7}}] = −1/2f_{7} · g_{8}.

It follows that [σ(f_{{1,3,5,7}}), σ(f_{J})] = σ([f_{{1,3,5,7}}, f_{J}]) for any f_{J} ∈ p′. Thus, if we fix u = f_{{1,3,5,7}}, we have [u, σ(v)] = [σ(u), σ(v)] = σ([u, v]) for any v ∈ p′. From the Jacobi identity and the facts that σ is a Lie algebra automorphism of k′ and a k′ intertwining operator on p′ we have [σ([x, u]), σ(v)] = σ([[x, u], v]) for any x ∈ k′ and any v ∈ p′, and the proof follows.

Our above discussion yields the following result.

Proposition 10.2. g′^{σ} = k′^{σ} + p′^{σ} is a Lie algebra.

Now k′^{σ} = ⟨f_{1}·g_{2}, f_{2} · g_{1}, h_{1} − h_{2}⟩ ⊕ ⟨ζ, η, ω⟩ where

ζ = ∑_{j=2}^{4} f_{2j−1} · g_{2j}, η ∑_{j=2} ·f_{2j} · g_{2j−1}, and ω = ∑_{j=2}^{4}(h_{2j−1} − h_{2j}).

The following elements are a basis for p′^{σ}. f_{{1,3,5,7}},f_{{1,4,5,7}} − f_{{1,3,6,7}} + f_{{1,3,5,8}}, f_{{1,3,6,8}} − f_{{1,4,5,8}} + f_{{1,4,6,7}},f_{{1,4,6,8}}, f_{{2,3,5,7}},f_{{2,4,5,7}} − f_{{2,3,6,7}} + f_{{2,3,5,8}}, f_{{2,3,6,8}} − f_{{2,4,5,8}} + f_{{2,4,6,7}},f_{{2,4,6,8}}.

So g′^{σ} is a Lie algebra of rank 2 and dimension 14. Since it is clearly simple we have a direct proof of the folowing.

Theorem 10.3. g′^{σ} = g_{2,C}.

Let L be the analytic subgroup of F̃_{4(4)} having Lie algebra g_{2(2)} = g_{2.C} ∩ f_{4(4)}. Since Z(G_{2,C}) = {1}, Z(G_{2(2)}) ≅ Z_{2}, and e_{1} ⋯ e_{16} ∈ Z(L), we see that L = G̃_{2(2)} and Z(G̃_{2(2)}) = Z(Ẽ_{8(8)}).

If g_{2(2)} = k_{2} + p_{2} is the corresponding Cartan decomposition of g_{2(2)},

a_{2} = ⟨X_{{1,3,5,7}}, X_{{1,3,6,8}} − X_{{1,4,5,8}} + X_{{1,4,6,7}}⟩

is a maximal abelian subalgebra of p_{2}. Let K̃_{2} be the maximal compact subgroup of G̃_{2(2)}, and set M̃_{2} = Z(a_{2}) ∩ K̃_{2}. Recall that M̃_{4} = ⟨τω_{1}, τω_{4}, ω_{2}, ω_{3}⟩. If G′ is the analytic subgroup of F̃_{4(4)} having Lie algebra g′ ∩f_{4(4)}, we have that M̃_{4} ⊂ G′, and so M̃_{4}^{σ} is a subgroup of M̃_{2}. It is easy to see that σ(ω_{2}) = ω_{3},σ(ω_{3}) = ω_{2}ω_{3}, σ(τω_{1}) = τω_{1}, and σ(τω_{4}) = τω_{4}. Hence M̃_{4}^{σ} = ⟨τω_{1}, τω_{4}⟩.

Proposition 10.4. M̃_{2} = M̃_{4}^{σ} is a nonabelian group with one – up to equivalence – irreducible representation of degree 2.

Proof. Since M̃_{4}^{σ} is of order 8, we have our equality. Since τω_{1}τω_{4} = −τω_{4}τω_{1}, M̃_{2} is not abelian. Moreover, as M̃_{2}/Z(G̃_{2(2)}) ≅ Z_{2} × Z_{2}, our result follows.

Remark. It is easy to see that M̃_{2} = D_{3}.

This concludes our examination of the universal covering groups of the split simple exceptional groups.

11. e_{8(−24)} and Ẽ_{8(−24)}

Recall e_{8,C} = so(16, C) + ⋀^{e} W, e_{7,C} = sl(8, C) + ⋀^{4} W, and sl(2, C) = ⟨∑_{j=1}^{8} h_{j}, f_{Ø}, f_{S}⟩. Set k̂_{C} = e_{7,C} ⊕ sl(2, C), and p̂_{C} = F + G + ⋀^{2} W + ⋀^{6} W, where F = ⟨f_{j} · f_{k} : j < k⟩, G = ⟨g_{j} · g_{k} : j < k⟩. We now have the following commutation relations.

[F, F] = 0, [F, ⋀^{j} W] ⊂ ⋀^{j+2} W, [F, G] ⊂ sl(8, C), [G, G] = 0,

[G, ⋀^{j} W] ⊂ ⋀^{j−2} W, [⋀^{2} W, ⋀^{2} W] = 0, [⋀^{6} W, ⋀^{6} W] = 0,

and

[⋀^{2} W, ⋀^{6} W] ⊂ sl(2, C) ⊕ sl(8, C).

Thus [p̂_{C}, p̂_{C}] ⊂ k̂_{C}, and [k̂_{C}, p̂_{C}] ⊂ p̂_{C}. Hence (k̂_{C}, p̂_{C}) is a symmetric pair for e_{8,C}. Finally, if k̂ = k̂_{C} ∩ (k_{R} + ip_{R}) and p̂ = p̂_{C} ∩ (ik_{R} + p_{R}), the Lie algebra ĝ = k̂+p̂ is a real form of e_{8,C}. If Ĝ is the simply connected analytic group with Lie algebra ĝ, Ĝ has maximal compact subgroup K̂ = E_{7(−133)}×SU(2). Thus e_{8(−24)} = ĝ and Ĝ = Ẽ_{8(−24)}. The subgroup of SU(2) having Lie algebra ⟨ih_{0}⟩ where h_{0} = ∑_{j=1}^{8} h_{j} is

⟨exp tih_{0} : t ∈ R⟩/⟨ω_{1}^{2}⟩(ω_{1} = exp πih_{0}),

and the subgroup of E_{7(−133)} having Lie algebra su(8) is SU(8)/{±I}. Now Z(SU(2)) ≅ Z_{2} and Z(E_{7(−133)}) ≅ Z_{2} (see [4]). A simple calculation thus shows that

Z(SU(2)) = ⟨ω_{1}⟩/⟨ω_{1}^{2}⟩ = {±I_{2}}, Z(E_{7(−133)}) = ⟨iI_{8}⟩/{±I_{8}},

and

Z(Ẽ_{8(−24)}) = ⟨(iI_{8}, ω_{1})⟩/⟨(−I_{8}, ω_{1}^{2})⟩ ≅ Z_{2}.

Thus Ẽ_{8(−24)} is not a linear group.

If â = ⟨X_{{1,2}}, X_{{3,4}}, X_{{5,6}}, X_{{7,8}}⟩, â is a maximal abelian subalgebra of p̂. Then k̂ ∩ z(â) = k′ + ip′ ≅ so(8) where k′ and p′ are as defined in section 10. Set M̂ = K̂ ∩ Z(â) and let M̂_{0} be the connected component of the identity. It is easy to see that the subgroup M̂_{0} of K̂ or of K̂/Z(Ĝ) having Lie algebra k̂ ∩ z(â) is Spin(8). Hence from [10] we see that if M̂ = Z(â) ∩ K̂, |M̂/M̂_{0}| = 8. Clearly, Z(Ẽ_{8(−24)}) ⊂ M̂ and Z(Ẽ_{8(−24)}) ∩ M̂_{0} = {1}. To analyze the group M̂ further we recall some results from [7] and [10].

Suppose Σ_{r} is the set of restricted roots and let Φ_{r} be the set of simple restricted roots. Then from [7] we have that M̂/Z(Ĝ) = Z_{1} · M̂_{0} where

Z_{1} = ⟨exp 2πiH_{α}/|α|^{2} : α ∈ Σ_{r}⟩ = ⟨exp 2πiH_{α}/|α|^{2} : α ∈ Φ_{r}⟩.

Moreover, since the Satake diagram of e_{8(−24)} has four white dots and only two white dots are not adjacent to any black dots, we have from [10] that Z_{1} = ⟨exp 2πiH_{α}/|α|^{2} : α ∈ Φ_{r} and α is long ⟩ ≅ Z_{2} × Z_{2}.

For j ≤ 4 let ϵ_{j} ∈ â^{∗} be the map
ϵ_{j}(a_{1}X_{{1,2}} + a_{2}X_{{3,4}} + a_{3}X_{{5,6}} + a_{4}X_{{7,8}}) = a_{j}.

An elementary but tedious calculation shows that

Σ_{r} = {± 4ϵ_{j} ± 4ϵ_{k} : j < k ≤ 4} ∪ {±8ϵ_{j} : j ≤ 4} ∪ {±4ϵ_{1} ± 4ϵ_{2} ± 4ϵ_{3} ± 4ϵ_{4}}

and Φ_{r} = {4ϵ_{1} − 4ϵ_{2} − 4ϵ_{3} − 4ϵ_{4}, 8ϵ_{4}, 4ϵ_{3} − 4ϵ_{4}, 4ϵ_{2} − 4ϵ_{3}}.

The 24 long restricted roots are all of multiplicity 1, and the 24 short restricted roots are all of multiplicity 8. Setting α_{1} = 4ϵ_{1} − 4ϵ_{2} − 4ϵ_{3} − 4ϵ_{4} another routine calculation yields H_{α_{1}}/|α_{1}|^{2} = 1/16(X_{{1,2}} − X_{{3,4}} − X_{{5,6}} − X_{{7,8}}) and H_{4ϵ_{4}}/|4ϵ_{4}|^{2} = 1/8X_{{7,8}}. If x is in the preimage of exp(1/8)πi(X_{{1,2}} − X_{{3,4}} − X_{{5,6}} − X_{{7,8}}) in M̂ and y is in the preimage of exp(1/4)πiX_{{7,8}}, we have that Ad(x) and Ad(y) are both non trivial on sl(2, C). Hence x^{2} = y^{2} is the nontrivial element of Z(Ẽ_{8(−24)}), and x^{4} = y^{4} = 1. Since Ad(x)16f_{Ø} = −f_{S}, Ad(x)f_{S} = −16f_{Ø}, Ad(y)f_{Ø} = −f_{Ø} and Ad(y)f_{S} = −f_{S}, we have xyx^{−1} = y^{−1}. If Ẑ_{1} is the preimage of Z_{1} in Ẽ_{8(−24)} Ẑ_{1} = ⟨x, y⟩ is a nonabelian group of order 8 centralized by M̂_{0}, and so we have the following proposition.

Proposition 11.1. M̂ = Ẑ_{1} · M̂_{0} and Ẑ_{1} ∩ M̂_{0} = {1}.

Corollary 11.2. M̂ = Ẑ_{1} × M̂_{0}. Remark. Note that Ẑ_{1} ≅ D_{3} is the quaternionic group.

12. The real form e_{7(−5)} and Ẽ_{7(−5)}

Recall e_{7,C} = sl(8, C) + ⋀^{4} W. Fix W_{1} = ⟨f_{1}, f_{2}, f_{3}, f_{4}, f_{5}, f_{6}⟩, W_{2} = ⟨f_{7}, f_{8}⟩, and let gl(6, C) be the Lie subalgebra of sl(8, C) generated by all elements of the form f_{j} · g_{k} where j ≠ k ≤ 6, and all sums of the form ∑_{j=1}^{8} a_{j}h_{j} where a_{7} = a_{8} and ∑_{j=1}^{8} a_{j} = 0. Now

⋀^{4} W = ⋀^{4} W_{1} ⊕ (⋀^{3} W_{1} ⊗ W_{2}) ⊕ (⋀^{2} W_{1} ⊗ ⋀^{2} W_{2}).

If we now set

k_{0,C} = gl(6, C) + ⋀^{4} W_{1} + ⋀^{2} W_{1} ⊗ ⋀^{2} W_{2},

we have that k_{0,C} is a simple Lie algebra of dimension 66 and rank 6. Hence k_{0,C} = so(12, C). Set also

sl(2, C) = ⟨f_{7} · g_{8}, f_{8} · g_{7}, h_{7} − h_{8}⟩, k_{C} = k_{0,C} ⊕ sl(2, C),

and

p_{C} = ⟨f_{j} · g_{k} : j ≤ 6 < k or j > 6 ≥ k > + ⋀^{3} W_{1} ⊗ W_{2}.

Then (k_{C}, p_{C}) is a symmetric decomposition of e_{7,C}, and if g = k + p is the corresponding real form, g = e_{7(−5)}. Let Ẽ_{7(−5)} be the corresponding real Lie group with maximal compact subgroup K = Spin(12) × SU(2). Now Z(K) = Z_{2}^{3} and by examining the representation of K on p, we see Z(E_{7(−5)} = Z_{2}^{2}. Thus E_{7(−5)} is not a linear group.

Setting

H_{1} = X_{{1,3,5,7}}, H_{2} = X_{{1,3,6,8}}, H_{3} = X_{{1,4,5,8}}, H_{4} = X_{{1,4,6,7}},

a = ⟨H_{1}, H_{2}, H_{3}, H_{4}⟩ is easily seen to be a maximal abelian subalgebra of p. For j ≤ 4 let ϵ_{j} ∈ a^{∗} be the map

ϵ_{j}(a_{1}H_{1} + a_{2}H_{2} + a_{3}H_{3} + a_{4}H_{4}) = a_{j}

Then, if Σ_{r} is the set of restricted roots and Φ_{r} is a set of simple restricted roots, Σ_{r} and Φ_{r} are exactly as in section 11; this time the short roots are all of multiplicity 4.

If I ⊂ S, set h_{I} = ∑_{j∈I} h_{j} − ∑_{j∈I′} h_{j}. Then, if |I| = 4, T r(h_{I}) = 0. We now see that

m_{C} = ⟨f_{{1,2,3,4}}, f_{{5,6,7,8}}, h_{{1,2,3,4}}⟩⊕
⟨f_{{1,2,5,6}}, f_{{3,4,7,8}}, h_{{1,2,5,6}}⟩ ⊕ ⟨f_{{1,2,7,8}}, f_{{3,4,5,6}}, h_{{1,2,7,8}}⟩.

Let M̂ = K ∩ Z(a) and let M̂_{0} be the connected component of the identity of M̂. Setting x_{1} = exp(πi/2)h_{{1,2,3,4}}, x_{2} = exp(πi/2)h_{{1,2,5,6}}, and x_{3} = exp(πi/2)h_{{1,2,7,8}},we see that

M̂_{0} = (SU(2) × SU(2) × SU(2))/L

where L is a finite subgroup of

Z(SU(2) × SU(2) × SU(2)) = ⟨x_{1}, x_{2}, x_{3}⟩.

A simple calculation on ⋀^{4} W shows that L is trivial. Furthermore, since the representation of M̂_{0} on any short restricted root space is the tensor pruduct of two irreducible 2-dimensional repesentations of two of the SU(2) factors of M̂_{0}, (−I_{2}, −I_{2}, −I_{2}) ∈ Z(E_{7(−5)}). Hence M̂_{0} ∩ Z(E_{7(−5)}) = Z_{2}. Setting α_{1} = 4(ϵ_{1} − ϵ_{2} − ϵ_{3} − ϵ_{4}), we have

H_{α_{1}}/|α_{1}|^{2} = 1/16(H_{1} − H_{2} − H_{3} − H_{4}) and H_{8ϵ_{4}}/|8ϵ_{4}|^{2} = (1/8)H_{4}.

If x is a preimage of exp(πi/8)(H_{1} − H_{2} − H_{3} − H_{4}) in K and y is a preimage of exp(πi/4)(H_{4}) in K, we have that M̂ = Ẑ_{1} · M̂_{0} where Ẑ_{1} = ⟨x, y⟩ is a group of order 8. Now Ad(x)(f_{7} · g_{8}) = −f_{8} · g_{7}, Ad(x)(f_{8} · g_{7}) = −f_{7} · g_{8}, Ad(y)(f_{7} · g_{8}) = −f_{7} · g_{8}, and Ad(y)(f_{8} · g_{7}) = −f_{8} · g_{7}. Hence we have x^{2} = y^{2} x^{4} = y^{4} = 1 and xyx^{−1} = y^{−1}.Finally, M̂_{0} centralizes Ẑ_{1}, and we have the following.

Proposition 12.1. M̂ = Ẑ_{1} · M̂_{0} and M̂_{0} ∩ Ẑ_{1} = {1}.

Corollary 12.2. M̂ = Ẑ_{1} × M̂_{0}. Remark. As in section 11, Ẑ_{1} ≅ D_{3} is the quaternionic group.

13. e_{6(2)} and Ẽ_{6(2)}

Set

sl(2, C) = ⟨f_{7} · g_{8}, f_{8} · g_{7}, h_{7} − h_{8}⟩ and sl(6, C) = ⟨f_{i} · g_{j} : i ≠ j ≤ 6⟩ + h_{0}

where h_{0} = {∑_{j=1}^{6} a_{j}h_{j} : ∑_{j=1}^{6} a_{j} = 0}. If W_{1} = ⟨f_{j} : 1 ≤ j ≤ 6⟩ and W_{2} = ⟨f_{7}, f_{8}⟩, p_{C} = ⋀^{3} W_{1}⊗W_{2} is an irreducible k_{C} = sl(6, C)⊕sl(2, C)-module. As before (k_{C}, p_{C}) is a symmetric pair for e_{6,C}. Now e_{6(2)} is the corresponding real form of e_{6,C}. Let Ẽ_{6(2)} be the real connected Lie group with Lie algebra e_{6(2)}and maximal compact subgroup K = SU(6) × SU(2).

Now p has a maximal abelian subalgebra a = ⟨H_{j} : 1 ≤ j ≤ 4⟩ is the same a as in section 12. Then m_{0} = ⟨h_{1} + h_{2} − h_{3} − h_{4}, h_{1} + h_{2} − h_{5} − h_{6}⟩, and M̂_{0} = S^{1} × S^{1}. Moreover, if M̂ = K ∩ Z(a) and Ẑ_{1} is the quaternionic group in section 12 we have the following result.

Proposition 13.1. M̂ = Ẑ_{1} ∩ M̂_{0} and M̂_{0} ∩ Ẑ_{1} = {1}.

Corollary 13.2. M̂ = Ẑ_{1} × M̂_{0}. Remarks.

1. Since Z(Ẽ_{6(2)}) = ⟨(ωI_{6}, ϵI_{2}) : ω^{3}ϵ = 1, ω^{6} = ϵ^{2} = 1⟩ ≅ Z_{6}, Ẽ_{6(2)} is not a linear group.

2. A simple calculation shows that M̂_{0} ∩ Z(E_{6(2)}) = Z_{3}

3. The short restricted roots are all of multiplicity 2.

14. The Hermitian real form e_{7(−25)} and the group Ẽ_{7(−25)}

Recall from sections 12 and 13 that e_{7,C} = sl(8, C) + ⋀^{4} W and

e_{6,C} = sl(6, C) + sl(2, C) + ⋀^{3} W_{1} ⊗ W_{2}.

If W_{1}^{∗} = ⟨g_{1}, g_{2}, g_{3}, g_{4}, g_{5}, g_{6}⟩, W_{2}^{∗} = ⟨g_{7}, g_{8}⟩ and h_{0} = ∑_{j=1}^{6} h_{j} − 3(h_{7} + h_{8}),

e_{7,C} = e_{6,C} + ⟨h_{0}⟩ + (W_{1} · W_{2}^{∗} + ⋀^{4} W_{1}) + (W_{1}^{∗} · W_{2} + ⋀^{2} W_{1} ⊗ ⋀^{2} W_{2})

where all the sums are direct as vector spaces. If

k_{C} = e_{6,C} + Ch_{0}, p_{+} = (W_{1} · W_{2}^{∗} + ⋀^{4} W_{1}), p_{−} = (W_{1}^{∗} · W_{2} + ⋀^{2} W_{1} ⊗ ⋀^{2} W_{2}),

ad(h_{0}) is 0 on k_{C}, 4 on p_{+} and -4 on p_{−}. It follows immediately that k_{C} is a Lie algebra, [p_{±}, p_{±}] = 0 and [p_{+}, p_{−}] ⊂ k_{C}. If p_{C} = p_{+} + p_{−}, (k_{C}, p_{C}) is a symmetric pair for e_{7,C}. If g = k + p is the corresponding real form, g = e_{7p(−25)}.

Now Ẽ_{7(−25)} is the simply connected analytic group with Lie algebra e_{7(−25)}. Let Φ : Ẽ_{7(−25)} ⟶ E_{7(−25)} be the covering homomorphism. The maximal compact subgroup of E_{7(−25)} is K = E_{6(−78)} · S^{1} ,and the pullback of K is K̃ = Φ^{−1}(K) = E_{6(−78)} × R. For (x, τ) ∈ K̃, Φ(x, τ) = x · exp(iτh_{0}).

Using the fact that Z_{3} ≅ Z(E_{6(−78)}) ⊂ SU(6) · SU(2), we see that Z(Ẽ_{7(−25)}) ⊂ SU(6) · SU(2) × R, and a direct calculation yields

Z(Ẽ_{7(−25)}) = {(ωI_{6}, ϵI_{2}, τ) : ω^{3}ϵ = 1, ω^{6} = 1, ω^{4} exp 4iτ = 1}/{±(I_{6}, I_{2}, 0)}.

If ω_{0} is a primitive sixth root of unity, we have that Z(E_{7(−25)}) is generated by (ω_{0}I_{6}, −I_{2}, π/6) and so Z(Ẽ_{7(−25)}) ≅ Z. Now a = ⟨X_{{1,2,3,4}}, X_{{1,2,5,6}}, X_{{1,2,7,8}}⟩ is a maximal abelian subalgebra of p, and from [10] we have that M̂ = K̃ ∩ Z(a) is generated by M̂_{0} = Spin(8) and the pullback of Ad(exp(πi/4)X_{{1,2,7,8}}). Since

Ad(exp(πi/4)X_{{1,2,7,8}}) = Ad(diag(1, 1, −1, −1, −1, −1, 1, 1))

and diag(1, 1, −1, −1, −1, −1, 1, 1) ∈ Spin(8), we have the following.

Proposition 14.1. M̂ = M̂_{0} · Z(Ẽ_{7(−25)}) ≅ M̂_{0} × Z.

Remark. From [10] and [13] we have that Φ(M̂) ≅ M̂_{0} × Z_{2}.

15. The Hermitian real form e_{6(−14)} and group Ẽ_{6(−14)}

Now

so(10, C) ≅ ⟨f_{i} · f_{j}, g_{i} · g_{j}, f_{i} · g_{j} : i, j ≤ 5, i ≠ j⟩ + ⟨h_{j} : j ≤ 5⟩,

and if h_{0} = h_{6} − h_{7} − h_{8}, k_{C} = so(10, C) ⊕ Ch_{0} is a reductive Lie algebra. Setting V = ⟨f_{1}, f_{2}, f_{3}, f_{4}, f_{5}⟩, p_{+} = ⋀^{o} V · f_{6}, p_{−} = ⋀^{e} V · f_{7} · f_{8}, and p_{C} = p_{+} + p_{−}, we have that k_{C} + p_{C} = e_{6,C} and (k_{C}, p_{C}) is a symmetric pair. The Lie algebra corresponding to the real form of this symmetric pair is e_{6(−14)}. Now K = Spin(10) · S^{1} is the maximal compact subgroup of E_{6(−14)} and K̃ = Spin(10) × R is the pullback of K in Ẽ_{6(−14)}. Note that ad(h_{0}) = ±3/2 on p_{±}.

Now a = ⟨X_{{1,2,5,6}}, X_{{1,2,7,8}}⟩ is a maximal abelian subalgebra of p. If M̂ = Z(a)∩K̃, we have from [10] and [12], that the image of M̂ in K is connected and m = su(4) ⊕ ⟨H_{0}⟩ where H_{0} = h_{5} − 1/3h_{0}. A simple calculation shows that Z(Ẽ_{6(−14)}) ≅ Z and has a generator the pullback of

exp πi(h_{1} + h_{2} + h_{3} + h_{4} + h_{5} − 1/3h_{0}).

Since h_{1} + h_{2} + h_{3} + h_{4} ∈ su(4), we have the following.

Proposition 15.1. The group M̂ = M̂_{0} ≅ SU(4) × R.

16. The groups E_{6(−26)} and F_{4(−20)}

The groups E_{6(−26)} and F_{4(−20)} are both simply connected linear groups with trivial centers. From [10] we see that for E_{6(−26)} the group M̂ is Spin(8), and for F_{4(−20)} the group M̂ is Spin(7). This concludes our analysis of the exceptional groups.

17. The Classical Groups

Throughout this section we will use the notation (G, K) to denote a pair where G is a simply connected real simple Lie group and K is the pullback in G of a maximal compact subgroup of Ad(G).

1. (S̃L(n, R), Spin(n)): Now p is the space of real n×n symmetric matrices of trace 0. If a is the space of diagonal matrices in p, a is a maximal abelian subalgebra of p. We then have the following.

Proposition 17.1. M̂ = Z(a) ∩ Spin(n) = D_{n}.

Remark. If P is a non minimal parabolic subgroup of S̃L(n, R), an easy calculation shows that {±I} = [D_{n}, D_{n}] ⊂ P_{0}.

2. (SU^{∗}(2n), Sp(n)): Since SU^{∗}(2n) is a linear group we have the following proposition.

Proposition 17.2. M̂ = (Sp(1))^{n}.

3. (S̃U(p, q), SU(p)×SU(q)×RH) with 1 ≤ p < q and H = ( \begin{matrix}qI_{p} & O \\ 0 & −pI_{q} \end{matrix} ),

and the map of K onto S(U(p) × U(q)) sends tH to exp(itH): Then a = ⟨E_{j,p+j} + E_{p+j,j} : 1 ≤ j ≤ p⟩ is a maximal abelian subalgebra of p. Setting M̂ = K ∩ Z(a) we now state.

Proposition 17.3. M̂ ≅ (S^{1})^{p−1} × SU(q − p) × R.

Proof. By a simple calculation, we have

M̂ = ⟨(α, β, θH) : α = diag(a_{1}, …, a_{p}), β = ( \begin{matrix}β_{1} & 0 \\ 0 & β_{2} \end{matrix} ) ,

β_{1} = diag(b_{1}, …, b_{p}), β_{2} ∈ U(q − p), a_{1} ⋯ a_{p} = 1, b_{1} ⋯ b_{p} · detβ_{2} = 1,

|a_{1}| = … = |a_{p}| = 1, b_{j} = a_{j}e^{i(p+q)θ)}⟩ ≅

⟨(b_{1}, …, b_{p}, β_{2}, θ) : b_{1}, …, b_{p} ∈ C, |b_{1}| = … = |b_{p}| = 1, β_{2} ∈ U(q − p),

b_{1} ⋯ b_{p} = e^{ip(p+q)θ} = (detβ_{2})^{−1}⟩ ≅
⟨(b_{1}, …, b_{p}, β_{2}, θ) : |b_{1}| = … = |b_{p}| = 1, β_{2} ∈ SU(q − p),
b_{1} ⋯ b_{p} = e^{ip(p+q)θ}⟩ ≅
⟨(b_{1}, …, b_{p−1}, β_{2}, θ) : |b_{1}| = … = |b_{p}| = 1, β_{2} ∈ SU(q − p), θ ∈ R⟩ ≅
(S^{1})^{p−1} × SU(q − p) × R.

4. (S̃U(p, p), SU(p) × SU(p) × RH), H = diag(I_{p}, −I_{p}): Taking a is as in 3, we obtain:

Proposition 17.4. M̂ ≅ (S^{1})^{p−1} × Z Proof. : A simple calculation yields

M̂ = ⟨(a_{1}, g …, a_{p}, b_{1}, …, b_{p}, θ) : |a_{1}| = … = |a_{p}| = 1, b_{j} = a_{j}e^{2iθ},
a_{1} ⋯ a_{p} = b_{1} ⋯ b_{p} = 1, e^{2piθ} = 1⟩ ≅
⟨(b_{1}, …, b_{p−1}, θ) : |b_{1}| = … = |b_{p}| = 1, θ ∈ (π/p)Z⟩ ≅
(S^{1})^{p−1} × Z.

Remarks. Note that Z(G) ≅ ⟨(z, (k/p)π) : z^{p} = 1, k ∈ Z⟩, and the kernel of the map of K onto S(U(p) × U(p)) is isomorphic to

⟨(exp(−2πi/p), 2π/p)⟩ ≅ Z.

Then the image of M̂ in S(U(p) × U(p)) is (S^{1})^{p−1} × Z_{2}, and the image of M̂ in AdG is (S^{1})^{p−1}.

5. (S̃pin_{0}(p, q), Spin(p) × Spin(q)), where 1 ≤ p ≤ q, p ≠ 2, and q ≠ 2: Taking a as in 3, and keeping our notation we obtain the following.

Proposition 17.5. M̂ = D_{p} × Spin(q − p) Note that D_{1} and Spin(1) are trivial.

a. If p = 1, S̃pin_{0}(p, q) is a linear group and M̂ = Spin(q − 1).

b. If q = p or if q = p + 1, M̂ = D_{p}.

6. (S̃pin_{0}(2, q), R × Spin(q)) with q ≠ 2: We have a map ϕ : R ⟶ Spin(q + 2) where ϕ(t) = exp(te_{1} · e_{2}). If a is as in 3, then M̂ = Z̃_{1} · M̂_{0} where M̂_{0} = Spin(q − 2) and

Z̃_{1} = {(kπ/2, ±e_{3} · e_{4}) : k odd} ∪ {(kπ/2, ±1) : k even}.

With addition in the first variable and multiplication in the second variable, Z̃_{1} is easily seen to be a group. Since (0, −1) ∈ M̂_{0}, if q > 3,this product is not direct. If Ẑ_{1} = {(kπ/2, (e_{3} · e_{4})^{k}) : k ∈ Z}, Ẑ_{1} is a group isomorphic to Z. Note that Z̃_{1} ≅ Ẑ_{1} × ⟨(0, ±1)⟩ ≅ Z × Z_{2}.

Proposition 17.6. If q > 3, M̂ ≅ Z × M̂_{0}. If q = 3, M̂ ≅ Z × Z_{2} Remarks. (a) Note that S̃pin_{0}(2, 3) ≅ S̃p(2, R).

(b) The image of Ẑ_{1} in Spin(q + 2) is ⟨e_{1} · e_{2} · e_{3} · e_{4}⟩ ≅ Z_{2}.

(c) If q > 3, the image of M̂ in Spin(q + 2) is isomorphic to Z_{2} × Spin(q − 2).

(d) If q = 3, the image of M̂ in Spin(5) is isomorphic to Z_{2} × Z_{2}.

(e) The image of M̂ in Ad(S̃pin_{0}(2, q)) is connected if and only if q is even.

(f) If p = q = 2, S̃pin_{0}(2, 2) ≅ S̃L(2, R) × S̃L(2, R) is not a simple group.

7. (S̃p(n, R), SU(n) × RH), where H = ( \begin{matrix}0 & I \\ −1 & 0 \end{matrix} ): Then

p = \{( \begin{matrix}A & B \\ B & −A \end{matrix} ) : A^{T} = A, B^{T} = B\}, with

a = {diag(x_{1}, …, x_{n}, −x_{1}, …, −x_{n}) : x_{1}, …, x_{n} ∈ R}

is a maximal abelian subalgebra of p.

314 Johnson Proposition 17.7. M̂ ≅ (Z_{2})^{n−1} × Z. Proof. Since the image of M̂ in U(n) is

M = ⟨diag(ϵ_{1}, …, ϵ_{n}) : ϵ_{j} = ±1⟩ ≅ (Z_{2})^{n},

M̂ ≅ ⟨(A, θ) : A ∈ SU(n), θ ∈ R, A exp(iθ) ∈ M⟩.

Hence M̂ ≅ (M ∩ SU(n)) × \overline{M} where M ∩ SU(n) ≅ Z_{2}^{n−1} and

\overline{M} = \{(A, θ) : A ∈ SU(n), θ ∈ R, A exp(iθ) = diag(±1, 1, …, 1)\} =

{(diag((−1)^{k}, 1, …, 1) exp(−ikπ/n), kπ/n) : k ∈ Z} ≅ Z.

Thus the result holds.

8. (S̃O^{∗}(2n), SU(n) × RH), H = ( \begin{matrix}0 & I \\ −I & 0 \end{matrix} ): Now

p = \{i ( \begin{matrix}A & B \\ B & −A \end{matrix} ) : A, B ∈ M_{n}(R), A^{T} = −A, B^{T} = −B\}.

If a is the set

\{i ( \begin{matrix}X & 0 \\ 0 & −X \end{matrix} ) ∈ p : X = ∑_{k=1}^{[n/2]} x_{k}(E_{2k−1,2k} − E_{2k,2k−1}), x_{k} ∈ R\},

a is a maximal abelian subalgebra of p. A simple calculation shows that ( \begin{matrix}A & B \\ −B & A \end{matrix} ) × θH ∈ M̂ if and only if Ze^{iθ}X = X\overline{Z}e^{−iθ} and XZe^{iθ} = \overline{Z}e^{−iθ}X where Z = A + iB.

a. If n = 2k + 1, Z = diag (Z(1), …, Z(k), e^{iϕ}) where Z(1), …, Z(k) ∈ M_{2}(C),

Z(j) = ( \begin{matrix}z(j)_{11} & z(j)_{12} \\ −\overline{z(j)}_{12}e^{−2iθ} & \overline{z(j)}_{11}e^{−2iθ} \end{matrix} ) ,

and det Z(1) ⋯ det Z(k) e^{iϕ} = 1. Setting Z_{o}(j) = ( \begin{matrix}z(j)_{11} & z(j)_{12} \\ −\overline{z(j)}_{12} & \overline{z(j)}_{11} \end{matrix} ), we have

det Z_{o}(1) = ⋯ = det Z_{o}(k) = 1

and exp(iϕ − 2kiθ) = 1. Hence we obtain the following. Proposition 17.8. If n = 2k + 1,then M̂ ≅ SU(2)^{k} × R.

b. If n = 2k, Z = diag(Z(1), …, Z(k)) where Z(1), …, Z(k) ∈ M_{2}(C),

Z(j) = ( \begin{matrix}z(j)_{11} & z(j)_{12} \\ −\overline{z(j)}_{12}e^{−2iθ} & \overline{z(j)}_{11}e^{−2iθ} \end{matrix} ) ,

and det Z(1) ⋯ det Z(k) = 1. Setting Z_{o}(j) = ( \begin{matrix}z(j)_{11} & z(j)_{12} \\ −\overline{z(j)}_{12} & \overline{z(j)}_{11} \end{matrix} ), we have

det Z_{o}(1) = ⋯ = det Z_{o}(k) = 1

and exp(−2kiθ) = 1.

Proposition 17.9. If n = 2k, then M̂ ≅ SU(2)^{k} × Z.

Remarks. For n = 2k the image of M̂ in Spin^{∗}(4k) is SU(2)^{k} × Z_{2}. The image of M̂ in SO^{∗}(4k) is SU(2)^{k}, and the image of M̂ in Ad(SO^{∗}(4k)) is SU(2)^{k}/{±I}.

9. (Sp(p, q), Sp(p)×Sp(q)) with p ≤ q: Then Sp(p, q) is a simply connected linear group and M̂ = Sp(1)^{p} × Sp(q − p).

18. Summary

Suppose G is a connected real simple Lie group with Lie algebra g contained in a simply connected group G_{C} having Lie algebra g_{C}. Let G̃ be the universal cover of G. Fix K, a maximal compact subgroup of G and denote the pullback of K in G̃ by K̃. Fixing a as in the previous sections, set M̂ = Z(a) ∩ K̃ and let M denote the image of M̂ in K.

1. Then M̂ = Ẑ_{1} × M̂_{0} where Ẑ_{1} is a discrete group and M̂_{0} is the identity component of M̂.

2. The group Ẑ_{1} is infinite if and only if G/K is a tube type domain.

3. Since, as topological spaces, K̃/M̂ = K/M,

π_{1}(K/ M) ≅ π_{0}(M̂) ≅ Ẑ_{1}.

Hence K/M is simply connected if and only if M̂ is connected. From the homotopy exact sequence, it follows that M̂ is connected if and only if M is connected.

4. Suppose l is the number of white dots of the Satake diagram of G that are not adjacent to a black dot nor connected to another white dot by an arrow.

(a) The group M̂ is connected if and only if l = 0.

(b) If l = 1, G/K is a tube type domain and Z_{1} = Z.

(c) If l > 1 and G/K is a tube type domain, G = S̃p(l, R) and

M̂ = Z_{1} = Z_{2}^{l−1} × Z.

(d) If l > 1 and G/K is a not tube type domain, Ẑ_{1} is a non abelian group of order 2^{l+1}.

5. If G is a split group, M̂ = Ẑ_{1}.

References

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[3]  Baez, J. C., The Octonions, Bulletin Amer. Math. Soc. 39 (2002), 145–205.
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[5] Cartan, E., “The Theory of Spinors,” Dover Publications, Inc., New York, 1981.

[6] Helgason, S., “Differential geometry and symmetric spaces, ” Academic Press, New York, 1962.

[7] —, A duality for symmetric spaces with applications to group representations,” Advances in Math. 5 (1970), 1–154.

[8] Jacobson, N., “Lectures in Abstract Algebra, Volume II–Linear Algebra, ” Van Nostrand, Princeton, 1953.

[9]  —, “Lie Algebras,” Interscience, New York, 1962.

[10] Johnson, K. D., A strong generalization of Helgason’s theorem, Trans. Amer. Math. Soc. 304 (1987), 171–192.

[11] —, A constructive approach to tensor product decompositions, J. für die reine und angewandte Mathematik 388 (1988), 129–148.

[12] Satake, I., On representations and compactifications of symmetric Riemann spaces, Ann. of Math. 72 (1960), 555–580.

[13] Schlichtkrull, H., One-dimensional K-types in finite dimensional representations of semisimple Lie groups: A generalization of Helgason’s theorem, Math. Scand. 54 (1984), 279–294.

Kenneth D. Johnson The University of Georgia Athens, GA 30602 ken@math.uga.edu

Received April 2, 2003 and in final form July 20, 2002