Journal of Lie Theory Volume 9 (1999) 203–213 C ◯1999 Heldermann Verlag

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Transformations Groups of the Andersson-Perlman Cone

Erhard Neher^{∗}

Communicated by J. Faraut

Abstract. An Andersson-Perlman cone is a certain subcone Ω(K) of the symmetric cone Ω of a Euclidean Jordan algebra. We exhibit a subgroup of the automorphism group of Ω which operates transitively on Ω(K) and show that Ω(K) is a simply-connected submanifold of Ω.

1. Introduction.

Andersson-Perlman cones in the setting of Euclidean Jordan algebras (henceforth abbreviated as AP cones) were introduced by H. Massam and the author in [MN] as a generalization of certain cones defined by the statisticians S. A. Andersson and M. D. Perlman for real symmetric matrices [AP]. All mathematical results in [AP] were generalized in [MN] to the setting of Euclidean Jordan algebras, except the existence of transitive transformation groups which play a predominant role in the development in [AP]. In fact, the paper [MN] stresses a different, perhaps more direct approach to the description of Andersson-Perlman cones by employing Peirce decompositions and Frobenius transformations.

In this note we show that one can also generalize the results of [AP] on transitive groups to the framework of Andersson-Perlman cones in Euclidean Jordan algebras. Our interest in these groups is explained in the following remarks. An Andersson-Perlman cone is a subcone Ω(K) of the cone Ω of an Euclidean Jordan algebra V defined in terms of a complete orthogonal system E = (e_{1}, …, e_{n}) of idempotents of V and a ring K of subsets of I = {1, …, n}, see 6.. If Ω_{i} denotes the symmetric cone of the Peirce-1-space V (e_{i}, 1) of e_{i} then always

Ω_{1} ⊕ Ω_{2} ⊕ ⋯ ⊕ Ω_{n} ⊂ Ω(K) ⊂ Ω,

and both upper and lower bounds can be obtained by varying K. Thus, one may consider Ω(K) as an interpolation between Ω and Ω_{1} ⊕ Ω_{2} ⊕ ⋯ ⊕ Ω_{n}. In the

ISSN 0949–5932 / $2.50 ^{_{C} ◯} Heldermann Verlag same spirit, the transitive transformation group T (denoted T_{E,⪯} in the paper) of Ω(K) interpolates various well-known subgroups of the automorphism group G(Ω) = {g ∈ GL(V); gΩ = Ω} of Ω. In general, T is a semidirect product of a unipotent subgroup N of G(Ω) (denoted N_{E,≺} in the paper) and the real reductive group

∗ Research partially supported by a research grant from NSERC (Canada)

M_{E} = {g ∈ G(Ω); gΩ_{i} = Ω_{i}} = P (Ω_{1} ⊕ Ω_{2} ⊕ ⋯ ⊕ Ω_{n}) · K_{E}  (1)

where K_{E} = {f ∈ Aut V; fe_{i} = e_{i} for 1 ≤ i ≤ n}. Observe that (1) is the Cartan decomposition of M_{E}. We always have

M_{E} ⊂ T = M_{E} · N ⊂ G(Ω),  (2)

and both bounds are attained. For example, if Ω(K) = Ω and E is a Jordan frame then N is the so-called strict triangular subgroup [FK], while if E = {e}(n = 1) then also Ω(K) = Ω, N = {Id} and M_{E} = G(Ω). In this case, (1) is just the standard Cartan decomposition of G(Ω).

2. Notation and review.

Our basic reference for Jordan algebras is [FK]. Some of the results and notations used are summarized below.

Throughout, V denotes a Euclidean Jordan algebra with identity element e, left multiplication L(u) defined by L(u)v = uv(u, v ∈ V) and quadratic representation P given by P (u)v = 2u(uv) − u^{2}v. The linearization of P is

\begin{aligned}\{u v w\} : & = P (u, w)v := P (u + w)v − P (u)v − P (w)v \\ & = 2u(vw) + 2w(uv) − 2(uw)v \end{aligned}

for (u, v, w ∈ V). The Jordan triple system left multiplication L(u, v) (denoted u□v in [FK]) is given by L(u, v) = 2(L(uv)+[L(u), L(v)]) and hence L(u, v)w = P (u, w)v. For any endomorphism φ of V , φ^{∗} is the adjoint of φ with respect to the positive definite trace form of V .

We will use the term “Lie group” and “Lie subgroup” as defined in [B]. In particular, any Lie subgroup of a Lie group is closed and has the induced topology. Closed subgroups of a Lie group are always Lie subgroups in a unique way.

We denote the symmetric cone of V by Ω = Ω(V). This is an open convex cone which is homogeneous with respect to the group G(Ω) = {g ∈ GL(V); gΩ = Ω}, the automorphism group of Ω. The group G(Ω) is a Lie subgroup of GL_{ℝ}(V). Its identity component will be denoted by G. Moreover, G(Ω) is an open subgroup of the structure group of V , defined as the group of all invertible endomorphisms g of V with the property

P (gx) = gP (x)g^{∗}  (1)

for all x ∈ V , or, equivalently,

gL(u, v)g^{−1} = L(gu, g^{∗−1}v)  (1′)

for all u, v ∈ V ([FK; III.5 and VIII.2]). The Lie algebra 𝔤(V) of the structure group of V coincides with the Lie algebra of G(Ω). It consists of all endomorphisms X of V satisfying for all u, v ∈ V

[X, L(u, v)] = L(Xu, v) − L(u, X^{∗}v)  (2)

([FK; VIII.2.6]). The group of automorphisms of V will be denoted Aut V . For any g ∈ G(Ω) one knows ([FK; III.5] and [FK; VIII.2.4]):

ge = e ⇔ gg^{∗} = Id ⇔ g ∈ Aut V  (3)

In particular, Aut V is a maximal compact subgroup of G(Ω).

Following [FK] we denote the Peirce spaces of an idempotent c ∈ V by V (c, i) = \{v ∈ V; cv = iv\}, i ∈ \{0, \frac{1}{2}, 1\}. The Peirce decomposition of an arbitrary y ∈ V is written in the form y = y_{1} + y_{12} + y_{0} where y_{i} ∈ V (c, i) for i = 0, 1 and y_{12} ∈ V (c, \frac{1}{2}). The symmetric cone of the Euclidean Jordan algebra V (c, 1) will be denoted Ω_{c}. For an idempotent c and z ∈ V (c, \frac{1}{2}) the Frobenius transformation on V is defined as τ_{c}(z) = exp (L(z, c))) ∈ G. It is straightforward to check that τ_{c} : V (c, \frac{1}{2}) → G is a homomorphism, thus τ_{c}(z + z′) = τ_{c}(z)τ_{c}(z′) and τ_{c}(−z) = τ_{c}(z)^{−1} . If x = x_{1} + x_{12} + x_{0} is the Peirce decomposition of x ∈ V with respect to c then

τ_{c}(z)x = x_{1} ⊕ 2zx_{1} + x_{12} ⊕ 2(e − c)[z(zx_{1}) + zx_{12}] + x_{0}  (4)
= x_{1} ⊕ 2zx_{1} + x_{12} ⊕ P (z)x_{1} + 2(e − c)(zx_{12}) + x_{0}.

The adjoint of the Frobenius transformation operates as follows [MN; 2.7]:

τ_{c}(z)^{∗}x = (x_{1} + 2c(zx_{12}) + P (z)x_{0}) ⊕ (x_{12} + 2zx_{0}) ⊕ x_{0}.  (5)

3. Frobenius transformations with respect to an orthogonal system.

Throughout, we fix a complete orthogonal system E = (e_{1}, …, e_{n}) of (arbitrary) idempotents of V . Thus, e_{i}e_{j} = δ_{ij}e_{i} and e_{1} + ⋯ + e_{n} = e. We denote by V_{ij} , 1 ≤ i, j ≤ n, the Peirce spaces of E [FK IV.2] and define, for 1 ≤ i < n, subspaces

V^{(i)} := ⊕_{k=i+1}^{n} V_{ik} = V (e_{i}, \frac{1}{2}) ∩ V (e_{i+1} + … + e_{n}, \frac{1}{2}).

For x ∈ V we let x = ∑_{i≤j} x_{ij} , x_{ij} ∈ V_{ij}, be the Peirce decomposition of x ∈ V . We abbreviate τ_{i} = τ_{e_{i}} and Ω_{i} = Ω_{e_{i}} = Ω(V_{ii}), 1 ≤ i ≤ n. By [MN; 2.8] the map

F : V^{(1)} × ⋯ × V^{(n−1)} × Ω_{1} × ⋯ × Ω_{n} → Ω

given by

\begin{matrix}F (z_{1}, ⋯, z_{n−1}, & y_{1}, ⋯ y_{n}) := τ_{1}(z_{1}) ⋯ τ_{n−1}(z_{n−1})(y_{1} ⊕ ⋯ ⊕ y_{n}) \\ = τ_{1}(z_{1})y_{1} + τ_{2}(z_{2})y_{2} + ⋯ + τ_{n−1}(z_{n−1})y_{n−1} + y_{n} \end{matrix}

is a bijection. Even more, we have:

Proposition 3. The map F is a diffeomorphism.

Proof. It follows from the definition of the Frobenius transformation that F is differentiable. Since both manifolds have the same dimension, it suffices to show that the tangent map T_{ζ}F of F in a point ζ = (z_{1}, ⋯, z_{n−1}, y_{1}, ⋯ y_{n}) ∈ M := V^{(1)} ×⋯×V^{(n−1)} ×Ω_{1} ×⋯×Ω_{n} is injective. For n = 2 and (u_{1} , v_{1}, v_{2}) ∈ V_{12} × V_{11} × V_{22} = T_{ζ}M , the tangent space of M at ζ, we have

T_{ζ}F (u_{1}, v_{1}, v_{2}) = v_{1} ⊕ 2(u_{1}y_{1} + z_{1}v_{1}) ⊕ P (z_{1})v_{1} + {z_{1} y_{1} u_{1}} + v_{2}.

Hence, if T_{ζ}F (u_{1}, v_{1}, v_{2}) = 0 we obtain v_{1} = 0, then u_{1} = 0 because 4y_{1}^{−1}(y_{1}u_{1}) = u_{1} by [MN; (2.6.7)] and finally v_{2} = 0. In general, if w = (u_{1}, ⋯, u_{n−1}, v_{1}, ⋯ v_{n}) ∈ V^{(1)} × ⋯ × V^{(n−1)} × V_{11} × ⋯ × V_{nn} = T_{ζ}M lies in the kernel of T_{ζ}F then, since τ_{2}(z_{2})y_{2} + ⋯ + τ_{n−1}(z_{n−1})y_{n−1} + y_{n} ∈ V (e_{1}, 0), it follows by considering the V_{11} - and V^{(1)} -component of T_{ζ}F w that v_{1} = 0 = u_{1} , but then w = 0 by induction.

Lemma 4. (a) For z_{ij} ∈ V_{ij} , i ≠ j, and x_{mn} ∈ V_{mn} the Frobenius transformation τ_{i}(z_{ij}) operates as follows

τ_{i}(z_{ij})(x_{mn})  (1)

= x_{mn} + \{ \begin{matrix}2x_{ii}z_{ij} ⊕ P (z_{ij})x_{ii} ∈ V_{ij} ⊕ V_{jj} & m = n = i \\ 2e_{j}(z_{ij}x_{ij}) ∈ V_{jj} & \{m, n\} = \{i, j\} \\ 2z_{ij}x_{ik} ∈ V_{jk} & \{m, n\} = \{i, k\}, i, j, k ≠ \\ 0 & i ∉ \{m, n\} \end{matrix}

(b) For z_{ij} ∈ V_{ij} and z_{kl} ∈ V_{kl} we have the following commutation formulas:

τ_{i}(z_{ij})τ_{k}(z_{kl}) = τ_{k}(z_{kl}) τ_{i}(z_{ij}) i ∉ {j, k, l} and k ∉ {l, i, j},  (2)
τ_{i}(z_{ij})τ_{k}(z_{ki}) = τ_{k}(z_{ki} + 2z_{ij}z_{ki})τ_{i}(z_{ij}) |{i, j, k}| = 3,  (3)
τ_{i}(z_{ij})τ_{j}(z_{jl}) = τ_{j}(z_{jl}) τ_{i}(z_{ij} − 2z_{ij}z_{jl}) |{i, j, l}| = 3.  (4)

Proof. (a) is immediate from (2.4). The formulas in (b) can be checked by using (1) and a case-by-case analysis. An alternative proof for (2) and (3) goes as follows. Since τ_{c}(z) = exp(L(z, c)) we have for any invertible endomorphism g of V

gτ_{k}(z_{kl})g^{−1} = exp(gL(z_{kl}, e_{k})g^{−1}).  (5)

By (2.1′)

τ_{i}(z_{ij})L(z_{kl}, e_{k})τ_{i}^{−1}(z_{ij}) = L(τ_{i}(z_{ij})z_{kl}, τ_{i}^{∗−1}(z_{ij})e_{k})

where τ_{i}(z_{ij})z_{kl} = z_{kl} + δ_{li}2z_{ij}z_{kl} by (1) and τ_{i}(z_{ij})^{∗−1}e_{k} = τ_{i}(−z_{ij})^{∗}e_{k} = e_{k} by (2.5). This, together with (5) for g = τ_{i}(z_{ij}) implies (2) and (3). One can prove (4) in a similar fashion:

τ_{j}(z_{jl})^{−1}τ_{i}(z_{ij})τ_{j}(z_{ij}) = exp L(τ_{j}(−z_{jl})z_{ij}, τ_{j}(z_{jl})^{∗}e_{i}) = exp L(z_{ij} − 2z_{ij}z_{jl}, e_{i}).

4. Transformation groups of Ω defined by E.

We define

Ω_{1} ⊕ Ω_{2} ⊕ ⋯ ⊕ Ω_{n} = ω_{1} +_{2} + ⋯ + ω_{n}; ω_{i} ∈ Ω_{i}, 1 ≤ i ≤ n} ⊂ Ω,
A_{E} = P (Ω_{1} ⊕ Ω_{2} ⊕ ⋯ ⊕ Ω_{n}) = exp L(V_{11} ⊕ V_{22} ⊕ ⋯ ⊕ V_{nn}),
K_{E} = {f ∈ Aut V ; fe_{i} = e_{i} , 1 ≤ i ≤ n},
M_{E} = {m ∈ G(Ω) ; mV_{ii} ⊂ V_{ii} , 1 ≤ i ≤ n}.

The second equality in the definition of A_{E} follows from P (exp x) = exp L(2x), see [FK; II.3.4], and Ω = exp V , see the proof of [FK; III.2.1]. Clearly, K_{E} and M_{E} are Lie subgroups of G(Ω).

Theorem 5. (a) M_{E} = {g ∈ G(Ω); gV_{ij} = V_{ij} for all i, j} = {g ∈ G(Ω); gL(e_{i})g^{−1} = L(e_{i}) for 1 ≤ i ≤ n}.

(b) M_{E} operates transitively on Ω_{1}⊕Ω_{2}⊕⋯⊕Ω_{n} ⊂ Ω. More precisely, A_{E} ⊂ M_{E} and for every ω ∈ Ω_{1} ⊕ Ω_{2} ⊕ ⋯ ⊕ Ω_{n} there exists a unique a ∈ A_{E} such that ω = a(e).

(c) K_{E} is a subgroup of M_{E} satisfying

K_{E} = M_{E} ∩ Aut V = {m ∈ M; mm^{∗} = Id}.  (1)

(d) Any m ∈ M_{E} can be uniquely written in the form m = ak where a ∈ A_{E} and k ∈ K_{E}. Thus, we have a decomposition

M_{E} = A_{E} · K_{E} ≈ (V_{11} ⊕ V_{22} ⊕ ⋯ ⊕ V_{nn}) × K_{E} (diffeomorphism). (2)

Proof. We abbreviate A = A_{E}, K = K_{E} and M = M_{E}.

(a) Let m ∈ M. Since m is invertible, we have mV_{ii} = V_{ii}. For i ≠ j and z_{ij} ∈ V_{ij} we have z_{ij} = {e_{i} z_{ij} e_{j}} and hence, by (2.2′) and the Peirce multiplication rules,

mz_{ij} = m{e_{i} z_{ij} e_{j}} = {me_{i} m^{∗−1}z_{ij} me_{j}} ∈ {V_{ii} V V_{jj}} ⊂ V_{ij} ,

whence the first equality in a). The second is then immediate since the Peirce spaces V_{ij} are the joint eigenspaces of the commuting endomorphisms L(e_{i}), 1 ≤ i ≤ n.

(b) Let ω = ω_{1} ⊕ ⋯ ⊕ ω_{n} ∈ Ω_{1} ⊕ ⋯ ⊕ Ω_{n}. Then, by the Peirce multiplication rules, P (ω)V_{ii} = P (ω_{i})V_{ii} ⊂ V_{ii} and hence A ⊂ M. Let \sqrt{ω} = \sqrt{ω_{1}} ⊕ ⋯ \sqrt{ω_{n}} where \sqrt{ω_{i}} ∈ Ω_{i} is the unique square root in Ω_{i} of ω_{i}. Then P (\sqrt{ω}) ∈ A and P (\sqrt{ω})e = ω. If there exist a, a′ ∈ A with ae = a′e and a = P (x), a′ = P (x′) for x, x′ ∈ Ω_{1} ⊕ ⋯ ⊕ Ω_{n} we get x^{2} = P (x)e = P (y)e = y^{2} , thus x = y by the uniqueness of the square root on Ω, and a = a′. Since gΩ̄ = Ω̄ for any g ∈ G(Ω), we have mΩ_{i} = m(Ω̄ ∩ V_{ii}) ⊂ Ω̄ ∩ V_{ii} = Ω_{i} for every m ∈ M. Therefore M(Ω_{1} ⊕ ⋯ ⊕ Ω_{n}) ⊂ Ω_{1} ⊕ ⋯ ⊕ Ω_{n}.

(c) For any m ∈ M ∩ Aut V we have m|V_{ii} ∈ Aut V_{ii} and hence me_{i} = e_{i}. Conversely, any f ∈ K ⊂ Aut V ⊂ G(Ω) has the property fV_{ii} = fV (e_{i}, 1) = V (fe_{i}, 1) = V_{ii} and thus lies in M ∩ Aut V . The equality M_{E} ∩ Aut V = {m ∈ M; mm^{∗} = Id} then follows from (2.3).

(d) For m ∈ M there exists a unique a ∈ A such that me = ae, i.e., k = a^{−1}m ∈ Aut V ∩ M = K in view of (2.3) and c). (2) follows from the fact that exp is a diffeomorphism.

Remarks 6. 1) Let Str(V) be the structure group of V . Since Str(V) = Str(V)^{∗}, it is the group of real points of a reductive algebraic group, and G(Ω) ⊂ Str(V) is a finite covering of the (topological) identity component Str(V)^{0} . More generally, Str(V)_{E} := {g ∈ Str(V); mV_{ij} = V_{ij} for all i, j} is invariant under ^{∗} and hence the group of real points of a reductive algebraic group. Since Str(V)_{E}^{0} ⊂ M_{E} ⊂ Str(V)_{E} it follows that M_{E} is a real reductive group in the sense of [W; 2.1]. The decomposition (2) is the Cartan decomposition of M_{E} in the sense of [W; 2.1.8]. In particular, K_{E} is a maximal compact subgroup of M_{E}.

2) If E = {e} then (2) specializes to the well-known Cartan decomposition G(Ω) = P (Ω) · Aut V ([BK; XI Satz 4.5]). The corresponding decomposition of the Lie algebra LieG(Ω) = 𝔤(V) is the Cartan decomposition 𝔤(V) = L(V) ⊕ Der V . If E is a Jordan frame, i.e., every e_{i} is primitive: V_{ii} = ℝe_{i}, A_{E} is an abelian group and coincides with the group A of [FK; VI.3, p. 112]. In this case 𝔞 = L(V_{11} ⊕ V_{22} ⊕ ⋯ ⊕ V_{nn}) is a maximal abelian subspace of L(V) ⊂ 𝔤(V) so that M_{E} coincides with the group M of [W; 2.2.4].

5. Transformation groups of Ω defined by E and a partial order.

We let ⪯ be a partial order on I = {1, …, n} which is weaker than the canonical order: i ⪯ j ⇒ i ≤ j. We put i ≺ j ⇔ i ⪯ j, i ≠ j and define

\begin{aligned}e_{⟨i⟩} & = ∑_{k≺i} e_{k} , \\ V_{⟨i]} & = ⊕_{k≺i} V_{ki} = V (e_{⟨i⟩}, \frac{1}{2}) ∩ V (e_{i}, \frac{1}{2}) , \\ V_{ij≺} & = (⊕_{j≺l} V_{il}) ⊕ (⊕_{i<k≤l} V_{kl}) , (1 ≤ i ≤ j ≤ n), \end{aligned}  \begin{aligned}τ_{⟨i⟩} & = τ_{e_{⟨i⟩}} , \\ V^{(i≺)} & = ⊕_{i≺j} V_{ij} , \\ V_{ij⪯} & = V_{ij} ⊕ V_{ij≺}. \end{aligned}

Thus, V^{(i≺)} = V^{(i)} in case ⪯ coincides with the canonical order. We will consider the following subgroups of G(Ω):

N_{E,≺} = {u ∈ G(Ω); (u − Id)V_{ij} ⊂ V_{ij≺} for all i ≤ j},
T_{E,⪯} = {t ∈ G(Ω) ; tV_{ij} ⊂ V_{ij⪯} for all i ≤ j}.

Theorem 7. (a) The group N_{E,≺} is a unipotent simply-connected Lie subgroup of T_{E,⪯} and has the descriptions

N_{E,≺} = {τ_{1}(z_{1}) ⋯ τ_{n−1}(z_{n−1}) ; z_{i} ∈ V^{(i≺)}, 1 ≤ i < n}  (1)
= {τ_{⟨n⟩}(z_{n}) ⋯ τ_{⟨2⟩}(z_{2}) ; z_{i} ∈ V_{⟨i]} , 1 < i ≤ n}.  (2)

The Lie algebra of N_{E,≺} is

𝔫_{E,≺} = ⊕_{i=1}^{n−1} {L(z_{i} , e_{i}) ; z_{i} ∈ V^{(i≺)}} = ⊕_{i≺j} L(V_{ij} , e_{i}).

(b) The group M_{E} ⊂ T_{E,⪯} normalizes N_{E,≺} , and T_{E,⪯} is a semidirect product: T_{E,⪯} = M_{E} · N_{E,≺} .

(c) K_{E} = T_{E} ∩ Aut V = {g ∈ T_{E}; ge = e} = {g ∈ T_{E}; gg^{∗} = Id}.

Proof. For easier notation we abbreviate K = K_{E} , M = M_{E} , N = N_{E,≺} and T = T_{E,⪯}.

(a) Any u ∈ N is of the form u = Id + n with n nilpotent, i.e., u is unipotent. Transitivity of ≺ implies that 𝔫 = {n ∈ End V; nV_{ij} ⊂ V_{ij≺} for all i ≤ j} is a nilpotent subalgebra of End V . Therefore, u^{−1} = Id + ∑_{i≥1} (−n)^{i} shows that N is closed under taking inverses. Similarly, N is also closed under products and therefore a subgroup of G(Ω). It is a closed subgroup of G(Ω) and therefore a Lie subgroup of G(Ω). It follows from (1) that N is simply-connected (This is not so surprising since, by [B; §9.5, Cor. 2 of Prop. 18], any unipotent group is simply-connected.) We are therefore left with proving (1) and (2).

Proof of (1): For any i ≺ j we have τ_{i}(z_{ij}) ∈ N by Lemma 4.a. Since τ_{i}(∑_{j≻i} z_{ij}) = ∏_{j≻i} τ_{i}(z_{ij}), we also have {τ_{1}(z_{1}) ⋯ τ_{n−1}(z_{n−1}) ; z_{i} ∈ V^{(i≺)}} ⊂ N. Conversely, let u ∈ N. By definition, there exist unique z_{1} ∈ V^{(1≺)} and v_{0} ∈ V (e_{1}, 0) such that ue_{1} = e_{1} + z_{1} + v_{0} . Observe that u^{∗}x_{11} = x_{11} for all x_{11} ∈ V_{11} since (u − Id)V ⊂ V_{11}^{⊥}. Hence, by (2.4) and the Peirce multiplication rules,

\begin{aligned}ux_{11} & = uP (e_{1})x_{11} = P (ue_{1})u^{∗−1}x_{11} = P (e_{1} + z_{1} + v_{0})x_{11} \\ & = x_{11} ⊕ \{e_{1} x_{11} z_{1}\} ⊕ P (z_{1})x_{11} = x_{11} ⊕ 2x_{11}z_{1} ⊕ P (z_{1})x_{11}. \end{aligned}

In view of (2.4) this shows ux_{11} = τ_{1}(z_{1})x_{11} . Let ũ = τ_{1}(z_{1})^{−1}u ∈ N and put c = e − e_{1} . Since V′ := V (c, 1) = V (e_{1}, 0) = ⊕_{2≤k≤l≤n} V_{kl} it follows that ũ leaves V′ invariant. Because ũΩ̄ = Ω̄ and Ω_{c} = Ω̄ ∩ V (c, 1) we see that ũ|V′ lies in the corresponding subgroup N′ of G(Ω_{c}) defined with respect to E ∩ V (c, 1) = (e_{2}, …, e_{n}) and the restriction of ⪯ to {2, …, n}. By induction, ũ|V′ = τ_{2}(z_{2}) ⋯ τ_{n−1}(z_{n−1})|V′ for suitable z_{i} ∈ V^{(i≺)} (= Id if n = 2). Then

û := (τ_{2}(z_{2}) ⋯ τ_{n−1}(z_{n−1}))^{−1}ũ = τ_{n−1}(−z_{n−1}) ⋯ τ_{2}(−z_{2})ũ ∈ N

has the property ûx_{ii} = x_{ii} for all 1 ≤ i ≤ n. Thus, û = M ∩ N = {Id}. Proof of (2): We have for k ≺ i

τ_{⟨i⟩}(z_{ki}) = exp L(z_{ki}, e_{⟨i⟩}) = exp L(z_{ki}, e_{k}) = τ_{k}(z_{ki}),  (4)

and hence for z_{i} ∈ V_{⟨i]}

τ_{⟨i⟩}(z_{i}) = ∏_{k≺i} τ_{⟨i⟩}(z_{ki}) = ∏_{k≺i} τ_{k}(z_{ki}).

This shows that

N′ := {τ_{⟨n⟩}(z_{n}) ⋯ τ_{⟨2⟩}(z_{2}) ; z_{i} ∈ V_{⟨i]}, 1 < i ≤ n} ⊂ N.

By (4), N′ contains the canonical generators of N. Hence N′ = N if N′ is a subgroup of N. To prove this, it suffices to show that for j < l and i ≺ j, k ≺ l we have τ_{⟨j⟩}(z_{ij}) τ_{⟨l⟩}(z_{kl}) ∈ N′. Since |{i, j, l}| = 3 and τ_{⟨j⟩}(z_{ij}) τ_{⟨l⟩}(z_{kl}) = τ_{i}(z_{ij}) τ_{k}(z_{kl}) there are two cases to be considered: if k = i or k ∉ {i, j, l} then, by Lemma 4.b, τ_{i}(z_{ij}) τ_{k}(z_{kl}) = τ_{k}(z_{kl})τ_{i}(z_{ij}) = τ_{⟨l⟩}(z_{kl})τ_{⟨j⟩}(z_{ij}) ∈ N′, while for k = j we have, by Lemma 4.b and (4)

\begin{aligned}τ_{i}(z_{ij})τ_{j}(z_{jl}) & = τ_{j}(z_{jl})τ_{i}(z_{ij} − 2z_{ij}z_{jl}) = τ_{⟨l⟩}(z_{jl})τ_{⟨l⟩}(−2z_{ij}z_{jl})τ_{⟨j⟩}(z_{ij}) \\ & = τ_{⟨l⟩}(z_{jl} − 2z_{ij}z_{jl}) τ_{⟨j⟩}(z_{ij}) ∈ N′. \end{aligned}

This finishes the proof of (2).

Since τ_{i}(z_{i}) = exp L(z_{i}, e_{i}) we have 𝔫′ := ∑_{i=1}^{n} L(V^{(i≺)}, e_{i}) ⊂ 𝔫 := LieN_{E,≺} by (1). That the sum is direct follows from L(z_{i} , e_{i})e_{j} = δ_{ij}z_{i}. To conclude 𝔫′ = 𝔫 it is sufficient to prove that 𝔫′ is a subalgebra. Indeed, the Lie subgroup N′ of N corresponding to 𝔫′ contains τ_{i}(V^{(i≺)}), hence N′ = N by (1) and therefore 𝔫′ = 𝔫. That 𝔫′ is a subalgebra of 𝔫 follows from the following calculations. Let z_{i} ∈ V^{(i)}, w_{j} ∈ V^{(j)} . If i = j then, by (2.2),

[L(z_{i} , e_{i}), L(w_{i} , e_{i})] = L({z_{i} e_{i} w_{i}}, e_{i}) − L(w_{i}, {e_{i} z_{i} e_{i}}) = 0

since {e_{i} z_{i} e_{i}} = 0, {z_{i} e_{i} w_{i}} ∈ V (e_{i}, 0) and L(V (e_{i}, 0), V (e_{i}, 1)) = 0. If i < j then w_{j} ∈ V (e_{i}, 0) and so {z_{i} e_{i} w_{j}} = 0. Hence, (2.2) shows

[L(z_{i} , e_{i}), L(w_{j} , e_{j})] = −L(w_{j} , {e_{i} z_{i} e_{j}}).

Here {e_{i} z_{i} e_{j}} = z_{ij} ∈ V_{ij} and so {e_{i} e_{i} z_{ij}} = z_{ij}. A second application of (2.2) then yields

−L(w_{j}, z_{ij}) = [L(e_{i}, e_{i}), L(w_{j}, z_{ij})] = −[L(w_{j}, z_{ij}), L(e_{i}, e_{i})] = −L({w_{j}z_{ij}e_{i}}, e_{i})

where {w_{j}z_{ij}e_{i}} = ∑_{j≺k}{w_{jk} z_{ij} e_{i}}. Each term {w_{jk} z_{ij} e_{i}} ∈ V_{ik} with i ≺ j ≺ k since z_{ij} = 0 unless i ≺ j. This proves [L(z_{i} , e_{i}), L(w_{j}, e_{j})] ∈ L(V^{(i≺)}, e_{i}).

(b) It follows from Theorem 5.a that M ⊂ T. Moreover, M normalizes N since for m ∈ M and u ∈ N we have

(mum^{−1} − Id)V_{ij} = m(u − Id)m^{−1}V_{ij} = m(u − Id)V_{ij} ⊂ mV_{ij≺} = V_{ij≺}.

Because M ∩ N = {Id} it is clear that MN = {mn; m ∈ M, n ∈ N} ⊂ T is a semidirect product. To prove the other inclusion, let t ∈ T. We will construct inductively an n ∈ N such that nt ∈ M. Assuming that tV_{jj} = V_{jj} for 1 ≤ j < i we will find n_{i} ∈ N such that n_{i}tV_{jj} = V_{jj} for 1 ≤ j ≤ i. Let te_{i} = x_{ii} + x_{i≺} + b where b is an element of

B = ⊕_{i<k≤l≤n} V_{kl} = V (e_{i+1} + ⋯ + e_{n}, 1) ⊂ V (e_{i}, 0).

We claim that x_{ii} ∈ Ω_{i}. Indeed, te = te_{1} + ⋯ + te_{i} + ⋯ + te_{n} = x_{11} + ⋯ + x_{ii} + x_{i≺}+b̃ for suitable x_{jj} ∈ V_{jj} and b̃ ∈ B, and therefore x_{ii} = P (e_{i})te ∈ P (e_{i})Ω = Ω_{i} by [MN; 3.2]. For any z ∈ V^{(i≺)} we have τ_{i}(z)te_{i} = x_{ii} ⊕2zx_{ii} +x_{i≺} ⊕b′ for a suitable b′ ∈ B. Since x_{ii} ∈ Ω_{i} is invertible in V_{ii} , we can find z′ ∈ V^{(i≺)} such that 2z′x_{ii} + x_{i≺} = 0. Thus, replacing t by τ_{i}(z′)t, we can assume te_{i} = x_{ii} + b′ and, by (2.4), still have tV_{jj} ⊂ V_{jj} for j < i. Let

C = (⊕_{i<l≤n} V_{il}) ⊕ (⊕_{i<k≤l} V_{kl}) = (⊕_{i<l≤n} V_{il}) ⊕ B.

Since t^{−1}C ⊂ C we have t^{∗−1}V_{ii} ⊂ D := C^{⊥} = V_{ii} ⊕ (⊕_{1≤k<i,k≤l} V_{kl}), the orthogonal complement of C with respect to the trace form. Because of P (B)D = 0 = {V_{ii} D B} it now follows for arbitrary v_{ii} ∈ V_{ii}

\begin{aligned}tv_{ii} & = tP (e_{i})v_{ii} = P (te_{i})t^{∗−1}v_{ii} ∈ P (x_{ii} + b′)D \\ & = P (x_{ii})D + P (b′)D + \{x_{ii} D b′\} = P (x_{ii})D = V_{ii}, \end{aligned}

which completes the induction process.

(c) With respect to a suitable orthonormal basis of V , any g ∈ T is represented by an upper triangular block matrix whose block structure is determined by the Peirce spaces V_{ij}. If such a g is also orthogonal, the matrix is in fact a diagonal block matrix. It follows that ge_{i} ∈ V_{ii} is an idempotent of the same rank as e_{i} and hence ge_{i} = e_{i}. Thus T ∩ Aut V ⊂ K, and the other inclusion is obvious. The remaining equalities then follow from (2.3).

Remarks 8. 1) Since N_{E,≺} is unipotent it does not contain any non-trivial compact subgroup, and thus K_{E} is also a maximal compact subgroup of T_{E,⪯} , see Remark 6(1).

2) The map

V^{(1≺)} × ⋯ × V^{(n−1≺)} → N_{E} : (z_{1} , …, z_{n−1}) ↦ τ_{1}(z_{1}) ⋯ τ_{n−1}(z_{n−1})

is in fact a diffeomorphism. Indeed, that the map is a bijection follows from (1) and Proposition 3. As a product of exponentials, it is obviously differentiable. That its inverse is differentiable too, can be shown inductively, following the method of the proof of (1). Of course, since N is nilpotent this is also a special case of a general result on canonical coordinates of solvable Lie groups ([B; §9.6, Prop. 20]).

3) If ⪯ is the minimal order, i.e., i ⪯ j ⇔ i = j, we have N_{E,≺} = {Id} and T_{E,⪯} = M_{E}. For example, this is the case if E = {e}. On the other extreme, if E is a Jordan frame and ⪯ is the canonical order, the group N_{E,≺} coincides with the so-called strict triangular subgroup N of [FK; VI.3]. By (3) it is also the group N of [W; 2.1.8]. In this case, A_{E} · N_{E,≺} is a subgroup of T_{E,≺} , the so-called triangular subgroup T of [FK; VI.3].

6. The AP cone ([MN]).

An AP cone Ω(K) ⊂ Ω is defined in terms of an orthogonal system (c_{1}, …, c_{s}) of primitive idempotents c_{i} ∈ V and a unital ring K, i.e., a set of subsets of {1, …, s} which is closed under union and intersection: K, L ∈ K ⇒ K ∪ L ∈ K and K ∩ L ∈ K, and which moreover has the property that Ø ∈ K and {1, …, s} ∈ K. To describe Ω(K) we need the following notations. For any K ⊂ {1, …, s} and x ∈ V we put c_{K} = ∑_{k∈K} c_{k} and x_{K} = P (c_{K})x, the V (c_{K}, 1)-component of x. If x ∈ Ω and K ≠ Ø then x_{K} ∈ P (c_{K})Ω, and one knows that this is the symmetric cone of the Euclidean Jordan algebra V (c_{K}, 1). In particular, x_{K} is invertible in V (c_{K}, 1). We denote by x_{K}^{−1} the inverse of x_{K} in V (c_{K}, 1) and view x_{K}^{−1} as an element of V . We note that in general x_{K}^{−1} ≠ P (c_{K})(x^{−1}). For K = Ø we put c_{Ø} = 0 and x_{K}^{−1} = 0^{−1} = 0. The AP cone Ω(K) is then defined as the set of all x ∈ Ω satisfying

x_{K∪L}^{−1} + x_{K∩L}^{−1} = x_{K}^{−1} + x_{L}^{−1}

for all K, L ∈ K. Equivalent characterizations of Ω(K) are given in [MN; Thm. 2.4].

The link with the results obtained so far in this paper is property (1) below. To explain it, we recall that Ø ≠ K ∈ K is join-irreducible if K is not a union of proper subsets of K belonging to K. Thus, if we put ⟨K⟩ := ∪{K′ ∈ K; K′ ⊊ K} and [K] := K \ ⟨K⟩ then K is join-irreducible if and only if [K] ≠ Ø. We denote by J (K) the set of all join-irreducible sets in K. One knows [AP; 2.1] that any K ∈ K is partitioned by {[L] ; L ∈ J (K) and L ⊂ K}. Moreover, by [AP; 2.7], one can always find a never-decreasing listing of J (K), i.e., an enumeration J (K) = (K_{1}, …, K_{n}) with the property i < j ⇒ K_{j} ⊄ K_{i}. We fix such a listing and define a partial order ⪯ on {1, …, n} by i ⪯ j ⇔ [K_{i}] ⊂ K_{j}. For 1 ≤ j ≤ s we put e_{j} = ∑_{i⪯j} c_{i} and obtain in this way an orthogonal system E = (e_{1}, …, e_{n}). After renumbering, we may assume that ⪯ is weaker than the canonical order, so that we are in the setting of 6. Then, by [MN; 2.14], the map

F_{K} : V^{(1≺)} × ⋯ × V^{(n−1≺)} × Ω_{1} × ⋯ × Ω_{n} → Ω(K)

given by

F_{K}(z_{1}, ⋯, z_{n−1}, y_{1}, ⋯, y_{n}) = τ_{1}(z_{1}) ⋯ τ_{n−1}(z_{n−1})(y_{1} ⊕ ⋯ ⊕ y_{n})

is a bijection. Thus,

Ω(K) = N_{E,≺}(Ω_{1} ⊕ ⋯ ⊕ Ω_{n})  (1)

We transport the obvious manifold structure of V^{(1≺)} ×⋯×V^{(n−1≺)} ×Ω_{1} ×⋯× Ω_{n} to Ω(K) via F_{K}. By Proposition 3, Ω(K) is then a simply-connected closed submanifold of Ω (with the induced topology). Also, Proposition 3 implies,

Ω(K) = Ω ⇔ ⪯ is the canonical order.  (2)
Ω(K) = Ω_{1} ⊕ ⋯ ⊕ Ω_{n} ⇔ ⪯ is the minimal order.  (3)

Theorem 9. T_{E,⪯} is a transitive Lie transformation group of Ω(K). For this operation, the isotropy group of e ∈ Ω(K) is K_{E}, and we have an isomorphism of manifolds

Ω(K) ≈ T_{E,⪯}/K_{E}.

Proof. For easier notation we abbreviate K = K_{E}, M = M_{E}, N = N_{E,≺} and T = T_{E,⪯}. By Theorem 5.b, we know that M operates transitively on Ω_{1} ⊕⋯⊕Ω_{n}. Thus, by (7.1), Ω(K) = NMe. But this implies that both M and N leave Ω(K) invariant: NΩ(K) = NNMe = Ω(K) and, since M normalizes N, MΩ(K) = MNMe = NMMe = Ω(K). Therefore, T operates transitively on Ω(K). By Theorem 6.c), the isotropy group of e in T is K_{E}, and hence the isomorphism follows from ([B; §1.7 Prop. 14]).

References

[AP] Andersson, S. A., and M. D. Perlman, Lattice Models for Conditional Independence in a Multivariate Normal Distribution, Ann. Statist. 21 (1993), 1318–1358.

[B] Bourbaki, N., “Groupes et Algèbres de Lie,” Chap. III, Hermann Paris 1972.

[BK] Braun, H., und M. Koecher, ,,Jordan-Algebren“, Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 128, Springer-Verlag, Berlin Heidelberg 1965.

[FK] Faraut, J., and A. Koranyi, “Analysis on Symmetric cones,” Clarendon Press, Oxford 1994.

[MN] Massam, H., and E. Neher, Estimation and testing for lattice conditional independence models on Euclidean Jordan algebras, Ann. Statist., to appear.

[W] Wallach, N., “Real reductive groups I,” Pure and Applied Mathematics 132, Academic Press 1988.

Department of Mathematics and Statistics University of Ottawa Ottawa, Ontario K1N 6N5 CANADA neher@uottawa.ca

Received March 4, 1998 and in final form May 19, 1998