Journal of Lie Theory Volume 10 (2000) 375–381 ◯^{C}2000 Heldermann Verlag

Compactification structure and conformal compressions of symmetric cones

Jimmie D. Lawson and Yongdo Lim

Communicated by K.-H. Neeb

Abstract. In this paper we show that the boundary of a symmetric cone Ω in the standard real conformal compactification M of its containing euclidean Jordan algebra V has the structure of a double cone, with the points at infinity forming one of the cones. We further show that \overline{Ω}^{M} admits a natural partial order extending that of Ω. Each element of the compression semigroup for Ω is shown to act in an order-preserving way on \overline{Ω}^{M} and carries it into an order interval contained in \overline{Ω}^{M}.

1. Introduction

Let V be a euclidean Jordan algebra with identity e and let Ω = exp V. Then Ω is a symmetric cone and T_{Ω} := V +iΩ is a symmetric tube domain with symmetry j(z) = −z^{−1} at ie. The biholomorphic automorphism group G(T_{Ω}) of T_{Ω} is generated by N^{+}, H and j, where N^{+} is the group of all real translations t_{x} and H is the automorphism group of the symmetric cone Ω. There exists a conformal compactification M := G(T_{Ω})/P with respect to the parabolic subgroup P = HN^{−}, where N^{−} = jN^{+}j, and the embedding of V into M given by x ∈ V → t_{x} · P ∈ M is an open dense embedding. A Lie semigroup that is naturally related to the symmetric cone Ω occurs as the semigroup of compressions of Ω in G(T_{Ω}):

Γ_{Ω} := {g ∈ G(T_{Ω}) | g · Ω ⊆ Ω}.

In this paper we investigate the structure of \overline{Ω}^{M}, particularly its boundary and the points at infinity, and we establish a natural order structure on this compactification of Ω. We show that the compression semigroup acts in an order-preserving way on \overline{Ω}^{M} and carries it to an internal order interval.

This paper is organized as follows. In section 2 we realize the ideal boundary (boundary points in M \ V ) of Ω as a cone in V, and in section 3 we show how to endow \overline{Ω}^{M} with a natural partial order. In section 4, we represent the compressed domain g(Ω) by g in Γ_{Ω} as an order interval which is determined by the points g(0) and gj(0).

ISSN 0949–5932 / $2.50 ◯^{C} Heldermann Verlag

2. Boundary structure of symmetric cones

We recall certain basic notions and well-known facts concerning Jordan algebras (see, for example, [3]). A commutative algebra V over the field ℝ or ℂ with product xy is said to be a Jordan algebra if for all elements x, y in V, x(x^{2}y) = x^{2}(xy). For x ∈ V, let L(x) be the linear map of V defined by L(x)y = xy, and let P (x) = 2L(x)^{2} − L(x^{2}). A finite dimensional real Jordan algebra V is called a euclidean Jordan algebra if it carries an inner product ⟨·|·⟩ such that ⟨xy| z⟩ = ⟨y| xz⟩, for all x, y, z ∈ V.

Let V be a euclidean Jordan algebra with identity e, and let Q = {x^{2} | x ∈ V } be the set of squares. Then the set Q is a self dual cone and the interior Ω of Q is a symmetric cone, i.e., Ω is a self-dual cone (with respect to the inner product) and the group G(Ω) := {h ∈ GL(V) | h(Ω) = Ω} acts on it transitively. The closure \overline{Ω} is a closed pointed generating cone in V with interior Ω.

A Jordan frame is a family c = {c_{1}, ⋯, c_{r}} of primitive idempotents c_{i} with

c_{i}c_{j} = 0, i ≠ j,
∑_{i=1}^{r} c_{i} = e.

The spectral theorem states that for x ∈ V, there is a Jordan frame c = {c_{1}, ⋯, c_{r}} (r fixed) and real numbers λ_{1}, ⋯, λ_{r} such that x = ∑_{i=1}^{r} λ_{i}c_{i}. The real numbers λ_{i} (with their multiplicities) are uniquely determined by x. Furthermore, x ∈ Ω (resp. x ∈ \overline{Ω}) if and only if λ_{i} > 0 (resp. λ_{i} ≥ 0) for each i.

The set T_{Ω} := V + iΩ ⊆ V + iV is a symmetric tube domain. Let G(T_{Ω}) be the Lie group of all biholomorphic automorphisms on the tube domain T_{Ω}. The group G(T_{Ω}) can be described in the following way: an element in G(Ω) acts on the tube domain T_{Ω} by z = x + iy ⟶ g(z) = g(x) + ig(y). For x in V , the translation by x, t_{x} : z ⟶ z + x is a holomorphic automorphism of T_{Ω} and the group of all real translations is an abelian group N^{+} isomorphic to the vector group V. The map (the symmetry of T_{Ω} at ie) j : z ⟶ −z^{−1} is in G(T_{Ω}). We set t̃_{x} = j ∘ t_{x} ∘ j and N^{−} = j ∘ N^{+} ∘ j. Then G(T_{Ω}) is generated by N^{+}, G(Ω) and j [3].

Let G be the identity component of G(T_{Ω}) and let H := G(Ω)_{0}. Let P = HN^{−}. Then P is a maximal parabolic subgroup of G. Then it is known that the homogeneous space M := G/P is a compact real manifold containing V as an open dense subset, i.e., a real conformal compactification of the Jordan algebra V [1]. The embedding is given by

V ⟶ M, x → t_{x}P,

and we henceforth identify V with its image in M. The space N^{+}HN^{−} can be characterized by the elements g ∈ G such that g(0) ∈ V (in this case, g(0) = g · 0, see Theorem 2.14 [1] and Corollary 2.20 [5]). One computes that j(x)(= jt_{x}P) = −x^{−1} for x an invertible element of V .

For a subset A of V , let \overline{A} denote its closure in V and let \overline{A}^{M} denote its closure in M. We are interested in studying the structure of the boundary \overline{Ω}^{M} \Ω of Ω, which consists of the set of “infinite” points in the closure ∂_{∞}Ω := \overline{Ω}^{M} \ \overline{Ω} and the set of “finite” points in the closure given by ∂Ω = \overline{Ω} \ Ω. This study is facilitated by considering an equivalent bounded symmetric domain that arises by applying the “real” version of the Cayley transform to Ω resp. \overline{Ω}^{M}. We recall that this Cayley transform is given by

c(x) = (x − e)(x + e)^{−1} = e − 2(x + e)^{−1} = t_{e} ∘ 2id_{V} ∘ j ∘ t_{e}(x);

note that the first description has singular points, but that the last extends it and is defined on all of M. The Cayley transform has a global inverse given by

d(x) = −(x + e)(x − e)^{−1} = −e + 2(e − x)^{−1} = t_{−e} ∘ j ∘ (1/2)id_{V} ∘ t_{−e}(x).

We define two useful relations on V , the partial order on V defined by x ≤ y if and only if y − x ∈ \overline{Ω}, and x ≪ y if and only if y − x ∈ Ω. The relation ≤ is a closed partial order (i.e., closed as a relation in V × V ) and is the closure of ≪. We have x ≤ w ≤ y (resp. x ≪ w ≪ y) if and only if w ∈ (x + \overline{Ω}) ∩ (y − \overline{Ω}) (resp. w ∈ (x + Ω) ∩ (y − Ω)). We note that if x ≪ y, then the closure of x + Ω ∩ y − Ω is equal to x + \overline{Ω} ∩ y − \overline{Ω} (if x ≤ w ≤ y for x ≪ y, then the sequence w_{n} = (1/n)((x + y)/2)) + ((n − 1)/n)w ∈ x + Ω ∩ y − Ω and converges to w). For x ≤ y, we define the order interval

[x, y] = \{w : x ≤ w ≤ y\} = y − \overline{Ω} ∩ x + \overline{Ω}.

If x = ∑_{i=1}^{r} λ_{i}c_{i} and y = ∑_{i=1}^{r} μ_{i}c_{i} are spectral decompositions for the Jordan frame {c_{1}, …, c_{r}}, then x ≤ y (resp. x ≪ y) if and only if λ_{i} ≤ μ_{i} (resp. λ_{i} < μ_{i}) for all i. In particular if e ≪ x = ∑_{i=1}^{r} λ_{i}c_{i}, then 1 < λ_{i} for all i, and thus x^{−1} = ∑_{i=1}^{r}(1/λ_{i})c_{i} ≪ e.

The following is a basic fact about the Cayley transform applied to Ω (see, for example, [2] for observation (1)).

Proposition 2.1. Let V be a euclidean Jordan algebra with symmetric cone Ω. Then we have the following:

(1) c(Ω) = e − Ω ∩ Ω − e.

(2) c(\overline{Ω}^{M}) = e − \overline{Ω} ∩ \overline{Ω} − e = [−e, e].

(3) x ∈ c(∂_{∞}(Ω)) if and only if x has a spectral decomposition of the form ∑_{i=1}^{r} λ_{i}c_{i} such that −1 ≤ λ_{i} ≤ 1 for all i and at least one λ_{j} = 1.

(4) x ∈ c(∂(Ω)) if and only if x has a spectral decomposition of the form ∑_{i=1}^{r} λ_{i}c_{i} such that −1 ≤ λ_{i} < 1 for all i and at least one λ_{j} = −1.

Proof. Let x ∈ Ω. Then e ≪ e + x = t_{e}(x). It follows that (e + x)^{−1} ≪ e and thus −e ≪ jt_{e}(x). Thus −e = e + 2(−e) ≪ e + 2jt_{e}(x) = c(x). From jt_{e}(x) ≪ 0 we derive c(x) ≪ e, and thus c(Ω) ⊆ e−Ω ∩(−e+Ω). Similar arguments applied to d and e − Ω ∩ (−e + Ω) yield d(e − Ω ∩ (−e + Ω)) ⊆ Ω. Since c and d are inverses on M, we conclude that c(Ω) = e − Ω ∩ (−e + Ω). This completes (1).

Assertion (2) follows from taking closure of both sides of (1) (note that the right-hand side is compact, hence closed in M). Assertion (4) follows from the straightforward observation that x ∈ ∂Ω if and only if all λ_{i} ≥ 0 and some λ_{j} = 0 in some spectral decomposition of x if and only if all μ_{i} satisfy −1 ≤ μ_{i} < 1 and some μ_{j} = −1 in some spectral decomposition of c(x). Assertion (3) then follows from the other three. □

We give a more explicit characterization of the boundary of [−e, e] and hence implicitly of the boundary in M of Ω. We set

E = c(∂_{∞}Ω) ∩ c(\overline{∂Ω}^{M}) = c(∂_{∞}Ω) ∩ \overline{c(∂Ω)}
= {x = ∑_{i=1}^{r} λ_{i}c_{i} : max{λ_{i}} = 1, min{λ_{i}} = −1

for some Jordan frame}

We note that second and third sets are equal since c is a homeomorphsim. To see that the third and fourth sets are equal, let x = ∑_{i=1}^{r} λ_{i}c_{i} with max{λ_{i}} = 1 and min{λ_{i}} = −1, then by the preceding proposition x ∈ c(∂_{∞}Ω). Let x_{n} have i − th coefficient the larger of −1 and λ_{i} − (1/n). Then x_{n} ∈ c(∂Ω) and converges to x, so x ∈ c(∂_{∞}Ω) ∩ \overline{c(∂Ω)}^{M}. Conversely let y ∈ c(∂_{∞}Ω) ∩ \overline{c(∂Ω)}^{M}. Then it must have an eigenvalue of 1 among the coefficients in a spectral decomposition with respect to a Jordan frame. Pick a sequence x_{n} ∈ c(∂Ω) converging to x, and choose a spectral decompostion for each x_{n}. Then for each n the 2n-tuple consisting of the i-th coefficient λ_{i} in the coordinate i and of c_{i} in coordinate r + i lies in the compact space [−1, 1]^{r} × [0, e]^{r}, and thus admits a convergence subsequence. Since each x_{n} has a coefficient of −1, there must be a entry of −1 in the limit, and the Jordan algebra components must converge to another Jordan frame. Thus y must also be in the terminal set.

Theorem 2.2. Let V be a euclidean Jordan algebra with symmetric cone Ω. Then we have the following:

(1) c(∂_{∞}Ω) = {(1 − t)e + tx : x ∈ E, t ∈ [0, 1]}, the cone over E with vertex e.

(2) c(\overline{∂Ω}^{M}) = \{(1 − t)(−e) + tx : x ∈ E, t ∈ [0, 1]\}, the cone over E with vertex −e.

Thus the boundary c(∂_{∞}(Ω)∪∂Ω) of c(\overline{Ω}^{M}) is the double cone over E with vertices e and −e. Proof. We use the characterization of c(∂_{∞}Ω) given in Proposition 2.1. For any point x besides the cone point e in this set, the spectral decomposition ∑_{i=1}^{r} λ_{i}c_{i} must have some λ_{i} = 1 and min{λ_{i}} = γ < 1. Then elementary calculations yield that the one and only possibility for obtaining x in the right-hand side of (1) is to choose y = ∑_{i=1}^{r} μ_{i}c_{i} ∈ E, where μ_{i} = (2λ_{i} − γ − 1)/(1 − γ), and t = 2/(1 − γ). Similar calculations hold for part (2), and the last assertion then follows directly from these. □

3. The orders ≪ and ≤

We continue in the same setting as the previous section: a euclidean Jordan algebra V with symmetric cone Ω. We consider for a ≪ b the order interval [a, b] = a + \overline{Ω} ∩ b − \overline{Ω}. We note that the translation t_{−a} is order-preserving for both ≤ and ≪ and carries the interval to [0, c], where c = b − a ∈ Ω. The linear transformation P (c^{− \frac{1}{2}} ) preseves Ω, hence is order-preserving, and carries c to e and 0 to 0. Thus we see that any interval [a, b] with a ≪ b can be carried onto [0, e] by an affine isomorphism that preserves both the orders ≪ and ≤. Thus we can without loss of generality restrict our attention to the study of one such interval.

We choose for consideration the order interval [−e, e] = e − \overline{Ω} ∩ (−e + \overline{Ω}). This set has dense interior c(Ω) = {w : −e ≪ w ≪ e} = e−Ω ∩ Ω−e (Proposition 2.1). Let x, y ∈ [−e, e] with x ≤ y. Then for 0 < t < 1, we have

−e ≪ −te ≤ tx ≤ ty ≪ ty + \frac{1}{2}(1 − t)e ≪ ty + (1 − t)e ≤ e.

Since as t → 1, tx → x and ty + (1/2)(1 − t)e → y, we conclude that there exist x_{n} → x and y_{n} → y such that −e ≪ x_{n} ≪ y_{n} ≪ e for each n. We have thus proved:

Proposition 3.1. Let V be a euclidean Jordan algebra with symmetric cone Ω. If a ≪ b, then the order relation ≤ on [a, b] is the closure of the order relation ≪ on {w : a ≪ w ≪ b} = a + Ω ∩ b − Ω, the interior of [a, b].

The next lemma is a known result (see, for example, Exercise 7 of Chapter III of [3]).

Lemma 3.2. Let a, b ∈ Ω. Then a ≪ b if and only if b^{−1} ≪ a^{−1}. Proof. Suppose that a, b ∈ Ω with a ≪ b. Since P (a^{− \frac{1}{2}} ) ∈ H, P (a^{− \frac{1}{2}} )(b−a) ∈ Ω. This implies that P (a^{− \frac{1}{2}} )(a) = e ≪ P (a^{− \frac{1}{2}} )b. Since a^{\frac{1}{2}} , b ∈ Ω, (P (a^{− \frac{1}{2}} )b)^{−1} = P (a^{\frac{1}{2}} )b^{−1} ≪ e (see Section II.3 of [3]). This implies that b^{−1} ≪ P (a^{− \frac{1}{2}} )e = a^{−1}. □

Corollary 3.3. Each of the following mappings preserve ≪.

(1) The mapping j : Ω → −Ω and j : −Ω → Ω;

(2) The Cayley transform c(x) = (x − e)(x + e)^{−1} = t_{e} ∘ 2id_{V} ∘ j ∘ t_{e}(x) from Ω to e − Ω ∩ Ω − e;

(3) The inverse Cayley transform d(x) = −(x + e)(x − e)^{−1} = −e + 2(e − x)^{−1} = t_{−e} ∘ j ∘ (1/2)id_{V} ∘ t_{−e}(x) from e − Ω ∩ Ω − e to Ω.

Proof. Item (1) is an immediate consequence of Lemma 3.2, since j(x) = −x^{−1} is the composition of two order-reversing transformations. It is immediate that the other mappings besides j in the right-most definitions of c and d are orderpreserving for ≪, and hence c and d are. □

Corollary 3.4. The closure of the relation {(x, y) ∈ Ω × Ω : x ≪ y} in \overline{Ω}^{M} × \overline{Ω}^{M} is a closed partial order on \overline{Ω}^{M} which extends the partial order ≤ on Ω. The mappings c and d are order-preserving inverses between \overline{Ω}^{M} and [−e, e] with respect to this order. We can then denote \overline{Ω}^{M} as the order interval [0, ∞], where ∞ := j(0). Proof. By Proposition 3.1 the closed partial order ≤ on [−e, e] is the closure of ≪ on the interior. We pull this closed order back to \overline{Ω}^{M} via the homeomorphism c^{−1} = d. Since by Corollary 3.3 d is an order isomorphism from the interior of [−e, e] to Ω with respect to ≪, it follows immediately that the pulled back order is the closure of ≪ on Ω. Since d is also an order isomorphism on the interior of [−e, e] with respect to ≤, it follows that the closure agrees with ≤ on Ω. We note that the largest point of \overline{Ω}^{M} will be d(e) = t_{−e} ∘ j ∘ (1/2)id_{V} ∘ t_{−e}(e) = t_{−e}j(0). Note also that −(1/n)e → 0 implies that ne = j(−(1/n)e) → j(0) = ∞, so that t_{−e}(∞) = lim t_{−e}(ne) = lim(n − 1)e = ∞. Thus d(e) = ∞ is the largest element of \overline{Ω}^{M}. □

4. The compression semigroup

Let V be a euclidean Jordan algebra with the corresponding symmetric cone Ω. In the action of G on M = G/P, we consider the compression semigroup of Ω ⊂ M

Γ_{Ω} = {g ∈ G | g · Ω ⊆ Ω}.

Since the closure \overline{Ω}^{M} of Ω in M is compact with Ω as its interior, the compression semigroup Γ_{Ω} is a closed subsemigroup of G [4].

Now let

Γ^{+} = \{t_{x} ∈ N^{+} | x ∈ \overline{Ω}\}, Γ_{∘}^{+} = \{t_{x} | x ∈ Ω\},
Γ^{−} = \{t̃_{−x} ∈ N^{−} | x ∈ \overline{Ω}\}, Γ_{∘}^{−} = \{t̃_{x} | x ∈ −Ω\}.

Then Γ^{+} and Γ^{−} are closed subsemigroups of N^{+} and N^{−}, respectively. The following appear as Theorem 4.9 in [6] and Corollary 7.7 in [7]:

Theorem 4.1. We have Γ_{Ω} = Γ^{+}HΓ^{−}. Furthermore, the interior Γ_{Ω}^{∘} of Γ_{Ω} is given by Γ_{Ω}^{∘} = Γ_{∘}^{+}HΓ_{∘}^{−}. In particular, Γ_{Ω}^{∘} = \{g ∈ G | g · \overline{Ω}^{M} ⊂ Ω\}.

Theorem 4.2. Let g ∈ Γ_{Ω}. Then g(\overline{Ω}^{M}) = [g(0), g(∞)], and the mapping g is an order isomorphism from the first to the second set. Proof. Let g = t_{x}ht̃_{−y} ∈ Γ_{Ω} (by Theorem 4.1). Since j is an order isomorphism for ≪ from Ω to −Ω, t_{−y} is order isomorphism from −Ω into −Ω, and j is again an order isomorphism from −Ω to Ω, we conclude that t̃_{−y} is an order isomorphism into Ω for ≪ on Ω. Since h is a linear mapping on V preserving Ω, it is an order isomorphism for ≪ on Ω, and t_{x} obviously is. Thus the composition g is an order isomorphism for ≪ from Ω onto the image of Ω, and then by continuity for it is also one for ≤ on \overline{Ω}^{M} (see Corollary 3.4). Thus g(\overline{Ω}^{M}) ⊆ [g(0), g(∞)], where g(0) = x and g(∞) = x + h(y^{−1}) if y ∈ Ω.

Consider the case that y ∈ Ω. Suppose that z in in the interior of [x, x + h(y^{−1})]. Then z = x + a = x + h(y^{−1}) − b for some a, b ∈ Ω. Note that a = h(y^{−1})−b, so a ≪ h(y^{−1}). Since the inversion −j is order decreasing on Ω (Lemma 3.2), w := ((h^{−1}(a))^{−1} − y)^{−1} ∈ Ω. This implies that z = x + a = g(w) ∈ g(Ω). Since the interior of [g(0), g(∞)] is dense, it follows that g(\overline{Ω}^{M}) = [g(0), g(∞)].

Finally consider the case that y ∈ ∂Ω. Let z ∈ [g(0), g(∞)]. Pick a sequence h_{n} in Γ_{Ω}^{∘} converging to the identity (for example, t_{ϵe} ∘ t̃_{−ϵe} for ϵ = 1/n). Then g_{n} := h_{n}g ∈ Γ_{Ω}^{∘}, since the latter is a semigroup ideal, and thus by the previous paragraph there exists w_{n} ∈ \overline{Ω}^{M} such that g_{n}(w_{n}) = h_{n}(z). By compactness and continuity the sequence w_{n} converges to w ∈ \overline{Ω}^{M} such that g(w) = z.

□

References

[1] Bertram, W., Un théorème de Liouville pour les algèbres de Jordan, Bulletin de la Soc. Math. Française 124 (1996), 299–327.

[2] Faraut, J., Fonctions de Legendre sur une algè bre de Jordan, CWI Quarterly 5 (1992), 309–320.

[3] Faraut, J., and A. Koranyi, “Analysis on symmetric cones,” Oxford University Press, Oxford, 1994.

[4] Hilgert, J., and G. Ólafsson, “Causal symmetric spaces,” Academic Press, San Diego, 1997.

[5] Kaup, W., A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces, Math. Z. 183 (1983), 503–529.

[6] Koufany, K., Semi-groupe de Lie associé á un cône symmétrique, Ann. Inst. Fourier, Grenoble 45 (1995),1–29.

[7] J. D. Lawson and Y. Lim, Lie semigroups with triple decompositions, Pacific J. of Math., to appear.

Department of Mathematics Louisiana State University Baton Rouge La 70803, U.S.A.

Topology and Geometry

Research Center

Kyungpook National University Taegu 702-701, Korea

Received June 7, 1999 and in final form February 1, 2000