Journal of Lie Theory Volume 11 (2001) 155–171 \begin{matrix}c \\ ◯ \end{matrix} 2001 Heldermann Verlag Berlin

Basic relative invariants associated to homogeneous cones and applications

Hideyuki Ishi^{∗}

Communicated by E. B. Vinberg

Abstract. In this paper, we determine the basic relative invariants on the ambient vector space of a homogeneous cone Ω under the action of the solvable linear Lie group acting on Ω simply transitively. The results are applied to a study of the Riesz distributions on Ω and to an algebraic description of the closure \overline{Ω} of Ω.

Introduction

In analysis on homogeneous cones, relatively invariant polynomials on the ambient vector spaces play significant roles. Among them, of particular importance are the basic relative invariants, that is, the polynomials by which every relatively invariant polynomial is expressed as a product of their powers. We determine in this paper the basic relative invariants under the action of the solvable Lie group acting simply transitively on a homogeneous cone. These are generalizations of principal minors of real symmetric matrices. As an application, we establish the condition that the Riesz distributions on the homogeneous cone are supported by the origin. We also give an algebraic description of the closure of the cone by introducing analogues of submatrices and minors.

Let Ω be a homogeneous cone in a real vector space V . Vinberg [14] shows that there exists a split solvable linear Lie group H ⊂ GL(V) which acts on Ω simply transitively. In order to describe Ω algebraically, he introduces in [14, Chapter 3] polynomials which we denote by D_{1}, D_{2}, …, D_{r} here, where r is the rank of the cone Ω (see also [10, p. 73]). These D_{k} are also treated in Gindikin [4] and called integral compound power functions there. As Gindikin observes, D_{k} may have extra factors. In the present work, we give a simple method to factorize D_{k} ’s into irreducibles in a general setting. As a result, exactly r polynomials Δ_{1}, Δ_{2}, …, Δ_{r} appear as irreducible factors, and we show that they are the basic relative invariants.

Our first application of the basic relative invariants is to a study of Riesz distributions. The Riesz distribution on Ω is characterized as the distribution

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag whose Laplace transform is a relatively invariant function on the dual cone Ω^{∗} (see (3.23)), where Ω^{∗} is regarded as a homogeneous cone by the adjoint action of H. Let Δ_{1}^{∗}, …, Δ_{r}^{∗} be the basic relative invariants associated to Ω^{∗}. We show that the Riesz distribution is supported by the one-point set {0} if and only if its Laplace transform equals a product of powers of Δ_{k}^{∗}. This is a refinement of Gindikin’s result [4, Proposition 3.4].

∗ Partially supported by JSPS Research Fellowships for Young Scientists.

Our second application is to describe the closure \overline{Ω} of the cone Ω. The motivation is the fact that a real symmetric matrix x = (x_{ij}) of size r is positive semi-definite if and only if the determinants of the submatrices (x_{ij})_{i,j∈I} are nonnegative for all non-empty subsets I ⊂ {1, …, r}. Note that the non-negativity of the principal minors alone is not sufficient (see the beginning of Section 4). We give a criterion for the elements in V to belong to \overline{Ω}. Namely, after introducing polynomials Δ^{I} on V which are analogues of the minors of matrices, we characterize \overline{Ω} by 2^{r} − 1 inequalities Δ^{I} ≥ 0.

Let us explain the organization of this paper. According to [14], we introduce a certain algebra structure on V , called the clan of Ω, and present Vinberg’s polynomials D_{k} in Section 1. In Proposition 1.4, we give recurrence formulas practical for the computation of D_{k}.

We consider H -relatively invariant polynomials in Section 2. The basic relative invariants are obtained in Theorem 2.2. We show in Proposition 2.3 that x ∈ Ω if and only if Δ_{k}(x) > 0 (k = 1, …, r). In Section 3, after describing the dual cone Ω^{∗} as a homogeneous cone, we prove the result on the Riesz distributions mentioned above. Section 4 is devoted to getting the description of \overline{Ω} by the polynomials Δ^{I}.

In Section 5, two examples are presented to write down D_{k} and to get Δ_{k} from D_{k} by factorization. The first example is the cone of real positive definite symmetric matrices, which is a typical homogeneous cone. It turns out that the polynomials Δ_{k} for this cone are equal to the principal minors. The second is so-called the Vinberg cone [14]. The cone and its dual cone are the lowest dimensional ones among the non-symmetric homogeneous cones. For these cones, our polynomials Δ_{k} coincide with Gindikin’s in [6, p. 98 (e), (f)] as a matter of fact.

The present author would like to express his sincere gratitude to Professor Takaaki Nomura for the encouragement and the advices in writing this paper. He is also grateful to Professor Soji Kaneyuki and Professor Simon Gindikin for the various comments about this work.

1. Preliminaries

Let V be a real vector space and Ω an open convex cone in V containing no line. We assume that Ω is homogeneous, that is, the linear automorphism group G(Ω) ⊂ GL(V) of Ω acts transitively on the cone Ω. Then it is known [14] that there exists a split solvable Lie subgroup H ⊂ G(Ω) which acts on Ω simply transitively. Let 𝔥 ⊂ End(V) be the Lie algebra of H and fix a point E in Ω. Then we have the linear isomorphism 𝔥 ∋ T ↦^{∼} T · E ∈ V obtained by differentiating the orbit map H ∋ t ↦ t · E ∈ Ω. Let j : V → 𝔥 be the inverse map of the isomorphism, and define a bilinear multiplication △ on V by x△y := (jx) · y ∈ V (x, y ∈ V). The algebra (V, △) is called the clan of the homogeneous cone Ω ([14, Chapter 2]). This algebra is left-symmetric:

x△(y△z) − (x△y)△z = y△(x△z) − (y△x)△z (x, y, z ∈ V),

which is equivalent to

[jx, jy] = j(x△y − y△x) (x, y ∈ V).  (1.1)

By [14, Chapter 2, Proposition 8] we have a normal decomposition of the space V :

V = \begin{matrix}∑ & ⊕ \\ 1≤k≤m≤r \end{matrix} V_{mk},  (1.2)

where V_{kk} (k = 1, …, r) is the one-dimensional subspace spanned by an idempotent E_{k}, and V_{mk} consists of the elements x ∈ V such that

y△x = (1/2)(y_{m} + y_{k})x and x△y = y_{k}x  (1.3)

for all y = ∑_{i=1}^{r} y_{i}E_{i} ∈ V (y_{1}, …, y_{r} ∈ ℝ). It is known [14, p. 376] that the following relations hold:

\begin{matrix}V_{mk}△V_{li} = \{0\} (if k ≠ l, i), V_{mk}△V_{ki} ⊂ V_{mi}, \\ V_{mk}△V_{lk} ⊂ V_{ml} or V_{lm}. \end{matrix}  (1.4)

Put 𝔥_{mk} := jV_{mk} ⊂ 𝔥 (1 ≤ k ≤ m ≤ r) and A_{k} := jE_{k} ∈ 𝔥_{kk} (k = 1, …, r). The space 𝔞 := ∑_{1≤k≤r}^{⊕} 𝔥_{kk} = ∑_{1≤k≤r}^{⊕} ℝA_{k} is a commutative subalgebra of 𝔥 by (1.1) and (1.4), and we get from (1.2)

𝔥 = 𝔞 ⊕\begin{matrix}∑ & ⊕ \\ 1≤k<m≤r \end{matrix} 𝔥_{mk}.  (1.5)

Remark 1.1. The semidirect product V ⋊ 𝔥 has a normal j -algebra structure and the decompositions (1.2) and (1.5) coincide with the root space decompositions of the normal j -algebra ([1], [7], [10]).

Now we introduce global coordinate systems on H and V . For each t ∈ H, there exist unique t_{kk} > 0 (k = 1, 2, …, r) and T_{mk} ∈ 𝔥_{mk} (1 ≤ k < m ≤ r) for which

t = exp T_{11} · exp L_{1} · exp T_{22} ⋯ exp L_{r−1} · exp T_{rr}  (1.6)

with T_{kk} := (2 log t_{kk})A_{k} ∈ 𝔞 and L_{k} := ∑_{m>k} T_{mk} (see [7, Proposition 2.1 (ii)] for the proof). On the other hand, for any x ∈ V , we take unique x_{kk} ∈ ℝ (k = 1, 2, …, r) and X_{mk} ∈ V_{mk} (1 ≤ k < m ≤ r) such that

x = ∑_{k=1}^{r} x_{kk}E_{k} + ∑_{1≤k<m≤r} X_{mk}  (1.7)

according to (1.2). We call t_{kk}, T_{mk} (resp. x_{kk}, X_{mk}) the coordinates of t ∈ H (resp. x ∈ V ). Let E^{∗} be the linear form on V given by

⟨x, E^{∗}⟩ := ∑_{k=1}^{r} x_{kk}  (x ∈ V).  (1.8)

It is deduced from [14, p. 376] that the bilinear form

(x|y) := ⟨x△y, E^{∗}⟩/2 (x, y ∈ V)  (1.9)

defines an inner product on V , and that the normal decomposition (1.2) of V is orthogonal with respect to this (·|·) (cf. [7, Lemma 2.3]). Let ∥ · ∥ be the norm on V defined by ∥x∥^{2} := (x|x) (x ∈ V). Transferring this norm by means of j, we also define a norm on 𝔥, that is, ∥jx∥ := ∥x∥ (x ∈ V). Then the coordinates of each element of Ω are expressed as follows:

Proposition 1.2. [7, Proposition 2.5] For t ∈ H let x be the element t · E of Ω. Then one has

x_{kk} = (t_{kk})^{2} + ∑_{i<k} ∥T_{ki}∥^{2} (k = 1, …, r),
X_{mk} = t_{kk}T_{mk} · E_{k} + ∑_{i<k} T_{mi}T_{ki} · E_{i} (1 ≤ k < m ≤ r).

Taking τ_{ki} ∈ V_{ki} for which T_{ki} = jτ_{ki}, we have T_{ki} · E_{i} = τ_{ki}△E_{i} = τ_{ki} by (1.3). Thus the equalities in Proposition 1.2 are rewritten as

\begin{aligned}x_{kk} & = (t_{kk})^{2} + ∑_{i<k} ∥τ_{ki}∥^{2} (k = 1, …, r), \\ X_{mk} & = t_{kk}τ_{mk} + ∑_{i<k} τ_{mi}△τ_{ki} (1 ≤ k < m ≤ r). \end{aligned}  (1.10)

Vinberg [14] solved t_{ii} and τ_{li} (l > i) in the equations (1.10) as functions of x_{kk} and X_{mk} (see also [4, p. 16]) by utilizing certain polynomials on V determined as follows: For x ∈ V and i = 1, …, r, define x^{(i)} = ∑_{k=1}^{r} x_{kk}^{(i)}E_{k} + ∑_{m>k} X_{mk}^{(i)} ∈ V by

\begin{aligned}x^{(1)} & := x, \\ x_{kk}^{(i+1)} & := x_{ii}^{(i)}x_{kk}^{(i)} − ∥X_{ki}^{(i)}∥^{2} (i < k ≤ r), \\ X_{mk}^{(i+1)} & := x_{ii}^{(i)}X_{mk}^{(i)} − X_{mi}^{(i)}△X_{ki}^{(i)} (i < k < m ≤ r), \end{aligned}  (1.11)

and put

D_{k}(x) := x_{kk}^{(k)} ∈ ℝ (k = 1, …, r), Y_{mk}(x) := X_{mk}^{(k)} ∈ V_{mk} (1 ≤ k < m ≤ r).

In particular, D_{1}(x) = x_{11} and Y_{m1}(x) = X_{m1} (m > 1). We call these D_{k}, Y_{mk} the determinant type polynomials associated to the cone Ω. We remark that D_{k} appears also in [4, Lemma 3.1] as an integral compound power function. It is easily seen from (1.11) that the degrees of D_{k} and Y_{mk} are 2^{k−1}. We write simply D for the polynomial D_{r} and call it the composite determinant. Then Vinberg’s results are stated as follows:

Proposition 1.3. [14, Chapter 3, Section 3] (i) If x = t · E ∈ Ω for t ∈ H, then

(t_{kk})^{2} = \frac{D_{k}(x)}{∏_{i<k} D_{i}(x)} (k = 1, …, r),  (1.12)
t_{kk}τ_{mk} = \frac{Y_{mk}(x)}{∏_{i<k} D_{i}(x)} (1 ≤ k < m ≤ r).  (1.13)

(ii) One has

Ω = { x ∈ V ; D_{k}(x) > 0 (k = 1, …, r) } .

Although the recurrence formula (1.11) is quite simple, actual calculation of the composite determinant D is hard to carry out because we must compute all the quantities x_{kk}^{(i)} and X_{mk}^{(i)} before getting to D(x) = x_{rr}^{(r)} for x ∈ V . The following proposition is useful for the computation of D_{k} and Y_{mk}.

Proposition 1.4. The determinant type polynomials D_{k} and Y_{mk} satisfy the following recurrence relations:

D_{k} = D_{1}D_{2} ⋯ D_{k−1}x_{kk} − ∑_{i<k−1}D_{i+1} ⋯ D_{k−1}∥Y_{ki}∥^{2}− ∥Y_{k,k−1}∥^{2}
(k = 2, …, r),
Y_{mk} = D_{1}D_{2} ⋯ D_{k−1}X_{mk} − ∑_{i<k−1}D_{i+1} ⋯ D_{k−1}Y_{mi}△Y_{ki} − Y_{m,k−1}△Y_{k,k−1}
(2 ≤ k < m ≤ r).

Proof. It is sufficient to show the relations in the case of x = t·E ∈ Ω (t ∈ H). Substituting (1.12) and (1.13) to (1.10), we have

x_{kk} = \frac{D_{k}}{D_{1} ⋯ D_{k−1}} + ∑_{i<k} \frac{∥Y_{ki}∥^{2}}{(t_{ii})^{2}(D_{1} ⋯ D_{i−1})^{2}} ,

X_{mk} = \frac{Y_{mk}}{D_{1} ⋯ D_{k−1}} + ∑_{i<k} \frac{Y_{mi}△Y_{ki}}{(t_{ii})^{2}(D_{1} ⋯ D_{i−1})^{2}} .

Since D_{i} = (t_{ii})^{2}D_{1} ⋯ D_{i−1} by (1.12), the above equalities are rewritten as

\frac{D_{k}}{D_{1} ⋯ D_{k−1}} = x_{kk} − ∑_{i<k} \frac{∥Y_{mi}∥^{2}}{D_{1} ⋯ D_{i−1}D_{i}} ,

\frac{Y_{mk}}{D_{1} ⋯ D_{k−1}} = X_{mk} − ∑_{i<k} \frac{Y_{mi}△Y_{ki}}{D_{1} ⋯ D_{i−1}D_{i}} .

Multiplying the both sides by D_{1} ⋯ D_{k−1}, we obtain the desired formulas.

2. Basic relative invariants

In this section, we determine all the polynomials on V relatively invariant under the action of H. Actually, we factorize the polynomials D_{1}, D_{2}, …, D_{r} into irreducibles to obtain the basic relative invariants, that is, the generators of the set of H -relatively invariant polynomials. Then the cone Ω is distinguished as the set on which all the basic relative invariants are positive.

For s = (s_{1}, …, s_{r}) ∈ ℂ^{r}, we define the one-dimensional representation χ_{s} of H by

χ_{s}(exp(∑_{k=1}^{r} c_{k}A_{k})) = e^{s_{1}c_{1}+···+s_{r}c_{r}} (c_{1}, …, c_{r} ∈ ℝ).  (2.14)

Let F be an H -relatively invariant rational function on V with χ_{s} the multiplier. Namely, F (t · x) = χ_{s}(t)F (x) for all t ∈ H and x ∈ V . Since H acts transitively on the open cone Ω in V , the rational function F is determined uniquely (up to constant) by χ_{s}. Throughout this section, we assume that rational functions f are normalized as f(E) = 1 unless otherwise stated.

Proposition 2.1. (i) Putting

μ(k) = \begin{cases}(1, 0, …, 0) & (k = 1), \\ (2^{k−2}, 2^{k−3}, …, 1, 1, 0, …, 0) & (k = 2, …, r), \end{cases}

one has

D_{k}(t · x) = χ_{μ(k)}(t)D_{k}(x) (t ∈ H, x ∈ V).

(ii) There exists an H -relatively invariant rational function corresponding to the multiplier χ_{s} if and only if s ∈ ℤ^{r}.

Proof. (i) Although the statement can be found in [4, Lemma 3.1], we give a proof here for completeness. The equality (1.12) implies that for t ∈ H

\begin{aligned}D_{1}(t · E) & = (t_{11})^{2}, \\ D_{k}(t · E) & = (t_{11})^{2^{k−1}}(t_{22})^{2^{k−2}} ⋯ (t_{k−1,k−1})^{2} · (t_{kk})^{2} (k = 2, …, r). \end{aligned}  (2.15)

In other words, we have D_{k}(t · E) = χ_{μ(k)}(t), whence the assertion (i) follows.

(ii) Take an arbitrary s ∈ ℤ^{r}. Since μ(k)_{k} = 1 and μ(k)_{m} = 0 (m > k) for k = 1, …, r, there exist integers a_{1}, …, a_{r} for which s = a_{1}μ(1) + a_{2}μ(2) + ⋯ + a_{r}μ(r). Then the assertion (i) tells us that the product (D_{1})^{a_{1}}(D_{2})^{a_{2}} … (D_{r})^{a_{r}} is an H -relatively invariant rational function corresponding to χ_{s}, which proves the “if” part of the statement. Next, we show the “only if” part. Let F be an H - relatively invariant rational function and χ_{s} its multiplier. For x_{11}, …, x_{rr} > 0, let t be the element of H given by t_{kk} := \sqrt{x_{kk}} (k = 1, …, r) and T_{mk} := 0 (1 ≤ k < m ≤ r). Then we have t · E = x_{11}E_{1} + ⋯ + x_{rr}E_{r} by Proposition 1.2, so that

\begin{aligned}F (x_{11}E_{1} + x_{22}E_{2} + ⋯ + x_{rr}E_{r}) & = (t_{11})^{2s_{1}}(t_{22})^{2s_{2}} ⋯ (t_{rr})^{2s_{r}}F (E) \\ & = (x_{11})^{s_{1}}(x_{22})^{s_{2}} ⋯ (x_{rr})^{s_{r}}. \end{aligned}  (2.16)

Therefore, since F is rational, we conclude that s_{1}, s_{2}, …, s_{r} are integers.

The relation (2.16) also implies that, if F is a polynomial, then s_{1}, …, s_{r} are non-negative integers. Now we give an algorithm to find the generators of H -relatively invariant polynomials. The algorithm is valid for any homogeneous cone.

Put Δ_{1} = D_{1}, and for k ≥ 2 let Δ_{k} be the polynomials on V determined by requiring that D_{k} are written as

D_{k} = Δ_{k} · (Δ_{1})^{a_{k1}}(Δ_{2})^{a_{k2}} ⋯ (Δ_{k−1})^{a_{k,k−1}}  (2.17)

with the following two conditions:

(i) a_{k1}, a_{k2}, …, a_{k,k−1} are non-negative integers,

(ii) Δ_{k} is not divisible by any of Δ_{1}, …, Δ_{k−1}.

Theorem 2.2. (i) The polynomial Δ_{k} is irreducible and relatively invariant under the action of H. Let σ(k) be the element of ℤ^{r} such that χ_{σ(k)} is the multiplier of Δ_{k}. Then σ(k)_{k} = 1 and σ(k)_{m} = 0 (m = k + 1, …, r).

(ii) Let σ be the r × r matrix whose (k, m)-component is σ(k)_{m}. For s ∈ ℤ^{r}, let (a_{1}, …, a_{r}) be the row vector (s_{1}, …, s_{r})σ^{−1}. Then, there exists an H -relatively invariant polynomial corresponding to the multiplier χ_{s} if and only if all a_{1}, …, a_{r} are non-negative integers. In this case, the polynomial equals (Δ_{1})^{a_{1}}(Δ_{2})^{a_{2}} … (Δ_{r})^{a_{r}} up to constant.

In the proof of Theorem 2.2, we quote a well-known fact about prehomogeneous vector spaces, noting the fact that the action of H on V is algebraic ([14]).

Proof. (i) We shall show the assertion by induction on k. The case k = 1 is obvious. For k ≥ 2, assume that the assertion holds for Δ_{1}, …, Δ_{k−1}. Let

D_{k} = φ_{1}φ_{2} ⋯ φ_{N}  (2.18)

be the factorization of D_{k} into irreducible polynomials. Then each φ_{n} (n = 1, …, N) is H -relatively invariant by [12, Proposition 2 (2)]. Let s^{n} = (s_{1}^{n}, …, s_{r}^{n}) be the r-tuple of non-negative integers such that χ_{s^{n}} is the multiplier of φ_{n}. By (2.18) and Proposition 2.1 (i) we have μ(k) = s^{1} + ⋯ + s^{N}. In particular, we have for m = k + 1, …, r

0 = μ(k)_{m} = s_{m}^{1} + s_{m}^{2} + ⋯ + s_{m}^{N},

which implies s_{m}^{1} = s_{m}^{2} = ⋯ = s_{m}^{N} = 0. Similarly, we have

1 = μ(k)_{k} = s_{k}^{1} + s_{k}^{2} + ⋯ + s_{k}^{N},

whence it follows that the only one of s_{k}^{1}, …, s_{k}^{N} equals 1 and the others are 0. Assume that s_{k}^{1} = 1. Put σ̃ := s^{2} + ⋯ + s^{N}. Then we have σ̃_{m} = 0 for m ≥ k, so that we can take integers a_{k1}, a_{k2}, …, a_{k,k−1} for which σ̃ = a_{k1}σ(1) + a_{k2}σ(2) + ⋯ + a_{k,k−1}σ(k − 1) thanks to the induction hypothesis on σ(i) (i < k). Then

φ_{2} … φ_{N} = (Δ_{1})^{a_{k1}}(Δ_{2})^{a_{k2}} ⋯ (Δ_{k−1})^{a_{k,k−1}}

because the both sides are H -relatively invariant polynomials corresponding to the same multiplier χσ̃ = (χ_{σ(1)})^{a_{k1}}(χ_{σ(2)})^{a_{k2}} … (χ_{σ(k−1)})^{a_{k,k−1}} . We see from the irreducibility of Δ_{1}, …, Δ_{k−1} that integers a_{k1}, …, a_{k,k−1} are non-negative. Therefore we have Δ_{k} = φ_{1} and σ(k) = s^{1}, whence the assertion follows. (ii) It is immediate from the definitions that s = a_{1}σ(1) + ⋯ + a_{r}σ(r), so that (Δ_{1})^{a_{1}} … (Δ_{r})^{a_{r}} is an H -relatively invariant function corresponding to χ_{s}. This function is polynomial if and only if a_{k} ’s are non-negative integers since Δ_{k} ’s are irreducible by (i). Hence the theorem follows from the one-to-one correspondence between relative invariants and multipliers.

We write Δ for Δ_{r} and call it the reduced determinant polynomial associated to the cone Ω. When Ω is an irreducible symmetric cone, the polynomials Δ_{1}, …, Δ_{r} coincide with the principal minors for the Euclidean Jordan algebra [3, p. 114]. ¿From (2.17) and Proposition 1.3 (ii), we obtain the following result.

Proposition 2.3. The cone Ω is described as

Ω = { x ∈ V ; Δ_{k}(x) > 0 (k = 1, …, r) } .

3. Application to Riesz distributions

For A ∈ End(V), we denote by A^{∗} the adjoint operator of A: A^{∗} · ξ := ξ ∘ A (ξ ∈ V^{∗}). Let H^{∗} be the subgroup { t^{∗}; t ∈ H } of GL(V^{∗}). Then H^{∗} acts simply transitively on the dual cone Ω^{∗} of Ω ([14, Chapter 1, Proposition 9]). We shall consider the determinant type polynomials on V^{∗} associated to Ω^{∗}.

Taking E^{∗} as the base point (see (1.8)), we define the clan (V^{∗}, △′) of the homogeneous cone Ω^{∗}. Let 𝔈_{k} (k = 1, …, r) be the element of V^{∗} given by ⟨x, 𝔈_{k}⟩ := x_{r+1−k,r+1−k} (x ∈ V). Then V^{∗} allows a normal decomposition with respect to the idempotents 𝔈_{k}, where the (m, k)-th component equals the dual space of V_{r+1−k,r+1−m} (see [6, p. 86], [14, Chapter 3, Section 6] and also [8, Section 2]). Using this algebra structure, we define the determinant type polynomials D_{k}^{∗}, Y_{mk}^{∗}, and Δ_{k}^{∗} on V^{∗}. Let us observe the relative invariance of Δ_{k}^{∗}. Since (1.3) leads us to

⟨x, A_{k}^{∗} · E^{∗}⟩ = ⟨A_{k} · x, E^{∗}⟩ = ⟨x_{kk}E_{k} + ∑_{m>k} X_{mk}/2, E^{∗}⟩
= x_{kk} = ⟨x, 𝔈_{r+1−k}⟩,

we have 𝔈_{k}△′ξ = A_{r+1−k}^{∗} · ξ (ξ ∈ V^{∗}). Thus, defining the one-dimensional representation χ_{s}^{∗} (s = (s_{1}, …, s_{r}) ∈ ℂ^{r}) of H^{∗} by

χ_{s}^{∗}(exp(∑_{k=1}^{r} c_{k}A_{r+1−k}^{∗})) = e^{s_{1}c_{1}+···+s_{r}c_{r}} (c_{1}, …, c_{r} ∈ ℝ),  (3.19)

we see from Theorem 2.2 (i) that there exists ρ(k) ∈ ℤ^{r} for which

Δ_{k}^{∗}(τ · ξ) = χ_{ρ(k)}^{∗}(τ)Δ_{k}^{∗}(ξ) (ξ ∈ V^{∗}, τ ∈ H^{∗}).  (3.20)

We see also from Theorem 2.2 (i) that ρ(k)_{k} = 1 and ρ(k)_{m} = 0 (m > k). Comparing (2.14) and (3.19), we have

χ_{s}^{∗}(t^{∗}) = χ_{s^{∗}}(t) (t ∈ H),  (3.21)

where s^{∗} := (s_{r}, …, s_{1}) ∈ ℂ^{r}.

Now we recall the definition of the Riesz distributions on the homogeneous cone Ω ([4], [7]). Let μ be an H -invariant measure on Ω, and Υ_{s} (s ∈ ℂ^{r}) the function on Ω given by Υ_{s}(t · E) := χ_{s}(t) (t ∈ H). Set p_{k} := ∑_{i<k} dim 𝔥_{ki} (1 ≤ k ≤ r). When ℜs_{k} > p_{k}/2 for all k = 1, …, r, we define a tempered distribution R_{s} on V by

⟨R_{s}, f⟩ := \frac{c}{∏_{k=1}^{r} Γ (s − p_{k}/2)} ∫_{Ω} f(x)Υ_{s}(x) dμ(x),

where f is a rapidly decreasing Schwartz function on V and c is the normalizing constant determined in such a way that ⟨R_{s}, e^{−⟨·,E^{∗}⟩}⟩ = 1. For general s ∈ ℂ^{r}, we define ⟨R_{s}, f⟩ by analytic continuation. The distribution R_{s}’s are called the Riesz distributions on Ω. The support of R_{s} is contained in the closure \overline{Ω} of Ω, and the distribution R_{s} is H -relatively invariant in the following sense:

⟨R_{s}, f ∘ t⟩ = χ_{−s}(t)⟨R_{s}, f⟩ (t ∈ H).  (3.22)

Using this relative invariance, we can easily compute the Laplace transforms of R_{s}. In fact, if ξ = t^{∗} · E^{∗} ∈ Ω^{∗} (t ∈ H), we see from (3.22) that

⟨R_{s}, e^{−⟨· ,ξ⟩}⟩ = χ_{−s}(t) ⟨R_{s}, e^{−⟨· ,E^{∗}⟩}⟩ = χ_{−s}(t).

This together with (3.21) gives us the H^{∗}-relative invariance of the Laplace transform of R_{s}:

⟨R_{s}, e^{−⟨· ,τ·ξ⟩}⟩ = χ_{−s^{∗}}^{∗}(τ) ⟨R_{s}, e^{−⟨· ,ξ⟩}⟩ (ξ ∈ Ω^{∗}, τ ∈ H^{∗}).  (3.23)

For each polynomial ϕ on V^{∗} let ϕ(∂_{x}) be the differential operator determined by

ϕ(∂_{x})e^{⟨x,ξ⟩} = ϕ(ξ)e^{⟨x,ξ⟩} (ξ ∈ V^{∗}).  (3.24)

Then we obtain the following theorem, which is a refinement of Gindikin’s result [4, Proposition 3.4]. Recall the element ρ(k) of ℤ^{r} in (3.20).

Theorem 3.1. Let ρ be the r × r matrix whose (k, m)-component is ρ(k)_{m}. For s ∈ ℂ^{r}, put (a_{1}, a_{2}, …, a_{r}) := (−s_{r}, −s_{r−1}, …, −s_{1})ρ^{−1}. Then the Riesz distribution R_{s} is supported by the origin if and only if all a_{1}, …, a_{r} are nonnegative integers, and in this case, one has

R_{s} = Δ_{1}^{∗}(∂_{x})^{a_{1}}Δ_{2}^{∗}(∂_{x})^{a_{2}} ⋯ Δ_{r}^{∗}(∂_{x})^{a_{r}}δ,

where δ is the Dirac distribution at {0}.

Proof. It is known that a distribution supported by {0} is of the form ϕ(∂_{x})δ with some polynomial ϕ ∈ P(V^{∗}), and by (3.24), its Laplace transform ⟨ϕ(∂_{x})δ, e^{−⟨· ,ξ⟩}⟩ equals ϕ(ξ). On the other hand, (3.23) says that the Laplace transform of R_{s} is H^{∗}-relatively invariant with multiplier χ_{−s^{∗}}^{∗}. Therefore, applying Theorem 2.2 (ii) to this situation, we conclude that a_{k} ’s are non-negative integers and the Laplace transform ⟨R_{s}, e^{−⟨· ,ξ⟩}⟩ equals the polynomial function Δ_{1}^{∗}(ξ)^{a_{1}}Δ_{2}^{∗}(ξ)^{a_{2}} ⋯ Δ_{r}^{∗}(ξ)^{a_{r}} of ξ ∈ Ω^{∗}. Hence the injectivity of the Laplace transform completes the proof.

4. Description of \overline{Ω}

Proposition 2.3 implies that, if x ∈ \overline{Ω}, then Δ_{k}(x) ≥ 0 (k = 1, …, r). However the converse is false. For instance, although the principal minors of ( {0 0 \atop 0 −1}) are all nonnegative, the matrix is not positive semi-definite. Thus we need more polynomials to describe the closure \overline{Ω} of Ω. In this section, we define a polynomial Δ^{I} for each non-empty subset I ⊂ \{1, …, r\}, so that \overline{Ω} is characterized by the inequalities Δ^{I} ≥ 0.

For integers a ≤ b, we denote by ⌊a, b⌋ the set {a, a + 1, …, b}. Let I = {i_{1}, i_{2}, …, i_{l}} (i_{1} < i_{2} < ⋯ < i_{r}) be a subset of ⌊1, r⌋. We put

V^{I} := \begin{matrix}∑ & ⊕ \\ 1≤α≤β≤l \end{matrix} V_{i_{β}i_{α}}.  (4.25)

Then we see from (1.3) and (1.4) that (V^{I}, △) is a subalgebra of (V, △) with E^{I} := ∑_{α=1}^{l} E_{i_{α}} ∈ V^{I} the unit element (cf. [14, p. 377]). Put

𝔥^{I} := jV^{I} = \begin{matrix}l \\ ∑ & ⊕ \\ α=1 \end{matrix} ℝA_{i_{α}} ⊕\begin{matrix}∑ & ⊕ \\ 1≤α<β≤l \end{matrix} 𝔥_{i_{β}i_{α}}.  (4.26)

Clearly V^{I} is stable under the action of 𝔥^{I}, and (1.1) implies that 𝔥^{I} is a Lie subalgebra of 𝔥. Let H^{I} be the Lie subgroup exp 𝔥^{I} of H. In view of [14, Chapter 2, Section 2], the orbit Ω^{I} := H^{I} · E^{I} ⊂ V^{I} is the homogeneous cone whose clan is (V^{I}, △). Let P_{I} : V → V^{I} be the orthogonal projection and denote by x_{I} the image P_{I}(x) of x ∈ V .

Lemma 4.1. (i) The projection P_{I} is H^{I} -equivariant:

P_{I}(t̃· x) = t̃· P_{I}(x) (x ∈ V, t̃∈ H^{I}).
(ii) The image of \overline{Ω} under P_{I} is equal to \overline{Ω^{I}}.

Proof. (i) Let A^{I} be the element jE^{I} = ∑_{α=1}^{l} A_{i_{α}} of 𝔥^{I}. For λ ∈ ℝ, let V (λ; A^{I}) be the eigenspace { x ∈ V ; A^{I} · x = λx } of the operator A^{I}. Then V = V (1; A^{I}) ⊕ V (1/2; A^{I}) ⊕ V (0; A^{I}). In fact, by (1.3) and (4.26) we have V (1; A^{I}) = V^{I}, V (0; A^{I}) = V^{⌊1,r⌋\I} and

V (1/2; A^{I}) = \begin{matrix}l \\ ∑ & ⊕ \\ α=1 \end{matrix}(\begin{matrix}∑ & ⊕ \\ m>i_{α},m∉I \end{matrix} V_{mi_{α}} ⊕\begin{matrix}∑ & ⊕ \\ k<i_{α},k∉I \end{matrix} V_{i_{α}k}).

The decomposition V = V^{I} ⊕ V (1/2; A^{I}) ⊕ V^{⌊1,r⌋\I} is orthogonal, and since [𝔥^{I}, A^{I}] = {0} by (1.1), (1.3) and (4.26), the action of H^{I} = exp 𝔥^{I} preserves each of the subspaces. Hence the assertion (i) holds. (ii) Clearly we have Ω^{I} = H^{I} · E^{I} ⊂ \overline{Ω}, so that \overline{Ω^{I}} = P_{I}(\overline{Ω^{I}}) ⊂ P_{I}(\overline{Ω}). Let us show the converse inclusion \overline{Ω^{I}} ⊃ P_{I}(\overline{Ω}). Let E_{I}^{∗} be the linear form on V^{I} given by

⟨x̃, E_{I}^{∗}⟩ := ∑_{k∈I} x̃_{kk} (x̃ ∈ V^{I}).  (4.27)

Then E_{I}^{∗} belongs to the dual cone (Ω^{I})^{∗} ([11, Theorem 4.15]). Thus [7, Lemma 1.2] implies that an element x̃ of V^{I} belongs to \overline{Ω^{I}} if and only if ⟨t̃· x̃, E_{I}^{∗}⟩ ≥ 0 for all t̃ ∈ H^{I}. Take x ∈ Ω and t̃ ∈ H^{I}. Then we have ⟨t̃· P_{I}(x), E_{I}^{∗}⟩ = ⟨P_{I}(t̃· x), E_{I}^{∗}⟩ by (i). Since t̃· x ∈ Ω, we can take t ∈ H for which t̃· x = t · E. We then obtain by Proposition 1.2 and (4.27)

⟨P_{I}(t̃· x), E_{I}^{∗}⟩ = ∑_{k∈I}((t_{kk})^{2} + ∑_{i<k} ∥T_{ki}∥^{2}) ≥ 0.

Thus we get P_{I}(x) ∈ \overline{Ω^{I}}, so that P_{I}(Ω) ⊂ \overline{Ω^{I}}. Hence the continuity of P_{I} completes the proof.

Making use of the algebra structures of (V^{I}, △), we define the determinant type polynomials D_{α}^{I} (α = 1, …, l) on V^{I} associated to Ω^{I}. Let μ(I; α) = (μ(I; α)_{1}, …, μ(I; α)_{r}) be the element of ℤ^{r} given by

μ(I; α)_{k} := \begin{cases}2^{α−β−1} & (if k = i_{β}, β < α), \\ 1 & (if k = i_{α}), \\ 0 & (otherwise). \end{cases}

In view of (2.15), we have for t ∈ H^{I}

\begin{aligned}D_{1}^{I}(t · E^{I}) & = (t_{i_{1}i_{1}})^{2}, \\ D_{α}^{I}(t · E^{I}) & = (t_{i_{1}i_{1}})^{2^{α−1}}(t_{i_{2}i_{2}})^{2^{α−2}} ⋯ (t_{i_{α−1},i_{α−1}})^{2} · (t_{i_{α}i_{α}})^{2} \\ & (α = 2, …, l), \end{aligned}  (4.28)

so that we obtain

D_{α}^{I}(t · x) = χ_{μ(I;α)}(t) D_{α}^{I}(x) (t ∈ H^{I}, x ∈ V^{I}).  (4.29)

We denote by D^{I} the composite determinant D_{l}^{I} associated to Ω^{I}, and extend this D^{I} to V by D^{I}(x) := D^{I}(x_{I}) (x ∈ V).

Applying Proposition 1.3 (ii) to the cone Ω^{I}, we see that the composite determinant D^{I}(x̃) is non-negative for x̃ ∈ \overline{Ω^{I}}. This together with Lemma 4.1 implies that, if x ∈ \overline{Ω}, then D^{I}(x) = D^{I}(x_{I}) ≥ 0 for all non-empty I ⊂ ⌊1, r⌋. Proposition 4.3 below states that the converse also holds. The proposition is proved by induction on the rank r, and the following lemma plays a substantial role in the induction.

Lemma 4.2. Assume that r ≥ 2. Let x be an element of V such that x_{11} ≠ 0 and L the element − ∑_{m=2}^{r} jX_{m1}/x_{11} of 𝔥. Then x′ := (exp L) · x belongs to x_{11}E_{1} + V^{⌊2,r⌋}. For a non-empty subset I of ⌊2, r⌋, one has

D^{{1}∪I}(x) = (x_{11})^{2^{l−1}} D^{I}(x′) (l := ♯I).

Proof. Since L · V_{11} ⊂ ∑_{m>1}^{⊕} V_{m1}, L · (∑_{m>1}^{⊕} V_{m1}) ⊂ V^{⌊2,r⌋} and L · V^{⌊2,r⌋} = 0 by (1.4), we have

(exp L) · x_{11}E_{1} = x_{11}E_{1} + L · x_{11}E_{1} + (1/2)L^{2} · x_{11}E_{1},
(exp L) · ∑_{m>1} X_{m1} = ∑_{m>1} X_{m1} + L · ∑_{m>1} X_{m1},  (4.30)
(exp L) · x_{⌊2,r⌋} = x_{⌊2,r⌋}.

By (1.3) we have

L · x_{11}E_{1} = − ∑_{m>1} X_{m1}△E_{1} = − ∑_{m>1} X_{m1},

so that

L^{2} · x_{11}E_{1} = −L · ∑_{m>1} X_{m1} ∈ V^{⌊2,r⌋}.

Using them, we sum up the both sides of the equalities in (4.30) to obtain

\begin{aligned}(exp L) · x & = x_{11}E_{1} + x_{⌊2,r⌋} + (1/2)L · ∑_{m>1} X_{m1} \\ & = x_{11}E_{1} + x_{⌊2,r⌋} − (2x_{11})^{−1}(∑_{m>1} X_{m1})△(∑_{m>1} X_{m1}) \\ & ∈ x_{11}E_{1} + V^{⌊2,r⌋}. \end{aligned}  (4.31)

For the proof of the second assertion, we show the following claim: Claim: For an element x̃ of V_{11} ⊕ V^{⌊2,r⌋}, one has

D(x̃) = (x̃_{11})^{2^{r−2}}D^{⌊2,r⌋}(x̃_{⌊2,r⌋}).

We first prove the claim for the case x̃ ∈ Ω. Take t ∈ H for which x̃ = t · E and express this t as in (1.6). Since 0 = X̃_{m1} = t_{11}τ_{m1} by (1.10), we have L_{1} = ∑_{m>1} jτ_{m1} = 0. Put t̃ := exp T_{22} exp L_{2} ⋯ exp T_{rr} ∈ H^{⌊2,r⌋}. Since A_{1} ·V^{⌊2,r⌋} = {0} and 𝔥^{⌊2,r⌋} ·V_{11} = {0} by (1.4), the elements exp T_{11} and t̃∈ H^{⌊2,r⌋} act trivially on the spaces V^{⌊2,r⌋} and V_{11} = ℝE_{1} respectively. Thus we have

x̃ = t · E = (exp T_{11})t̃· (E_{1} + E^{⌊2,r⌋}) = (exp T_{11}) · (E_{1} + t̃· E^{⌊2,r⌋})
= (exp T_{11}) · E_{1} + t̃· E^{⌊2,r⌋} = (t_{11})^{2}E_{1} + t̃· E^{⌊2,r⌋},

which implies that x̃_{11} = (t_{11})^{2} and x̃_{⌊2,r⌋} = t̃· E^{⌊2,r⌋} ∈ Ω^{⌊2,r⌋}. In particular, we have

x̃ = x̃_{11}E_{1} + x̃_{⌊2,r⌋} ∈ Ω ⟺ x̃_{11} > 0, x̃_{⌊2,r⌋} ∈ Ω^{⌊2,r⌋}.  (4.32)

By (4.28), we get

D^{⌊2,r⌋}(x̃_{⌊2,r⌋}) = (t_{22})^{2^{r−2}}(t_{33})^{2^{r−3}} ⋯ (t_{r−1,r−1})^{2}(t_{rr})^{2},

so that by (2.15) we obtain

D(x̃) = (t_{11})^{2^{r−1}} · (t_{22})^{2^{r−2}} ⋯ (t_{r−1,r−1})^{2}(t_{rr})^{2} = (x̃_{11})^{2^{r−2}}D^{⌊2,r⌋}(x̃_{⌊2,r⌋}).

Namely, the claim is verified in the case that x̃_{11} > 0 and x̃_{⌊2,r⌋} ∈ Ω^{⌊2,r⌋}. Because (0, +∞) and Ω^{⌊2,r⌋} are open in the spaces ℝ and V^{⌊2,r⌋} respectively, the claim holds for all x̃ ∈ V_{11} ⊕ V^{⌊2,r⌋}.

Now we come back to the proof of the lemma. Put J := {1} ∪ I. Applying the claim with x̃ and V^{⌊2,r⌋} replaced by x′_{J} = x_{11}E_{1} + x′_{I} and V^{I} respectively, we obtain D^{J}(x′_{J}) = (x_{11})^{2^{l−1}}D^{I}(x′_{I}), that is,

D^{J}(x′) = (x_{11})^{2^{l−1}}D^{I}(x′).  (4.33)

On the other hand, replacing x and L by x_{J} and L_{J} := − ∑_{m∈I} jX_{m1}/x_{11} respectively in (4.31), we get

(exp L_{J}) · x_{J} = x_{11} + x_{I} − (2x_{11})^{−1}(∑_{m∈I} X_{m1})△(∑_{m∈I} X_{m1}).

Comparing this with (4.31), we have x′_{J} = P_{J}((exp L) · x) = (exp L_{J}) · x_{J}, so that

D^{J}(x′) = D^{J}((exp L_{J}) · x_{J}).

Since exp L_{J} ∈ H^{J} and χ_{μ(J; l+1)}(exp L_{J}) = 1, the right-hand side of the above is equal to D^{J}(x_{J}) = D^{J}(x) by (4.29). This fact together with (4.33) completes the proof.

We now give an algebraic description of \overline{Ω}.

Proposition 4.3. One has

\overline{Ω} = \{ x ∈ V ; D^{I}(x) ≥ 0 (I ⊂ ⌊1, r⌋, I ≠ Ø) \} .  (4.34)

Proof. Thanks to the observation preceding Lemma 4.2, it is sufficient to prove that the right-hand side of (4.34) is contained in \overline{Ω}. We shall show this by induction on the rank r. The case r = 1 is clear. For r ≥ 2, assume that the claim holds for the rank r − 1 cone Ω^{⌊2,r⌋} ⊂ V^{⌊2,r⌋}, and let x be an element of V such that D^{I}(x) ≥ 0 for all non-empty I ⊂ ⌊1, r⌋. Since D^{{1}}(x) = x_{11} ≥ 0, we have the following two cases (1) and (2).

(1) The case x_{11} = 0.

In this case, we have for k = 2, …, r

0 ≤ D^{{1,k}}(x) = x_{11}x_{kk} − ∥X_{k1}∥^{2} = −∥X_{k1}∥^{2},

so that X_{k1} = 0 and x ∈ V^{⌊2,r⌋}. Therefore, since D^{I}(x) ≥ 0 for all non-empty I ⊂ ⌊2, r⌋, the induction hypothesis tells us that x ∈ \overline{Ω^{⌊2,r⌋}} ⊂ \overline{Ω}.

(2) The case x_{11} > 0.

We set L := − ∑_{m>1} jX_{m1}/x_{11} and x′ := (exp L) · x. Then Lemma 4.2 tells us that x′ = x_{11}E_{1} + x′_{⌊2,r⌋}, and that for all non-empty I ⊂ ⌊2, r⌋,

D^{I}(x′_{⌊2,r⌋}) = D^{I}(x′) = D^{{1}∪I}(x)/(x_{11})^{2^{l−1}} ≥ 0 (l := ♯I).

Therefore x′_{⌊2,r⌋} ∈ \overline{Ω^{⌊2,r⌋}} by the induction hypothesis. On the other hand, since (4.32) tells us that x_{11}E_{1} + Ω^{⌊2,r⌋} ⊂ Ω for x_{11} > 0, we have x_{11}E_{1} + \overline{Ω^{⌊2,r⌋}} ⊂ \overline{Ω}. Therefore x′ = x_{11}E_{1} + x′_{⌊2,r⌋} ∈ \overline{Ω}, so that we conclude x ∈ \overline{Ω}.

For non-empty I = {i_{1}, i_{2}, …, i_{l}} ⊂ ⌊1, r⌋, we denote by Δ_{1}^{I}, Δ_{2}^{I}, …, Δ_{l}^{I} the basic relative invariants on V^{I} associated to the cone Ω^{I}, and extend these polynomials to V by Δ_{α}^{I}(x) := Δ_{α}^{I}(x_{I}) (x ∈ V). We write Δ^{I} for the reduced determinant Δ_{l}^{I}. Then Δ_{α}^{I} = Δ^{I_{α}} with I_{α} := {i_{1}, …, i_{α−1}, i_{α}}, and we see from (2.17) that

D^{I} = D_{l}^{I} = Δ^{I} · (Δ^{I_{1}})^{c_{1}}(Δ^{I_{2}})^{c_{2}} ⋯ (Δ^{I_{l−1}})^{c_{l−1}}  (4.35)

for some non-negative integers c_{1}, c_{2}, …, c_{l−1}.

Now that the polynomials Δ^{I} are introduced, we arrive at our goal, another description of \overline{Ω} with lower degree than (4.34).

Theorem 4.4. The closure \overline{Ω} of Ω is described as

\overline{Ω} = \{ x ∈ V ; Δ^{I}(x) ≥ 0 (I ⊂ ⌊1, r⌋, I ≠ Ø) \} .  (4.36)

Proof. Similarly to the argument preceding Lemma 4.2, we see from Proposition 2.3 and Lemma 4.1 that \overline{Ω} is contained in the right-hand side of (4.36). For the proof of the converse inclusion, let x be an element of V such that Δ^{I}(x) ≥ 0 for all non-empty I ⊂ ⌊1, r⌋. Then (4.35) implies that D^{I}(x) ≥ 0. Therefore Proposition 4.3 tells us that x ∈ \overline{Ω}, which completes the proof.

5. Examples

In this section, we write a′ for the transpose of a matrix a, and denote by E_{ij} the matrix unit. For a symmetric matrix x = (x_{mk})_{m,k=1}^{r}, let x̌ = (x̌_{mk}) be the lower triangular matrix defined by

x̌_{mk} := \begin{cases}x_{mk} & (m > k), \\ x_{kk}/2 & (m = k), \\ 0 & (m < k), \end{cases}

and x̂ the upper triangular matrix (x̌)′ (cf. [14, p. 381]).

5.1. The cone of real positive definite symmetric matrices Let V be the space of real symmetric matrices of size r, Ω the subset of V consisting of the positive definite elements, and H ⊂ GL(r, ℝ) the group of lower triangular matrices with positive diagonals. Define the action of H on V by t · x := txt′ (t ∈ H, x ∈ V). Then Ω is a homogeneous cone on which H acts simply transitively. As the base point E of Ω, we choose the unit matrix. Then the multiplication of the clan (V, △) is given by x△y := x̌y+yx̂. The determinant type polynomials D_{k} are calculated as

D_{k}(x) = \begin{cases}det x_{[k]} & (k = 1, 2), \\ (det x_{[1]})^{2^{k−3}}(det x_{[2]})^{2^{k−4}} ⋯ (det x_{[k−2]}) · (det x_{[k]}) & (k ≥ 3), \end{cases}

where x_{[k]} (k = 1, …, r) are the submatrices (x_{ij})_{1≤i,j≤k} of x = (x_{ij}) ∈ V . Hence the basic relative invariants Δ_{k}(x) are nothing but the principal minors det x_{[k]} by (2.17). More generally, Δ^{I}(x) (I ⊂ ⌊1, r⌋, I ≠ Ø) is equal to the minor det((x_{ij})_{i,j∈I}).

5.2. The Vinberg cone Let V be the vector space

\{x = (x_{(1)}, x_{(2)}) ; x_{(k−1)} :=(\begin{matrix}x_{11} & x_{k1} \\ x_{k1} & x_{kk} \end{matrix}), x_{11}, x_{k1}, x_{kk} ∈ ℝ, k = 2, 3\},

and Ω the set of elements (x_{(1)}, x_{(2)}) ∈ V such that the both of the components are positive definite. Then Ω is an open convex cone, called the the Vinberg cone ([14]). Define the group H to be the set

\{t = (t_{(1)}, t_{(2)}) ; t_{(k−1)} :=\binom{t_{11}}{t_{k1} t_{kk}}, t_{11}, t_{kk} > 0, t_{k1} ∈ ℝ, k = 2, 3\}

with the multiplication of each component. The group H acts on V as

t · x := (t_{(1)}x_{(1)}t′_{(1)}, t_{(2)}x_{(2)}t′_{(2)}) ∈ V (t ∈ H, x ∈ V),

so that H acts simply transitively on Ω. Setting E := (({1 0 \atop 0 1}), ({1 0 \atop 0 1})) ∈ Ω, we obtain the clan (V, △) of the homogeneous cone Ω:

x△y = (x̌_{(1)}y_{(1)} + y_{(1)}x̂_{(1)}, x̌_{(2)}y_{(2)} + y_{(2)}x̂_{(2)}) (x, y ∈ V),

and have the normal decomposition of V with respect to the idempotents E_{1} = (E_{11}, E_{11}), E_{2} = (E_{22}, 0) and E_{3} = (0, E_{22}). Using Proposition 1.4, we calculate the composite determinants D^{I} (I ⊂ {1, 2, 3}, I ≠ Ø) as follows:

D^{{1}}(x) = x_{11},  D^{{2}}(x) = x_{22},  D^{{3}}(x) = x_{33},
D^{{1,2}}(x) = x_{11}x_{22} − x_{21}^{2}, D^{{1,3}}(x) = x_{11}x_{33} − x_{31}^{2}, D^{{2,3}}(x) = x_{22}x_{33},
D^{{1,2,3}}(x) = (x_{11}x_{22} − x_{21}^{2})(x_{11}x_{33} − x_{31}^{2}),

so that the reduced determinants are obtained as

\begin{aligned}Δ^{\{1\}}(x) & = x_{11}, & Δ^{\{2\}}(x) & = x_{22}, \\ Δ^{\{1,2\}}(x) & = x_{11}x_{22} − x_{21}^{2}, & Δ^{\{3\}}(x) = Δ^{\{2,3\}}(x) & = x_{33}, \\ Δ^{\{1,3\}}(x) = Δ^{\{1,2,3\}}(x) & = x_{11}x_{33} − x_{31}^{2}. \end{aligned} (5.37)

Here Δ^{{1}}, Δ^{{1,2}} and Δ^{{1,2,3}} are the basic relative invariants. These are the same as Gindikin’s observed in [6, p. 98 (e)]. We note that the degree of Δ^{I} is sometimes less than ♯I. For example, the degree of Δ = Δ^{{1,2,3}} is equal to 2.

Now let us consider the dual cone of Ω. We realize the dual vector space of V as

V^{∗} := \{ξ = (\begin{matrix}ξ_{11} & ξ_{21} & ξ_{31} \\ ξ_{21} & ξ_{22} & 0 \\ ξ_{31} & 0 & ξ_{33} \end{matrix}) ; ξ_{11}, ξ_{21}, ξ_{31}, ξ_{22}, ξ_{33} ∈ ℝ\}

with the dual coupling ⟨ · , · ⟩ defined by

⟨(x_{(1)}, x_{(2)}), ξ⟩ := ∑_{k=1}^{3} x_{kk}ξ_{kk} + 2 ∑_{k=2}^{3} x_{k1}ξ_{k1}.

For t = (t_{(1)}, t_{(2)}) ∈ H and ξ ∈ V^{∗}, we have t^{∗} · ξ = t̃ξ(t̃)′ with

t̃:= (\begin{matrix}t_{11} & t_{21} & t_{31} \\ 0 & t_{22} & 0 \\ 0 & 0 & t_{33} \end{matrix}) .

Then the clan (V^{∗}, △′) of Ω^{∗} (see Section 3) is given by

ξ△′η = ξ̂η + ηξ̌ (ξ, η ∈ V^{∗}),

and the idempotents 𝔈_{k} (k = 1, 2, 3) are equal to E_{4−k,4−k}. We obtain the composite and the reduced determinants on V^{∗} as

D_{{1}}^{∗}(ξ) = ξ_{33},  D_{{2}}^{∗}(ξ) = ξ_{22},  D_{{3}}^{∗}(ξ) = ξ_{11},
D_{{1,2}}^{∗}(ξ) = ξ_{22}ξ_{33}, D_{{1,3}}^{∗}(ξ) = ξ_{11}ξ_{33} − ξ_{31}^{2}, D_{{2,3}}^{∗}(ξ) = ξ_{11}ξ_{22} − ξ_{21}^{2},
D_{{1,2,3}}^{∗}(ξ) = ξ_{11}ξ_{22}ξ_{33}^{2} − ξ_{22}ξ_{33}ξ_{31}^{2} − ξ_{33}^{2}ξ_{21}^{2},

and

Δ_{{1}}^{∗}(ξ) = ξ_{33}, Δ_{{3}}^{∗}(ξ) = ξ_{11}, Δ_{{1,3}}^{∗}(ξ) = ξ_{11}ξ_{33} − ξ_{31}^{2},
Δ_{{2}}^{∗}(ξ) = Δ_{{1,2}}^{∗}(ξ) = ξ_{22}, Δ_{{2,3}}^{∗}(ξ) = ξ_{11}ξ_{22} − ξ_{21}^{2},  (5.38)
Δ_{{1,2,3}}^{∗}(ξ) = ξ_{11}ξ_{22}ξ_{33} − ξ_{22}ξ_{31}^{2} − ξ_{33}ξ_{21}^{2} = det ξ,

respectively. The basic relative invariants are Δ_{{1}}^{∗}, Δ_{{1,2}}^{∗} and Δ_{{1,2,3}}^{∗} (see also [6, p. 98 (f)]). By Proposition 2.3 and (5.38), the dual cone Ω^{∗} is described as

Ω^{∗} = { ξ ∈ V^{∗}; ξ_{33} > 0, ξ_{22} > 0, ξ_{11}ξ_{22}ξ_{33} − ξ_{22}ξ_{31}^{2} − ξ_{33}ξ_{21}^{2} > 0 }

= { ξ ∈ V^{∗}; ξ is positive definite } .

The expressions in (5.38) are quite different from the ones in (5.37), which reflects the non-symmetry of the Vinberg cone Ω.

References

[1] D’Atri, J. E., and I. D. Miatello, A characterization of bounded symmetric domains by curvature, Trans. Amer. Math. Soc. 276 (1983), 531–540.

[2] Dorfmeister, J., and M. Koecher, Relative Invarianten und nicht-assoziative Algebren, Math. Ann. 228 (1977), 147–186.

[3] Faraut, J., and A. Korányi, “Analysis on symmetric cones, ” Clarendon, Oxford, 1994.

[4] Gindikin, S. G., Analysis in homogeneous domains, Russian Math. Surveys 19 (1964), 1–89.

[5] —, Invariant generalized functions in homogeneous domains, Funct. Anal. Appl. 9 (1975), 50–52.

[6] —, “Tube domains and the Cauchy problem,” Transl. Math. Monogr. 111, Amer. Math. Soc., 1992.

[7] Ishi, H., Positive Riesz distributions on homogeneous cones, J. Math. Soc. Japan 52 (2000), 161–186.

[8] —, Representations of the affine transformation groups acting simply transitively on homogeneous Siegel domains, J. Funct. Anal. 167 (1999), 425– 462.

[9] Kaneyuki, S., On the automorphism groups of homogeneous bounded domains, J. Fac. Sci. Univ. Tokyo Sect. I 14 (1967), 89–130.

[10] Piatetskii-Shapiro, I. I., “Automorphic functions and the geometry of classical domains,” Gordon and Breach, New York, 1969.

[11] Rossi, H., and M. Vergne, Representations of certain solvable Lie groups on Hilbert spaces of holomorphic functions and the application to the holomorphic discrete series of a semisimple Lie group, J. Funct. Anal. 13 (1973), 324–389.

[12] Sato, M., T. Shintani, and M. Muro, Theory of prehomogeneous vector spaces (algebraic part)—English translation of Sato’s lecture from Shintani’s notes, Nagoya Math. J. 120 (1990), 1–34.

[13] Vinberg, E. B., Homogeneous cones, Soviet Math. Dokl. 1 (1960), 787–790.

[14] —, The theory of convex homogeneous cones, Trans. Moscow Math. Soc. 12 (1963), 340–403.

[15] —, The structure of the group of automorphisms of a homogeneous convex cone, Trans. Moscow Math. Soc. 13 (1965), 63–93.

Hideyuki Ishi Department of Mathematical Sciences Yokohama City University Yokohama 236-0027, Japan hideyuki@math.yokohama-cu.ac.jp

Received March 23, 2000 and in final form May 9, 2000