Clans Defined by Representations of Euclidean Jordan Algebras and the Associated Basic Relative Invariants

Kyushu J. Math. 67 (2013), 163–202 doi:10.2206/kyushujm.67.163

CLANS DEFINED BY REPRESENTATIONS OF EUCLIDEAN JORDAN ALGEBRAS AND THE ASSOCIATED BASIC RELATIVE INVARIANTS

Hideto NAKASHIMA and Takaaki NOMURA (Received 15 March 2012 and revised 4 June 2012)

Abstract. Starting with a representation ϕ of a Euclidean Jordan algebra V by selfadjoint operators on a real Euclidean vector space E, we introduce a clan structure in VE := E ⊕ V . 0 By the adjunction of a unit element to VE, we obtain a clan VE with unit element. By 0 computing the determinant of the right multiplication operators of VE, we get an explicit 0 expression of the basic relative invariants of VE in terms of the Jordan algebra principal 0 minors of V and the quadratic map associated with ϕ. For the dual clan of VE, we also obtain an explicit expression of the basic relative invariants in a parallel way.

0. Introduction

Vinberg [15] established a one-to-one correspondence, up to isomorphisms, between homogeneous convex domains in Euclidean spaces and certain non-associative algebras called clans. Homogeneous open convex cones containing no entire line (homogeneous cones for short in what follows) correspond to clans with unit element. Among homogeneous cones, selfdual cones (called symmetric cones) form a nice class, and have been extensively studied from various points of view as presented in the book by Faraut and Kor´anyi [7]. Jordan algebras serve efﬁciently as an algebraic tool there. In this way the ambient vector space V of a symmetric cone has two algebraic structures, clan and Jordan. At the very beginning of this work, we were interested in the interplay of these two structures. However, with the progress of the work we recognize that this interplay is just a special case (zero representation case) of clans obtained by selfadjoint representations of the Jordan algebra V . The key is that any selfadjoint Jordan algebra representation ϕ of V is automatically a representation of the clan V in the sense of Ishi [10] (see Proposition 3.3 of this paper). Then ϕ is a representation of in the sense of Rothaus [14](see[10]), so that it is a J -morphism of in the sense of Dorfmeister [6]. Thus, our construction of a homogeneous cone from a Jordan algebra representation could be included in a more general scheme developed by Rothaus [14] or Dorfmeister [5, 6]. However, we would like to emphasize that since Jordan algebra representations are well-studied, we are able to make everything explicit. In particular, our objective is to obtain an explicit expression of the basic relative invariants in terms of the ingredients of the original Jordan algebra V and its representation ϕ.

2010 Mathematics Subject Classiﬁcation: Primary 17C50; Secondary 16W10, 43A85. Keywords: basic relative invariants; clans; homogeneous cones; Jordan algebras.

c 2013 Faculty of Mathematics, Kyushu University 164 H. Nakashima and T. Nomura

Moreover, the homogeneous cones and their dual cones that we obtain are quite interesting in themselves. For example, the degrees of the basic relative invariants associated with the resulting dual cone are always 1, 2,...,r (r is the rank of the cone), whatever the representations ϕ. Now we describe the body of this paper in more detail. Let V be a simple Euclidean Jordan algebra of rank r with an inner product x | y=tr(xy) defined by the trace function tr(x) of V . We fix a Jordan frame c1,...,cr of V ,sothatwehavec1 +···+cr = e0, where e0 is the unit element of V .Let be the symmetric cone of V . We consider a representation ϕ of V by selfadjoint operators on a real Euclidean vector space E with inner product · | ·E.Ifϕ is a zero representation, then we are led to consider V as a clan (V, ), the product of which comes through an Iwasawa solvable subgroup of the linear automorphism group G() of . Note here that in this case G() is a reductive Lie group. Suppose next that ϕ is non-trivial. Then our task is to introduce a clan structure in the vector space VE := E ⊕ V . By using the Jordan frame, we can define the lower triangular part ϕ(x) of the selfadjoint operator ϕ(x) for x ∈ V (see (3.3)). The operators ϕ(x) actually represent a triangular action of the clan (V, ) on E. On the other hand, we have a symmetric bilinear map Q : E × E → V associated with ϕ defined by

ϕ(x)ξ | ηE =Q(ξ, η) | x (x ∈ V,ξ,η∈ E).

We note that Q is -positive, that is, we have Q(ξ, ξ) ∈ \{0} for any 0 = ξ ∈ E.Using these materials, and recalling that the space V , originally a Jordan algebra, is now considered as a clan (V, ) deﬁned in a canonical way as mentioned above, we introduce a product in VE by

(ξ + x) (η + y) := ϕ(x)η + (Q(ξ, η) + x y) (ξ, η ∈ E, x, y ∈ V).

We see in Theorem 3.2 that the algebra (VE, ) is indeed a clan. Since ϕ is supposed to be non-trivial, VE does not have a unit element. The corresponding homogeneous convex domain is the real Siegel domain D(, Q) defined by the data and Q (see (3.9)). 0 := R ⊕ We now make an adjunction of a unit element e to VE, and get a clan VE e VE to 0 which corresponds a homogeneous cone . Noting the fact that the unit element e0 of the Jordan algebra V is also a unit element of the clan (V, ), we put u := e − e0.Thenwealso 0 = R ⊕ 0 + + ∈ R have VE u VE, and we will write general elements of VE as λu ξ x with λ , ξ ∈ E and x ∈ V . The Siegel domain D(, Q) appears as the cross-section of 0 with the hyperplane u + VE. Since the irreducible factors of the determinant of the right multiplication operators are the basic relative invariants by Ishi and Nomura [11], we calculate Det R0(v) ∈ 0 0 0 (v VE) in Proposition 4.1 for the right multiplication operators R (v) of VE.Thisisa 0 preliminary step to obtain an explicit expression of the basic relative invariants of VE. To go further, we need to separate the cases according to the classification of simple Euclidean Jordan algebras (see [7], for example). Clearly the exceptional Jordan algebra Herm(3, O) does not have a non-trivial representation. Thus, we have two cases: (i) rank-two case, where V =∼ R ⊕ W and W is a real vector space (Lorentzian type); (ii) the case Herm(r, K),wherer ≥ 3andK = R, C or H (Hermitian type). We first describe the results for the Hermitian case. Let V = Herm(r, K) and (ϕ, E) be a selfadjoint representation of the Jordan algebra V .ByClerc[3] we necessarily have Clans defined by Jordan algebras 165

∗ E = Mat(r × p, K) with the canonical real inner product ξ | ηE = Re Tr(ξη ),andϕ(x) is ∈ 0 simply the left multiplication by x to ξ E. We see in Proposition 5.2 that the clan VE is isomorphic to a subclan X of Herm(r + p, K) deﬁned by λI ξ ∗ X := p ; λ ∈ R,ξ∈ E, x ∈ V . ξx

0 Our explicit expression of the basic relative invariants of the clan VE requires some care if p

K8 := O. Although we have two inequivalent irreducible representations ϕ if n = 3, 5, 9, 0 = K the resulting clans VE, including the case n 2, are isomorphic to Herm(3, n−1).Then 0 the above expression for 2(v) is equal to the determinant function of the Jordan algebra √1 Herm(3, Kn−1) up to some minor adjustments like ξ in (0.1). Although the isomorphism 2 0 =∼ K VE Herm(3, n−1) can be detected in view of Vinberg [16, Proposition 3] by just noting the equidimensionality of the off-diagonals, we prefer in this paper an explicit map that gives an actual isomorphism. ∗ Our ﬁnal objective is to realize the dual cone (0) of 0, and to get an explicit

expression of the associated basic relative invariants. By ﬁxing an appropriate inner product ∗ 0 0 0 t 0

in VE, the clan product of VE associated with ( ) is given by using the transpose Lv (v ∈ V 0) of the left multiplication operators L0 of V 0 as v v = tL0v (v, v ∈ V 0). Then, E ∗v E v E we show in Proposition 6.1 that the cone (0) is described as ∗ 0 ={ = + + ∈ 0; ∈ 1 −1 | } ( ) v λu ξ x VE x and λ> 2 ϕ(x) ξ ξ E . This description corresponds to what Rothaus calls in [14] the extension of by the representation ϕ. For the Hermitian case V = Herm(r, K), we prove in Theorem 6.3 that ∗ (0) is linearly isomorphic to the following cone : ∗ rp μη μ ∈ R,η∈ K ,y∈ V, := Y = ∈ Herm(rp + 1, K); . (0.2) ηy⊗ Ip Y is positive-deﬁnite Here note that Krp is the space of K-column vectors of size rp, and not considered as a space ∗ = of matrices. Let j (y) (j 1,...,r)be the principal minors from the right lower corner of the matrix y ∈ V . We denote by coy the cofactor matrix of y. Thus, coy = (Det y)y−1 if y is invertible in the case where K = R or C.ForK = H, we consider coy in the Jordan algebra sense (cf. [7, Proposition II.2.4]). Then the basic relative invariants Pj (Y ) (k = 1,...,r,r+ 1) associated with is given by (Corollary 6.4) ∗(y) (j = 1,...,r), P (Y ) := j j ∗ co μ det y − η ( y ⊗ Ip)η (j = r + 1).

We would like to emphasize that deg Pj = j for any j = 1,...,r,r+ 1, so that these cones generalize the non-symmetric cone of rank three that appeared in [11, Section 3] with the basic relative invariants of degrees one, two and three. The Lorentzian case will be treated separately, and we get Theorem 6.6 which says ∗ 0 that the basic relative invariants associated with ( ) are given by the polynomials Pj (v) ∈ 0 (v VE) defined by ∗(x) (j = 1, 2), P (λu + ξ + x) = j j − 1 | = λ det x 2 ϕ(x)ξ ξ E (j 3), ∗ ∗ where 1(x), 2(x) are the Jordan algebra principal minors of V dual to 1(x), 2(x),and x → x is the restriction to V of the canonical automorphism of the Clifford algebra Cl(W) extending the isometry w →−w of W (cf. [8], for example). Here also we emphasize that ∗ 0 deg Pj = j for any j = 1, 2, 3, although ( ) is not symmetric in general. Low-dimensional Lorentzian cases V = R ⊕ W with dim W = 4, 6, 7 and 8 for irreducible ϕ will be further Clans defined by Jordan algebras 167 observed by using explicit realizations of W as real subspaces of H or O.Thenϕ is realized as in Clerc [4, Section 6], although we will use left multiplication operators. The organization of this paper is as follows. Section 1 collects definitions and basic theorems on which this paper is grounded. In Section 2 we study a canonical clan structure (V, ) that is introduced in a simple Euclidean Jordan algebra V . We investigate the inductive structure of the right multiplication operators R(v) (v ∈ V) in (V, ).This is our Theorem 2.7. As a result we obtain the irreducible factorization of Det R(v) in Theorem 2.9. This section gives details of the results announced in the paper [13]by the second author. Section 3 is devoted to describing the clan VE = E ⊕ V defined by a selfadjoint representation (ϕ, E) of V . In Section 4, we compute the determinant of the 0 = R ⊕ right multiplication operators in VE e VE obtained from VE by the adjunction of a unit element e. In Section 5, we give an explicit expression to the basic relative invariants 0 0 of VE. The final section, Section 6, deals with the dual clan of VE and we obtain an explicit expression of the basic relative invariants of this clan. The contents of this paper extend significantly the first author’s master thesis submitted to Kyushu University in February 2011.

Notation For the determinant (respectively the trace) of operators and real or complex matrices we use the uppercase version Det (respectively Tr). For the determinant function and the trace function of a Jordan algebra we use det and tr as in the book [7]. Thus, when a Hermitian matrix X might not be real or complex, we write det X to avoid any misuse of the multiplicative property.

1. Preliminaries

1.1. Clans We begin this paper with the definition of clans following Vinberg [15], where it is shown that the correspondence is one-to-one up to isomorphisms between the class of homogeneous convex domains and the class of clans. Let V be a finite-dimensional real vector space with a bilinear product . We assume neither the associativity of the product nor the existence of unit element. For x ∈ V , we denote by Lx the left multiplication operator Lxy = x y(y∈ V). The pair (V, ) (or simply V ) is called a clan if the following three conditions are satisfied: (1) (V, ) is left-symmetric: Lx Ly − Ly Lx = Lx y−y x for all x, y ∈ V ; (2) there exists s ∈ V ∗ such that s(x y) defines an inner product in V ; (3) for each x ∈ V , the operator Lx has only real eigenvalues. Linear forms s with the property (2) are said to be admissible. Let V be a clan. By Vinberg [15, p. 369], V has a principal idempotent c by which V can be decomposed as V = V(1) ⊕ V(1/2), where := { ∈ ; = } := { ∈ ; = 1 } V(1) x V Lcx x ,V(1/2) x V Lcx 2 x . 168 H. Nakashima and T. Nomura

Denoting by Rx the right multiplication operator Rx y = y x(y∈ V),wealsohave

V(1) ={x ∈ V ; Rcx = x},V(1/2) ={x ∈ V ; Rcx = 0}.

We note here that if V has a unit element e,thenc = e and evidently we have V(1) = V and V1/2 ={0}. The following multiplication rules hold:

V(1) V(1) ⊂ V(1),V(1) V(1/2) ⊂ V(1/2), (1.1) V(1/2) V(1) ={0},V(1/2) V(1/2) ⊂ V(1).

Clearly V(1) itself is a clan with unit element c.Letr be the rank of the clan V(1) and let c1,...,cr be a complete set of primitive idempotents in V(1),sothatwehavec1 +···+ cr = c. Then, by [15, Proposition 8, p. 374] we have a normal decomposition (1.2) below of V(1) after relabeling c1,...,cr if necessary. Let Vjj be the one-dimensional subspace Rcj , and for k>jput := { ∈ ; = 1 + = = } Vkj x V(1) Lci x 2 (δij δik)x, Rci x δij x for i 1,...,r .

Then we have V(1) = Vkj, (1.2) 1≤j≤k≤r where the multiplication rules are

Vji Vlk ={0} (if i = k, l), Vkj Vji ⊂ Vki, (1.3) Vji Vki ⊂ Vjk or Vkj (according to j ≥ k or j ≤ k).

Moreover, the space V(1/2) is decomposed as

r V(1/2) = Vk0 k=1 := { ∈ ; = 1 = } with Vk0 x V(1/2) Lci x 2 δikx for i 1,...,r .

1.2. Homogeneous convex cones

Let V be a ﬁnite-dimensional real vector space and an open convex cone in V containing no entire line. Let G() be the linear automorphism group of :

G() := {g ∈ GL(V ); g() = }.

As a closed subgroup of GL(V ), the group G() is a Lie group. We assume that is homogeneous, that is, G() acts on transitively. We know by Vinberg [15, Theorem 1, p. 359] that there exists a split solvable Lie subgroup H of G() acting on simply transitively. A function f on is said to be relatively invariant under the action of H if there exists a one-dimensional representation χ of H with which we have f(hx)= χ(h)f(x)for all h ∈ H and x ∈ . Clans deﬁned by Jordan algebras 169

THEOREM 1.1. (Ishi [9]) Let r be the rank of the cone . Then there exist irreducible relatively H -invariant polynomial functions 1,...,r , by which any relatively H - invariant polynomial function p(x) on V is written as

= · m1 ··· mr ∈ Zr p(x) const 1(x) r (x) ((m1,...,mr ) ≥0). Moreover, one has ={x ∈ V ; 1(x) > 0,...,r (x) > 0}.

The polynomials 1(x), . . . , r (x) are called the basic relative invariants of the cone .Nowlet(V, ) be the clan associated with the cone . We also call 1(x), . . . , r (x) the basic relative invariants of the clan V .SinceV x → Det Rx is a relatively H -invariant polynomial function (see the proof of [12, Lemma 2.7]), it is written as in Theorem 1.1. Moreover the following stronger fact is true.

THEOREM 1.2. (Ishi and Nomura [11]) The polynomials 1(x), . . . , r (x) are the irreducible factors of Det Rx .

2. Jordan algebras and the associated clan structure

Let V be a real vector space equipped with a bilinear product ◦.ThenV is called a Jordan algebra if the following two conditions hold:

x ◦ y = y ◦ x, x2 ◦ (x ◦ y) = x ◦ (x2 ◦ y) (x, y ∈ V).

We note that the product ◦ need not be associative. We often drop the symbol ◦ for simplicity when there is no risk of confusion. Our reference for Jordan algebras is the book by Faraut and Kor´anyi [7]. A Jordan algebra V with unit element is said to be Euclidean if there exists a positive-deﬁnite symmetric bilinear form on V which is associative, that is to say, if there exists an inner product in V with respect to which the Jordan multiplication operators M(x) : y → xy are selfadjoint for all x ∈ V . A Jordan algebra V is said to be simple if V has only trivial ideals. It is known that Euclidean Jordan algebras correspond in a one-to-one way to symmetric cones up to isomorphisms. Since symmetric cones are a special kind of homogeneous open convex cones, we can introduce a clan structure canonically in Euclidean Jordan algebras as described below. Let V be a simple Euclidean Jordan algebra of rank r with unit element e0. We equip V with the trace inner product x |y :=tr(xy), where tr is the trace function on the Jordan algebra V .Letc1,...,cr be a Jordan frame, so that we have c1 +···+cr = e0.TheJordan frame c1,...,cr yields an orthogonal decomposition† V = Vkj, (2.1) 1≤j≤k≤r where Vjj = Rcj (j = 1,...,r)and ={ ∈ ; = 1 + = } ≤ ≤ Vkj x V M(ci)x 2 (δij δik)x for i 1,...,r (1 j

†Our Vkj are the Vjk in [7]. We have changed the notation in order to be consistent with the normal decomposition (1.2). 170 H. Nakashima and T. Nomura

The decomposition (2.1) is called the Peirce decomposition relative to the Jordan frame c1,...,cr . Let := Int{x2; x ∈ V } be the symmetric cone of the Euclidean Jordan algebra V and G() the linear automorphism group of the cone . We know that G() is reductive in this case. Let G be the identity component of G(),andg its Lie algebra. Let k be the derivation algebra Der(V ) of the Jordan algebra V and put p := {M(x); x ∈ V }.Then g = k + p is a Cartan decomposition of g with the usual Cartan involution θX =−tX,where tX denotes the transpose of the operator X relative to the trace inner product ·|·.Put a := RM(c1) ⊕···⊕RM(cr ).Thena is a maximal abelian subspace of p.Letα1,...,αr ∗ be the basis of a dual to M(c1),...,M(cr ). We know that the positive a-roots are 1 − 2 (αk αj )(j

nkj := {z 2 cj ; z ∈ Vkj}, (2.2) 2 := +[ ] := where a b M(ab) M(a),M(b) . Summing up all of the nkj as n j

PROPOSITION 2.1. = = (1) Lci M(ci) for i 1,...,r. (2) If x ∈ Vkj (1 ≤ j

Proof. (1) Clear from M(ci)e0 = ci and M(ci) ∈ a ⊂ h. (2) If x ∈ Vkj (1 ≤ j

2(x 2 cj )e0 = 2(M(xcj ) + M(x)M(cj ) − M(cj )M(x))e0 = x.

Hence, the proposition follows. 2

Using Proposition 2.1, we see easily that the Jordan frame c1,...,cr forms a complete set of primitive idempotents of the clan (V, ), and that the corresponding normal decomposition coincides with the Peirce decomposition (2.1): indeed, if x ∈ Vkj (1 ≤ j< k ≤ r), then it holds that = = 2 = 2 = Rci x Lx ci 2(x cj )ci 2(ci cj )x δij x. We thus have the multiplication rules (1.3). Proposition 2.1 also yields the relation 1 + t = ∈ 2 (Lx Lx ) M(x) (x V) (2.3) Clans deﬁned by Jordan algebras 171

(see also (2.5) below). In particular, we have

Tr Lx = Tr M(x) (x ∈ V). (2.4) The following proposition tells us the relation between the canonical inner products in V deduced from the two algebraic structures.

PROPOSITION 2.2. One has Tr Lx y = Tr M(xy), so that n Tr L = x |y x y r holds for all x, y ∈ V ,wheren := dim V . ∈ = + ∈ R ∈ Proof. Let x V and we write x i λi ci j

:= Vr1 ⊕···⊕Vr,r−1,W:= ⊕ Rcr . Then (1.3) immediately implies that W is a two-sided ideal in the clan V .Inotherwords,we have Lv(W) ⊂ W and Rv(W) ⊂ W for any v ∈ V . In view of this we put for v ∈ V W := | W := | Lv Lv W ,Rv Rv W , and V := Vji. 1≤i≤j≤r−1 172 H. Nakashima and T. Nomura

Then V is a Euclidean Jordan algebra of rank r − 1, and thus has the canonical clan structure. In addition, V is a subalgebra of V as a clan. The right multiplication operator by v ∈ V in ∈ = + ∈ ∈ the clan V is denoted by Rv . Then, by writing v V as v v w with v V and w W, the operator Rv is of the form R O R = v . v ∗ W Rv W ∈ We now analyze the operator Rv . First, we note that (1.3) implies that if v V ,then ⊂ := | we have Rv ( ) . We put Rv Rv . To see what Rv looks like, we deﬁne operators φ(v)(v ∈ V ) on by φ(v )ξ := 2v ξ(ξ∈ ). 1 Since V (respectively ) is the Peirce 0 (respectively Peirce 2 ) space for the idempotent cr , we know that the map φ : v → φ(v ) is a Jordan algebra representation of V to End( ) (cf. [7, Proposition IV.4.1]) = ∈ PROPOSITION 2.3. We have Rv φ(v ) for any v V . To prove this proposition, we need two lemmas.

LEMMA 2.4. Let c ∈ V be a non-zero idempotent in the Jordan algebra V , and consider the = 1 ∈ ∈ corresponding Peirce spaces Vj (c) (j 0, 2 , 1).Ifx V0(c) and y V1(c), then one has [M(x),M(y)]=0. Proof. This follows from [7, Proposition II.1.1(ii)] by putting z = c there. 2

LEMMA 2.5. If ξ ∈ , then one has Lξ = 2ξ 2 (e0 − cr ). = +···+ ∈ = Proof. We write ξ as ξ ξ1 ξr−1 with ξj Vrj . Proposition 2.1(2) tells us that Lξj 2(ξj 2 cj ).Ifi = j, r,wehaveξj ∈ V0(ci), so that Lemma 2.4 says that [M(ξj ), M(ci)]=0. Therefore, = 2 +···+ = 2 − Lξj 2(ξj (c1 cr−1)) 2ξj (e0 cr ). This clearly implies the lemma. 2 Now we are ready to prove Proposition 2.3. Let v ∈ V and ξ ∈ . By deﬁnition and Lemma 2.5 we have = = 2 − = 2 − Rv ξ Lξ v 2(ξ (e0 cr ))v 2(v (e0 cr ))ξ. Since v ∈ V0(cr ), Lemma 2.4 implies v 2 cr = M(v cr ) +[M(v ), M(cr )]=0. Hence, by the above we obtain = 2 = Rv ξ 2(v e0)ξ φ(v )ξ. The proof is complete. 2 PROPOSITION 2.6. By writing v ∈ V as v = v + ξ + vr cr with v ∈ V ,ξ∈ , vr ∈ R,the W operator Rv is of the form φ(v) 1 ·|c ξ RW = 2 r . v ·| ξ cr vr idVrr Clans deﬁned by Jordan algebras 173

= + + ∈ ∈ R Proof. Since Rv Rv Rξ vr Rcr , we look at each term separately. Let η and yr . We have by Proposition 2.3 and (1.3) Rv (η + yr cr ) = φ(v )η + yr cr v = φ(v )η. Next, by Lemma 2.5 we have

Rξ η = Lηξ = 2(η 2 (e0 − cr ))ξ

= 2M(η(e0 − cr ))ξ + 2[M(η),M(e0 − cr )]ξ

= 2M(η)ξ − 2(e0 − cr )(ηξ) = 2cr (ξη).

Since 2cr (ξη) ∈ Vrr = Rcr , we put 2cr (ξη) = acr with a ∈ R. Then the number a can be obtained as follows:

Thus, we get Rξ η =ξ |η cr . Moreover, since Proposition 2.1(1) says Rξ cr = cr ξ = = 1 = 1 | M(cr )ξ 2 ξ 2 cr cr ξ, we arrive at + = | + 1 | Rξ (η yr cr ) η ξ cr 2 yr cr cr ξ. = + = Finally, since η cr 0 by (1.3), we have Rcr (η yr cr ) yr cr . Now we get the operator matrix as described in the proposition. 2

In view of Proposition 2.6, we introduce the following inner product in W = + Rcr : + | + = | + 1 ∈ ∈ R η yr cr η yr cr W η η 2 yr yr (η, η and yr ,yr ). (2.6) W By using this inner product, the operator Rv expressed in Proposition 2.6 is rewritten in a more symmetric way as φ(v ) ·|cr W ξ RW = (v = v + ξ + v c ). (2.7) v ·| r r ξ W cr vr idVrr

Then we obtain the following inductive structure for Rv. THEOREM 2.7. Decomposing v ∈ V as v = v + ξ + vr cr with v ∈ V , ξ ∈ and vr ∈ R, one has ⎛ ⎞ R O ⎜ v ⎟ R = ⎝ ·| ⎠ . v ∗ φ(v ) cr W ξ ·| ξ W cr vr idVrr W In order to compute the determinant of Rv in (2.7), we recall the following elementary determinant formula: AB − Det = (Det A) Det(D − CA 1B) (Det A = 0). (2.8) CD

We suppose for the moment that v is invertible in the Jordan algebra V , so that the operator φ(v ) is also invertible. Recalling (2.6) for the inner product ·|·W ,wegetby(2.8) W = − 1 −1 | Det Rv (vr 2 φ(v ) ξ ξ ) Det φ(v ). 174 H. Nakashima and T. Nomura

Denoting by detV the determinant function on the Jordan algebra V , we put co −1 v := (detV v )(v ) . (2.9) We know by [7, Proposition II.2.4] that cov is polynomial in v. It is a Jordan algebra variant of the cofactor matrix. Using cov,wehave −1 −1 −1 co φ(v ) = φ((v ) ) = (detV v ) φ( v ).

Then, denoting by d the common dimension of Vkj (j < k), we arrive at − W = 1 · − 1 co | Det Rv (detV v ) Det φ(v ) (vr detV v 2 φ( v )ξ ξ ) d−1 co = (detV v ) (vr detV v −( v )ξ |ξ ), (2.10) d wherewehavemadeuseofthefactthatDetφ(v ) = (detV v ) (see [7, Proposition IV.4.2]). To continue we need the following lemma.

LEMMA 2.8. One has co detV (v + ξ + vr cr ) = vr detV v −( v )ξ |ξ . = +···+ ∈ Proof. Let v λ1c1 λr−1cr−1 be the spectral decomposition of v V ,where c1,...,cr−1 form a Jordan frame of V and the real numbers λj are allowed to be equal. Then we have a Jordan frame c1,...,cr−1,cr of V , and accordingly we decompose ξ as = +···+ ∈ ξ ξ1 ξr−1 with ξj j ,where := { ∈ ; = 1 = − } j ξ ci ξ 2 δij ξ for i 1,...,r 1 . Then (2.9) gives r−1 − co = 1 v (detV v ) (λj ) cj . j=1 Thus, what we have to prove is r−1 r−1 2 1 ξj detV λ c + ξ + vr cr = (det v ) vr − . (2.11) i i V 2 λ i=1 j=1 j Now by using the Frobenius transformations τ(z) as in [7, Proposition VI.3.2], simple computations yield that r−1 r−1 r−1 r−1 2 ξ 1 ξ τ − j λ c + ξ + v c = λ c + v − j c . λ i i r r j j r 2 λ r j=1 j i=1 j=1 j=1 j Then (2.11) follows from [7, Proposition VI.3.10]. 2

Let 1(x), . . . , r (x) be the Jordan algebra principal minors associated with the Jordan frame c1,...,cr of V . Then, (2.10) and Lemma 2.8 show that W = d−1 Det Rv r−1(v) r (v). Since both sides are polynomial in v, the equality holds without the restriction that v is invertible. Summing up all of the above discussions and using Theorem 2.7, we obtain by induction the following theorem. Clans deﬁned by Jordan algebras 175

d d THEOREM 2.9. For v ∈ V , one has Det Rv = 1(v) ···r−1(v) r (v). Remark 2.10. We have

−1 Det Rhv = χ(h)Det Rv (h ∈ H, v ∈ V), where χ(h) := (DetV h)(Det Ad h) .

For this we refer the reader to the proof of [12, Lemma 2.7]. The one-dimensional r + − representation χ of H comes from the linear form on a given by j=1(1 d(r j))αj . From this we also obtain Theorem 2.9.

3. Clans deﬁned by representations of Jordan algebras and the corresponding real Siegel domains

In this section, we deﬁne a clan starting from a representation of a Jordan algebra. Let V be a simple Euclidean Jordan algebra of rank r with unit element e0. We keep to the Jordan algebra notation used in Section 2. Let (ϕ, E) be a selfadjoint representation of the Jordan algebra V on a real Euclidean vector space E with inner product ·|·E . Throughout the following, we always assume that dim E>0. The case dim E = 0 has been treated in Section 2. By assumption every ϕ(x) (x ∈ V)is a selfadjoint operator on E,thatisϕ(x)∗ = ϕ(x) for any x ∈ V , and we have = 1 + ∈ ϕ(xy) 2 (ϕ(x)ϕ(y) ϕ(y)ϕ(x)) (x, y V). (3.1)

In the current case dim E>0, we always require that ϕ(e0) is the identity operator. Associated with ϕ we have a symmetric bilinear map Q : E × E → V deﬁned by

ϕ(x)ξ |η E =Q(ξ, η)|x (x ∈ V,ξ,η∈ E). (3.2)

Put VE := E ⊕ V and we will deﬁne a clan structure on VE. To do so, we need the ‘lower triangular part’ ϕ(x) of the selfadjoint operator ϕ(x). First of all we note by [7,p.76]thatϕ(c1), . . . , ϕ(cr ) are mutually orthogonal projection operators with equal rank. Thus, putting Ei := ϕ(ci)E for i = 1,...,r,wesee that dim Ei are all equal, and we have an orthogonal decomposition E = E1 ⊕···⊕Er . Before proceeding further we record the following basic facts.

LEMMA 3.1. Let x ∈ Vkj (1 ≤ j

ϕ(ci)ϕ(x)ϕ(cl) = ϕ(cl)ϕ(x)ϕ(ci) = 0 (l = 1,...,r);

(3) ϕ(cj )ϕ(x)ϕ(cj ) = ϕ(ck)ϕ(x)ϕ(ck) = 0; (4) ϕ(x) = ϕ(cj )ϕ(x)ϕ(ck) + ϕ(ck)ϕ(x)ϕ(cj ).

Proof. (1) Since cix = 0, we have by (3.1)

0 = 2ϕ(cix) = ϕ(ci)ϕ(x) + ϕ(x)ϕ(ci).

2 (2) Since ϕ(ci) = ϕ(ci),wehaveby(1)

ϕ(ci)ϕ(x)ϕ(cl) =−ϕ(ci)ϕ(x)ϕ(ci)ϕ(cl) =−δilϕ(ci)ϕ(x)ϕ(ci). 176 H. Nakashima and T. Nomura

If l = i, the last term equals zero. If l = i,wegetϕ(ci)ϕ(x)ϕ(ci) =−ϕ(ci)ϕ(x)ϕ(ci),which also gives ϕ(ci)ϕ(x)ϕ(ci) = 0. Hence, ϕ(ci)ϕ(x)ϕ(cl) = 0foranyl. Just taking the adjoints gives the remaining equality. (3) Since x = 2cj x, we have by (3.1)

ϕ(x) = 2ϕ(cj x) = ϕ(cj )ϕ(x) + ϕ(x)ϕ(cj ).

Multiplying both sides from the left by ϕ(cj ),weget

ϕ(cj )ϕ(x) = ϕ(cj )ϕ(x) + ϕ(cj )ϕ(x)ϕ(cj ).

From this we get immediately ϕ(cj )ϕ(x)ϕ(cj ) = 0. We can prove ϕ(ck)ϕ(x)ϕ(ck) = 0inthe same way. (4) We have ϕ(e0) = I and c1 +···+cr = e0. By (2) and (3), we get

ϕ(x) = ϕ(e0)ϕ(x)ϕ(e0)

= (ϕ(c1) +···+ϕ(cr ))ϕ(x)(ϕ(c1) +···+ϕ(cr ))

= ϕ(cj )ϕ(x)ϕ(ck) + ϕ(ck)ϕ(x)ϕ(cj ).

Thus, the proof is completed. 2 = + ∈ R ∈ In view of Lemma 3.1, we now set for x i λi ci j

We take an orthonormal basis of E by first taking the one from E1, then from E2,...,and finally from Er . By this ordering of orthonormal basis we see that every operator ϕ(x) for x ∈ Vkj (j < k) is represented by a strictly lower triangular matrix. On the other hand it is clear that each ϕ(ci ) is diagonal. Denoting the clan product of V simply by we now define a product of VE by

(ξ + x) (η + y) := ϕ(x)η + (Q(ξ, η) + x y) (x, y ∈ V,ξ,η∈ E). (3.5)

The aim of this section is to prove the following theorem. THEOREM 3.2. The algebra (VE, ) is a clan with an admissible linear form s given by s (ξ + x) = Tr Lx (ξ ∈ E, x ∈ V). Before proving Theorem 3.2, we need further observations about the Jordan algebra representation ϕ.

PROPOSITION 3.3. The map ϕ is also a representation of the clan V in the sense of Ishi [10], that is, ϕ satisﬁes

ϕ(x y) = ϕ(x)ϕ(y) + ϕ(y)ϕ(x)∗ (x, y ∈ V). Clans deﬁned by Jordan algebras 177

Proof. By linearity it is enough to prove the equality in the proposition separately for the cases x = ci (i = 1,...,r)and x ∈ Vkj (j < k). If x = ci, then for y ∈ V we have by Proposition 2.1(1), (3.1) and (3.3) = = 1 + 1 ϕ(ci x) ϕ(cix) 2 ϕ(ci)ϕ(x) 2 ϕ(x)ϕ(ci) ∗ = ϕ(ci)ϕ(x) + ϕ(x)ϕ(ci) .

Suppose next that x ∈ Vkj (j < k). In this case we consider the operator P(a,b)(a,b∈ V)on V deﬁned by P(a,b)z= (a 2 z)b (z ∈ V).Sinceϕ is a Jordan algebra representation we have (cf. [7, Lemma XVI.2.2]) = 1 + ∈ ϕ(P(a, b)z) 2 (ϕ(a)ϕ(z)ϕ(b) ϕ(b)ϕ(z)ϕ(a)) (a, b, z V). (3.6) Now for any y ∈ V , we have by Proposition 2.1, (3.6) and Lemma 3.1(4)

ϕ(x y) = 2ϕ((x 2 cj )y) = 2ϕ(P(x, y)cj )

= ϕ(x)ϕ(cj )ϕ(y) + ϕ(y)ϕ(cj )ϕ(x)

= (ϕ(ck)ϕ(x)ϕ(cj ) + ϕ(cj )ϕ(x)ϕ(ck))ϕ(cj )ϕ(y)

+ ϕ(y)ϕ(cj )(ϕ(ck)ϕ(x)ϕ(cj ) + ϕ(cj )ϕ(x)ϕ(ck)).

Since ϕ(ci) are mutually orthogonal projection operators, we obtain by (3.3)

ϕ(x y) = ϕ(ck)ϕ(x)ϕ(cj )ϕ(y) + ϕ(y)ϕ(cj )ϕ(x)ϕ(ck) ∗ = ϕ(x)ϕ(y) + ϕ(y)ϕ(x) .

Hence, the proposition is proved. 2 E := From now on we put Lx ξ ϕ(x)ξ to emphasize the V -action on E.

PROPOSITION 3.4. For x ∈ V and ξ, η ∈ E, one has = E + E Lx Q(ξ, η) Q(Lx ξ, η) Q(ξ, Lx η). t Proof. We see from (2.3) that if x ∈ V ,then Lx = 2M(x)− Lx . This together with Proposition 3.3, (3.1) and (3.4) gives for x, y ∈ V

t ϕ( Lx y) = 2ϕ(M(x)y)− ϕ(Lxy) = ϕ(x)ϕ(y) + ϕ(y)ϕ(x) − (ϕ(x)ϕ(y) + ϕ(y)ϕ(x)∗) ∗ = ϕ(x) ϕ(y) + ϕ(y)ϕ(x). (3.7)

Using this, we obtain for all y ∈ V E + E | = | + | Q(Lx ξ, η) Q(ξ, Lx η) y ϕ(y)ϕ(x)ξ η E ϕ(y)ξ ϕ(x)η E ∗ =(ϕ(y)ϕ(x) + ϕ(x) ϕ(y))ξ |η E t t =ϕ( Lx y)ξ |η E =Q(ξ, η)| Lx y

=Lx Q(ξ, η)|y . Therefore, we obtain the proposition. 2 178 H. Nakashima and T. Nomura

In what follows we put [x y]:=x y − y x. ∈ E =[ E E] LEMMA 3.5. For x, y V one has L[x y] Lx ,Ly . Proof. By Proposition 3.3 and (3.4), we have by a simple computation

ϕ([x y]) = ϕ(x)ϕ(y) + ϕ(y)ϕ(x)∗ − ϕ(y)ϕ(x) − ϕ(x)ϕ(y)∗ =[ E E]+ [ E E] ∗ Lx ,Ly ( Lx ,Ly ) . Taking the lower triangular part, we obtain the lemma. 2 We are now able to prove Theorem 3.2. By (3.5), the left multiplication operator Lξ+x of VE by ξ + x ∈ VE is written as an operator matrix E = Lx 0 Lξ+x , Q(ξ, ·)Lx ∈ E = where Lx is the left multiplication operator of (V, ) by x V . Since both of Lx ϕ(x) and Lx are lower triangularizable, Lξ+x is also lower triangularizable. Moreover, using Proposition 3.4 and Lemma 3.5, we obtain − Lξ+x Lη+y Lη+y Lξ+x [ E E] = Lx ,Ly 0 E · + · − E · − · [ ] Q(ξ, Ly ( )) LxQ(η, ) Q(η, Lx ( )) Ly Q(ξ, ) Lx ,Ly E L[x y] 0 = = L[ + + ]. E − E · (ξ x) (η y) Q(Lx η Ly ξ, )L[x y] Hence, (VE, ) is a left symmetric algebra. Now we deﬁne a linear form s on VE as in the statement of Theorem 3.2. Put s(x) = Tr Lx (x ∈ V) for simplicity. We know that s is an admissible linear form of the clan V . Then (3.5) gives

s((ξ + x) (η + y)) = s(Q(ξ, η)) + s(x y) (ξ, η ∈ E, x, y ∈ V), so that s is symmetric. Further letting n = dim V , we get by Proposition 2.2 n n s(Q(ξ, η)) = Tr L = Q(ξ, η)|e = ξ |η . Q(ξ,η) r 0 r E This together with Proposition 2.2 gives

n n s ((ξ + x) (η + y)) = ξ |η + x |y . (3.8) r E r Hence, s is an admissible linear form on VE, showing ﬁnally that VE is a clan. 2 Since the symmetric bilinear map Q is -positive, that is, since we have

Q(ξ, ξ) ∈ \{0} for all ξ ∈ E \{0},

(see [7, p. 74]), we can deﬁne a real Siegel domain D(, Q) in VE from the data and Q by := { + ∈ ; − 1 ∈ } D(, Q) ξ x VE x 2 Q(ξ, ξ) . (3.9) Clans deﬁned by Jordan algebras 179

Proposition 3.4 gives rise to = E E ∈ ∈ (exp Lx )Q(ξ, η) Q((exp Lx )ξ, (exp Lx )η) (x V,ξ,η E). (3.10) In order to see that D = D(, Q) is homogeneous, we ﬁrst recall that the split solvable group H ={exp Lz; z ∈ V } acts simply transitively on . Then we introduce afﬁne transformations nζ (ζ ∈ E) and hz (z ∈ V)on VE, respectively, by 1 nζ (ξ + x) := (ξ + ζ)+ (x + Q(ξ, ζ ) + Q(ζ, ζ )), 2 (ξ ∈ E, x ∈ V). (3.11) + := E + hz(ξ x) (exp Lz )ξ (exp Lz)x, Put ξ + x = nζ hz(ξ + x) (ξ ∈ E, x ∈ V). Then by (3.10) and (3.11), we obtain − 1 = − 1 x 2 Q(ξ ,ξ ) (exp Lz)(x 2 Q(ξ, ξ)). (3.12)

This shows that both nζ and hz leave D(, Q) stable. Put

ND := {nζ ; ζ ∈ E},HD := {hz; z ∈ V }.

Clearly HD is isomorphic to H .BothND and HD are subgroups of the afﬁne automorphisms − 1 = = E of VE leaving D(, Q) stable. Since hznζ hz nζ with ζ (exp Lz )ζ as can be easily veriﬁed by using (3.10), we form a semidirect product group H(D):= ND HD.Given ξ + x ∈ D(, Q), the equation

nζ hz(0 + e0) = ξ + x (3.13) has a unique solution ζ = ξ ∈ E and z ∈ V takeninsuchawaythat(exp Lz)e0 = x − 1 ∈ 2 Q(ξ, ξ) . This shows that H(D)acts on D(, Q) simply transitively.

4. The clan obtained by the adjunction of a unit element

Let us keep to the notation of the previous sections. In particular let VE = E ⊕ V be the clan obtained by means of a selfadjoint representation ϕ of a simple Euclidean Jordan algebra 0 := R ⊕ V on a Euclidean vector space E. In this section, we consider the algebra VE e VE obtained by the adjunction of a unit element e to the clan VE. As before we write e0 for the unit element of the Jordan algebra (and of the associated clan) V . We put u := e − e0.Then 0 = R ⊕ 0 we also have VE u VE. By this decomposition, the clan product of VE is written as (λu + ξ + x) (μu + η + y) = + + 1 + + + (λμ)u (μξ 2 λη ϕ(x)η) (Q(ξ, η) x y), (4.1) where λ, μ ∈ R, ξ, η ∈ E and x, y ∈ V . We use the symbol R0 for the right multiplication 0 operators of the clan VE. The elements u, c1,...,cr form a complete set of primitive 0 idempotents of VE, and the corresponding normal decomposition is described as 0 = VE Vji, 0≤i≤j≤r where V00 := Ru, Vj0 := ϕ(cj )E (j = 1,...,r)and the normal decomposition of the clan V is given by (2.1). 180 H. Nakashima and T. Nomura

0 Let us compute the determinant of the right multiplication operators Rλu+ξ+x,where ∈ R ∈ ∈ 0 λ , ξ E and x V . By (4.1), the operator Rλu+ξ+x is written as an operator matrix ⎛ ⎞ λ 00 ⎜ ⎟ 0 = 1 0 Rλu+ξ+x ⎝ 2 ξλidE Rξ ⎠ . 0 0 Rξ Rx 0 ⊂ 0 ⊂ Here Rx is the right multiplication operator in V , and we note that Rξ (V ) E and Rξ (E) V . = = + + ∈ 0 = PROPOSITION 4.1. Let N dim E and n dim V . Then for λu ξ x VE with λ 0 one has 0 = 1+N−n · Det Rλu+ξ+x λ Det(R − 1 ). λx 2 Q(ξ,ξ) Proof. Let R stand for the right multiplication operators in VE.Thenwehave λ R 0 = · idE ξ Det Rλu+ξ+x λ Det . Rξ Rx Since the bilinear map Q is symmetric, equation (3.5) and Proposition 3.4 yield for all y ∈ V = = E = 1 = 1 Rξ Rξ y (y ξ) ξ Q(Ly ξ, ξ) 2 Ly Q(ξ, ξ) 2 RQ(ξ,ξ)y. Using this and (2.8), we obtain for λ = 0 λ R idE ξ = N · − 1 = N−n · Det λ Det Rx RQ(ξ,ξ) λ Det(Rλx− 1 Q(ξ,ξ)). Rξ Rx 2λ 2 This completes the proof. 2 Before closing this section, we give a quick review of Vinberg’s argument [15]which tells us that the real Siegel domain D(, Q) in VE introduced by (3.9) appears as the cross- 0 section of the homogeneous cone corresponding to the clan VE with a hyperplane. To see 0 0 + ⊂ 0 0 this, first let be the cone in VE generated by u D(, Q) VE and the origin of VE. Then 0 ={ + + ∈ 0; + ∈ } λu λξ λx VE λ>0,ξ x D(, Q) ={ + + ∈ 0; − 1 ∈ } λu ξ x VE λ>0,λx 2 Q(ξ, ξ) . (4.2) ∈ ∈ 0 0 0 For ζ E and z V , we next consider the linear operators nζ and hz on VE defined respectively by ⎛ ⎞ ⎛ ⎞ 100 10 0 0 = ⎝ ⎠ 0 = ⎝ E ⎠ nζ ζ idE 0 ,hz 0expLz 0 , (4.3) 1 · 2 Q(ζ, ζ ) Q( ,ζ) idV 00expLz 0 = R ⊕ ⊕ where the matrix blocks follow the decomposition VE u E V . By (3.11) we have 0 + + = + + 0 + + = + + nζ (u ξ x) u nζ (ξ x), hz(u ξ x) u hz(ξ x). (4.4) Clans defined by Jordan algebras 181

∈ ∈ 0 0 + Hence, we see that for all ζ E and z V , both nζ and hz leave u D(, Q) stable and, hence, belong to G(0). 0 Now given λu + λξ + λx ∈ , we choose ζ and z as in (3.13), and recall e = u + e0. Then we obtain by (4.4) · 0 0 + = + + λ (nζ hz)(u e0) λu λξ λx. This shows that 0 is homogeneous. Moreover (4.1) shows that the left multiplication 0 ∈ R ∈ ∈ 0 operators Lλe+ξ+x for λ , ξ E and x V of the clan VE are of the form ⎛ ⎞ 00 0 0 ⎝ E ⎠ L + + = λ · id 0 + ξL 0 . λe ξ x VE x 0 Q(·,ξ) Lx Since we have 0 = 0 0 = 0 nζ exp Lζ ,hz exp Lz, 0 0 it holds that is indeed the homogeneous cone corresponding to the clan VE.

5. The basic relative invariants associated with 0

0 Let VE be the clan introduced in Section 4. Then it has a unit element e, so that Theorem 1.1 0 + 0 = tells us that VE has r 1 basic relative invariants i (v) (i 0,...,r). In this section we 0 = express i (v) in terms of the Jordan algebra principal minors j (x) (j 1,...,r)of the Euclidean Jordan algebra V . We put v = λu + ξ + x(λ∈ R,ξ∈ E, x ∈ V), and use this expression of v for a variable 0 = in VE. Theorem 2.9 and Proposition 4.1 tell us that for λ 0 r−1 d 0 1+N−n 1 1 Det R = λ j λx − Q(ξ, ξ) r λx − Q(ξ, ξ) . (5.1) v 2 2 j=1 − 1 We note here that the polynomials j (λx 2 Q(ξ, ξ)) are not always irreducible. To clarify the matters, we introduce the notion of regularity for selfadjoint representations ϕ of V after Clerc [3]. We say that ϕ is regular if there exists ξ ∈ E such that Q(ξ, ξ) = e0. Thus, r (Q(ξ, ξ)) does not vanish identically if ϕ is regular. In the regular case we have the following proposition. − 1 PROPOSITION 5.1. Suppose that ϕ is regular. Then the polynomial r (λx 2 Q(ξ, ξ)) is irreducible. := − 1 − r Proof. Consider the polynomial F(λ,ξ,x) r (λx 2 Q(ξ, ξ)) λ r (x). As we noted = − 1 above, F(0,ξ,x) r ( 2 Q(ξ, ξ)) is a non-zero polynomial by the regularity assumption of ϕ,sothatF(λ,ξ,x) does not have the factor λ. We next show quite explicitly that the degree of x in F(λ,ξ,x) is strictly lower than r. To do so we suppose that x ∈ .Then x has the square root x1/2 ∈ .LetP(y):= 2M(y)2 − M(y2)(y∈ V) be the quadratic representation of the Jordan algebra V .Wehave − 1 = 1/2 − 1 −1/2 λx 2 Q(ξ, ξ) P(x )(λe0 2 P(x )Q(ξ, ξ)), 182 H. Nakashima and T. Nomura so that [7, Proposistion III.4.2] gives − 1 = − 1 −1/2 r (λx 2 Q(ξ, ξ)) r (x)r (λe0 2 P(x )Q(ξ, ξ)). Let r k r−k r (λe0 − y) := (−1) ak(y)λ (5.2) k=0 be the Jordan algebra variant of the expansion of the characteristic polynomial given in [7, Proposition II.2.1] (see also the last line on page 29 of that book). Here the polynomial function ak (k = 0, 1,...,r)is homogeneous of degree k and invariant under the automorphism group Aut(V ) of the Jordan algebra V . Thus, r 1 k 1 −1/2 r−k r λx − Q(ξ, ξ) = r (x) (−1) ak P(x )Q(ξ, ξ) λ . 2 2 k=0

1/2 By [7, Lemma XIV.1.2], bk(x, y) := ak(P (x )y) is a polynomial in x, y such that bk(y, x) = bk(x, y) for every k. Therefore, we get r 1 k −1 1 r−k r λx − Q(ξ, ξ) = r (x) (−1) bk x , Q(ξ, ξ) λ , 2 2 k=0 so that noting a0(y) = 1foranyy ∈ V , we arrive at r − 1 − F(λ,ξ,x)= (x) (−1)kb x 1, Q(ξ, ξ) λr k. r k 2 k=1 From this we conclude that the degree of x in the polynomial F(λ,ξ,x) is strictly lower than r. Since r (x) is an irreducible polynomial of degree r, the properties of F(λ,ξ,x)shown above tell us that the polynomial − 1 = r + r (λx 2 Q(ξ, ξ)) λ r (x) F(λ,ξ,x) is irreducible. 2 To treat the case where the representation ϕ is not regular, more details are necessary about selfadjoint representations of Euclidean Jordan algebras as described by Clerc [3] (see also [4]). Thus, we need to separate the cases according to the classification of simple Euclidean Jordan algebras. Let K = R, C,orH. Simple Euclidean Jordan algebras V are classified according to the rank r of V as follows (cf. [7]): ⎧ ⎪r = 2 ⇒ V =∼ R ⊕ W (W is a real vector space) Lorentzian type, ⎨ ∼ V = Herm(r, K) Hermitian type, ⎩⎪r ≥ 3 ⇒ V =∼ Herm(3, O) Exceptional type. In this paper, we call V the Lorentzian type when r = 2, and the Hermitian type when r ≥ 3. Moreover we call Herm(3, O) the exceptional type. Since only the zero representation exists for the exceptional type, it is excluded from the consideration below. Clans defined by Jordan algebras 183

5.1. The case of Hermitian type

Let us put V = Herm(r, K),andd = dimR K.Thisd coincides with the one that appeared in (5.1). The Jordan product ◦ in V is deﬁned by ◦ := 1 + ∈ x y 2 (xy yx)(x,y V), where the products on the right-hand side are the usual matrix multiplication. The corresponding symmetric cone is given by ++ = Herm(r, K) ={x ∈ V ; x is positive-deﬁnite}.

We ﬁx a Jordan frame given by ci = Eii,the(i, i)-matrix unit for i = 1,...,r.The associated clan structure of V is described as

x y := xy + y(x)∗ (x, y ∈ V), where for x = (xij ) ∈ V , the symbol x denotes the matrix ⎛ ⎞ 1 x11 ⎜ 2 ⎟ ⎜ 1 ⎟ ⎜ x21 x22 0 ⎟ ⎜ 2 ⎟ x := ⎜ ⎟ . ⎜ . .. .. ⎟ ⎝ . . . ⎠ ··· 1 xr1 xr,r−1 2 xrr

Then obviously x + (x)∗ = x. In the present case the Jordan algebra principal minors 1(x), . . . , r (x) associated with the Jordan frame c1,...,cr is nothing other than the left upper corner principal minors det(k)x(k= 1,...,r)of a matrix x ∈ V ,whereifK = H, then det(k)x is taken as the Jordan algebra determinant function on Herm(k, H). Let (ϕ, E) be a selfadjoint representation of the Jordan algebra V .Clerc[3] tells us that ∗ E = Mat(r × p, K) with the canonical real inner product ξ |η E := Re Tr(ξη ),and

ϕ(x)ξ = xξ (x ∈ V,ξ ∈ E). (5.3)

Moreover, we see easily that

ϕ(x)ξ = xη(x∈ V,ξ ∈ E). (5.4)

Let Q be the symmetric bilinear map associated with the representation ϕ deﬁned by (3.2). Then Q is given by = 1 ∗ + ∗ ∈ Q(ξ, η) 2 (ξη ηξ )(ξ,ηE). Now for λ = 0, the formula (5.1) is rewritten as

0 1+N−n ∗ d ∗ d ∗ = − 1 ··· − − 1 − 1 Det Rv λ 1(λx 2 ξξ ) r 1(λx 2 ξξ ) r (λx 2 ξξ ), (5.5) wherewehaveN = dim E = drp and n = dim V = r + (d/2)r(r − 1). In the current case the regularity of ϕ is equivalent to the inequality p ≥ r (see [3,Th´eor`eme 3]). The following 0 + K proposition that realizes the clan VE in Herm(r p, ) equipped with the canonical clan structure is useful to have a better expression of the basic relative invariants. 184 H. Nakashima and T. Nomura

0 + K PROPOSITION 5.2. The clan VE is isomorphic to a subclan of Herm(r p, ).The embedding is given by ⎛ ⎞ 1 ∗ ⎜ λIp √ ξ ⎟ 0 + + −→ ⎜ 2 ⎟ ∈ + K VE λu ξ x ⎝ 1 ⎠ Herm(r p, ). √ ξx 2

Proof. Let X be a subspace of Herm(r + p, K) given by λI ξ ∗ X := X(λ, ξ, x) := p ; λ ∈ R,ξ∈ E, x ∈ V . (5.6) ξx

Since a direct computation shows

= + 1 + + X(λ, ξ, x) X(μ, η, y) X(λμ, μξ 2 λη xη, 2Q(ξ, η) x yg), (5.7) + K : 0 → X is a subclan of Herm(r p, ). Deﬁne a linear map VE X by + + = √1 + + ∈ 0 (λu ξ x) X λ, ξ, x (λu ξ x VE). 2 Comparing (4.1) with (5.7), we see that is a homomorphism. Since is visibly bijective, 0 2 VE is isomorphic to X.

LEMMA 5.3. For λ = 0, one has ⎛ ⎞ 1 ∗ ⎜ λIp √ ξ ⎟ (p+j)⎜ 2 ⎟ = p−j · − 1 ∗ = det ⎝ 1 ⎠ λ j λx ξξ (j 1,...,r). (5.8) √ ξx 2 2

Proof. We note the following decomposition: λI ζ ∗ I 0 λI 0 I λ−1ζ ∗ p = p p p −1 −1 ∗ , ζz λ ζIj 0 z − λ ζζ 0 Ij where ζ ∈ Mat(j × p, K) and z ∈ Herm(j, K). Considering the right-hand side as the image of diag[λI ,z− λ−1ζζ∗] by a unipotent transformation on the Jordan algebra Herm(p + p Ip 0 j, K) given by the matrix −1 ,wehave(cf.[7, Proposition III.4.3]) the formula in the λ ζIj lemma. 2

= + + ∈ 0 THEOREM 5.4. Let v λu ξ x VE. (1) If ϕ is regular, that is, if p ≥ r, then one has 0 = 0(v) λ, 0 = − 1 ∗ = j (v) j (λx 2 ξξ )(j 1,...,r). Clans deﬁned by Jordan algebras 185

(2) If ϕ is not regular, that is, if p

Let ξj ∈ Mat(j × p, K) denote the upper j × p submatrix of ξ ∈ E. Then we see easily that ∗ ∗ (ξξ )j = ξj (ξj ) . Therefore, we get − 1 ∗ = − 1 ∗ j (λx 2 ξξ ) j (λxj 2 ξj (ξj ) ). Since j ≤ p, we just repeat the argument of Proposition 5.1 in the Euclidean Jordan algebra K − 1 ∗ Herm(j, ), and obtain the irreducibility of j (λx 2 ξξ ). The rest is the case where p

√1 is irreducible. In what follows we put η := ξj and y := xj for simplicity. Suppose that 2 y ∈ Herm(j, K) is positive-deﬁnite. We have ∗ λI η∗y−1/2 λIp η = Ip 0 p Ip 0 1/2 −1/2 1/2 , ηy 0 y y ηIj 0 y so that regarding the right-hand side as the image of the central matrix under the linear map 1/2 given by the quadratic representation P(Ip + y ) of the Jordan algebra Herm(p + j, K), we get ∗ λI η∗y−1/2 λIp η = p det (det y) det −1/2 . ηy y ηIj Furthermore, we have λI η∗y−1/2 ∗ −1/2 − ∗ −1 I 0 p = Ip η y λIp η y η 0 p −1/2 −1/2 , y ηIj 0 Ij 0 Ij y ηIj 186 H. Nakashima and T. Nomura which shows, as in the proof of Lemma 5.3, and through the expansion (5.2) of the characteristic polynomial in the Jordan algebra Herm(p, K) that ∗ λI η ∗ − det p = (det y) det(λI − η y 1η) ηy p p p k ∗ −1 p−k = λ (det y) + (det y) (−1) ak(η y η)λ . k=1 0 η∗ = − p ∗ −1 = Since det ηy ( 1) (det y)ap(η y η) is a non-zero polynomial (just putting y Ij , we have a non-zero polynomial (−1)p det(η∗η) unlike det(ηη∗)), the argument parallel to ∗ λIp η Proposition 5.1 guarantees the irreducibility of det ηy. We now consider the irreducible factorization of (5.5). Suppose ﬁrst that p ≥ r. Then, noting that d 1 1 N − n ≥ dr2 − r − r(r − 1) = dr2 − r + dr ≥ r(d − 1) ≥ 0, 2 2 2 we see by the above that (5.5) is valid also for λ = 0 and is already the irreducible 0 factorization of Det Rv . Hence Theorem 1.2 gives the assertion of (1). Next suppose that p

r−1 Np := N − n + d(j − p) + r − p j=p+1 d d = drp − r − r(r − 1) + (r − p − 1)(r − p) + r − p 2 2 1 1 = dp2 + dp − p ≥ p(d − 1) ≥ 0, 2 2 we see by Lemma 5.3 that (5.5) is rewritten as ⎡ ⎛ ⎞⎤ d 1 ∗ p d r− λI √ ξ 1 ⎢ 1 ⎜ p ⎟⎥ 0 1+Np ∗ ⎢ (p+j)⎜ 2 ⎟⎥ Det R = λ j λx − ξξ ⎣ det ⎝ ⎠⎦ v 2 1 j=1 j=p+1 √ ξx 2 ⎛ ⎞ 1 ∗ ⎜ λIp √ ξ ⎟ × (p+r) ⎜ 2 ⎟ det ⎝ 1 ⎠ . √ ξx 2 This gives the assertion (2) by Theorem 1.2. 2

COROLLARY 5.5. Suppose that p

(2) The polynomials ⎛ ⎞ 1 ∗ ⎜ λIp √ ξ ⎟ (p+j) ⎜ 2 ⎟ det ⎝ 1 ⎠ √ ξx 2 for j = 1,...,p− 1 are reducible and the formula (5.8) gives their irreducible factorizations.

COROLLARY 5.6. The cone 0 is linearly isomorphic to the cone

{X ∈ X; X is positive-deﬁnite}.

5.2. The case of Lorentzian type

Let W be an n-dimensional real vector space and B a positive-deﬁnite symmetric bilinear form on W. We put V = R ⊕ W and deﬁne a product ◦ by

(a, w) ◦ (a,w) := (aa + B(w, w), aw + aw) ((a, w), (a,w) ∈ V).

Then (V, ◦) is a Jordan algebra, and any simple Euclidean Jordan algebra of rank two arises in this way, cf. [7, Corollary IV.1.5] for example. Moreover, we know by [2, Kapitel VI, §6] that this Jordan algebra has a canonical injection into the Clifford algebra Cl(W) defined by B. The injection is given by V (a, w) → a1 + w ∈ Cl(W),where1 denotes the unit element of Cl(W).SinceCl(W) is an associative algebra, we can turn it into a Jordan algebra 1 + by defining the Jordan product by 2 (xy yx),wherexy denotes the Clifford multiplication. Then V can be regarded as a Jordan subalgebra of Cl(W). In what follows we put e0 := (1, 0) ∈ V and let e1,...,en be an orthonormal basis of = 1 + = 1 − W relative to B.Wesetc1 2 (e0 en), c2 2 (e0 en). Then the elements c1,c2 define a Jordan frame of V . The corresponding Peirce spaces of V are given by

V11 = Rc1,V22 = Rc2,V21 = Re1 ⊕ Re2 ⊕···⊕Ren−1.

We write every x ∈ V as

x = x0e0 + x1e1 +···+xnen = x0e0 + xnen + x21 with xj ∈ R (j = 0, 1,...,n)and x21 ∈ V21.Then

x = x0(c1 + c2) + xn(c1 − c2) + x21 = (x0 + xn)c1 + (x0 − xn)c2 + x21.

Therefore, we have (cf. [7,p.31]) = = + = 2 − 2 −···− 2 tr x 2x0,1(x) x0 xn,2(x) x0 x1 xn. (5.9) 2 = 2 + 2 +···+ 2 ∈ In particular, we have tr(x ) 2(x0 x1 xn) for x V . Concerning irreducible selfadjoint representations of V , we have the following proposition (cf. Clerc [3,Th´eor`eme 2] and [4, p. 122, Footnote ()] and Harvey [8, Theorem 11.3 and Remark 11.11]). 188 H. Nakashima and T. Nomura

PROPOSITION 5.7. Any irreducible selfadjoint representation ϕ on a real Euclidean vector space E of the Jordan algebra V is obtained by the restriction to V of an irreducible Clifford algebra representation of Cl(W) on E. The irreducible representations of Cl(W) on E are listed in the following table.

n := dim W Cl(W) E dim E k 8k Mat(16k, R) R16 16k k 8k + 1Mat(16k, R) ⊕ Mat(16k, R) R16 16k · k 8k + 2Mat(2 · 16k, R) R2 16 2 · 16k · k 8k + 3Mat(2 · 16k, C) C2 16 4 · 16k · k 8k + 4Mat(2 · 16k, H) H2 16 8 · 16k · k 8k + 5Mat(2 · 16k, H) ⊕ Mat(2 · 16k, H) H2 16 8 · 16k · k 8k + 6Mat(4 · 16k, H) H4 16 16 · 16k · k 8k + 7Mat(8 · 16k, C) C8 16 16 · 16k

Up to equivalence, the number of irreducible representations is one for even n, and two for odd n. Now let (ϕ, E) be a selfadjoint representation of the Jordan algebra V . We assume from now on that n = dim W ≥ 2, since n = 0givesV = R, and since n = 1 gives a non-simple Jordan algebra√ R ⊕ R.Sincee0,e1,...,en form an orthogonal basis of V with norms all equal to 2, the map Q in (3.2) associated with ϕ is expressed as n n ej ej 1 2 1 Q(ξ, ξ) = Q(ξ, ξ) √ √ = ξ e0 + ϕ(ej )ξ |ξ Eej , (5.10) 2 E 2 j=0 2 2 j=1 where ξ ∈ E. Polarization gives Q(ξ, η) for ξ, η ∈ V . The clan structure of VE := E ⊕ V 0 is given by (3.5). Let VE be the clan obtained by the adjunction of a unit element to the clan VE. By Proposition 4.1 and (5.1) the determinant of the right multiplication operator 0 ∈ R ∈ ∈ := = + Rλu+ξ+x (λ ,ξ E, x V ) is described as (with N dim E, and note dim V n 1, d = n − 1 in the current case) 0 = N−n − 1 n−1 − 1 Det Rλu+ξ+x λ 1(λx 2 Q(ξ, ξ)) 2(λx 2 Q(ξ, ξ)). (5.11)

It should be noted by (5.9) that 2(x) is equal to the Lorentz metric x, x1,n, so that (5.11) is written as 0 = N−n − 1 n−1 − 1 − 1 Det Rλu+ξ+x λ 1(λx 2 Q(ξ, ξ)) λx 2 Q(ξ, ξ), λx 2 Q(ξ, ξ) 1,n. (5.12) = + + ∈ 0 0 = THEOREM 5.8. Let v λu ξ x VE and k(v) (k 0, 1, 2) be the basic relative invariants associated with 0. Then one has ⎧ ⎪0(v) = λ, ⎪ 0 ⎨ 0 = − 1 1(v) 1(λx 2 Q(ξ, ξ)), ⎪ ⎪ λ2(x) −x, Q(ξ,ξ)1,n (ϕ is irreducible and n = 2, 3, 5, 9), ⎩⎪0(v) = 2 − 1 2(λx 2 Q(ξ, ξ)) (otherwise). Clans deﬁned by Jordan algebras 189

− 1 Proof. We first note that the polynomial 1(λx 2 Q(ξ, ξ)) is irreducible whether ϕ is regular or not. In fact, equations (5.9) and (5.10) yield that − 1 = + − 1 2 − 1 | 1(λx 2 Q(ξ, ξ)) λ(x0 xn) 4 ξ E 4 ϕ(en)ξ ξ . Since the last two terms represent a non-zero polynomial function of degree two in the sole − 1 variable ξ, the irreducibility of 1(λx 2 Q(ξ, ξ)) follows immediately. Suppose now that one of the following two conditions is satisfied: (1) ϕ is irreducible and n = 2, 3, 5, 9; (2) ϕ is reducible. By Clerc [3,Théorème 3] this is equivalent to the condition that the representation ϕ is regular in the present case. By Clerc [3,p.59],wehavedimE1 > dim V21, where recall = ⊕ = − 1 that E E1 E2 with dim E1 dim E2. Then, since the polynomial 2(λx 2 Q(ξ, ξ)) is irreducible by Proposition 5.1, and since our assumption n ≥ 2 yields

N − n>2dimV21 − n = n − 2 ≥ 0, = 0 we see that (5.11), now valid also for λ 0, is the irreducible factorization of Det Rλu+ξ+x. Therefore, 0 = 0 = − 1 = 0(v) λ, j (v) j (λx 2 Q(ξ, ξ)) (j 1, 2). Suppose next that ϕ is irreducible and n = 2, 3, 5or9.ThenwehaveN = dim E = 2n − 2 = 2dimV21 by checking the four cases separately through the table given in Proposition 5.7. Since the representation ϕ is not regular, 2(Q(ξ, ξ)) vanishes identically (cf. Clerc [3, Section 4]). Therefore, by 2(x) =x, x1,n, a simple expansion gives − 1 = − 2(λx 2 Q(ξ, ξ)) λ(λ x, x 1,n x, Q(ξ,ξ) 1,n).

Since the polynomial λx, x1,n −x, Q(ξ,ξ)1,n is visibly irreducible, we have by (5.12) 0 the following irreducible factorization of Det Rv : 0 = n−1 − n−1 − Det Rv λ 1(λx Q(ξ, ξ)) (λ x, x 1,n x, Q(ξ,ξ) 1,n). The proof is now complete. 2 We now treat separately in more detail the case where the representation ϕ of V is not regular. This implies, as already mentioned in the proof of Theorem 5.8, that ϕ is irreducible and that n = dim W = 2, 3, 5 or 9. In each of these cases V is canonically isomorphic to Herm(2, Kn−1), where we put

K1 := R, K2 = C, K4 := H, K8 := O.

To describe the isomorphism, we regard V21 as the underlying real vector space of Kn−1,and we assume that the restriction of B to V21 = Re1 ⊕···⊕Ren−1 coincides with the quadratic form that makes V21 a Hurwitz algebra. Let z1 | z2=Re(z¯2z1) denote the real inner product in the Hurwitz algebra Kd ,whered = n − 1. We put e1 = 1forn = 2; for n = 3, in addition to e1 = 1, we put e2 = i;forn = 5, in addition to e1 = 1,e2 = i, we put e3 = j, e4 = k in the standard notation for H; and ﬁnally we let e1,...,e8 be the basis of O such that e1 := 1and e2,...,e8 have the following multiplication table, which describes the result of multiplying 190 H. Nakashima and T. Nomura the element in the pth row by the element in the qth column:

e2 e3 e4 e5 e6 e7 e8 e2 −e1 e5 e8 −e3 e7 −e6 −e4 e3 −e5 −e1 e6 e2 −e4 e8 −e7 e −e −e −e e e −e e 4 8 6 1 7 3 5 2 (5.13) e5 e3 −e2 −e7 −e1 e8 e4 −e6 e6 −e7 e4 −e3 −e8 −e1 e2 e5 e7 e6 −e8 e5 −e4 −e2 −e1 e3 e8 e4 e7 −e2 e6 −e5 −e3 −e1

Thus, our e1,...,e8 are Baez’ 1,e1,...,e7 in [1,Table1]forO. In particular, we see that Re1 ⊕ Re2 ⊕ Re3 ⊕ Re5 is isomorphic to H. Let z →¯z denote the standard conjugation of Kd . Thus, e1 = e1 and em =−em if m ≥ 2. Then the isomorphism V → Herm(2, Kn−1) is given by x0 + xn x21 x0e0 + xnen + x21 → , x21 x0 − xn as is easily veriﬁed. To continue we ﬁrst record the following lemma. See the book [8, Ch. 6], for example, for a proof.

LEMMA 5.9. Let z ∈ Kd . Denote by Lz and Rz the left and the right multiplication by z in Kd respectively. Then: ∗ 2 (1) (Lz) = Lz¯ and LzLz¯ =|z| I; ∗ 2 (2) (Rz) = Rz¯ and RzRz¯ =|z| I; ∈ K ⊥ =− =− (3) if a, b d satisfy a b,thenLaLb¯ LbLa¯ , and RaRb¯ RbRa¯ .

We now consider the real linear maps ϕ1,ϕ2 : V → SymR(Kd ⊕ Kd ) given by (x0 + xn)I Lz ϕ1(x0e0 + xnen + z) = , Lz¯ (x0 − xn)I (5.14) (x0 + xn)I Lz¯ ϕ2(x0e0 + xnen + z) = , Lz (x0 − xn)I ∈ R ∈ K 2 +| |2 = + + where x0,xn and z d .Sincexn z B(xnen z, xnen z), Lemma 5.9(1) 2 yields ϕm(w) = B(w, w)I for any w ∈ W (m = 1, 2). Thus, each of ϕm extends to a homomorphism Cl(W) → EndR(Kd ⊕ Kd ).Sinceϕ1 and ϕ2 are completely reducible, just comparing the dimensions of the representation spaces with those mentioned in Proposition 5.7 convinces us that they are irreducible. (i) If n = 2, that is, if V21 = R, then obviously we have ϕ2 = ϕ1. ∼ (ii) If n = 3, then V21 = C and Proposition 5.7 says Cl(W) = Mat(2, C) = Herm(2, C)C. Since the central element iI2 ∈ Mat(2, C) is sent to an operator with square equal to −I by any representation ψ of Cl(W), we see that the representation space for ψ has a complex structure, and that ψ(u) is a complex linear operator for any u ∈ Cl(W). If, moreover, ψ is irreducible, then ψ(iI2) =±iI. This implies that u → ψ(u) is C-linear or conjugate C-linear. On the other hand, the above ϕ1 is nothing other than the restriction of the standard Clans deﬁned by Jordan algebras 191

2 representation ρ of Mat(2, C) on C ,andϕ2 the restriction of the conjugate ρ¯. Thus, ϕ1 and ϕ2 are the only mutually inequivalent irreducible representations of V by complex linear operators (cf. Harvey [8, Ch. 8] for example). Note that ϕ1 and ϕ2 are equivalent if one forgets the complex structures in the representations spaces. (iii) For the case n = 5 or 9, we have the following proposition to which we give a direct proof for the inequivalency.

PROPOSITION 5.10. For n = 5 or 9, ϕ1 and ϕ2 are the only mutually inequivalent irreducible representations of V by real linear operators. T T Proof. Suppose that there exists T = 11 12 ∈ EndR(K ⊕ K ) (d = n − 1) such that T21 T22 d d

ϕ1(v)T = Tϕ2(v) for any v ∈ V. (5.15)

The purpose is to show that T is a zero operator. Choosing v = α0e0 + αnen with αn = 0 in (5.15), we see that both T12 and T21 are zero operators. Then T is of the form diag [T11,T22].Setv = e1 in (5.15). Then we get T11 = T22. Thus T is of the form diag [T11,T11]. Hence, (5.15) for v = z ∈ V21 gives

zT¯ 11(w) = T11(zw) for any z, w ∈ Kd .

Put γ := T11(1).ThenwegetT11(z) =¯zγ (z ∈ Kd ). (1) The case n = 5. In this case we have V21 = H. Recalling e2 = i, e3 = j and e4 = k, we get kγ =−T11(k) =−T11(ij) = iT11(j) =−ijT11(1) =−kγ. Hence, γ = 0, so that T is a zero operator. Note that we have used the associative law in the last equality. (2) The case n = 9. We have for any z, z ∈ O (z¯ z)γ¯ = (zz )γ = T11(zz ) =¯zT11(z ) =¯z(z γ). (5.16)

Then, take the inner product in O of (5.16) with e1 = 1. Then Lemma 5.9(1) gives γ | zz=zz | γ .

Now for any ek (k = 2,...,8), we can always ﬁnd ei,ej such that eiej = ek (see the multiplication table (5.13)). Then we get

γ | ek=γ | eiej =ej ei | γ =−ek | γ .

This implies γ | ek=0fork = 2,...,8, so that γ ∈ Re1. But now (5.16) is valid only for γ = 0. Hence, T11 is a zero operator, so that T is a zero operator. 2

In what follows it is convenient to identify V with Herm(2, Kd ) (d = 1, 2, 4, 8). Thus, setting αz v(α, β, z) := (α, β ∈ R,z∈ K ), zβ¯ d we have αI Lz t ϕ1(v(α, β, z)) = ,ϕ2(v) = ϕ1( v) (v ∈ Herm(2, Kd )). (5.17) Lz¯ βI 192 H. Nakashima and T. Nomura

:= K ⊕ K → K Let Q1,Q2 be the quadratic maps E d d Herm(2, d ) associated with ϕ1,ϕ2, respectively. Then for ξ = ξ1 ∈ E we have ξ2

2 2 ϕ1(v(α, β, z))ξ | ξE = α|ξ1| +zξ2 | ξ1+¯zξ1 | ξ2+β|ξ2| 2 ¯ ¯ 2 = α|ξ1| +z | ξ1ξ2+¯z | ξ2ξ1+β|ξ2| by Lemma 5.9(2). Since z1 | z2=Re(z¯2z1) by deﬁnition, we arrive at | |2 ¯ | = ξ1 ξ1ξ2 αz ϕ1(v(α, β, z))ξ ξ E Re Tr ¯ 2 ξ2ξ1 |ξ2| zβ¯ ∗ = Re Tr((ξξ )v(α, β, z)).

∗ t Hence, Q1(ξ, ξ) = ξξ .Sinceϕ2(v) = ϕ1( v) and since | |2 ¯ ¯ | |2 ¯ ξ1 ξ1ξ2 α z = ξ1 ξ2ξ1 αz Tr ¯ 2 Tr ¯ 2 , ξ2ξ1 |ξ2| zβ ξ1ξ2 |ξ2| zβ¯ we get t ∗ t Q2(ξ, ξ) = (ξξ ) = (Q1(ξ, ξ)). (5.18)

Before going further, we note here that it is not true in general that t(A2) = ( tA)2 for × amatrix A with entry not coming from a commutative ring; just consider the 2 2matrix = ik H A 0 j with the standard notation for (cf. [17]). Thus, we need a direct proof of the following proposition.

PROPOSITION 5.11. The map τ : v → tv deﬁnes a Jordan and a clan automorphism of Herm(2, Kd ). Proof. Since v(α, β, z)2 = v(α2 +|z|2,β+|z|2,(α+ β)z), it is easy to see that τ(v2) = τ(v)2. Polarization shows that τ is a Jordan automorphism. We next show that τ is a clan isomorphism. First of all we have for v = v(α, β, z) and v = v(α,β,z) 1 1 = 2 α 0 α z + α z 2 αz v v ¯ 1 ¯ ¯ 1 z 2 β z β z β 0 2 β + 1 + = αα α z 2 (α β)z ¯ + 1 + ¯ ¯ + . (5.19) α z 2 (α β)z 2Re(zz ) ββ

This implies ¯ + 1 + ¯ = αα α z 2 (α β)z τ(v v ) + 1 + ¯ + . α z 2 (α β)z 2Re(zz ) ββ On the other hand, replacing z with z¯ and z with z¯ in (5.19), we obtain ¯ + 1 + ¯ = αα α z 2 (α β)z τ(v) τ(v ) + 1 + ¯ + . α z 2 (α β)z 2Re(zz ) ββ Since Re(zz¯ ) = Re(zz¯),wegetτ(v v) = τ(v) τ(v). 2 Clans deﬁned by Jordan algebras 193

Using ϕ1 and ϕ2, whether they are equivalent or not, we can introduce two clan structures 0 = R ⊕ ⊕ in VE u E V by (4.1). The corresponding clan multiplications are denoted by 1 and 2, respectively. 0 0 PROPOSITION 5.12. The clans (VE, 1) and (VE, 2) are always isomorphic, whether the representations ϕ1 and ϕ2 are equivalent or not. : 0 → 0 Proof. Consider the linear map VE VE deﬁned by (λu + ξ + x) := λu + ξ + tx(λ∈ R,ξ∈ E, x ∈ V).

0 → Clearly is a linear isomorphism. Let us show that is a clan homomorphism (VE, 1) 0 ∈ R ∈ ∈ (VE, 2). Suppose that λ, μ , ξ, η E and x, y V . Then by (4.1) we have

(λu + ξ + x)2(μu+ η + y) = + + 1 + t + + t t (λμ)u (μξ 2 λη ϕ2( x)η) (Q2(ξ, η) x y).

t Since ϕ2( x) = ϕ1(x), and since (5.18) and Proposition 5.11 imply

t t t t Q2(ξ, η) + x y = Q2(ξ, η) + (x y) = (Q1(ξ, η) + x y), we arrive at

(λu + ξ + x) 2 (μu + η + y) = ((λu + ξ + x) 1 (μu + η + y)).

The proof is now complete. 2

THEOREM 5.13. Let V = Herm(2, Kd )(d= 1, 2, 4, 8). In either case of ϕ = ϕ1,ϕ2,the 0 K clan VE is isomorphic to Herm(3, d ). Proof. By Proposition 5.12, it is enough to treat the case of ϕ .Foreveryλ ∈ R, ξ = ξ1 ∈ E 1 ξ2 and x ∈ V = Herm(2, Kd ) we set λξ∗ X(λ, ξ, x) := ∈ Herm(3, K ). (5.20) ξx d

The computation same as in (5.7) yields √ √ = 1 + + + X(λ, ξ, x) X(μ, η, y) X(λμ, 2 λη μξ xη, Q1( 2ξ, 2η) x y).

By (3.3), we have 1 = 2 αI 0 = ϕ1(x) 1 for x v(α, β, z), Lz¯ 2 βI so that xη = ϕ1(x)η. Hence, comparison with (4.1) shows us that the mapping 0 + + → √1 ∈ K VE λu ξ x X λ, ξ, x Herm(3, d ) 2 is a surjective clan isomorphism. 2 194 H. Nakashima and T. Nomura

0 0 By Theorem 5.13, the homogeneous convex cone corresponding to VE is the 0 symmetric cone consisting of the positive-deﬁnite ones in VE. We close this section by noting the following fact. Identiﬁcation of V with Herm(2, Kd ) says that the basic relative invariants 1(v) and 2(v) of V are nothing other than the principal minors in Herm(2, Kd ) from the upper left corner. Thus, we have

2 1(v(α, β, z)) = α, 2(v(α, β, z)) = αβ −|z| .

Then, we see that 1 1 1 λ, λx − ξξ∗ , λx − ξξ∗ 1 2 λ 2 2 are the principal minors of the Hermitian matrix X(λ, √1 ξ, x). 2

0 6. Dual clan of VE

0 = R ⊕ 0 Let VE u VE betheclandeﬁnedinSection4.As before general elements of VE will be written as v = λu + ξ + x with λ ∈ R,ξ∈ E, x ∈ V without comments. With s as in 0 0 Theorem 3.2 we deﬁne a linear form s on VE by r s0(λu + ξ + x) = λ + · s(ξ + x). n By (3.8) and (4.1) we have

0 s ((λu + ξ + x) (μu + η + y)) = λμ +ξ |η E +x |y .

This means that s0 is indeed an admissible linear form on V 0. We use the symbol ·|·0 for E ∗ the inner product induced by s0. Then we deﬁne the dual cone (0) of 0 by ∗ 0 := { ∈ 0; | 0 ∈ 0 \{ }} ( ) v VE v v > 0forallv ( ) 0 . ∗ By (4.2) the unit element e sits in 0, and it is clear from deﬁnition that e ∈ (0) .For g ∈ G(0), we denote by tg the transpose of a linear operator g relative to the inner product ·|·0.Thenweset tG(0) := { tg ; g ∈ G(0)}. ∗

General theory [15] tells us that (0) is the tG(0)-orbit through e. In particular, the clan ∗ 0 t 0 product associated with ( ) with unit element e is given by the transpose Lv of the left 0 ∈ ∈ 0 multiplication operators Lv (v V).Infact,ifv, v VE,thenwehave t 0 | 0 = | 0 = 0 = | 0

Lve v e v v s (v v ) v v . t 0 = t 0 This implies Lve v. In what follows we write this clan product Lvv as v v . Now by using (4.1), a simple computation yields ⎛ ⎞ λ ξ |·E 0 t 0 = ⎝ 1 + ∗ · ⎠ Lv 0 2 λ idE ϕ(x) ϕ( )ξ , (6.1) t 00 Lx Clans deﬁned by Jordan algebras 195

so that we have

(λu + ξ + x) (μu + η + y) = + | + ∗ + 1 + +

(λμ ξ η E)u (ϕ(x) η 2 λη ϕ(y)ξ) x y, (6.2)

where x y is the dual clan product of V . Denoting by R the right multiplication operators for the clan product , we obtain ⎛ ⎞ λ ·|ξ E 0 = ⎝ 1 · ∗ ⎠ Rλu+ξ+x 2 ξϕ(x)ϕ( ) ξ , V 00Rx

V where Rx is the right multiplication operator in the dual clan product of V .Let ∗ ∗

1(x), . . . , r (x) be the Jordan algebra principal minors defined by the Jordan frame cr ,...,c1 of V (cf. [7, Section VII]). Then (the -version of) Theorem 2.9 tells us that ∗ ∗ ∗ V = d ··· d Det Rx 1(x) r−1(x) r (x). From this we arrive at the following formula: ·| = ∗ d ··· ∗ d ∗ λ ξ E Det Rλu+ξ+x 1(x) r−1(x) r (x) Det 1 . 2 ξϕ(x) Using the formula AB − Det = (Det D) Det(A − BD 1C) (Det D = 0), (6.3) CD which is similar to (2.8), we obtain = ∗ d ··· ∗ d ∗ − 1 co | Det Rλu+ξ+x 1(x) r−1(x) r (x)(λ Det ϕ(x) 2 ϕ(x)ξ ξ E), (6.4) where coT is the cofactor operator of an operator T , that is, for invertible T we have coT := (Det T)T−1. Thus, if T is a positive-definite selfadjoint operator, then so is coT . ∗ = + + ∈ 0 ∈ 0 PROPOSITION 6.1. Let v λu ξ x VE.Thenv ( ) if and only if ∈ 1 −1 | x and λ> 2 ϕ(x) ξ ξ E. (6.5) Here note that ϕ(x) is positive-definite for x ∈ , and that λ is necessarily positive under (6.5). ∗ Proof. Suppose that λu + ξ + x ∈ (0) . Any non-zero y ∈ is an element of 0,sothat

x |y =λu + ξ + x |y 0 > 0.

Hence, x ∈ ,andϕ(x) is positive-definite. Moreover, Theorems 1.1 and 1.2 tell us that Det Rλu+ξ+x > 0. Then, (6.4) gives (6.5). + + ∈ 0 To see the reverse implication, we assume that λu ξ x VE satisfies (6.5). Let us 0 ∈ 0 ∈ 0 recall the operators nζ (ζ E) and hz (z V)defined by (4.3). Since they are exp Lζ and 196 H. Nakashima and T. Nomura

0 exp Lz respectively, the formula (6.1) gives ⎛ ⎞ ⎛ ⎞ |· 1 · | 1 ζ E 2 ϕ( )ζ ζ E 10 0 t 0 = ⎝ · ⎠ t 0 = ⎝ E ∗ ⎠ nζ 0idE ϕ( )ζ , hz 0 (exp Lz ) 0 . t 00 idV 00expLz t Now ﬁrst pick a unique z ∈ V such that (exp Lz)e0 = x. With this z we put := − 1 −1 | := −μ/2 E ∗ μ log(λ 2 ϕ(x) ξ ξ E), ζ e (exp L−z) ξ. Then we have t 0 t 0 t 0 + = t 0 t 0 + 1 2 + + (exp μ Lu) hz nζ (u e0) (exp μ Lu) hz((1 2 ζ E)u ζ e0) = t 0 + 1 2 + E ∗ + (exp μ Lu)((1 2 ζ E)u (exp Lz ) ζ x) = μ + 1 2 + μ/2 −μ/2 + e (1 2 ζ E)u e (e ξ) x. Here we observe that μ 2 = E ∗ 2 = − − ∗ | e ζ E (exp L−z) ξ E (exp ϕ( z))(exp ϕ( z)) )ξ ξ E −1 =ϕ((exp Lz) e0)ξ | ξE , where the last equality follows from Proposition 3.3. By [7, Theorem III.5.3], we have −1 t −1 −1 (exp Lz) e0 = ( (exp Lz)e0) = x , μ 2 = −1 | μ + 1 2 = so that e ζ E ϕ(x) ξ ξ E. Hence, e (1 2 ζ E) λ,whichshows + + = t 0 t 0 t 0 λu ξ x (exp μ Lu) hz nζ (e). ∗ t 0 t 0 t 0 t 0 + + ∈ 0 2 Since (exp μ Lu), hz and nζ are in G( ), we conclude that λu ξ x ( ) .

6.1. The case of Hermitian type We make use of the notation in Section 5.1. Let us ﬁrst note that the dual clan product of

V = Herm(r, K) is given by x y := (x)∗y + yx (x, y ∈ V). rp Let Ip denote the unit matrix of order p. We consider the space K of column vectors of size rp. We do not consider Krp as the space of r × p matrices here. Then we define a real vector space Y consisting of matrices Y(μ,η,y)of the form μη∗ Y(μ,η,y):= (μ ∈ R,η∈ Krp,y∈ Herm(r, K)). (6.6) ηy⊗ Ip Before defining a duality pairing between X in (5.6) and Y above, we first define a duality pairing between E := Mat(r × p, K) and Krp. To do so, we write any ξ ∈ Mat(r × p, K), and any η ∈ Krp as ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ξ1 η1 ηj1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = . := ; = . = . ξ ⎝ . ⎠,ξj (ξj1,...,ξjp) η ⎝ . ⎠,ηj ⎝ . ⎠. ξr ηr ηjp Clans defined by Jordan algebras 197

Thus, ξj are row vectors and ηj are column vectors. With this notation we now deﬁne a real bilinear form ξ, η by r r ¯ ξ, η:=Re ξj η¯j = Re ξj ηj . j=1 j=1 Clearly ξ, η is a real bilinear duality pairing for Mat(r × p, K) and Krp. Furthermore, for rp any ξ ∈ E = Mat(r × p, K),letι(ξ) be the element of K such that ι(ξ)|η Krp =ξ, η rp rp holds for any η ∈ K ,where·|·Krp is the canonical real inner product of K .Thenwe have an obvious equality ξ |ξ E =ι(ξ)|ι(ξ )Krp for any ξ, ξ ∈ E. For any real linear operator T on Mat(r × p, K), we denote by T the dual linear operator of T on Krp deﬁned by the usual fashion Tξ, η=ξ, T η. Suppose that x ∈ Mat(r × r, K).Then r r p r xξ, η=Re (xξ)i ηi = Re (xij ξjk)ηik i=1 i=1 k=1 j=1 r p ¯ t ∗ = Re ξjk(( x¯ ⊗ Ip)η)jk =ξ, (x ⊗ Ip)η. (6.7) j=1 k=1 In particular, for ϕ(x) (x ∈ Herm(r, K)) in (5.3), we have ∗ ϕ(x) η = (x ⊗ Ip)η, (ϕ(x)) η = ((x) ⊗ Ip)η. Then proceeding in a way parallel to Proposition 5.2 we obtain the following.

PROPOSITION 6.2. The space Y is a subclan of Herm(rp + 1, K) with the dual clan product described above. Moreover, the clan (V 0, ) is isomorphic to Y by the map E 1 (λu + ξ + x) := Y λ, √ ι(ξ), x . 2 Now we consider Y as the dual vector space of X through the duality pairing defined by X(λ,ξ,x),Y(μ,η,y)=λμ + 2ξ, η+Re Tr(xy). (6.8) Let be the cone of positive-definite ones in Y: := {Y ∈ Y; Y is positive-definite}. Let 0 be realized as in Corollary 5.6. THEOREM 6.3. The cone is the dual cone of 0 with respect to the duality pairing (6.8). ∗ ∗ ∈ Proof. By considering the principal minors 1(y), . . . , r (y) of y V starting from the lower right corner, it is clear that Y = Y(μ,η,y)is positive-definite if and only if ∗ ∗ 1(y) > 0,...,r (y) > 0, det Y>0. Supposing y invertible, we consider the following decomposition of the matrix Y : ∗ ∗ −1 ⊗ − ∗ −1 ⊗ μη = 1 η (y Ip) μ η (y Ip)η 0 ηy⊗ Ip 0 Ip 0 y ⊗ Ip × 10 −1 . (y ⊗ Ip)η Ip 198 H. Nakashima and T. Nomura

∗ −1 We regard the right-hand side as the image of diag[μ − η (y ⊗ Ip)η, y ⊗ Ip] by a unipotent transformation on the Jordan algebra Herm(rp + 1, K) givenbythematrix ∗ −1⊗ 1 η (y Ip) .Thenwehave 0 Ip ∗ −1 det Y = (det y ⊗ Ip)(μ − η (y ⊗ Ip)η) p −1 = (det y) (μ −η |(y ⊗ Ip)η Krp ). Now putting ξ = ι−1(η) ∈ Mat(r × p, K),wehaveby(6.7) −1 −1 η |(y ⊗ Ip)η Krp =ξ, (y ⊗ Ip)η −1 −1 =ϕ(y) ξ, η=ϕ(y) ξ |ξ E . In this way can be described as −1 −1 −1 ={Y = Y(μ,η,y); y ∈ and μ>ϕ(y) (ι (η))|ι (η)E}. ∗ The right-hand side is equal to ((0) ) in view of Proposition 6.1. 2 COROLLARY 6.4. The basic relative invariants associated with the cone are given by the following polynomials Pj (Y ) (j = 1,...,r,r+ 1; Y = Y(μ,η,y)): ∗(y) (j = 1,...,r), P (Y ) := j j ∗ co μ det y − η ( y ⊗ Ip)η (j = r + 1).

Remark 6.5. (1) If K = R or C, the polynomial Pr+1(Y ) can be interpreted in the following way. Consider a formal matrix ⎛ ⎞ ∗ ··· ∗ μη ηr ⎜ 1 ⎟ ⎜η1 ⎟ ⎜ . ⎟ , ⎝ .y. ⎠ ηr where we do not mind whether the entries are scalars or vectors. Form a polynomial according to the deﬁnition formula of the ordinary matrix determinant, and when one meets a product of the vectors ηi and ηj , the product is taken to be the canonical inner product ηi |ηj Kp of p K .ThenwegetPr+1(Y ). (2) Observe that the degree of the polynomial Pj (Y ) in Corollary 6.4 is j.Ifp>1, then the cone is not symmetric, and generalizes the rank-three cone introduced in [11, Section 3] to higher ranks.

6.2. The case of Lorentzian type Here we continue to use the notation of Section 5.2. Let us consider the polynomial − 1 co | λ Det ϕ(x) 2 ϕ(x)ξ ξ E that appeared in (6.4). By [7, Proposition IV.4.2], we have Det ϕ(x) = (det x)N/2, where N is the dimension of the representation space of ϕ. Note that in the rank-two case, N is always even. Next we note that if x ∈ V is invertible, then we have by [7, p. 31] −1 −1 co co x = (det x) · x with x := −x + (tr x)e0. (6.9) Clans deﬁned by Jordan algebras 199

The deﬁnition of cox in (6.9) is just a Jordan algebra variant of the cofactor matrix (cf. (2.9)). If we write x = λe0 + w with λ ∈ R and w ∈ W,thenwehave

co x = λe0 − w.

Therefore, the map x → cox extends to an automorphism of the Clifford algebra Cl(W).This automorphism is called the canonical automorphism by Harvey [8, Deﬁnition 9.21], and we denote it by x → x instead of cox. Thus, we have x−1 = (det x)−1 · x,sothatweget − − coϕ(x) = (Det ϕ(x))ϕ(x) 1 = (det x)N/2 1ϕ(x).

This yields that − 1 co | = N/2−1 − 1 | λ Det ϕ(x) 2 ϕ(x)ξ ξ E (det x) (λ det x 2 ϕ(x)ξ ξ E ). Since det x is a degree-two irreducible polynomial, we see that the polynomial − 1 | λ det x 2 ϕ(x)ξ ξ E is irreducible. We thus arrive at the following theorem. ∗ ∗ THEOREM 6.6. Let (x), (x) be the Jordan algebra principal minors of V dual to 1 2 ∗ 0 1(x) and 2(x). Then the basic relative invariants associated with ( ) are given by the following polynomials Pj (v) (j = 1, 2, 3): ∗(x) (j = 1, 2), P (λu + ξ + x) = j j − 1 | = λ det x 2 ϕ(x)ξ ξ E (j 3). Remark 6.7. If n = 2, 3, 5or9,andifϕ is irreducible, we know by Theorem 5.13 that 0 =∼ K = co VE Herm(3, n−1). A direct computation using the fact that x x is equal to the ordinary cofactor matrix under the identiﬁcation of V with Herm(2, Kn−1) shows that 1 1 λ det x − ϕ(x)ξ |ξ E = det X λ, √ ξ, x 2 2 with the notation of (5.20). Finally we take a closer look at the low-dimensional cases where n := dim W = 2, 3,...,9 with irreducible representations ϕ. The cases n = 2, 3, 5, 9 have been already done in the previous section and commented on in Remark 6.7, so that the cases n = 4, 6, 7, 8 remain to be treated. (i) The case n = 4. In this case the Euclidean Jordan algebra V can be realized as a Jordan subalgebra of Herm(2, H) given by

{v = v(α, β, z); α, β ∈ R,z∈ H ∩ (Rk)⊥}.

The unique irreducible selfadjoint representation ϕ := ϕ1 : V → SymR(H ⊕ H) is given by the same formula as (5.17). Note that in this case ϕ2 := ϕ1 ◦ τ is equivalent to ϕ1. In fact, let T1 be a real linear operator on H such that

T11 = 1,T1i =−i, T1j =−j, T1k = k, 200 H. Nakashima and T. Nomura and set T = diag [T1,T1]∈GLR(H ⊕ H).ThenT is an intertwining operator (see the proof of Proposition 5.10; the lack of k in V21 makes T1 survive). Let π be the orthogonal projection operator Herm(2, H) → V . Then the quadratic map Q associated with the representation ϕ is given by Q(ξ, ξ) = π(ξξ∗). (ii) The case n = 8. For the remaining cases, we borrow a realization of V and its irreducible selfadjoint representation from Clerc [4] (but we use left multiplication operators). For n = 8, we realize V as V = Re0 ⊕ W by setting W = O. Any element in V will be written as v(α, w) = αe0 + w(α∈ R,w∈ W). The unique irreducible selfadjoint representation ϕ is given by αI Lw ϕ(v(α, w)) := ∈ SymR(O ⊕ O). (6.10) Lw αI The representation ψ(v(α, w)) := ϕ(v(α, w)) is equivalent to ϕ. Indeed if we set 0 I T = ∈ GLR(O ⊕ O), (6.11) I 0 then T is an intertwining operator. The quadratic map Q associated with ϕ is given by 2 2 ¯ ξ1 Q(ξ, ξ) = v(|ξ1| +|ξ2| ,ξ1ξ2), ξ = ∈ O ⊕ O. ξ2 ⊥ (iii) The case n = 7. In this case we take W = Im O := O ∩ (Re1) , and form V = Re0 ⊕ W. Continuing to write v(α, w) for elements of V , we have the irreducible representation ϕ given by the same formula as (6.10). However, note in the present case that w =− wfor w ∈ W. Moreover the representation space has a complex structure J given = 0 I ∈ by J −I 0 , and every operator ϕ(v) (v V) commutes with J . In fact, the origin of the operator J is as follows. Let e1 = 1ande2,...,e8 ∈ Im O be as in the multiplication := ··· table (5.13). Then Lemma 5.9(3) shows that S Le2 Le8 commutes with Lej for all j = 1, 2,...,8. Hence, we get SLz = LzS for any z ∈ O. Applying this operator equality to e1,wegetSz = zγ with γ := Se1. However, a simple computation using (5.13) shows that Se1 =−e1. Hence, we arrive at S =−I,and 0 −S J = = ϕ(v(0,e )) ···ϕ(v(0,e )). S 0 2 8 In this case the representation ψ defined as in (ii) is not equivalent to ϕ as a representation by complex linear operators (relative to J ). This is a reflection of the fact that the operator T in (6.11) does not commute with J . A direct proof like Proposition 5.13 for this inequivalency is also possible, by using the fact that for any k = 2,...,8, there is a non-associative triple ei ,ej ,ek ∈ Im O such that (ei ej )ek =−ei(ej ek) (see the table (5.13); one such triple is e2,e3,e4, and to look for other triples, the index cycling identity is useful (cf. Baez [1, p. 151]). The details are left to the reader. However, Proposition 5.12 applies also to the present case, so that we only have to do with ϕ, and the quadratic map Q associated with the representation ϕ is given by 2 2 ¯ ξ1 Q(ξ, ξ) = v(|ξ1| +|ξ2| ,πIm(ξ1ξ2)), ξ = ∈ O ⊕ O, ξ2 where πIm denotes the orthogonal projection operator O → Im O. Clans defined by Jordan algebras 201

⊥ (iv) The case n = 6. In this case we take W = O ∩ (Re1 ⊕ Re2) , and form V = Re0 ⊕ W. As before we write every element of V as v(α, w) with α ∈ R and w ∈ W.The unique irreducible selfadjoint representation ϕ is given by the same formula as (6.10). The similarly deﬁned representation ψ is equivalent to ϕ by the operator T in (6.11). The quadratic map Q associated with ϕ is given by 2 2 ¯ ξ1 Q(ξ, ξ) = v(|ξ1| +|ξ2| ,π1(ξ1ξ2)), ξ = ∈ O ⊕ O, ξ2

⊥ where π1 is the orthogonal projection O → O ∩ (Re1 ⊕ Re2) .

Note added in proof

Following the referee’s ﬁnal comments, we would like to add brieﬂy that the discussion at the beginning of Section 6.2 holds in the general case, too. In fact, with the notation used at the beginning of Section 6, we have, thanks to [7, Proposition IV.4.2], − 1 co | = N/r−1 − 1 co | λ Det ϕ(x) 2 ϕ(x)ξ ξ E (det x) (λ det x 2 ϕ( x)ξ ξ E). ∗ Then (6.4) tells us that the basic relative invariants associated with (0) are given by (as in Theorem 6.6) ∗ ∗ − 1 co | 1(x), . . . , r (x), λ det x 2 ϕ( x)ξ ξ E. The presentation of Section 6.1 is considered as a situation dual to Section 5.1, where we had 0 → + K to use the injective isomorphism VE Herm(r p, ) given by Proposition 5.2.

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Hideto Nakashima Graduate School of Mathematics Kyushu University 744 Motooka Nishi-ku Fukuoka 819-0735 Japan (E-mail: [email protected])

Takaaki Nomura Faculty of Mathematics Kyushu University 744 Motooka Nishi-ku Fukuoka 819-0735 Japan (E-mail: [email protected])