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26
MSC 32M15, 22E30, 43A85
Right multiplication operators in the clan structure of a Euclidean Jordan algebra
© T. Nomura
Kyushu University, Fukuoka, Japan
This is a summary of the talk given in the workshop held at Tambov University in April, 2009.
Keywords: Euclidean Jordan algebras, homogeneous cones, symmetric cones, clan
§ 1. Preliminaries about Euclidean Jordan algebras
Vinberg’s theory [3] tells us that associated to a homogeneous open convex cone containing no entire line, we have a clan structure in the ambient vector space. In this note we deal with the symmetric cone in a Euclidean Jordan algebra, and describe the associated clan structure.
Let V be a simple Euclidean Jordan algebra of rank r with unit element e. For x E V, we denote bv M(x) the multiplication operator1 bv x, so that M(x)y = xy for any y E V. Let tr denote the trace function on the Jordan algebra V, and define
an inner product in V by (x|y) := tr(xy). Let us fa a Jordan frame c1,..., cr, We
have c1 + ■ ■ ■ + cr = e. The Jordan frame c1,..., cr yields the Peirce decomposition
V = ©Kk Vjk-, where Vjj = Rcj (j = 1,..., r), and
Vjk := | x E V ; M (ci) x =\^(^ij + $ik )x (i =1,...,r)J (1 ^ j<k ^ r).
Let Q := Int{x2 ; x E V}, the interior of squares in V, be the symmetric cone in V. The linear automorphism group of the cone Q is denoted by G(Q):
G(Q) := {g E GL(V) ; g(Q) = Q}.
We know that G(Q) is reductive. Let g ^e the Lie algebra of G(Q), and t the derivation algebra Der(V) of Jordan algebra ^^ut p := {M(x) ; x E V}, Then g = t + p is a Cartan decomposition of g with the corresponding Cartan involution 0X = — tX. Let
a := RM(ci) © ... © RM(cr).
1The notation in the book [1] is L(x). Since we use left multiplication operators in clans, we have chosen a different symbol to avoid any confusion.
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Then a is an abelian subalgebra which is maximal in p. Let a1,..., ar be the basis of a* dual to M(c1),..., M(cr). We know that the positive a-roots are (ak — aj)/2, k > j, and the corresponding root spaces 0(«fc-aj)/2 =: nkj are described as
nkj := {z □ cj ; z E Vjk},
where a □ b := M(ab) + [M(a), M(b)]. Summing up all of the nkj as n := ^j<k nkj, we have an Iwasawa decomposition g = t + a + n. Let A := exp №d N := exp n, the G(Q) a n
group H := N x A acts on Q simply transitively, so that the orbit map H 3 h M he E Q is a diffeomorphism. Then its derivative at the unit element of H gives rise to a linear isomorphism h := Lie (H) 3 X M Xe E V, Its inverse map is denoted as
V 3 v M Xv E h- We have by definition Xve = v.
§ 2. Clan structure of a Euclidean Jordan algebra
We keep to the notation in § 1, Let us introduce a bilinear product A in V by
V1AV2 := XW1V2 (v1,V2 E V).
AV
(Cl) [XV1 , X^V2] -^Vl Av2-V2AV1 for all V1 , V2 E V;
(C2) there is s E V* such that (v1 Av2, s) defines an inner product in V;
(C3) the operators Xv (v E V) have only real eigenvalues.
We note that for (C2), it suffices to take s = tr(-) in this case. For Xv we have the
following lemma.
Lemma 2.1 (1) If v = a1c1 + ... + arcr (a1 E R,..., ar E R), one has Xv = M(v). (2) If v E Vj-k, then Xv = 2(v □ cj).
In what follows, we write the right multiplication operator by v E V:
v' := v'Av (v' E V).
c1 , . . . , cr
Peirce spaces Vjk, j ^ k, are the spaces for the normal decomposition relative to them:
j = {x E V ; Xc,x = i(iij + fa)x, x = &jx d = 1,..., r)}.
Thus the general clan multiplication rule is applied to the Peirce spaces, and we have
VklAVfk c j, if k = i, j, then VklAVj = 0, (2,1)
VklAVkm c or VJm, according tom ^ / or I ^ m.
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We put
e := V1r © ... © Vr-1,r, W := e © Rcr.
Then (2,1) immediately implies
Proposition 2.2 W is a two-sided ideal in the clan V. In other words, one has for vEV
Xv (W) c W, Rv (W) c W.
vEV
nW R '
: RV
\W'
Let us set
V' := © Vj.
1<i<i<r-1
Then V = V' © W, and V' is a Euclidean Jordan algebra of rank r — 1, and thus
v' E V' V'
denoted by RV'■
Corollary 2.3 By writing v E V as v = v' + w with v' E V' and w E W, the operator Rv is of the form
R =( RV' o Rv = ^ * RW
We next analyze the operator R^ First, (2,1) implies that if v' E V', we have Rv/(E) c E, We put R;, := Rv/ '„, To see what R;, looks like, we define operators 0(v') (v' E V') on E by
0(v')£ := 2v'£ (£ E E).
Since V' (resp, E) is the Peirce 0 (resp, the Peirce 1/2) space for the idempotent cr, we know that the map 0 : v' M 0(v') E End(E) is a unital Jordan algebra V'
Proposition 2.4 R; = 0(v') for any v' E V'.
Proposition 2.5 By writing v E V as v = v' + £ + vrcr with v' E V', £ E E, vr E R, RvW
W
Rv
2
V (‘| £) cr Vr /Vrr /
We now renormalize the inner product (■ | ■) in W = E © Rcr bv
(n + yrcr1 n' + yrcr)w := (n 1 n') + 2 yr yr (n n' E E and yr,yr E R).
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Then the operator RW expressed in Proposition 2,5 is written in a more symmetric wav:
0(vO H cr)w £
R
W
(v = v' + £ + Vr Cr).
(2.2)
; | £)w Cr Vr /yrr
In summary we obtain the following inductive structure for the right multiplication operators Rv:
R
2
\ (• 1 £) Cr Vr /Vrr /
Theorem 2.6 Decomposing v G V as v = v' + £ + vrcr with v' G V', £ G 5 and vr G R, one has
Rv
( RV
V
0
0(v')
1 £)W Cr
0
■ 1 Cr)W £ vr /
To get a “standard form” of the operator matrix RW in (2,2), we first take k G Aut(V') so that we have
v = k(AiCi + ... + Ar-icr-i) (Ai,..., Ar—i G R).
We have Aut(V') = expDer(V'), and we know that elements in Der(V') are all inner. Thus we write k = exp T', where T' is a sum of operators of the form [M(a'),M(b')] with a',b' G V', In this way, we see that Der(V') C Der(V), so that we have Aut(V') C Aut(V), Hence we regard k as an element in Aut(V) such that kcr = cr and k5 C 5, For n G 5, we have
0(v')n = 2v'n
= 2k {(AiCi + ... + Ar-icr-i)(k in)}
= k (AiPi + ... + Ar-iPr-i) k in,
where Pj denotes the orthogonal projection 5 ^ Vjr (j = 1,..., r — 1), Hence we obtain with £' = k-i£ G 5,
AiPi + ... + Ar_ iPr-1
R
W
(- 1 £')W Cr
1 Cr)W £'
vr Iyr,
i
(2.3)
Finally we compute det Rv (v G V) as an application of Theorem 2,6 and the expression (2,3), To do so, we recall the following obvious formula: if det A = 0, then
A B C D
A
0
C D-CA-iB
I A-iB
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*
*
k
0
SO that
det ^ A D) = det A ■ det (D — CA-iB) .
Ai ■ ■ ■ Ar-i = 0
£' G 5 as £' = £i + ... + £r-i with £j G Vjr, we have by a simple computation det RW = (Ai ■■■ Ar-i)d-i x
X |Ai ■ ■ ■ Ar-ivr — 2 (A2 ■ ■ ■ Ar-i||£l ||2 + ... + Ai ■ ■ ■ Ar-2 ll^-ill2) | , (2-4)
where d is the common dimension of Vjk j < k. Since both sides are polynomials in Ai,..., Ar-i, the equality holds without the restriction Ai ■ ■ ■ Ar-i = 0, Now we have the following lemma gotten bv applying [1, Proposition VI.3,2] to c = ci.
Lemma 2.7 One has
Ar (AiCi + ... + Ar-iCr-i + £' + vr Cr)
= Ai ■ ■ ■ Ar-ivr — 2 (A2 ■ ■ ■ Ar-i||£! ||2 + ... + Ai ■ ■ ■ Ar-2 ||£r— i 12) .
This lemma together with (2,4) shows that det RW = Ar-i(v)d-iAr(v). Now summing up all the above discussions and using Theorem 2,6, we obtain by induction the following proposition:
Proposition 2.8 For v G V, one has det Rv = A ^v)d ■ ■ ■ Ar- i(v)dAr(v).
Remark 2.9 We have det Rhv = x(h) det Rv, h G H, v G V, where x(h) := (detV h)(detAdh)- i. For this we refer the reader to the proof of [2, Lemma 2,7], The one-dimensional representation x of H comes from the linear form on a given by Y^r=i[1 + d(r — j)]aj. From this we also obtain Proposition 2,8,
References
1, J, Faraut and A, Koranvi, Analysis on symmetric cones, Clarendon Press, Oxford, 1994,
2, T, Nomura, On Penney’s Cayley transform of a homogeneous Siegel domain, J. Lie Theory, 2001, vol. 11, 185-206.
3, E. B. Vinberg, The theory of convex homogeneous cones, Trans, Moscow Math, Soe., 1963, vol. 12, 340-403.
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