ESI The Erwin Schr�odinger International Boltzmanngasse 9Institute for Mathematical Physics A-1090 Wien, AustriaRepresentations in L2{Spaceson In�nite{Dimensional Symmetric ConesKarl-Hermann NeebBent �rsted

Vienna, Preprint ESI 750 (1999) September 10, 1999Supported by Federal Ministry of Science and Transport, AustriaAvailable via http://www.esi.ac.at

Representations in L2 -spaces on in�nite-dimensional symmetric cones 1Representations in L2 -spaceson in�nite-dimensional symmetric conesKarl-Hermann Neeb, Bent �rstedIntroductionIn a previous paper [N�98] we gave an algebraic classi�cation of the unitary highest weightrepresentations for the in�nite-dimensional analogs of the automorphism groups of the clas-sical bounded symmetric domains. We also realized these representations globally in cer-tain Hilbert spaces of holomorphic functions on certain corresponding domains in (in�nite-dimensional) Hilbert spaces. This was accomplished using the Harish-Chandra realizations asbounded domains, whereas the Cayley transformed Siegel domains did not enter. Since in the�nite-dimensional case, it is exactly in the Siegel domains that one applies Laplace transformmethods to study the �ner analytical properties of the unitary highest weight representations, itis natural to see to what extent such tools make sense in the in�nite-dimensional case as well.Hence in particular we want to understand as explicitly as possible the action of the a�ne partof the automorphism group of an in�nite dimensional tube domain on L2 -realizations of unitaryhighest weight modules. In the �nite-dimensional case the L2 -picture for unitary highest weightrepresentations of automorphism groups of tube domains has �rst been described by J. L. Clercin [Cl95]. It has been re�ned in several respects in [HN97].We shall construct in�nite-dimensional analogs of the Wishart distributions on symmetriccones, perhaps of independent interest in probability theory (see also [OV96]); the most explicitresults in this paper are on the case of scalar measures, but there are extensions to the case ofvector-valued measures as well (i.e. where the measure takes as values positive operators in aHilbert space). The di�culties we have to face in this project are twofold. First, there are thesubtleties of in�nite-dimensional measure theory. There is no unique canonical dual space onwhich a measure with a given Laplace transform should live. This is a well known phenomenonin the theory of Gaussian measures. In our context we will see that there are three di�erentrealizations of the measures that we are interested in, and each of these three realizations hasits di�erent merits. The second problem is that the in�nite-dimensionality of the domains weare working with causes a certain type of singularity of holomorphic functions in the sense thatif they are naturally de�ned on a domain in a space of trace class operators, then they do notextend to the corresponding domain in the space of Hilbert-Schmidt operators. Neverthelessin our situations this type of singularity can always be removed by multiplying with a certainfunction, so that we obtain functions on domains in l2 -spaces.Since the representations we consider are de�ned most naturally before the removal ofthe singularities takes place, the representations one obtains after this process become morecomplicated. Furthermore, we will see that this process often leads to multiplier representationswhich are non-trivial in the sense that, even though we obtain a projective representation ofthe small group, to obtain a unitary representation of the larger group, we have to pass to anon-trivial central extension.The main achievements of this paper are the following:(1) the construction of the L2 -realization of all unitary highest weight representations of auto-morphism groups of classical in�nite-dimensional tube domains; in particular the vector-valuedL2 -spaces seem to be completely new.

2 September 8, 1999(2) A projective representation of the full identity component of the a�ne automorphism groupof the Hilbert-Schmidt version �2 of the tube domain. The cocycle is trivial on the subgroupcorresponding to the trace class version �1 , but non-trivial on the large group. The structure ofthe cocycle gives a geometric explanation for some of the di�culties we encountered in [N�98](Section IV).(3) We show that the operator-valued measures corresponding to the vector valued highest weightrepresentations have densities of a rather weak type with respect to Wishart distributions whichmakes it possible to determine their \supports" (Section VI).In Section I we give the necessary background on measures and Laplace transforms in thein�nite-dimensional setting, and in Section II we recall the Wishart distributions and the Euclid-ian Jordan algebra theory, which is then extended to the case of in�nite-dimensional symmetriccones. Section III relates our previous construction of unitary highest weight representationsand their reproducing kernels to positive de�nite functions on cones; this is done via the Cayleytransform, and gives a correspondence between the highest weight representations of automor-phisms groups of bounded domains and admissible representations of L , the structure groupassociated to the cone. The main technical result here is Theorem III.9, which asserts the exis-tence of an extension of the representations to L1 , a certain trace-class completion of L , usingresults in [Ne98b]. Furthermore, we show that the corresponding projective representation evenextends to the considerably larger group Lb of all bounded invertible operators on the Hilbertspace on which the Jordan algebra is represented in a natural way. In Section IV we constructthe corresponding projective unitary representations of the a�ne automorphism group of theHilbert-Schmidt version �2 of the tube domain and get the existence of the measures givingthe unitary structure. Finally in Sections V, VI, and VII we give some more details on themeasures, in particular (in the scalar case) ergodicity for the \compact" structure group andhence irreducibility for the corresponding motion group, see Corollary V.2. Note the example inSection VI, where the vector-valued measure is made explicit in the case of the odd metaplecticrepresentation.One may think of our results as treating natural analogs of the metaplectic representation,and at the same time giving a way of understanding the role played by cocycles in the theoryof representations of in�nite-dimensional groups. At the same time we have seen ways ofunderstanding the restrictions of the unitary highest weight representations to subgroups, viz.the a�ne groups of the Jordan algebra (a maximal parabolic), and the analog of the maximalcompact subgroup (without making that completely explicit, though). It would be interesting toconsider the spectrum of other subgroups in these representations. For example, one expects aunitary highest weight representation to decompose as a direct sum of such when restricted to asubgroup also of hermitian type.I. Measures and Laplace Transforms in an In�nite Dimensional SettingIn this section we describe preliminary material on measures and Laplace transforms in thein�nite-dimensional setting and discuss Gaussian measures as an important class of exampleswhich plays a crucial role in this paper. The main result is Theorem I.3, an existence theoremfor projective limits of operator-valued measures. In Theorem I.7 we explain the relation to thecorresponding space of holomorphic functions.De�nition I.1. (a) Let (J;�) be a directed set. A projective family of measurable spaces is afamily (Xj ;Sj)j2J of measurable spaces together with measurable maps 'j;k:Xk ! Xj , j � k ,satisfying 'jj = idXj and 'j;k �'k;l = 'j;l for j � k � l . The projective limit measurable spaceX := lim �Xj is the set X = nx 2Yj2JXj : (8j < k)'j;k(xk) = xjoendowed with the smallest � -algebra S for which all the projections pj:X ! Xj are measurablemaps.

Representations in L2 -spaces on in�nite-dimensional symmetric cones 3(b) If ': (X;SX )! (Y;SY ) is a measurable map between measurable spaces and � is a measureon X , then we obtain a measure '�� on Y by putting '��(E) := �('�1(E)) for E 2 SY .(c) If V is a real topological vector space, V 0 its topological dual space, and � is a probabilitymeasure on V for which all continuous linear functionals are measurable, then we de�ne itsFourier transform by b�:V 0 ! C ; b�(�) := ZV ei�(v) d�(v)and its Laplace transform byL(�):V 0 ! [0;1]; � 7! ZV e��(v) d�(v)De�nition I.2. Let H be a Hilbert space.(a) We write B(H) for the space of bounded operators on H and Herm(H) for the spaceof hermitian operators. Furthermore we denote the cone of positive hermitian operators byHerm+(H). For p 2 [1;1[ the ideal Bp(H) := fX 2 B(H): tr((XX�) p2 ) < 1g is called thepth Schatten ideal. Here B1(H) is the space of trace class operators and B2(H) the space ofHilbert-Schmidt operators. We also write B�n(H) for the space of �nite rank operators on H .We put Hermp(H) := Herm(H)\Bp(H) and recall the Banach Lie groups GLp(H) := GL(H)\(1+Bp(H)) (cf. [Mi89]). We further recall the dual pairing B1(H)�B(H) ! C ; (x; y) 7! tr(xy)which in turn induces a pairing Herm1(H)� Herm(H)! R.(b) If (X;S) is a measurable space, then a �nite Herm+(H)-valued measure � is a countablyadditive function �:S! Herm+(H)satisfying �(�) = 0 . We call such a measure inner regular if for each positive trace class operatorS 2 Herm+1 (H) the positive measure �S de�ned by �S(E) = tr(�(E)S) is inner regular. Formore on not necessarily �nite measures with values in cones we refer to [Ne98a].(c) We recall the construction of the space L2(X;�) for a Herm+(H)-valued measure � onX . First we consider the space L2(X;�) of all those measurable functions f :X ! H with�nite-dimensional range endowed with the positive semide�nite hermitian formhf; gi := ZX Pf(�);g(�) d�(�);where Px;y denotes the rank-1-operator on H given by Px;y(v) = hv; yix and the integral of theB1(H)-valued function � 7! Pf(�);g(�) is de�ned as in [Ne98a]. To obtain L2(X;�), one factorsby the subspace N := ff 2 L2(X;�): hf; fi = 0g and passes to the completion with respect tokfk2 :=phf; fi .The following theorem gives a su�cient criterion for the existence of a limit of a projectivefamily of operator-valued measures.Theorem I.3. Let (Xj ;Sj)j2J be a projective family of measurable spaces, 'j;k:Xk ! Xj ,j � k , the corresponding measurable maps and �j:Sj ! Herm+(H) operator valued measuressatisfying �j(Xj) = 1 and forming a consistent family, i.e.,'�j;k�k = �j for j � k:If the projective limit measurable space (X;S) := lim �(Xj ;Sj) carries a Herm+(H)-valuedmeasure � for which the natural maps 'j:X ! Xj satisfy '�j� = �j for all j 2 J , then itis unique. It exists if the following conditions are satis�ed:(1) Each measure �j is inner regular with respect to some topology �j on Xj .(2) If J is uncountable, then for every increasing sequence M := fjn:n 2 Ng � J the rangeof the natural mapping 'M :X ! XM := lim �n!1Xjn

4 September 8, 1999is thick in the sense that its complement contains no measurable subset of non-zero measure.Moreover, one of the following two conditions has to be satis�ed:(3a) (Xj ;Sj) is countably separated for each j 2 J , or(3b) the maps 'j;k: (Xk; �k)! (Xj ; �j) are continuous for j � k .Proof. If H is one-dimensional, i.e., the �j are probability measures, then the assertionfollows from the following results in [Ya85]: Theorem 8.2 (reduction to the countable case) andTheorem 7.2 (the countable case).For the general case, let S 2 Herm+1 (H) be a positive trace class operator. Then themeasures �Sj :E 7! tr ��j(E)S� form a consistent family of measures on the Xj which are innerregular with respect to the topology �j on Xj . Hence the scalar case applies and we �nd aunique �nite positive measure �S on X with '�j�S = �Sj for each j 2 J . Furthermore theassignment S 7! �S(E) is additive for each measurable subset E 2 S . Hence we �nd for eachE 2 S a unique operator �(E) 2 Herm+(H) withtr ��(E)S� = �S(E)for all S 2 Herm+1 (H). It follows from [Ne98a, Prop. I.7, Th. I.10] that we thus obtain aHerm+(H)-valued measure on X .Remark I.4. (a) If (Xi;Si)i2I is a family of measurable spaces and J the directed set of �nitesubsets of I , then we assign to each F 2 J the �nite product measurable space XF :=Qi2F Xi .With the natural projections 'F1;F2 :XF2 ! XF1 ; F1 � F2we obtain a projective system of measurable spaces. If for each i 2 I we are given a probabilitymeasure �j , then the projective family of measures �F = i2F�i de�nes a probability measureon the product space X = lim �XF =Yi2IXi:The main point is that this requires no additional regularity condition ([Ya85, Th. 12.1]).(b) Suppose that � is a measure on a projective limit measurable space (X;S) := lim �(Xj ;Sj)and write S0 for the set algebra of all those subsets of X which can be written as '�1j (Ej) forsome Ej 2 Sj . We claim that the characteristic functions �E , E 2 S0 , form a total subset ofL2(X;�).In fact, let V � L2(X;�) denote the closed subspace generated by the characteristicfunctions �E , E 2 S0 . Then E := fE � X:�E 2 V g is a monotone system containingS0 . Hence it contains the monotone system generated by the set algebra S0 and therefore the� -algebra S . We conclude that V = L2(X;�) which proves our claim. It follows in particularthat L2(X;�) = lim�!j2J L2(Xj ; �j)is a direct limit of Hilbert spaces because the right hand side can be identi�ed in a natural waywith a subspace of L2(X;�) which, in view of the observation above, is dense. This observationshows in particular that whether a measure exists or not, one always can consider the direct limitHilbert space lim�!j2J L2(Xj ; �j) as a substitute of L2(X;�) if � does not exist.Lemma I.5. If ':E ! F is a continuous linear mapping between locally convex spaces,'0:F 0 ! E0 its adjoint map, and �E a probability measure on E (de�ned on the smallest� -algebra for which all continuous functionals are measurable), then L('��E) = L(�E) � '0 .Proof. For � 2 F 0 we haveL('��E)(�) = ZF e��(x)d('��E)(x) = ZE e��('(y))d�E(y) = L(�E)('0:�):

Representations in L2 -spaces on in�nite-dimensional symmetric cones 5De�nition I.6. (a) If X is a set and V is a complex Hilbert space, then a function K:X�X !B(V ) is called a positive de�nite kernel if for each �nite sequence (x1; v1); : : : ; (xn; vn) 2 X � Vwe have nXi;j=1hK(xi; xj):vj; vii � 0:This condition is equivalent to the existence of a Hilbert subspace HK � V X such that theevaluation functions Kx:HK ! V satisfy K(x; y) = KxK�y (cf. [Ne99, Sect. I]).(b) If U is a real vector space, UC its complexi�cation, and � U a convex set, then we callT := +iU � UC the corresponding tube domain. Note that T is invariant under the complexconjugation z = x + iy 7! z := x � iy . If V is a Hilbert space, then a function ':T ! B(V )is said to be positive de�nite if the kernel K(z; w) := '( z+w2 ) is positive de�nite which in turnis equivalent to the positive de�niteness of the restriction to � (cf. [Gl99]). If ' is positivede�nite, then we write H' � V T for the corresponding Hilbert space.(c) We say that a subset in the real vector space U is �nitely open if for each �nite-dimensionalsubspace U0 � U the intersection U \ is open in U . If V is a Hilbert space, then a functionf :T ! V on the corresponding tube domain is called Gateaux holomorphic if the restriction toeach tube domain TU0\ is holomorphic. We write HolG(T; V ) for the space of all Gauteauxholomorphic V -valued functions on T . For the theory of holomorphic functions on domainsin in�nite-dimensional spaces we refer to the book of Herv�e [He89] (see also Appendix III in[Ne99]).Theorem I.7. Let U be a real vector space and � U a non-empty convex �nitely opensubset. Let V be a Hilbert space and ':T ! B(V ) a positive de�nite (G)-holomorphic function.Then the following assertions hold:(i) There exists a unique Herm+(V )-valued measure � on the smallest � -algebra on U� forwhich all point evaluations are continuous such that L(�) = ':(ii) The map F :L2(U�; �)! HolG(T; V ); f 7! bfwith h bf (z); vi := hf; e� z2 :vi is an isometry onto the reproducing kernel space H' corre-sponding to the kernel associated to ' .(iii) Let U0 � U be a subspace with 0 := \ U0 6= � and �0 the unique Herm+(V )-valuedmeasure on U�0 with L(�0) = ' j T0 . Then the restriction map r:U� ! U�0 satis�esr�� = �0 and induces an isometric embedding�:L2(U�0 ; �0)! L2(U�; �); f 7! f � r:If, in addition, there exists a topology on U for which 0 is dense in and ' is continuous,then � is surjective.Proof. (i) (cf. [Gl99]) The case where U is �nite-dimensional so that is open and ' isholomorphic is covered by [Ne98a, Th. III.5]. For the general case we consider the directed familyF = fUj: j 2 Jg of all those �nite-dimensional subspaces Uj � U with j := Uj \ 6= �. Weapply the �nite-dimensional case to each function 'j := ' jTj and obtain a unique Herm+(V )-valued measure �j on U�j with L(�j) = 'j . The uniqueness of these measures implies that if�j;k:U�k ! U�j are the restriction maps, then ��j;k�k = �j whenever Uj � Uk .The projective limit of the measurable spaces U�j is the algebraic dual space U� endowedwith the smallest � -algebra S for which all evaluations in elements of U are measurable.Since the conditions in Theorem I.3 are trivially satis�ed in this situation, there exists a uniqueHerm+(V )-valued measure � on S with L(�) = ' .(ii) Here we simply observe that with the appropriate modi�cations, the proof of [Ne98a, Th.III.9] also applies in the in�nite-dimensional situation.(iii) First we note that the uniqueness of �0 and L(r��) = ' jT0 = '0 implies that �0 = r�� .For f 2 L2(U�0 ; �0) (cf. De�nition I.2(c)) and Px := Px;x we havekf � rk22 = ZU� Pf(r(�)) d�(�) = ZU�0 Pf(�) d(r��)(�) = kfk22;

6 September 8, 1999showing that the map f 7! f � r induces an isometric embedding �:L2(U�0 ; �0)! L2(U�; �).To prove the remaining assertion, we assume that ' is continuous on for a topology forwhich 0 is dense. For z 2 T and v 2 V we consider the function fz;v(�) = e� 12�(z):v on U�and for z 2 (U0)C we write f0z;v for the corresponding function on U�0 . Then f0z;v � r = fz;vshows that the range of the isometric map � contains all functions fz;v , z 2 T0 , v 2 V (cf.[Ne98a, Lemma 3.8]). Now for each v 2 V the function v:T ! L2(U�; �); z 7! fz;vsatis�es k v(z)k2 = kfz;vk2 = h'(z):v; vi , hence is locally bounded (from [Ne99, Prop. A.III.10,11] it even follows that it is holomorphic). Furthermore it is weakly continuous and z 7! kfz;vkis continuous. This proves that v is continuous. Now v(T0 ) � im� implies v(T) � im� ,so that � is surjective because the functions fz;v , z 2 , v 2 V form a total subset of L2(X;�),as we have observed in (ii).Example I.8. (Gaussian measures) (a) An interesting particular case, where Theorem I.7applies is the setting where V is a real pre-Hilbert space, = V and '(z) = e 12 hz;zi . Hereh�; �i denote the hermitian extension of the scalar product on V to the complexi�cation VC ,and x+ iy = x� iy , x; y 2 V . Then the corresponding measure on V � is called the Gaussianmeasure of V and is denoted V . It is uniquely determined by L( V )(z) = e 12 hz;zi for z 2 VC .(b) If V ! W is a dense isometric embedding, then 'V jW = 'W , and the uniqueness of themeasure implies that the restriction map r:W � ! V � satis�es r� W = V . Moreover, TheoremI.7(iii) implies that �:L2(V �; V )! L2(W �; W ); f 7! f � ris isometric. This means that the two L2 -spaces are essentially the same, even though they aremodeled on di�erent measurable spaces. These observations apply in particular to the case whereW is a completion of V .(c) From now on we assume that V is complete, i.e., a real Hilbert space. To each �nite rankoperator A 2 B�n(VC ) we associate linearly a function qA on V � in such a way that for arank-1-operator Px;y with Px;y(v) = hv; yix we haveqPx;y (�) = 12�(x)�(y):Let � := fX 2 B(VC ):X +X� >> 0g . We claim that for 1 +A 2 � and A 2 B�n(A) wehave e�qA 2 L1(V �; V ) with(1:1) ZV � e�(z)�qA(�) d V (�) = det(1+ A)� 12 e 12 h(1+A)�1 :z;zi;where det� 12 : �! C � corresponds to the unique branch of the square root on the open convexdomain � which is uniquely determined by det(1)� 12 = 1.Since the real part of qA only depends on the hermitian part 12(A + A�) of A , we mayw.l.o.g. assume that A is hermitian. Then the general assertion follows from the holomorphy ofboth sides as functions on �, and the assertion for a hermitian A . In view of the assumption thatrkA <1 , we even may assume that dimV <1 . Choosing an orthonormal basis of eigenvectorsfor A , we observe that both sides decompose accordingly as a product. Hence we may assumethat dimV = 1. Then we have to show thatZRe�(z)� 12a�2 d R(�) = 1p2� ZRe�(z)�12a�2e� �22 d� = (1 + a)� 12 e 12 (1+a)�1z2 :The integral exists if and only if 1 + a > 0, and in this case the veri�cation of the formula is asimple exercise. This proves (1.1).The right hand side of (1.1) still makes sense for 1 + A 2 �1 := � \ (1 + B1(V )). Toprove the assertion in this case, one also reduces matters to the case where A is hermitian. Then

Representations in L2 -spaces on in�nite-dimensional symmetric cones 7we consider a sequence An ! A with An = A�n 2 B�n \ �. Furthermore we may assume thatall eigenvector for A are eigenvectors of each An . Now the corresponding formula for the �niterank case shows that the sequence e�qAn , n 2 N , is a convergent sequence in L1(V �; V ). Inthis sense the function e�qA makes sense as an element of L1(V �; V ) and (1.1) holds. By abuseof notation we also write qA(�) = 12 hA:�; �i for A 2 �1 , � 2 V � , even though hA:�; �i is notde�ned literally.(d) We keep the assumption that V is complete. We consider the action of GL(V ) (the groupof continuous linear operators on V ) on the algebraic dual V � by g:� := � � g�1 .It is shown in [Se58, Th.3] that the Gaussian measure g� V on V � is absolutely continuouswith respect to V if and only if g is contained in the restricted general linear grouprGL(V ) := fg 2 GL(V ): g>g � 1 2 B2(V )g;the subgroup of GL(V ) consisting of those operator g with polar decomposition g = up andp� 1 2 B2(V ). For g 2 rGL we writeX(g)(�) := d V (g�1:�)d V (�) 2 L1(V �; V )for the corresponding Radon-Nikodym derivative. To obtain an explicit formula, one has toconsider elements of a smaller subgroup. In the sense discussed in (c), we have for gg> 2 GL1(V )the explicit formula (using the notation g�> for the inverse transpose of g )X(g)(�) = det(gg>) 12 e 12 h(1�g�>g�1):�;�i([Se58, p.23], [Se65, pp.463{466]). We recall from [Sh62, Cor. 3.1.2] that the functionX: GL2(V )! L1(V �; V )is continuous. The natural unitary action of rGL(V ) on L2(V �; E) is given by��(g):f�(�) =pX(g)(�)f(g�1:�);and for gg> 2 GL1(V ) we obtain in particular��(g):f�(�) = (det gg>) 14 e 14 h(1�g�>g�1):�;�if(g�1:�)(cf. [Sh62, Th.3.1]).II. Wishart Distributions and Their GeneralizationsIn this section we will study certain L2 -spaces of measures living on the algebraic dual spaceof the in�nite-dimensional classical euclidean Jordan algebra Herm(J;K), the �nite hermitianJ � J -matrices, where J is an in�nite set and K 2 fR; C ;H g . These Jordan algebras areof in�nite rank and direct limits of �nite-dimensional euclidean Jordan algebras. First we willrecall the construction of the Wishart distributions in the �nite-dimensional case, and then wewill see how we can use Gaussian measures of in�nite-dimensional Hilbert spaces to see that theconstruction from the �nite-dimensional case essentially carries over to the in�nite-dimensionalcase. In Section III we will see that the scalar-valued measures considered in this section corre-spond to the scalar type highest weight representations of the corresponding conformal Koecher-Tits groups. This implies that these representations have natural realizations in L2 -spaces onthe set ?k of �nite rank positive semide�nite matrices in the dual space Herm(J;K)� consistingof all hermitian J � J -matrices. The measures showing up in this picture are natural in�nite-dimensional analogs of Wishart distributions.In Section IV we carry out the next step of this process which consists in extendingthis picture to obtain L2 -realizations of general unitary highest weight representations of theconformal groups in vector-valued L2 -spaces. In this case the measures we have to consider areoperator-valued measures and we will use Theorem I.7 to prove their existence.

8 September 8, 1999Wishart distributionsIn this small subsection U is a �nite-dimensional euclidean Jordan algebra, is the opencone of invertible squares in U , L � GL(U ) the identity component of the structure group,� = �U :U ! R the determinant function of U , and ?k � ? � U� the set of elements ofrank � k in the dual cone (cf. [FK94]). It is instructive for the construction in the in�nite-dimensional case that we �rst recall the �nite-dimensional results. The notation is the usual, ndenotes dimension, and d the \�eld" dimension.Proposition II.1. Let U be a simple euclidean Jordan algebra of rank r . For� 2 f0; : : : ; (r � 1)d2g or � > (r � 1)d2there exists a so called Riesz measure R� on ? such that L(R�) = ���U . Moreover, thefollowing assertions hold:(i) For k � r � 1 the measure Rk d2 is supported by ?k and satis�es Rk d2 (?k�1) = 0:(ii) The measures R� are L-semiinvariant with l�R� = det(l)�rn R� for all l 2 L .(iii) The function ���U on is positive de�nite if and only if � 2 f0; : : : ; (r � 1)d2g or� > (r � 1)d2 .Proof. [FK94, Prop. VII.2.3]De�nition II.2. (a) Let U be a euclidean Jordan algebra and E a euclidean vector space.A symmetric representation of U on E is a linear map ':U ! Sym(E) satisfying'(x � y) = 12�'(x)'(y) + '(y)'(x)�:(b) If (';E) is a symmetric representation of the euclidean Jordan algebra, then the associatedquadratic form Q:E ! U� is de�ned byhQ(v); xi = 12 h'(x):v; vi for x 2 U; v 2 E:Let (';E) be a symmetric representation of U on E and E the Gaussian measure on E givenby d E(v) = 1(2�)N2 e� 12kvk2d�E(v);where N = dimE and �E is Lebesgue measure on U . Then the measure Q� E on U is calledthe Wishart distribution associated to the representation ' .Remark II.3. It is easy to identify the Wishart distributions in terms of Riesz measures. Todo this, we calculate the Laplace transforms in y 2 using formula (1.1):L(Q� E)(y) = ZU� e�hx;�i d(Q� E )(�) = ZE e�hQ(v);yi d E(v)= ZE e� 12 h'(y):v;vi d E(v) = det �1+ '(y)�� 12 :If, in addition, '(1) = 1 , then [FK94, Prop. IV.4.2] shows thatdet �1+ '(y)�� 12 = det �'(1+ y)�� 12 = �(1+ y)� N2r = L(e�1�RN2r )(y);where 1�:U� ! R denote the evaluation in the unit element 1 2 U . ThusQ� E = e�1�R N2r :We note that one similarly shows that the image of �E under Q is a multiple of R N2r (cf. [FK94,Prop. VII.2.4]).

Representations in L2 -spaces on in�nite-dimensional symmetric cones 9Example II.4. Let r 2 N and U = Herm(r;K). We consider E := M (r; k;K) as a euclideanvector space, where the scalar product is given by hA;Bi := Re tr(AB�): Then we obtain asymmetric representation ' of U on E by '(A)(B) := AB: In this caseh'(A):B;Bi = Re tr(ABB�) = hA;BB�iU ;so that the corresponding quadratic form Q:E ! U� is given by Q(B) = 12BB�:Let d = dimRK . Then N = drk , so that N2r = k d2 . This shows that for each Riesz measureRk d2 , k 2 N0 , there exists a symmetric Jordan algebra representation ' with Q� E = e�1�Rk d2 .This covers in particular all singular Riesz measures, i.e., those with k � r � 1.Below we will see how the preceding example can be generalized to an in�nite dimensionalsetting. The in�nite-dimensional settingNow we turn to the in�nite-dimensional versions of the classical euclidean Jordan algebrasHerm(n;K) . We �rst introduce the notation that we will use throughout this paper.Notation: Let K 2 fR;C ;H g , d = dimRK , and J be an arbitrary in�nite set. We writeU = Herm(J;K) for the locally �nite euclidean Jordan algebra of �nite J � J -matrices, i.e.,�nitely supported functions J �J ! K . Here the Jordan product is given by x�y := 12(xy+yx)and the scalar product by hx; yi := Re tr(xy) = 1d trR(xy);where trR(x) is the trace of x as an operator on the real vector space K(J) �= R(Jd) . We writeH := l2(J;K) for the real Hilbert space of all square summable functions J ! K with the scalarproduct hx; yi :=Xj2J Re(xjy�j ):Then B(H;K) � B(H) denotes the space of all K-linear bounded operators on H , Sym(H) �B(H) the space of symmetric operators on H , Ub := Herm(H) := Sym(H) \ B(H;K) thespace of bounded K-hermitian operators (the \bounded version of U ") and Up := Hermp(H) :=Herm(H) \Bp(H) the space of hermitian operators of Schatten class p 2 [1;1[ . For p = 1 weobtain the trace class operators and for p = 2 the Hilbert-Schmidt operators.In B(H) we have the Siegel tube �b := fZ 2 B(H):Z + Z> >> 0g , its restrictedversions �p := �b \ �1 + Bp(H)� , p 2 [1;1[ , � := �b \ UC , as well as the convex domainsp := �p \Herm(H) and := �b \ (1 + U ). We have the natural inclusionsU � U1 � U2 and � 1 � 2:The dual space U� can be identi�ed in a natural way with the space of all hermitianJ � J -matrices, where the real bilinear pairing U � U� ! R is given byU � U� ! R; (x�) 7! Re tr(x�)which makes sense since the matrix x� has �nitely many non-zero rows. Even though we canidentify U in a natural way with a subspace of U� , we prefer not to do so because this oftenleads to conceptual confusion and certain groups act in di�erent ways.Let gl(J;K) denote the Lie algebra of �nite J � J -matrices and GL(J;K) be the groupof all those invertible J � J -matrices g for which g � 1 is �nite, where 1 denotes the identitymatrix. Further let L := GL(J;K)0denote the identity component of the group GL(J;K) . Note that GL(J;K) is connected forK = C and H , but not for K = R. Here our notation is slightly inconsistent with the notation

10 September 8, 1999for the �nite-dimensional case because the action of L on U is not faithful, but it will turn outthat it is much more convenient to work directly with the group GL(J;K) and its completions.For g 2 GL(H) we write g> for the adjoint operator on H (considered as a real Hilbert space)and observe that for g 2 GL(H;K) we have g> = g� . The groupsLp := fg 2 GL(H;K): g>g � 1 2 Bp(H)g0; p = 1; 2act in a natural way on Up by g:x = gxg> , and this action preserves the domains p becausefor g 2 Lp and A 2 p we haveg:A � 1 = gAg> � 1 = g(A � 1)g> + gg> � 1 2 Up + Up � Up:The action on Up induces a natural action on the algebraic dual space U�p by (g:�)(x) := �(g�1:x)for x 2 Up . Likewise we have an action of the small group L on the algebraic dual space U� .With respect to these actions we have p = Lp:1 and = L:1 . Moreover, the action of L onU� preserves for each k 2 N0 the set?k := f� 2 ?: rk� � kg � U�of matrices of rank � k .Example II.5. We generalize Example II.4 to the in�nite-dimensional setting. We have aJordan algebra representation of U on the space M (J; k;K) of all J �k -matrices, but this spaceis in some sense too big to carry the structure of a euclidean vector space. On the subspace ofall �nite matrices we have the natural real bilinear scalar producthA;Bi := Re tr(AB�)which also makes sense if only one of the two arguments has only �nitely many non-zero entries.Nevertheless, for each A 2 M (J; k;R) and x 2 U the matrix '(x)A = xA has only �nitelymany non-zero rows, hence is a �nite J � k -matrix. Thereforeh'(x)A;Ai = Re tr �('(x)A)A��makes sense, and we thus obtain a quadratic mapQ:M (J; k;K) ! U�; A 7! 12AA�with hQ(A); xi = 12 h'(x):A;Ai for x 2 U;A 2 M (J; k;K):Note that, even though the matrix product A�A makes no sense, the product AA� makes senseand is a hermitian J � J -matrix.We think of the space M (J; k;K) as the dual of the euclidean vector space V of all �niteJ � k -matrices with the real bilinear scalar product hA;Bi = Re tr(AB�). We thus obtain aGaussian measure V on M (J; k;K) and �k := Q� V is a certain probability measure on U� forwhich the subset ?k of all those positive semide�nite hermitian matrices in U� of rank � k isthick in the sense that its complement contains no measurable subset of non-zero measure.If 1 + x 2 is positive de�nite, then we obtain as in the �nite-dimensional case:L(�k)(x) = ZU� e��(x) d(Q� V )(�) = ZV � e�hQ(A);xi d V (A)= ZV � e� 12 h'(x):A;Ai d V (A) = detV �1+ '(x)�� 12 = detV �'(1+ x)�� 12 :In our case we have detV �'(1+ x)� = detR(1+ x)k and thereforeL(�k)(x) = detR(1+ x)�k2 :

Representations in L2 -spaces on in�nite-dimensional symmetric cones 11On the other hand detR(1+ x) = �(1+ x)d holds for the Jordan determinant of U (see below)and each x 2 U . We thus conclude in particular that for each k 2 N0 the function��kd2 : ! Ris the Laplace transform of a positive measure on ?k .If we replace U by the bigger Banach Jordan algebra U1 , then the determinant functionstill makes sense on the domain 1 , where we have �(x)d = detR(x). Using Theorem I.7, weobtain a probability measure �1k on U�1 withL(�1k)(x) = �(1+ x)� kd2for x 2 1 . Moreover, the natural map r:U�1 ! U� satis�es r��1k = �k and induces an isometricbijection L2(U�; �k)! L2(U�1 ; �1k); f 7! f � r(Theorem I.7).Applying Proposition IV.1 below to the representation of L given by �(g) = detR(g)� k2 ,we see that we obtain a unitary representation of the semidirect product group U1 o L1 on theHilbert space L2(U�1 ; �1k) by((u; g):f)(�) := ei�(u)e 12�(1�gg>) detR(gg>) k4 f(g�1:�):We note that the isometric embeddingL2(U�; �k)! L2(V �; V ); f 7! f �Qis equivariant with respect to the action of the group L on L2(U�; �k) �= L2(U�1 ; �1k) because wehave for A 2 V � and g 2 L the relation Q(g:A) = Q(g�>A) = 12g�>AA�g�1 = g:Q(A), wherethe action on the right hand side refers to the action of L on U� . Hence(g:f)(Q(A)) = detR(gg>) k4 e 12Q(A)(1�gg>)f(g�1:Q(A))= detV;R(gg>) 14 e 14 h(1�gg>)�A;Aif(Q(g�1:A)) = (g:(f �Q))(A);where the last term has to be understood in the sense of Example I.8(d).Appendix II: The quaternionic caseFor the convenience of the reader we brie y recall the convenient description of the Hilbertspace H and the operators thereon for the quaternionic case K = H . Quaternionic Hilbertspaces are most conveniently described as follows. Let H0 be a complex Hilbert space endowedwith an antilinear isometric involution v 7! v . Then the antilinear operator on H0 �H0 givenby �:(x; y) = (�y; x) satis�es �2 = �1 . We thus obtain on H := H0 � H0 the structure of aquaternionic Hilbert space with HC �= H0 �H0 (by restriction of scalars to C ).We write a complex linear operator X on H as a 2� 2-matrix of the formX = �A BC D� ; A;B;C;D 2 B(H0)and for A 2 B(H0) we de�ne the operator A by A:v := A:v . We see that X� = �X is equivalentto �A:y +B:x = �C:x+D:y = �C:x�D:y; �C:y +D:x = A:x+ B:y = A:x+B:y

12 September 8, 1999for x; y 2 H0 . This means that D = A and C = �B: Therefore the H -linear operators on Hcorrespond to the matrices of the formX = � A B�B A� ; A;B 2 B(H0):Let J := � 0 1�1 0� and observe that JX = XJ if and only if X is H -linear. We identifyB(H0) with a subspace of B(H) by the mapping A 7! �A 00 A� : In this sense an operatorX = A+ JB as above is hermitian if and only ifA + JB = X = X� = A� +B�J� = A� +B�(�J) = A� � JB>:In this case we put B> := B� . HenceHerm(H;H ) = fA+ JB:A� = A;B> = �Bg �= Herm(H0; C ) � J Skew(H0):We have �J(A + JB) = B � JA , and(B � JA)> = B> � A>J> = B> + A>J = B> + JA�:We conclude that �J(A + JB) is skew-symmetric if and only if B is skew-symmetric and A ishermitian. This shows that J Herm(H;H ) � Skew(H):One easily checks that J Herm(H;H ) is a real form of the space Skew(H), i.e., Skew(H) =J Herm(H;H ) � iJ Herm(H;H ).III. Relations to Unitary Highest Weight RepresentationsThe main objective of this paper is to provide an L2 -realization of the unitary highest weightrepresentations of the automorphism groups of in�nite-dimensional Hilbert domains which havebeen classi�ed in [N�98]. In this section we explain how one associates to a unitary highest weightrepresentation of such a group a positive de�nite function on the domain 1 , resp. T1 . Usingthe tools described in Section I, we can represent these positive de�nite functions by operator-valued measures on the algebraic dual space U�1 . In view of the �nite-dimensional situation,one expects to obtain an explicit picture for the action of the group U1 o L1 acting by a�neautomorphisms of the domain T1 on the corresponding Hilbert spaces. For the highest weightrepresentations of scalar type, this construction leads to the measures �1k on U�1 that we havediscussed in Example II.5.On the other hand, the construction of the measure �1k as a pushforward of a Gaussianmeasure shows that not only the group L1 but also the group L2 acts on the correspondingHilbert space. To make the action of L2 and also of the a�ne group U2 o L2 more explicit, we�rst use the regularized determinant det2 on the space 1 + B2(H) �rst to modify the positivede�nite function on the domain 1 in such a way that it extends to a positive de�nite functionon the bigger domain 2 and thus to obtain a measure on U�2 . Using this new realization ofour Hilbert space, we obtain an explicit formula for the representation of the group L2 . Itturns out that this representation extends in a natural way to a multiplier representation of thegroup U2 o L2 , but that the corresponding cocycle is not trivial, so that this representation canbe viewed as a unitary representation of a non-trivial central extension, but not as a unitaryrepresentation of the group U2 o L2 itself.

Representations in L2 -spaces on in�nite-dimensional symmetric cones 13Unitary highest weight representationsIn this subsection we consider the conformal Lie algebra gRassociated to the Jordan algebraU , resp. its complexi�cations g which is the conformal Lie algebra of the complex Jordan algebraUC . We explain how the unitary highest weight representations of the conformal algebra g arerelated to unitary highest weight representations of the Lie algebra l of L .De�nition III.1. (a) An involutive Lie algebra is a pair (g; �) of a complex Lie algebra gand an antilinear involutive antiautomorphism g! g; X 7! X� . The subspacegR:= fX 2 g:X� = �Xgis a real form of g which determines the involution uniquely.(b) We say that the involutive Lie algebra (g; �) has a root decomposition if there exists a maximalabelian �-invariant subalgebra h such that g = h+P�2� g�; whereg� = fZ 2 g: (8X 2 h)[X;Z] = �(X)Zg;and � = f� 2 h� n f0g: g� 6= f0gg is the corresponding root system.(c) A subset �+ � � is called a positive system if � = �+ [��+ and no sum of positive rootsvanishes.(d) Let V be a g-module and v 2 V � an h-weight vector of weight � . We say that v is aprimitive element of V (with respect to the positive system �+ ) if g�:v = f0g holds for all� 2 �+ . A g-module V is called a highest weight module with highest weight � (with respectto �+ ) if it is generated by a primitive element of weight � .(e) We call a hermitian form h�; �i on a g-module V contravariant if hX:v;wi = hv;X�:wi holdsfor all v; w 2 V , X 2 g . A g-module V is said to be unitary if it carries a contravariant positivede�nite hermitian form.Proposition III.2. Let g be an involutive complex Lie algebra with root decomposition and�+ a positive system. Then the following assertions hold:(i) For each � 2 h� there exists a unique irreducible highest weight module L(�) and eachhighest weight module of highest weight � has a unique maximal submodule.(ii) Each unitary highest weight module is irreducible.(iii) If L(�) is unitary, then � = �� .(iv) If � = �� and v� 2 L(�) is a primitive element, then L(�) carries a unique contravarianthermitian form h�; �i with hv�; v�i = 1: This form is non-degenerate.Proof. [Ne99, Props. IX.1.10/14]Example III.3. A typical example of an involutive complex Lie algebra is the Lie algebrag := gl(J; C ) endowed with the natural involution given by E�jk = Ekj for the matrix unitsEjk , j; k 2 J . The diagonal operators in g form an abelian subalgebra h and we have a rootdecomposition g = h� Mi6=j2J g"i�"j ;where "i(Ejj) = �ij and g"i�"j = C Eij . The positive systems in � are in one-to-one correspon-dence with the linear partial orderings � of the set J . This correspondence is established byassigning to � the positive system �+� := f"j � "k: j � kg(cf. [Ne98b, Lemma II.1]). A linear functional � 2 h� can be identi�ed in a natural way withthe function �: J ! C ; j ! �j := �(Ejj). It has been shown in [Ne98b, Lemma II.4] that thehighest weight module L(�;�+) of gl(J; C ) is unitary if and only if � is real and dominantintegral in the sense that �j � �k 2 N0 for j � k:

14 September 8, 1999Let gRdenote the conformal algebra of U and note that for U = Herm(J;K) we havegR=8<: sp(J;R) for K = R,su(J; J) for K = C ,o�(2J) for K = H .Accordingly we have g := (gR)C �= sp(J; C ); sl(J; C ); o(2J; C ): This is a complex involutive Liealgebra with the following properties. There exists a maximal abelian subalgebra h contained inthe subalgebra k := lC � g so that we obtain a root decomposition g = h +P�2� g� . We call�k := f� 2 �: g� � kg the set of compact roots and �p := � n�k the set of non-compact roots.In [N�98] we have classi�ed all the unitary highest weight representations of these Lie algebras.For a characterization of the Lie algebras gRas the hermitian real forms of simple complex locally�nite Lie algebras with a root decomposition, we refer to [NeSt99]. In the following we explainhow this classi�cation looks like and what this means for the corresponding measures on U� .One important observation is that if �+ � � (the root system of g) is a positive system forwhich there exists a non-trivial unitary highest weight module L(�), then the set �+p of positivenon-compact roots is invariant under the Weyl group Wk of k , and for the simple hermitian Liealgebras this condition determines �+p up to sign. A necessary condition for the unitarity ofL(�;�+) is that � is dominant integral with respect to �+k and antidominant with respect to�+p (cf. [N�98, Prop. I.5]). In the following the numberk := �2d supf�(� ): 2 �+p gthat we associate to a unitary highest weight module L(�;�+) of g will play a crucial rolebecause it describes the location of the corresponding operator-valued measure (cf. Section V).Theorem III.4. The classi�cation of the unitary highest weight modules for the Lie algebrasg and the corresponding numbers k 2 N0 are as follows. We assume that � 2 h� is real-valuedon hermitian elements, that it is dominant integral with respect to �+k and antidominant withrespect to �+p .(1) K = R: Here gR= sp(J;R) and k �= gl(J; C ) so that we may identify � with a functionon J . We assume that �+p = f"i + "j : i; j 2 Jg and put M := supf�j: j 2 Jg . Then thehighest weight module L(�;�+) is unitary if and only if2M + jfj 2 J :�j < M � 1gj+ jfj 2 J :�j < Mgj � 0; and in this case k = �2M:(2) K = C : Here gR= u(J�; J+) , where J� are copies of one set J , and we furthermorehave k �= gl(J�; C ) � gl(J+; C ) . We assume that �+p = f"i � "j : i 2 J�; j 2 J+g , write� = (��; �+) according to the decomposition of k and put m := sup�� and M := inf �+ .Then the highest weight representation L(�;�+) is unitary if and only ifM �m � jfj 2 J�:�j 6= mgj+ jfj 2 J+:�j 6=Mgj; and in this case k =M �m:(3) K = H : Here gR= o(2(J+ [ J�); C ) , where J� are copies of the set J , and k �= gl(J+ [J�; C ) . We assume that �+p = f"i � "j : i 2 J�; j 2 J+g and put M := sup� . Then thehighest weight module L(�;�+) is unitary if and only ifM + jfj 2 J+ [ J�:�j < Mgj � 0; and in this case k = �M:Proof. (1) We recall that�k = f"i � "j : i 6= j 2 Jg and �p = f�("i + "j): i; j 2 Jg:A positive system is given by a partial order � on the set J and for this order we may w.l.o.g.assume that �+k = f"i � "j: i � jg and �+p = f"i + "j: i; j 2 Jg

Representations in L2 -spaces on in�nite-dimensional symmetric cones 15(cf. [N�98]).Now the antidominance of � with respect to �+p implies that �j � 0 for each j 2 J ,so that M = maxf�j : j 2 Jg exists because the integrality with respect to �k means that alldi�erences �i � �j are integral. The characterization of the unitary modules has been obtainedin [N�98, Prop. I.9]. In view of d = 1, we further have�k2 = �dk2 = supf�(��):� 2 �+p g =Mbecause there exists an � = 2"j with M = �j = �(��). This proves (1).(2) We have �k = f"i � "j: i 6= j 2 J�g and �p = f�("i � "j): i 2 J�; j 2 J+g:In this case a positive system is given by a pair of partial orders � on the sets J+ and J� . Forthis order we may w.l.o.g. assume that�+k = f"i � "j 2 �k: i � jg and �+p = f"i � "j : i 2 J�; j 2 J+g(cf. [N�98]). The characterization of the unitary modules has been obtained in [N�98, Prop.I.7]. We further have �k = �dk2 = supf�(��):� 2 �+p g = m �M;and this proves (2).(3) We have�k = f"i � "j : i 6= j 2 J+ [ J�g and �p = f�("i + "j): i 6= j 2 J+ [ J�g:In this case a positive system is given by a partial order � on the set J+ [ J� and for this orderwe may w.l.o.g. assume that�+k = f"i � "j 2 �k: i � jg and �+p = f"i � "j : i 2 J�; j 2 J+g:The characterization of the unitary modules has been obtained in [N�98, Prop. I.11]. If thehighest weight representation L(�;�+) is unitary, then jfj 2 J :�j =Mgj � 2, so that�2k = �dk2 = supf�(��):� 2 �+p g = 2M;and this proves (3). Admissible representations of LTo relate unitary highest weight representations of the conformal algebras to positivede�nite functions on , resp. 1 , we �rst use the correspondence to positive de�nite kernelson the bounded domains explained in [N�98] and then an appropriate Cayley transform toobtain a correspondence between the bounded and the unbounded picture.Proposition III.5. (The Cayley transform) Let H be a real Hilbert space and de�ne theCayley transform c: GL(H) � 1! B(H); Z 7! (1� Z)(1 + Z)�1:Then the following assertions hold:(i) c2 = idGL(H)�1 , i.e., c is a biholomorphic involution of the domain GL(H) � 1 .(ii) c(�Z) = c(Z)�1 and c(Z�) = c(Z)� for 1+ Z 2 GL(H) .

16 September 8, 1999(iii) For Z;W 2 GL(H)� 1 we havec(Z) + c(W )�2 = (1+W �)�1(1 �W �Z)(1 + Z)�1 = (1+ Z)�1(1� ZW �)(1+W �)�1:(iv) The Cayley transform maps the ball Db := fZ 2 B(H): kZk < 1g di�eomorphically ontothe domain �b := fW 2 B(H):W +W � >> 0g:Moreover, for p 2 [1;1] and Dp := Db \ Bp(H) , D := UC \ Db we have �p := c(Dp) =�b \ �1+Bp(H)� and c(D) = � .Proof. (i) For 1+Z 2 GL(H) and W := c(Z) the operator W +1 = 2(1+Z)�1 is invertible,showing that c(GL(H)�1) � GL(H)�1 . One easily checks that c2(Z) = Z for Z 2 GL(H)�1 ,and this proves (i).(ii) These are trivial veri�cations.(iii) The �rst equality follows from(1+W �)�c(Z) + c(W )��(1+ Z) = (1+W �)(1� Z) + (1�W �)(1+ Z) = 2(1�W �Z):The second one follows from the observation that the left hand side does not change if we exchangeZ and W � .(iv) For kZk < 1 the operator 1 � Z�Z is positive de�nite, so that (iii) implies that the sameholds for c(Z) + c(Z)� , i.e., c(Z) 2 �b .Next we show that 1 + �b � GL(H). Since 1 + �b � �b , it su�ces to show that�b � GL(H). So let W 2 �b . Then there exists c > 0 with h(W + W �):v; vi � ckvk2 forall v 2 H . Hence(3:1) ckvk2 � hW:v; vi + hv;W:vi � 2kWkkvk2implies that W:v = 0 entails v = 0, hence that W is injective. The same argument shows thatW � is injective, so that (W:H)? = kerW � = f0g implies that W has dense range. Further(3.1) implies that W has closed range so that W is surjective, hence invertible. We conclude inparticular that for each W 2 �b the operator 1+W is invertible. Now Proposition III.1 showsthat Z := c(W ) satis�es 1� Z�Z = (1+ Z�) (W +W �)2 (1 + Z) >> 0;i.e., kZk < 1, and therefore c(Db) = �b . Finallyc(Z) � 1 = (1 � Z)(1 + Z)�1 � 1 = (1 � Z � 1� Z)(1+ Z)�1 = 2Z(1 + Z)�1shows that Z 2 Bp(H) is equivalent to c(Z) � 1 2 Bp(H), and this proves the second part.Now we consider the real Hilbert space H = l2(J;K) for an in�nite set J and K 2fR;C ;H g . For K = H we write HC for the H -vector space H viewed as a complex vectorspace with respect to the inclusion C ! H . We de�neLC :=8<:GL(J; C ) � GL(HC ) for K = R,GL(J; C )� GL(J; C ) for K = C ,GL(2J; C ) � GL(HC ) for K = H .De�nition III.6. (a) If G is a group endowed with an involutive antiautomorphism � , thenan involutive representation of G on a Hilbert space V is a group homomorphism �:G! GL(V )with �(�:g) = �(g)� for all g 2 G .(b) On the group L we consider the involutive antiautomorphism � (g) = g> . We say that aninvolutive representation �:L = GL(J;K) ! GL(V ) on the Hilbert space V is a unitary highestweight representation if there exists a dense subspace V0 � V on which the derived representationof lC is a highest weight representation in the sense of De�nition III.1.(c) An involutive representation � of L is said to be admissible if it is a unitary highest weightrepresentation and if the restriction � j: ! B(V ) is a positive de�nite function.

Representations in L2 -spaces on in�nite-dimensional symmetric cones 17Example III.7. We have already seen in Example II.5 above that for the one-dimensionalrepresentations �k of L given by�k(g) = detR(gg>)� k4 = j detR(g)j� k2the corresponding function on ! R; x 7! detR(x)� k2 is positive de�nite, so that the repre-sentations �k , k 2Z, are admissible since one-dimensional representations are trivially highestweight representations.The following proposition can be applied in all those cases that we have in mind because if': ! B(V ); x 7! �(x) is positive de�nite, then the condition below is automatically satis�ed([HN97, Lemma V.11, Cor. V.13] or [Cl95]). We call a function on a �nitely open domain inan �nite-dimensional vector space �nitely continuous, if its restrictions to all intersections of with �nite-dimensional subspaces are continuous.The following proposition is a special case of far more general extension results for positivede�nite functions by H. Gl�ockner (cf. [Gl99]).Proposition III.8. Each �nitely continuous positive de�nite function ': ! B(V ) satisfying'(x+ y) � '(x) for x 2 ; y 2 C := Herm+(J;K)extends in a unique way to a Gateaux-holomorphic positive de�nite function on the tube domainT .Proof. Instead of , we may also consider the �nitely open domain U := � 1 � U whichcontains in particular the cone C . We note that since U is a directed union of the subspacesUF = Herm(F;K), F � J a �nite subset, it su�ces to prove the assertion for the domainsU \ UF = fx 2 UF :1+ x >> 0g = F � 1 . This reduces the assertion to the case where J is�nite and � U is the open cone of positive de�nite operators. This case is covered by [Ne98a,Th. IV.2].Our next step consists in a description of the admissible representations of the group L .This will be achieved by showing that they correspond exactly to the unitary highest weightrepresentations of the locally �nite hermitian Lie algebras of tube type that have been classi�edin [N�98].The following result will be one of our main tools throughout the remainder of this paper.Theorem III.9. (Factorization Theorem) If � is an admissible representation of L , then thereexist k 2 N0 , c 2 Z, and a continuous involutive representation �0:Lb = GL(H;K) ! GL(V )such that �(g) = detR(gg>)� k4 detR(gg�>)c�0(g) for all g 2 L:It follows in particular that � extends uniquely to a continuous representation � of the BanachLie group L1 . For K = R;H we always have c = 0 .Proof. For the proof we consider the three di�erent cases for K separately.(1) K = R. Here K = GL(J; C ) and the kernel on D = Db \ UC associated to the holomorphichighest weight representation �K of K in [N�98] is given byK(Z;W ) = �K(1� ZW �):Here we simply have K = LC and we put �L := �K . In view of Proposition III.5(iii) we havefor Z;W 2 D the relation�L�c(Z) + c(W )�2 � = �L(1+ Z)�1�L(1� ZW �)�L(1+W )��:This implies that the function �L j� is positive de�nite if and only if the kernel K on D ispositive de�nite.

18 September 8, 1999(2) K = C . Here K = GL(J; C )�GL(J; C ) and the kernel on D associated to the holomorphichighest weight representation �K = �� �+ of K in [N�98] is given byK(Z;W ) = ��(1� ZW �) �+(1�W �Z)�1:The natural inclusion L ! K is given by g 7! (g; g��), so that the corresponding map ! Kis given by Z 7! (Z;Z�1). This map in turn extends to a holomorphic map � ! K given bythe same formula. In this sense we have�L(g) = �K(g; g��) = ��(g)�+(g�)�1; g 2 L:Using Proposition III.5(iii), we now obtain�L�c(Z) + c(W )�2 �= ���(1 + Z)�1(1� ZW �)(1 +W �)�1� �+�(1+W �)(1 � ZW �)�1(1 + Z)�= ���(1 + Z)�1(1� ZW �)(1 +W �)�1� �+�(1+ Z)(1 �W �Z)�1(1+W �)�= �L(1+ Z)�1���(1 � ZW �) �+(1 �W �Z)�1��L(1+W �)�1:(3) K = H . Here K = GL(2J; C ) and the kernel on D associated to the holomorphic highestweight representation �K of K in [N�98] is given byK(Z;W ) = �K(1� (JZ)(JW )�) = �K (1� JZW �J�) = �K(1� JZW �J�1) = �K (1� ZW �):Here the group K acts on JUC � Skew(H) by k:Z = kZk> . On the other hand L acts on Uby g:Z = gZg� . In view of J:(g:Z) = JgZg� = g(JZ)g� , this means that the natural embeddingL! K is given by g 7! g , and this shows that �L(g) = �K(g), g 2 L , de�nes the representation�L of L corresponding to the representation �K of the complex group K = LC .Finally we obtain with Proposition III.5(iii) the relation�L�c(Z) + c(W )�2 � = �L(1+ Z)�1�L(1� ZW �)�L(1+W �)�1= �L(1+ Z)�1�K(1 � ZW �)�L(1+W �)�1:This proves that the function �L on T is positive de�nite if and only if the kernel K is positivede�nite on D .Thus we have seen that in all three cases the function � on is positive de�nite if andonly if the kernel K on D is positive de�nite. If � is a unitary highest weight representationwith highest weight � , this is equivalent to the unitarity of the highest weight module L(�;�+)of the conformal Lie algebra g of UC .Again we have a look at all three cases to see that the assertions of the theorem hold if Kis positive de�nite. We use the description of the unitary highest weight module of g in TheoremIII.4 to see that in all cases the highest weight functionals � on the Lie algebra gl(J; C ) can bewritten as � = �0 + c tr, where �0 has �nitely many non-zero entries:(1) K = R: Here we have for g 2 K :��(g) = ��0(g) det(g)M = ��0 (g) det(g)� k2and for g 2 L we have detR(g) = det(g).(2) K = C : Here we have for g = (g+; g�) 2 K :��(g�; g+) = det(g�)m det(g+)M��0� (g�) ��0+(g+)= det(g�g�1+ )m�M2 det(g�g+)m+M2 ��0� (g�) ��0+(g+):

Representations in L2 -spaces on in�nite-dimensional symmetric cones 19Note that to assign the appropriate sense to the �rst two factors separately, we have to consideran appropriate covering group of K or simply the universal covering group fGL(J; C )� fGL(J; C ).For g 2 L the corresponding element in K is (g; g��), so thatdet(g)m det(g��)M = det(gg�)m�M2 det(gg��)m+M2 = det(gg>)� k2 det(gg�>)m+M2 :Here both factors make sense since the �rst one is de�ned on L , and so that the second one isalso de�ned. It follows in particular that m +M 2 2Zwhenever we start with a representation�L of L .(3) K = H : First we note that for g 2 L = GL(J;K) we havedetR(g) = detC (gg�) = j detC (g)j2:Hence we obtain for g 2 K the relation ��(g) = det(g)M��0(g) and for g 2 L we havedet(g)M = j detC (g)jM = detR(g)M2 = detR(g)� k2 :Since a representation ��0 of GL(J; C ) with supp�0 �nite extends in a canonical way toa holomorphic representation of the group GL �l2(J)� ([Ne98b, Prop. III.10, Cor. III.11, LemmaIII.12]), the proof is complete.Remark III.10. (a) It is interesting to observe that, even though there exist unitary highestweight representations of covering groups of K = LC which do not factor to the group K(this happens if the numbers M or m are not integral which is possible in case (1) and (2)),the corresponding representation of L is almost always well de�ned in the sense that it factorsthrough the inclusion L! K . The only exception is the case K = C with M +m 6= 0.(b) Suppose that K = C . Then 2c = m +M may be any real number. Then g 7! det(gg��)cis not de�ned as a representation of L , and we have to consider its universal covering groupeL �= fGL(J; C ) instead. Only for c 2Zthis representation factors over the group L .IV. The Representation on the L2 -spacesIn this section we eventually come to the description of the representation of the group U1oL1 ,resp. U2oL2 , on the vector-valued L2 -spaces on U�1 , resp. U�2 , that we assign to an admissiblerepresentation of L . First we explain that the realization on U�1 leads in a quite straightforwardmanner to a representation of U1 oL1 . The interesting behavior shows up if one wants to makethe projective representation of the bigger group U2 o L2 on the L2 -space explicit. To assigna sense to the cocycles for this group, one encounters for the realization on U�1 , we �rst haveto pass to a realization on U�2 . The existence of the measures on U�2 is obtained by extendinga modi�ed representation �2 from the domain �1 to the bigger domain �2 . This is done byreplacing the determinant factor in the representation (cf. the Factorization Theorem) by theregularized Hilbert-Schmidt determinant. After working out the cocycles for the action of thegroup U2 o L2 , we show that the unitary multiplier representation of U2 o L2 obtained thatway leads to a non-trivial central extension of the group, hence cannot be modi�ed to a unitaryrepresentation of the group itself.The following proposition generalizes the discussion in Example II.5 to the case of higher-dimensional representations of L .Proposition IV.1. If �:L! GL(V ) is an admissible representation, then it uniquely extendsto a continuous representation � of L1 and there exists a unique Herm+(V )-valued measure �1on U�1 with L(�1)(x) = �(1 + x) for x 2 1 � 1 � U1:

20 September 8, 1999The prescription(4:1) ��1(u; g):f)(�) := ei�(u)e 12�(1�gg>)�(g�>):f(g�1:�)de�nes a unitary representation of the group U1oL1 on the V -valued L2 -space L2(U�1 ; �1) . Thecorresponding representation on the reproducing kernel Hilbert space HK � Hol(�1; V ) , whereK(z; w) = �� z+w�2 � and the isometry L2(U�1 ; �1)!HK is given by f 7! bf withh bf (z); vi = hf; e� z�12 vi; z 2 �1is given by (�1(u; g):f)b(z) = �(g): bf�g�1:(z � 2iu)� for g 2 L1; u 2 U1; z 2 �1:Proof. First we use Theorem III.9 to see that � extends to a continuous representation of theBanach Lie group L1 . In view of the continuity of � , the function1 � 1! B(V ); x 7! �(1+ x)is positive de�nite as a function on the convex open domain 1 . Now Theorem I.7 implies theexistence of a unique Herm+(V )-valued measure �1 on U�1 with the required properties.One easily checks that (4.1) de�nes a representation on the space V U�1 of all V -valuedfunctions on U�1 . We claim that the measure �1 transforms under L1 according tod�1(g�1:�) = �(g�1) � �e�(1�gg>)d�1(�)� � �(g�>):This will be done by showing that both measures have the same Laplace transforms. For the lefthand side we getZU�1 e��(x)d�1(g�1:�) = ZU�1 e�(g:�)(x)d�1(�) = ZU�1 e��(g�1 :x)d�1(�) = �(1 + g�1:x)and for the right hand side�(g�1)�ZU�1 e��(x)e�(1�gg>) d�1(�)��(g�>) = �(g�1)�ZU�1 e��(x�1+gg>) d�1(�)��(g�>)= �(g�1)�(x + gg>)�(g�>) = �(g�1xg�> + 1) = �(1 + g�1:x):That the action of L1 on V U�1 induces a unitary action on the space L2(U�1 ; �1) can now beseen as follows. If f 2 L2(U�1 ; �1), thenk�1(g):fk22 = ZU�1 P(g:f)(�) d�1(�) = ZU�1 e�(1�gg>)�(g�>)Pf(g�1 :�)�(g�1)d�1(�)= ZU�1 Pf(g�1 :�)�e�(1�gg>)�(g�1)d�1(�)�(g�>)� = ZU�1 Pf(g�1 :�)d�1(g�1:�) = kfk22:To obtain the formula for the action on HK , we �rst observe that for each v 2 V we haveh(eiuf)b(z); vi = heiuf; e� z�12 vi = hf; e�iue� z�12 vi = hf; e� z�2iu�12 vi = h bf (z � 2iu); vi:For g 2 L1 and z 2 �1 we obtain(g:(e� z�12 v))(�) = e 12�(1�gg>)e� z�12 (g�1:�)�(g�>):v = e 12�(1�gg>)e� (g:(z�1))2 (�)�(g�>):v= e� gg>�1+g:(z�1)2 (�)�(g�>):v = e� g:z�12 (�)�(g�>):v:For g 2 L1 and z 2 �1 we thus geth(g:f)b(z); vi = hg:f; e� z�12 vi = hf; g�1:e� z�12 vi = hf; e� g�1:z�12 �(g>):vi= h bf (g�1:z); �(g>):vi = h�(g): bf (g�1:z); vi:Putting these formulas together, the assertion follows.In the following it will be important to go one step further and to realize the measure onthe space U�2 instead of U�1 . To obtain this realization, we need the generalized determinant.

Representations in L2 -spaces on in�nite-dimensional symmetric cones 21De�nition IV.2. Let H be a Hilbert space and X 2 B2(H). Then (1+X)e�X �1 2 B1(H)follows from 1+X � eX = X2(� � �). Hence the generalized determinantdet2(1 +X) := det �(1 +X)e�X �makes sense for X 2 B2(H) (cf. [Mi89, Prop. 6.2.3]). This means that for g 2 GL2(H) we havedet2(g) = det(ge1�g):For g 2 GL1(H) this simpli�es to det2(g) = det(g)etr(1�g):Lemma IV.3. The generalized determinant satis�es for x; y 2 1+B2(H) the relationdet2(xy) = det2(yx) = det2(x) det2(y)e� tr((x�1)(y�1)):Proof. (cf. [Mi89, p.138]) First we observe that both sides are continuous functions on(1 + B2(H)) � (1 + B2(H)). Hence it su�ces to prove the equality for (x; y) in the densesubset (1 +B1(H)) � (1 +B1(H)), where we havedet2(xy) = det(x) det(y) det(e1�xy) = det(xe1�x) det(ye1�y)etr(x�1+y�1+1�xy)= det2(x) det2(y)etr(x�1+y�xy) = det2(x) det2(y)e� tr((x�1)(y�1)):Let � be an admissible representation of L1 , and for K = C we assume that c = 0 in thefactorization from Theorem III.9:�(g) = detR(g�>g�1) k4 �0(g) for all g 2 L1with k 2 N0 . It is clear that this expression does not make sense for g 2 L2 . Since �0 extendsin a canonical way to L2 , we de�ne a continuous map�2:L2 ! GL(V ); g 7! detR;2(g�>g�1) k4 �0(g)and note that for g 2 L1 we have �2(g) = e k4 tr(1�g�>g�1)�(g) (cf. De�nition II.6). Thus �2 jL1 isa multiplier representation which de�nes the same projective representation as �0 . The followinglemma displays the corresponding cocycle.Lemma IV.4. For x; y 2 L2 we have�2(xy) = �2(x)�2(y)e� k4 trR((x�>x�1�1)(y�>y�1�1)) and �2(x>) = �2(x)�:Proof. The �rst formula is an immediate consequence of Lemma IV.3, and the second onefollows from �2(g>) = detR;2(g�1g�>) k4 �0(g>) = detR;2(g�>g�1) k4 �0(g)� = �2(g)�:Even though tr does not makes sense as a linear functional on U2 , we want to assign asense to the functional (g: tr� tr) for g 2 L2 . Here one has to be careful because naively onewould consider (g�1: tr� tr)(x) = tr(gxg> � x)which in general does not makes sense for g 2 L2 because there exist orthogonal operators uand Hilbert-Schmidt operators x for which uxu�1 � x is not of trace class. Nevertheless, theapproach described in the following lemma works.

22 September 8, 1999Lemma IV.5. For g 2 L2 and x 2 U2 we de�ne c(g) 2 U�2 byc(g)(x) := kd2 tr �(g�>g�1 � 1)x�:(i) We have the cocycle identity c(g1g2) = g1:c(g2) + c(g1) for g1; g2 2 L .(ii) �:g := g�1:��� c(g)� de�nes an a�ne right action of L2 on U�2 .Proof. (i) For each x 2 U2 we have the relationc(g1g2)(x) = kd2 tr �(g�>1 g�>2 g�12 g�11 � 1)x�= kd2 tr �(g�>1 g�>2 g�12 g�11 � g�>1 g�11 )x�+ kd2 tr �(g�>1 g�11 � 1)x�= kd2 tr �(g�>2 g�12 � 1)g�11 xg�>1 �+ c(g1)(x)= (g1:c(g2))(x) + c(g1)(x):(ii) This is an immediate consequence of (i).Proposition IV.6. If � is an admissible representation of L1 , �2:L2 ! GL(V ) is de�ned by�2(g) = detR;2(g�>g�1) k4 �0(g);and ': 2 ! B(V ) by '(x) := detR;2(x)� k2 �0(x) , then there exists a unique Herm+(V )-valuedmeasure �2 on U�2 with L(�2)(x) = '(1 + x) for 1+ x 2 2:Moreover ��2(g):f�(�) = e 12�(1�gg>)�2(g�>):f�g�1:(�� c(g))�(4:2) = det2;R(gg>) k4 e 12�(1�gg>)�0(g�>):f�g�1:(�� c(g))�de�nes a unitary representation of L2 on the space L2(U�2 ; �2) .Proof. For � 2 U�1 we write �� for the corresponding point measure and note that theconvolution �� � �1 is a measure on U�1 withL(�� � �1)(x) = e��(x)�(1+ x):The function ' is positive de�nite because it is continuous, and for 1+ x 2 1 we have'(1+ x) = e k2 trRx�(1+ x) = e k2 trRxL(�1)(x) = L(�� k2 trR� �1)(x):Using Theorem I.7(iii), we therefore obtain a unique Herm+(V )-valued measure �2 on U�2 withL(�2)(x) = '(1+ x) for 1+ x 2 2 .To see that the action (4.2) induces a unitary representation of L2 on the space L2(U�2 ; �2);we �rst show that(4:3) d�2(g�1:�) = �2(g�1)�e(�+c(g))(1�gg>)d�2��+ c(g)���2(g�>):This will be done by showing that both have the same Laplace transforms. First we observe thatdetR;2(1+ g�1xg�>)� k2 = detR;2(1+ g�>g�1x)� k2= detR;2(gg>) k2 detR;2(gg> + x)� k2 e� kd2 tr((gg>�1)g�>g�1x)= detR;2(gg>) k2 detR;2(gg> + x)� k2 e� kd2 tr((1�g�>g�1)x)= detR;2(gg>) k2 ec(g)(x) detR;2(gg> + x)�k2 :(4:4)

Representations in L2 -spaces on in�nite-dimensional symmetric cones 23For the Laplace transform of the left hand side of (4.3) we now getZU�2 e��(x)d�2(g�1:�) = detR;2(1+ g�1:x)� k2 �0(1+ g�1:x)= detR;2(1+ g�1xg�>)� k2 �0(1+ g�1xg�>)(4:4)= detR;2(gg>) k2 ec(g)(x) detR;2(gg> + x)� k2 �0(g�1)�0(gg> + x)�0(g�>)= detR;2(gg>) k4 detR;2(g>g) k4 ec(g)(x)�0(g�1)'(gg> + x)�0(g�>)= ec(g)(x)�2(g�1)'(gg> + x)�2(g�>)and for the Laplace transform of the right hand side of (4.3) we obtain�2(g�1) ZU�2 e��(x)e(�+c(g))(1�gg>) d�2(�+ c(g)) �2(g�>)= �2(g�1) ZU�2 e�(��c(g))(x)e�(1�gg>) d�2(�) �2(g�>)= �2(g�1)ec(g)(x) ZU�2 e��(x�1+gg>) d�2(�) �2(g�>)= �2(g�1)ec(g)(x)'(x+ gg>)�2(g�>):That the action of L2 on V U�2 induces a unitary action on the space L2(U�2 ; �2) now follows forf 2 L2(U�2 ; �2) fromkg:fk22 = ZU�2 P(g:f)(�) d�2(�) = ZU�2 e�(1�gg>)�2(g�>)Pf(g�1 :(��c(g))�2(g�1) d�2(�)= ZU�2 e(�+c(g))(1�gg>)�2(g�>)Pf(g�1 :�)�2(g�1) d�2(�+ c(g))= ZU�2 Pf(g�1 :�)e(�+c(g))(1�gg>)��2(g�1) d�(�+ c(g))�2(g�>)�= ZU�2 Pf(g�1 :�)d�2(g�1:�) = kfk22:It is instructive to see how the cocycle representation �2 of U2oL2 on L2(U�2 ; �2) looks likein the realization of the space L2(U�2 ; �2) as a reproducing kernel Hilbert space HK � Hol(�2; V ),where the kernel is given by K(z; w) = �2� z+w�2 � , and the isometry is given by f 7! bf withh bf (z); vi = hf; e� z�12 vi; z 2 �2(cf. Theorem I.7).Proposition IV.7. For g 2 L2 and z 2 �2 we have(�2(g):f)b(z) = e� 12 c(g)(z�1)�2(g): bf (g�1:z):Proof. For g 2 L2 and z 2 �2 we have, in view of Proposition IV.6,(g:(e� z�12 v))(�) = e 12�(1�gg>)e� z�12 (g�1:(�� c(g)))�2(g�>):v= e 12�(1�gg>)e� g:(z�1)2 (�� c(g))�2(g�>):v= e 12 c(g):(g:(z�1))e� g:(z�1)+gg>�12 (�)�2(g�>):v= e 12 c(g):(g:(z�1))e� g:z�12 (�)�2(g�>):v

24 September 8, 1999In view ofc(g)(g:(z � 1)) = k2 tr((g�>g�1 � 1)g(z � 1)g>)= k2 tr(g>(g�>g�1 � 1)g(z � 1)) = k2 tr((1� g>g)(z � 1)) = �c(g�1)(z � 1);this simpli�es to (g:(e� z�12 v))(�) = e� 12 c(g�1):(z�1)e� g:z�12 (�)�2(g�>):v:For g 2 L2 and z 2 �2 we thus obtain with Lemma IV.4:h(g:f)b(z); vi = hg:f; e� z�12 vi = hf; g�1:e� z�12 vi = hf; e� 12 c(g):(z�1)e� g�1:z�12 �2(g>):vi= e� 12 c(g):(z�1)h bf (g�1:z); �2(g>):vi = e� 12 c(g):(z�1)h�2(g): bf (g�1:z); vi:One motivation to look for a realization on the bigger space U�2 is that this realization �tswell with the action of the larger group L2 suggested by the discussion in Example I.8 on thespaces L2(U�2 ; �2k), resp. L2(U�1 ; �1k). The natural isomorphism between these two spaces is givenby(4:5) �:L2(U�1 ; �1k)! L2(U�2 ; �2k); �(f)(�) = f(kd2 tr+� jU1):Remark IV.8. (a) Let �1 be the measure on U�1 with L(�1)(x) = �(1 + x) for 1+ x 2 1 .Since the right hand side is a di�erentiable function on the open domain 1 � 1 � U1 , we cancalculate its derivative as follows:d�(1)(y) = dL(�1)(0)(y) = � ZU�1 �(y) d�1(�):Hence we can view the linear function �d�(1):U1 ! Herm(V ) as the expectation value E(�1)of the measure �1 on U�1 . For the special case �(g) = detR(g)� k2 we obtain in particularE(�1k) = k2 trR and E(�� k2 trR� �1) = E(�1k)� k2 trR= 0:In general E(�� k2 trR� �1) = E(�1)� k2 trR= �d�0(1)extends to a continuous linear function on Ub . Since the expectation value of the shifted measureis a linear function U1 ! Herm(V ) that extends continuously to the bigger space U2 , this shiftis the most natural one, to obtain a related measure on the space U�2 . For the scalar case wehave in particular E(�2) = 0.The projective representation of the a�ne groupIn Proposition IV.6 above we have constructed a natural representation of the group L2on the space L2(U�2 ; �) given by��2(g):f�(�) = e 12�(1�gg>)�2(g�>):f�g�1:(�� c(g))�:On the other hand we have seen in Proposition IV.1 how L1 acts on the space L2(U�1 ; �1). Onthe group L1 both representations are compatible in the sense that the unitary map�:L2(U�1 ; �1)! L2(U�2 ; �2); �(f)(�) = f�kd2 tr+� jU1�

Representations in L2 -spaces on in�nite-dimensional symmetric cones 25is L1 -equivariant.If we consider the representation of U1 on L2(U�2 ; �2) obtained by the requirement that �is equivariant, we �nd the formula��1(u):f�(�) = ei( kd2 tr +�)(u)f(u):Since the functional tr cannot be extended in a canonical way to U2 , this formula cannot beused to obtain a representation of U2 . Thus we simply put��2(u):f�(�) = ei�(u)f(�); u 2 U2; � 2 U�2 ;and note that for u 2 U1 we have �2(u) = e�i kd2 tr�1(u); so that the corrosponding projectiverepresentations are the same. For (u; g) 2 U2 o L2 we de�ne�2(u; g) := �2(u)�2(g)and claim that �2(g)�2(u)�2(g�1) = e�ic(g)(g:u)�2(g:u):In fact, for f :U�2 ! V and h:U�2 ! C we have��2(g):(hf)�(�) = h(g�1:(�� c(g))) � ��2(g):f�(�)and therefore��2(g)�2(u):f�(�) = ei(g�1 :(��c(g)))(u)��2(g):f�(�) = ei(��c(g))(g:u)��2(g):f�(�)so that ��2(g)�2(u)�2(g�1):f�(�) = ei(��c(g))(g:u)f(�) = e�ic(g)(g:u)��2(g:u):f�(�):More explicitly we havec(g)(g:u) = kd2 tr �(g�>g�1 � 1)gug>)� = kd2 tr �g>(g�>g�1 � 1)gu� = kd2 tr �(1 � g>g)u�:On the group G := U2 o L2 we now consider the multiplier given bym�(u1; g1); (u2; g2)� = e�ic(g1)(g1:u2):This multiplier de�nes a central extension q: bG! G with bG = G�T and the multiplication(u1; g1; z1) � (u2; g2; z2) = �u1 + g1:u2; g1g2; z1z2e�ic(g1)(g1:u2)�;and we obtain a unitary representation of bG on L2(U�2 ; �2) by b�2(u; g; z) := z�2(u; g):Since the multiplier m:G�G!T is smooth, it is clear that bG is a Banach Lie group.Proposition IV.9. The central extension bG ! G de�ned by the multiplier m is non-trivialas a central extension of Banach Lie groups.Proof. To see that the central extension bG of G is non-trivial, it su�ces to show that itis non-trivial on the Lie algebra level. According to [Va85, Th. 7.37], the Lie algebra cocyclede�ned by m is given byc(X;Y ) = @2@t@s t;s=0 m(exp tX; exp sY ) � @2@t@s t;s=0 m(exp sY; exp tX):It follows immediately that c(X;Y ) = 0 for X;Y 2 u2 �= U2 or X;Y 2 l2 . For u 2 U2 andX 2 l2 we have m(tu; exp sX) = 1 andm(exp sX; tu) = e�ic(exp sX)(exp sX:u2) = e�i k2 tr�(1�exp(sX>) exp sX)tu�:Hence c(X;u) = dds s=0 � ik2 tr �(1� exp(sX>) exp sX)u� = ik2 tr �(X +X>)u�:For X = X> we obtain in particular c(X;u) = ik tr(Xu): On the other hand [X;u] = X:u =Xu+ uX> = Xu+ uX . If c is a trivial cocycle, then there exists a continuous linear functionalf on g = U2 o l2 with �([X;Y ]) = c(X;Y ) for X;Y 2 g and in particular�(Xu + uX) = ik tr(Xu) = ik2 tr(Xu + uX):This implies that � = ik2 tr on the space U1 of all trace class operators in U2 , contradicting thecontinuity on U2 .

26 September 8, 1999Problem IV. Describe the space H1c (L2; U�2 ) = Z1c (L2; U�2 )=U�2 of equivalence classes ofcontinuous 1-cocycles of L2 with values in U�2 . The same calculation as in the proof of LemmaIV.5 shows that for each A 2 Ub the prescription cA(g)(x) := tr((g�>Ag�1 � A)x) de�nes anelement in Z1c (L2; U�2 ), and this cocycle is trivial if and only if A 2 U2 . We thus obtain aninclusion Ub=U2 ,!H1c (L2; U�2 ). Is this map surjective?V. Ergodicity of the scalar measures on U�As before, let K 2 fR; C ;H g and consider the special case where the set J is countable, sothat we may w.l.o.g. assume that J = N . Then U = Herm(N;K) the Jordan algebra of �niteN� N-matrices and the algebraic dual space U� is the space of all in�nite hermitian matrices.As a subgroup of L , the connected group LK := U (N;K)0 = lim�! U (n;K)0 acts in a natural wayon U� . In this subsection we explain how a recent result of G. Ol'shanski�� and A. Vershik onLK -ergodic measures on these spaces is related to the measures considered in Section II.Theorem V.1. (Ol'shanski��{Vershik) Let � be an LK -invariant probability measure onHerm(N;K)� and b�: Herm(N;K) ! C its Fourier transform. Then � is ergodic if and onlyif � is multiplicative in the sense that there exists a function F :R ! C such that if A =diag(�1; : : : ; �r) 2 Herm(r;K) � Herm(N;K) , thenb�(A) = rYj=1F (�j):In particular the measures � on Herm(N;K)� whose Fourier transform is given by thefollowing formula are ergodic:(3:1) b�(A) = ei�1 trA��2 trA22 1Yk=1det �e�ixkA(1� ixkA)�1�;where det denotes the holomorphic extension of the real determinant function of the Jordanalgebra Herm(N;K) to Herm(N;K)C , �1 2 R, �2 � 0 , and P1k=1 x2k <1 .Note that the factor ei�1 trA corresponds to the point measure in �1 tr, and that e��2 trA22 corre-sponds to a Gaussian measure U with respect to the real scalar product hA;Bi = �2 trR(AB).The corresponding measures are supported by the positive cone if and only if �2 = 0, xj � 0for all j and P1k=1 xk � �1 ([OV96, Rem. 2.11]). The paper [OV96] also contains the resultthat for K = C (the argument also works for K = R;H ) these are all the ergodic LK -invariantprobability measures measures.Proof. (cf. [OV96]) The �rst assertion (cf. [OV96, Th. 2.1]) is proved in [Ol90] for sphericalfunctions of the pairs �GL(N;K); U (N;K)0� . Here we need the analog result for the correspond-ing motion groups G = Herm(N;K) o U (N;K)0:Nevertheless, [Ol90, Th. 23.6] also covers this case, and a closer inspection of the proof of [Ol90,Th. 23.8] shows that it carries over from GL(N;K) to the motion groups G .The second assertion is also discussed extensively in [OV96]. To explain why formula (3.1) isthe Fourier transform of a probability measure, we explain how such measures can be constructedif xn = 0 for n > k .Let z 2 R and E 2 Herm(k;K). We consider the map�z;E : (KN)k �= M (N; k;K) ! U�; v = (v1; : : : ; vk) 7! z1+ 12vEv�;where vEv� should be read as a matrix product. We may w.l.o.g. assume thatE = diag(x1; : : : ; xk):

Representations in L2 -spaces on in�nite-dimensional symmetric cones 27Then �z;E is LK -equivariant and the image of the Gaussian measure on (KN)k is an analog of aWishart distribution (cf. De�nition II.2). Let denote the Gaussian measure on (KN)k and put�z;E := ��z;E . If E = E1+E2 and z = z1+ z2 , then we pointwise have �z;E = �z1;E1 +�z2;E2 :Therefore �z;E = �z1;E1 � �z2 ;E2 :To calculate the Fourier transforms it therefore su�ces to do that for the special case where Eis a rank-1-matrix, say E = x1E11 and z = 0. Then�z;E(v) = x12 vE11v� = x12 v1v�1 ;so that, up to the factor x1 , the measure �0;E is a Wishart distribution, and the same calculationas in Example II.5 yields b�0;E(A) = det(1� ix1A)�1:This shows that in general we have the formulab�z;E(A) = eiz trA kYj=1 det(1 � ixjA)�1 = ei(z+Pnk=1 xk) trA kYj=1det �e�ixjA(1� ixjA)�1�:The general formula (3.1) is obtained for the in�nite convolution product� = ��11 � U � ��x1;x1E11 � ��x2;x2E22 � � � � :Corollary V.2. The measures �k on the positive cone in Herm(N;K)� are ergodic for theaction of the group LK = U (N;K) . The representation of the semidirect product group U o LKon L2(U�; �k) is irreducible.Proof. For z = 0 and Ek := E11 + : : :+ Ekk we obtain in particularb�0;Ek(A) = kYj=1 det(1 � iA)�1 = det(1� iA)�k;i.e., �k = �0;Ek is the measure from Example II.5. Now the ergodicity follows from TheoremV.1. To see that the natural rerpresentation of the group G := U o LK on H := L2(U�; �k)is irreducible, we �rst observe that the von Neumann algebra of the multiplication operatorscoming from L1(U�; �k) is a maximal abelian subalgebra of B(H) which is generated by theimage of the representation of U . Hence the commutant of the representation of G consists ofthe LK -invariant elements in L1(U�; �k), which, in view of the ergodicity of the measure, forma one-dimensional space. Now the irreducibility of the representation is a consequence of Schur'sLemma.Note that Corollary V.2 is false in the �nite-dimensional case. In this sense the irreducibilityof the action of U oLK on L2(U�; �k) is a new phenomenon in the in�nite-dimensional setting.Problem V. Can anything comparable to Corollary V.2 be said about operator-valued mea-sures? VI. Additional Information on the MeasuresLet � be an admissible representation of L and k the corresponding number which can becomputed as explained in Theorem III.4 from the highest weight � of the corresponding rep-resentation of lC . We further write � for the measure on U� with L(�)(x) = �(1 + x) forx 2 . For a trace class operator S 2 Herm1(V ) we write �S for the scalar measure obtainedby �S(E) = tr ��(E)S� .

28 September 8, 1999Lemma VI.1. The measure �S is absolutely continuous with respect to the measure �k onU� .Proof. We have to show that �k(M ) = 0 implies that �S(M ) = 0 holds for a measurablesubset M � U� . Since each S can be approximated by a positive linear combination of rank oneoperators, we may w.l.o.g. assume that S = Pv for some v 2 V . We put �v := �Pv . Then foreach M 2 S the mapping V ! R; v 7! �v(M ) = h�(M ):v; vi is continuous, so that it su�ces toprove �v(M ) = 0 for each v in a dense subspace V0 � V .To see how we can obtain a suitable subspace, we recall that the representation of L1 on Vis a direct limit of representations of �nite-dimensional subgroups LE � L corresponding to �nite-dimensional Jordan algebras Herm(E;K), where E � J is a �nite subset. The correspondingsubspace V0 := SVE of V is dense.If �E is the corresponding representation of LE on VE , then the corresponding positivede�nite function on E is given by 'E(x) = PE'(x + e � eE)P �E , where PE :V ! V is theorthogonal projection onto VE . Let �E denote the corresponding Herm+(VE)-valued measureon ?E . ThenL(�E)(x) = 'E (x+ eE ) = PE'(x+ e)P �E = PEL(�)(x)P �E = L(�PE )(x)for x 2 E implies that the restriction map �E :U� ! U�E , which is the adjoint of the inclusionmap E :UE ,! U , satis�es ( �E )��PE = �E :According to [HN97, Th. V.17] (or [Cl95]), the measure �E has a density �E with respectto the measure �E;k := ( �E)��k , i.e., �E = �E � �E;k . For UE � UF we likewise have( �E;F )��F;PE = �Ewhich implies that the density �F satis�es �E � �E;F = PE�FP �E (cf. [Ne98a, Lemma I.18]).We further conclude that�E � �E = �E � �E;F � �F = PE(�F � �F )P �E :This shows that the function � := �E � �E :U� ! Herm(VE ) is measurable with( �F )�(��k) = (PE�FP �E)�F;k = �F;PEwhenever UE � UF . This proves that ��k = �PE ; so that �PE has a density for each E . Thiscompletes the proof of the lemma.Theorem VI.2. If � is the unique measure on U� with L(�)(x) = �(1 + x) for x 2 , andk 2 N0 as in Theorem III.4, then the set ?k of elements of rank � k in the cone ? � U� isthick with respect to the measure � . If, in addition, J is countable, then ?k is a measurablesubset of U� .Proof. If E � U� is a measurable subset not intersecting the set ?k , then Lemma VI.1 showsthat for each positive trace class operators S on V we have �S (E) = tr(�(E)S) = 0 becausethe set ?k contains the range of the map Q (cf. Example II.5), hence is thick with respect tothe measure �k = Q� V . We conclude that �(E) = 0, and this means that ?k is thick withrespect to � .If, in addition, J is countable, then the weak-�-topology turns U� into a metrizableseparable topological vector spaces. >From that it follows easily that all closed subsets of U� aremeasurable and hence that ?k is measurable for each k 2 N0 .Remark VI.3. (a) Theorem VI.2 implies in particular that if S denotes the � -algebra on ?kthat we obtain by intersecting measurable subsets of U� with ?k , then we obtain a measuree� on S by e�(E \ ?k) := �(E). This is compatible with the observation that one could alsoapply Theorem I.3 directly to the projective system of the domains ?k;F corresponding to the�nite-dimensional Jordan algebras Herm(F;K), F � J �nite.

Representations in L2 -spaces on in�nite-dimensional symmetric cones 29(b) Let r:U�1 ! U� denote the restriction map. Then r��1 = � and therefore each measurablesubset E � U� with E \?k = � satis�es�1(r�1(E)) = �(E) = 0:But this does not necessarily imply that r�1(?k) is thick in U�1 with respect to �1 because ifE � U�1 is measurable with E \ r�1(?k) = �, then r(E) need not be measurable in U1 .(c) One could also try to see that the measure �1k can also be realized on the cone of positivefunctionals in U�1 which can be identi�ed in a canonical way with the cone of all bounded positiveoperators on H (cf. [Ne98a]).If we consider the Jordan algebra U1 as the direct limit of the set of all those �nite-dimensional subspaces UF ; F � J; with UF = UF \U+1 �UF \U+1 , then the algebraic dual U�1 isthe projective limit of the dual spaces U�F and likewise the dual cone (U+1 )? can be viewed as theprojective limit of the �nite-dimensional cones (U+F )? , where U+F := UF \ U+1 . Unfortunatelythe corresponding restriction maps (U+F )? ! (U+E )? are not always surjective if UE � EF , sothat the assumption (2) of Theorem I.3 is not satis�ed in an obvious fashion.Example VI.4. We have seen above that for each admissible representation � of L there existsa k 2 N0 such that the set ?k is thick for the corresponding operator-valued measure � on U� .Moreover, for each positive trace class operator S 2 Herm+1 (V ) the positive measure �S on U�has a density with respect to the measure �k . To see how one can make this weak type of adensity more explicit, we consider the particular case where K = R and�(g) = det(g)� 12 g�>which corresponds to the odd part of the metaplectic representation of the two-fold covering ofSp(J;R). On k = gl(J; C ) the highest weight is given by� = �12 tr�"j0 ;where j0 2 J is maximal with respect to the ordering de�ning the positive system �+k . Thisshows that M = �12 and therfore that k = 1 (cf. Theorem III.4). The decomposition � =det� 12 �0 is given by �0(g) = g�> , and for the measure �11 corresponding to det� 12 we haveL(�11)(x) = det(1+ x)�12 ; 1+ x 2 1:The determinant function det:GL1(H) ! C � is a homomorphism of Lie groups, so that itsdi�erential is given by d det(g)(g:X) = det(g)d det(1)(X) = det g � trX:Thus dL(�11)(x)(S) = �12 det(1+ x)� 12 tr((1 + x)�1S):This means that the continuous linear functional dL(�11)(x) on U1 is represented by the boundedhermitian operator �12 det(1+ x)�12 (1+ x)�1 = �12�(1 + x):We conclude that �(1+ x) = �2dL(�11)(x)and therefore that for each S 2 Herm+1 (V ) we haveL(�S)(x) = tr(�(1 + x)S) = �2dL(�11)(x)(S) = 2 ZU�1 tr(�S)e��(x) d�11(�):This means that d�(�) = 2� � d�11(�)holds in the weak sense that for each S 2 Herm+1 (V ) we have d�S(�) = 2�(S) � d�11(�):Similar methods can be used to make other operator-valued measures � more concrete inthe same sense. If k is larger than 1, then one has to use higher derivatives.

30 September 8, 1999References[Cl95] Clerc, J.-L., Laplace transform and unitary highest weight modules, J. Lie The-ory 5:2 (1995), 225{240.[FK94] Faraut, J., and A. Koranyi, \Analysis on symmetric cones," Oxford Mathemat-ical Monographs, Oxford University Press, 1994.[Gl99] Gl�ockner, H., \In�nite-dimensional Complex Groups and Semigroups: Repre-sentations of Cones, Tubes and Conelike Semigrouops," Dissertation, Universityof Technology Darmstadt, 1999.[He89] Herv�e, M., \Analyticity in in�nite-dimensional spaces," de Gruyter, Berlin,1989.[HN97] Hilgert, J., K.{H. Neeb, Vector Valued Riesz Distributions on Euclidian JordanAlgebras, J. Geom. Analysis, to appear.[Mi89] Mickelsson, J., \Current algebras and groups," Plenum Press, New York, 1989.[Ne98a] Neeb, K.{H., Operator valued positive de�nite kernels on tubes, Monatshefte f�urMath. 126 (1998), 125{160.[Ne98b] |, Holomorphic highest weight representations of in�nite-dimensional complexclassical groups, J. reine ang. Math. 497 (1998), 171{222.[Ne99] |, \Holomorphy and Convexity in Lie Theory," book in preparation.[N�98] Neeb, K.{H., and B. �rsted, Unitary Highest Weight Representations in HilbertSpaces of Holomorphic Functions on In�nite Dimensional Domains, J. Funct.Anal. 156 (1998), 263{300.[NeSt98] Neeb, K.{H., and N. Stumme, On the Classi�cation of Locally Finite SplitSimple Lie Algebras, submitted.[Ol90] Ol'shanski��, G. I., Unitary representations of in�nite-dimensional pairs (G;K)and the formalism of R. Howe, in \Representations of Lie groups and relatedtopics," Eds. A. M. Vershik and D. P. Zhelobenko, Gordan and Breach SciencePubl., New York etc., 1990.[OV96] Ol'shanski��, G. I., and A. Vershik, Ergodic Unitarily Invariant Measures on theSpace of In�nite Hermitian Matrices, Amer. Math. Soc. Translations (2) 175(1996), 137{175.[Sh62] Shale, D., Linear symmetries of the free boson �elds, Trans. of the Amer. Math.Soc. 103 (1962), 146{169.[Se58] Segal, I. E., Distributions in Hilbert Space and Canonical Systems of Operators,Transactions of the Amer. Math. Soc. 88 (1958), 12{41.[Se65] |, Algebraic integration theory, Bull. Amer. Math. Soc. 71 (1965), 419{489.[Va85] Varardarajan, V. S., \Geometry of Quantum Theory," Second Edition, SpringerVerlag, New York etc., 1985.[Ya85] Yamasaki, Y., \Measures on in�nite dimensional spaces," Series in Pure Math.5, World Scienti�c, Singapore, 1985.Karl-Hermann NeebTechnische Universit�at DarmstadtSchlossgartenstrasse 7D-64289 DarmstadtDeutschlandneeb@mathematik.tu-darmstadt.de Bent �rstedMatematisk InstitutOdense UniversitetCampusvej 55DK-5230 OdenseDanmarkorsted@imada.ou.dk