TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 351, Number 9, Pages 3771–3792 S 0002-9947(99)02101-7 Article electronically published on March 1, 1999

CAUSAL COMPACTIFICATION AND HARDY SPACES

G. OLAFSSON´ AND B. ØRSTED

Abstract. Let = G/H be a irreducible symmetric space of Cayley type. M Then is diffeomorphic to an open and dense G-orbit in the Shilov boundary M of G/K G/K. This compactification of is causal and can be used to give answers× to questions in harmonic analysisM on . In particular we relate the Hardy space of to the classical Hardy spaceM on the bounded symmetric M domain G/K G/K. This gives a new formula for the Cauchy-Szeg¨okernel for . × M

Introduction In this paper we shall begin a study of the harmonic analysis of a certain class of affine symmetric spaces by introducing a compactification of the space. We shall call these causal compactifications. The idea of using such compactifications has been studied before in special cases, see for example [1], [9] and [10], where Cayley type spaces are treated as well as the case of the metaplectic group, and also recently by V. Molchanov for the case of the orthogonal group of Hermitian type and the corresponding hyperboloid [12]. The causal compactification is especially convenient for studying the noncommutative analogue of the Hardy space, since it is a Shilov boundary of a classical bounded domain. Such non-classical Hardy spaces were introduced by Stanton in [20] in the group case, also studied by Ol’shanskii, and later constructed for symmetric spaces of Hermitian type in [6]. One may trace some of these ideas to the celebrated Gelfand-Gindikin program of understanding the harmonic analysis on a symmetric space by studying the holomorphic extension of functions into domains in the complexification of the space. One of the first questions was to calculate the reproducing kernel for the Hardy space, in the sense of finding the orthogonal projection from L2 onto the Hardy space, i.e. to find explicitly the Cauchy-Szeg¨o kernel which reproduces the holomorphic functions in the Hardy space. One of our aims in this paper is, in addition to the harmonic analysis treated, to make precise the notion of causal compactification. For symmetric spaces of Cayley type there is, as we shall see in this paper, a natural way to obtain a causal compactification and to use it to identify the Hardy space and its Cauchy-Szeg¨o kernel explicitly in terms of classical Hardy spaces, namely a tensor product of two such spaces corresponding to the associated tube domain. By specializing our results we obtain as a corollary an explicit formula for the Cauchy-Szeg¨okernel

Received by the editors April 29, 1996 and, in revised form, March 10, 1997. 1991 Mathematics Subject Classification. Primary 43A85, 22E46; Secondary 43A65, 53C35. Key words and phrases. Causal compactification, Hardy spaces, holomorphic discrete series. The first named author was supported by NSF grant DMS-9626541, LEQSF grant (1996-99)- RD-A-12 and the Danish Research Council.

c 1999 American Mathematical Society

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for the two-dimensional hyperboloid of one sheet - namely a special case of an old formula of Heine: 1 ∞ = (2n +1)P (z)Q (t), t z n n n=0 − X see e.g. [21] p. 322. (We thank J. Faraut for this observation.) Here the functions in the sum are Legendre functions of the first and second kind. Our formula is obtained by setting z = 1 and is the expression of the Cauchy-Szeg¨okernelasa sum of the invariant distributions which define the projections onto the holomorphic discrete series representations for the symmetric space. The numerical coefficients in the sum are due to the standard normalization of the Legendre functions, but as shown in [6] the multiplicity is one in general, so in principle the coefficients are all 1. Although in this paper we deal only with affine symmetric spaces of Cayley type, it is plausible that the ideas and constructions carry over to quite general (reductive) affine symmetric spaces G/H of Hermitian type. Namely, for such a (general) space there will be a causal compactification into the Shilov boundary of a Hermitian symmetric space G1/K1,whereG1 is the group of local causal transformations on G/H. This compactification map extends to a holomorphic imbedding of Ξ, the domain in a suitable complexification of G/H for a Hardy space of functions on G/H. In this way it is possible to directly compare the Hardy space on G/H with the classical one on the Shilov boundary of G1/K1;to have an isomorphism between the two Hardy spaces will require some control over the asymptotics at infinity in Ξ of the functions in the Hardy space. This is an interesting program, and several researchers are presently studying special cases; we hope to pursue it with regard to calculating the Cauchy-Szeg¨o kernels, and also to utilize the compactifications to study the other series of representations as in the original program of Gelfand and Gindikin. The main result in this paper is Theorem 6.6, complemented by Theorem 6.9, giving an explicit isomorphism between the classical Hardy space and the one for the symmetric space. Note that in this case the causal group is G1 = G G,so that the corresponding Hardy space is actually a tensor product of the Hardy× space on G/K tensor itself. Hence we may interpret the result as calculating this tensor product and identifying it with the direct sum of all holomorphic discrete series representations for G/H (it is also possible to do this tensor product calculation di- rectly by algebraic means - this we did first - and to compare it with the parameters in the holomorphic discrete series for G/H). We start by recalling the structure theory of Cayley type spaces where a number of new technical aspects have to be dealt with. One is the normalization of various measures and corresponding Ja- cobians, as we do in Sections 2 and 5. Another is the introduction of a double covering, necessary in two of the series of spaces, similar to the one appearing in [10]. It seems a natural feature of the geometry that one should consider a double covering of Ξ, defined much in the spirit of the Riemann surface for √z, but with z replaced by the complex Jacobian of the (extended) causal compactification. It is a pleasure to thank P. Torasso for the invitation to Poitiers, where we first lectured on these results, J. Faraut and K. Koufany for remarks and discussions, and Odense University and the Mittag-Leffler Institute, where this work was finished. We also thank the referee and the communicating editor for many suggestions to improve the presentation.

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1. Symmetric Spaces of Cayley Type In this section we recall some facts about causal symmetric spaces. We refer to the original paper [14] and the monograph [5] for proofs and more information. Let V be a Euclidean space. A subset C in V is called a cone if C is convex and R+C C. The closed cone C is called nontrivial if C = C and regular if it is pointed⊂ (C C = 0 ) and generating (C C = V). Suppose6 − that the Lie group L acts on V.∩− We say that{ } the cone C is L-invariant− if a(C) C for all a L.Denote ⊂ ∈ by ConeL(V) the set of regular L-invariant cones in V.Let be a manifold. A cone field or causal structure on is a map M M C : m C(m) Tm( ), M3 7→ ⊂ M where C(m) is a closed nontrivial cone in Tm( ), m . The cone field is said to be differentiable if there exists a differentiableM atlas∈MA on T (M) such that for (ϕ, U) A there exits an open V Rm and a closed cone C Rm, m =dim , ∈ ⊂ m ⊂m M such that pr2(ϕ(C(m))) = C, m U.Herepr2:V R R is the projection ∈ × → onto the second factor. Suppose that the Lie group G acts on .Let`g(m):=gm. M · The cone field is said to be invariant if (d`g)(C(m)) = C(g m) Tg m( ). If the action is transitive, then is said to be causal if there exists· a smooth⊂ · GM-invariant cone field on .If Mis causal, then C(m) is invariant under Gm = g G M M m { ∈ | g m = m . On the other hand if C Tm( )isaG invariant cone, then the · } ⊂ M assignment g m (d`g)m(C) defines an invariant causal structure on . Hence the G-invariant· causal7→ structures on correspond bijectively to the GmM-invariant M cones in Tm( ). M Definition 1.1. Let and be manifolds with causal structure m C(m) respectively n MD(n).N Let Φ : be smooth. The pairM3 ( , Φ)7→ is called a causal compactificationN3 7→ of if Φ isM→N a diffeomorphism onto an openN dense subset M of ,and(dΦ)m(C(m)) = D(Φ(m)) for all m .LetGbe a Lie group acting onN such that C is G-invariant. The causal∈M compactification ( , Φ) is called G-equivariantM if is a G-space, Φ is G-equivariant, and D is G-invariant.M N Let G be a connected simple Lie group and g be its Lie algebra. Let GC be the connected simply connected complex Lie group with Lie algebra g := g C. C ⊗R For simplicity we will assume that G GC.Letτbe a nontrivial involution on G commuting with a (fixed) Cartan involution⊂ θ. We assume that τ = θ.Wedenote by the same letters the corresponding involutions on G , g, g , g6 ,andg .Let C C ∗ C∗ H:= Gτ H := Gτ ,andK:= Gθ K := Gθ .BothH and K are connected ⊂ C C ⊂ C C C C as GC is simply connected, cf. [11], p. 171. K is a connected maximal compact subgroup of G and := G/K is a Riemannian symmetric space. Let := G/H. is a pseudo-RiemannianD symmetric space. M M For a vector space V over the field K, T End(V)andλ Klet V(λ, T )= v V T(v)=λv .Leth:= gτ = g(1,τ)and∈ k:= g(1,θ).∈ Then h is the Lie algebra{ ∈ of| H and k is} the Lie algebra of K. Finally we define q := g( 1,τ)and p:= g( 1,θ). Then, as τ and θ commute, − − g = h q = k p = h k q k h p q p . ⊕ ⊕ ∩ ⊕ ∩ ⊕ ∩ ⊕ ∩ Let o = H .Thenqis isomorphic to To( )insuchawaythat(d`h)o corre- sponds to Ad({ h}) q, h H. Thus the causal structuresM on correspond bijectively to the H-invariant| nontrivial∈ cones in q. There are threeM different types of causal structures on : M

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o (CC) Compactly causal: there exists a C ConeH (q) such that C k = . ∈ ∩ 6o ∅ (NCC) Noncompactly causal: there exists a C ConeH (q) such that C p = . (CT) Cayley type: is both compactly causal∈ and noncompactly causal.∩ 6 ∅ We say that Mis irreducible if there are no τ-stable ideals in g that are not contained in h.M From now on we assume that is irreducible. We will also assume that [q, q]=h. This is no loss of generalityM because [q, q] q is an ideal in g and there exists an ideal l contained in h such that g = l ([⊕q, q] q). Let ha := k h p q = gτθ. Denote the center of ha by z(ha). Denote⊕ the center⊕ of k by z(k).∩ We⊕ have,∩ cf. [5, 14]:

H K Theorem 1.2. Let q ∩ = X q k K H : Ad(k)X = X . { ∈H |∀K ∈ ∩ } 1) is causal if and only if q ∩ = 0 . MH K H K H K 6 { } H K 2) q ∩ =(q ∩ k) (q ∩ p)and dim q ∩ 2. H K ∩ ⊕ ∩ ≤ 3) dim q ∩ =2if and only if is of Cayley type. 4) The following statements areM equivalent: i) is compactly causal. MH K ii) q ∩ k = 0 . iii) qz(k) =∩ 06 . { } 5) The following6 { } statements are equivalent: i) is noncompactly causal. MH K ii) q ∩ p = 0 . iii) q p ∩z(h6 a){=} 0 . ∩ ∩ 6 { } Assume that is compactly causal. Then (4) part (iii) implies that is a bounded symmetricM domain and that we can choose Z0 q z(k) such thatD ad Z0 ∈ 0∩ has exactly the eigenvalues 0, +i, i.Wehavek=g(0, ad Z )andpC decomposes + − + 0 0 into two KC-modules pC = p p− with p = gC(i, ad Z )andp− =gC( i, ad Z ). + ⊕ − Both p and p− are abelian. The Cartan involution is an inner automorphism π 0 π 0 2ad Z given on the Lie algebra by θ =Ad(exp2Z)=e and on the group by 0 1 θ = Int(exp([π/2]Z )), where Int(a) stands for the inner automorphism x axa− . + + + 1 7→+ + Let P =expp and P − =expp−. Let log := (exp p )− : P p . + + | → P KCP − is open and dense in GC and P KC P − (p, k, q) pkq + × × 3 7→+ ∈ P KCP − GC is a diffeomorphism. Write g = p(g)k(g)q(g)forg P KCP−. ⊂+ + ∈ Define ζ : P KCP − p by ζ(g):=log(p(g)). The bounded realization of G/K, cf. [2] or [3], is given→ by

b = ζ(g) g G . D { | ∈ } + + + For g GC and Z p with g exp Z P KCP −, define g Z p and k(g,Z) KC by ∈ ∈ ∈ · ∈ ∈

(1.1) g exp Z exp(g Z)k(g,Z)P − . ∈ · The universal factor of automorphy k(g,Z) satisfies (1.2) k(ab, Z)=k(a, b Z)k(b, Z) . · If is noncompactly causal, then there is Y 0 q p such that g = g(0, ad Y 0) g(1, adMY 0) g( 1, ad Y 0)and ∈ ∩ ⊕ ⊕ − (1.3) ha =z(Y0):=g(0, ad Y 0) .

Denote the Killing form on g by B.Set(X Y)θ:= B(X, θ(Y )). Then ( )θ is an inner product on g. | − ·|·

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Lemma 1.3. g(1, ad Y 0) g( 1, ad Y 0)=g( 1,θτ). ⊕ − − Proof. ad Y 0 is semisimple, symmetric and with real spectrum as Y 0 p. Hence 1 0 ∈ g = j= 1 g(j, ad Y ) is an orthogonal direct sum. But the same holds for the decomposition− g = ha g( 1,θτ). As ha = g(0, ad Y 0) the lemma follows. L ⊕ − 0 Let aq be a maximal abelian subalgebra of p containing Y . Because of (1.3) aq is contained in q p and aq is abelian in q.WehaveH=ZK H(aq)Ho, cf. [5, 16]. From now on we∩ will - if not otherwise stated - assume that∩ is an irreducible semisimple symmetric space of Cayley type. Let Y 0 and Z0 beM as above. Define πi (1.4) c =exp Y 0 G . 2 ∈ C Lemma 1.4. Ad(c) g(0, ad Y 0)=id,andAd(c) g( 1, ad Y 0)= iid.Further- | 0 | ± ± more τθ =Ad(c2)=eπi ad Y . Proof. The first part is obvious from the definition of c. By Lemma 1.3 we have Ad(c2) g( 1,θτ)= id. The claim follows now as g(0, ad Y 0)=g(1,θτ). | − − 1 3 Lemma 1.5. 1) Ad(c− )=Ad(c )=τθ Ad(c)=Ad(c) τθ. 2) τ Ad(c)=Ad(c) θ. ◦ ◦ ◦1 ◦ 3) c− HCc = KC. Proof. (1) This follows immediately from Lemma 1.4. For (2) we notice first that 0 0 0 0 1 3 τ(Y )=θ(Y )= Y as Y q p.Thusτ(c)=θ(c)=c− =Ad(c). From − ∈ ∩ 1 2 this we get τ Ad(c)=Ad(τ(c))τ =Ad(c− )θ=Ad(c)Ad(c) τ =Ad(c)θas Ad(c)2 = θτ. (3)◦ follows from (2) as H = Gτ and K = Gθ . C C C C Lemma 1.6. Let X 0 = i Ad(c)Z0. Then the following hold: − 1) X0 =[Y0,Z0] h p. 2) ad(X0) is semisimple∈ ∩ with spectrum 1, 0, 1 . 3) h = g(0, ad X0). {− } 4) Let q+ := g(1, ad X0).Thenq+is contained in q and q+ = g Ad(c)p+. 0 ∩ 5) Let q− := g( 1, ad X ).Thenq−is contained in q and q− = g Ad(c)p−. + − + ∩+ 6) q and q− are abelian Lie algebras, θ(q )=q−,andq=q q− is a decomposition of q into irreducible nonequivalent H-modules. ⊕ 0 7) The involution τ is given by τ = eπi ad(X ). 8) Let Q± =expq±.ThenPmax := HQ− is a maximal parabolic subgroup of G with Levi part H.

0 0 0 Proof. (1) Let Z = Z+ + Z with Z+ g(+1, ad Y )andZ g( 1,ad Y ). Then − ∈ − ∈ −

0 0 πi πi 0 0 X = i Ad(c)Z = i e 2 Z+ + e− 2 Z = Z+ Z =[Y ,Z ]. − − − − −   ad(X0) is semisimple with real spectrum because of (1). (2) follows from the fact that ad Z0 has spectrum i, 0,i . (3)-(5) follow from Lemma 1.5 and (7) is obvious. {− } + + (6) That q and q− are abelian follows from (5). That θ(q )=q− follows by θ(X0)= X0.Let 0 =sbe an H-invariant subspace in q+.Letu:= X q+ + − {}6 { ∈ | Y q :(X Y)θ =0 ,q := s θ(s), and g := [q , q ] q . g is a τ-and ∀ ∈ | } 1 ⊕ 1 1 1 ⊕ 1 1

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h-stable subalgebra of g by the Jacobi identity. Furthermore u θ(u)ish-stable. Let X u and Y θ(s). Then ⊕ ∈ ∈ ([Y,X] [Y,X])θ = ([θ(Y ), [Y,X]] X)=0. | − | Hence [u,θ(s)] = 0 . Applying θ to this it also follows that [θ(u),s]= 0.As + { } {} [q,u]=[q−,θ(u)] = 0 it now follows that [g1, u θ(u)] g1. Hence g1 is a τ-stable ideal in g.Thus{ }V=q+. ⊕ ⊂ 0 (8)WehavethatZG (X) is connected and contained in H by (7). By (3) C C 0 both groups have the same Lie algebra. As H is connected H = ZG (X ). Hence C C C 0 H = ZG(X ). From this the claim follows easily.

2. The Function Ψm + + Lemma 2.1. Let g GC and Z p be such that g exp( Z) P KCP −.Then + ∈ ∈ − ∈ θ(g)exp(Z) P KCP− and ζ(θ(g) Z)= ζ(g ( Z)). In particular ζ(θ(g)) = ∈ + · − · − ζ(g) for g P K P −. − ∈ C Proof. Write g exp Z =exp[g ( Z)]kp, with k K and p P −.Then − · − ∈ C ∈ 1 1 θ(gexp Z)=θ(g)expZ =exp(θ(g ( Z)))kp− =exp( g ( Z))kp− − · − − · − as g ( Z) p+. Hence the lemma. · − ∈ + The Shilov boundary of b is = G E, E := ζ(c) p . We view G a subgroup of G = G G by the diagonalD embeddingS · g (g,g).∈ By restriction a G -space is 1 × 7→ 1 therefore also a G-space. Similarly we view GC G1C := GC GC. ⊂ + × In [7] it was proved that G/HQ− G/HQ . The following lemma gives a geometric interpretation ofM' this fact. ×

Lemma 2.2. 1) The stabilizer of E in G is Pmax. 1 + 2) We have E = ζ(c− )=ζ(θ(c)) and the stabilizer of E is HQ . − + − 3) 1 := G/HQ− G/HQ . 4) SLet ξ S×S'=(E, E) .Then× G ξ as a G-space. 0 − ∈S1 M' · 0 Proof. (1) We have H Q− G = HQ and gc cK P if and only if g C C − C − 1 ∩ ∈ ∈ cK P c = H Q−. This implies (1). (2) follows from (1) and Lemma 2.1 as C − − C C 1 + c− = θ(c)andθ(HQ−)=HQ . (3) and (4) are trivial in view of (1) and (2) and Lemma 2.1.

Let t be a Cartan subalgebra of k (and g) such that a := t q is maximal abelian in q, cf. [17]. Then Z0 a.Let∆=∆(g,t)bethesetofrootsof∩ t in g.Then ∈ C C C 0 ∆k := α ∆ α(Z )=0 =∆(kC,tC), { ∈ | 0 } ∆n := α ∆ α(Z )= i =∆(p ,t ). { ∈ | ±} C C + + + + + + Choose a positive system ∆k and let ∆ := ∆n ∆k ,where∆n = α gCα p . ∪ { |+ ⊂ } Let Γ = γ1,... ,γr be a maximal set of strongly orthogonal roots in ∆n such that { } + γ1 is the maximal root in ∆n and γj+1 is the maximal root strongly orthogonal to γ1,... ,γj. By the Theorem of Moore [8] and [17] we have the following facts: 1) Let H ,... ,H ia be given by H [g , g ] and Lemma 2.3. 1 r j Cγj C( γj ) r ∈ ∈ − γj (Hj )=2.Thena=i j=1 RHj. 2) Z0 = i r H . 2 j=1 j L P

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+ + 3) We can choose ∆k and ∆n such that

+ 1 + 1 ∆ = (γi + γj ) 1 i j r and ∆ = (γi γj) 1 i

4) The root spaces gCγj are one dimensional and all the root spaces corresponding 1 toarootoftheform (γi γj)have the same dimension d. 2 ± By Proposition 4.2.11 in [5] we have Lemma 2.4. Suppose that C is generated by Ad(H)Z0.Theni[Co a]= r − ∩ tjHj tj > 0 . { j=1 | } PWe list here the irreducible symmetric spaces G/H of Cayley type, cf. [5], the Shilov boundary of G/K, together with the real rank r of G/K, and the common dimension d.Herek 3, T is the one dimensional torus, Qn is the real quadric in ≥ the real projective space RPn defined by the quadratic form of signature (1,n)and the subscript + means positive determinant: Irreducible Spaces of Cayley Type = G/H = K/K Hr d M S ∩ Sp(n, R)/ GL(n, R)+ U(n)/O(n) n 1 SU(n, n)/ GL(n, C)+ U(n) n 2 + SO∗(4n)/ SU∗(2n)R U(2n)/ Sp(2n)2n 4 + SO(2,k)/SO(1,k 1)R Qn 2 k 2 +− − E7( 25)/E6( 26)R E6T/F4 38 − − The following lemma and theorem are well known, but we prove them here for lack of a good reference.

Lemma 2.5. Let ρn be the half sum of positive noncompact roots with multiplici- ties. 1 d(r 1) 1) ρn = 1+ − (γ + +γr). 2 2 1 ···   1 2(ρn γj) d(r 1) 2(ρn (γi + γj)) 2) | =1+ − and | 2 =2+d(r 1). γ 2 2 1 2 j 2 (γi + γj ) − 2(ρ| | γ ) | | 3) n j + if and only if g = sp(2n, ) or g = so(2, 2n +1). |2 Z R γj ∈ 6 6 | | 4) γ1 is a long root. Proof. (1) follows from Lemma 2.3, part 3. (2) is a direct consequence of (1), and (3) is clear from the last column of the above list of Cayley type spaces. (4) follows by inspection of the Dynkin diagrams.

+ + + Theorem 2.6. Fix the positive system ∆ (gC, tC)=∆n ∆k . Then the follow- ing hold: ∪−

1) There exists an irreducible finite dimensional representation of GC with lowest weight 2ρn. − 2) Assume that g = sp(2n, R), so(2, 2k +1), n, k 1. Then there exists a finite 6 ≥ dimensional irreducible representation of G with lowest weight ρn. C − + Proof. Let α ,... ,αk bethesimplerootsfor∆ (g ,t ). Then α = γ is the { 1 } C C 1 1 unique simple noncompact root. Furthermore (ρn α)=0forα ∆k.Ifγis an | ∈

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arbitrary noncompact positive root, then γ = γ1 + j>1 nαj αj. By Lemma 2.5: 2 2 2(kρn γ) 2(kρn γ1) γ1 Pd(r 1) γ1 | = | | | = k 1+ − | | . γ 2 γ 2 γ 2 2 γ 2 | | | 1| | |   | | 2 2 + d(r 1) As γ1 is a long root γ1 / γ Z .But 1+ − Z if and only if d is even | | | | ∈ 2 ∈ d(r 1)  or r isodd,andinallcases2 1+ − Z. The claim now follows from the 2 ∈ list of Cayley type space above.  Let m Z+ be such that there exists a finite dimensional irreducible represen- ∈ tation (πm, Vm)ofG with lowest weight mρn. Choose an inner product on Vm C − such that πm is unitary on the maximal compact subgroup of GC corresponding to the Lie algebra k ip.Letσ:GC GCbe the involutive conjugation with differential X + iY ⊕X iY , X, Y g→.Then 7→ − ∈ 1 πm(g)∗=πm(σθ(g)− ).

Let u0 be a lowest weight vector of norm 1, and let c be as before. Define Φm : + + + p C and Ψm : p p C by → × → 2 (2.1) Φm(Z):=(πm(c− exp Z)u u )andΨm(Z, W):=Φm(Z W). 0 | 0 − 01 Let G =SU(1,1) G =SL(2,C). Let τ(X)=X.Thenτ=Ad and ⊂ C 10 the holomorphic extension of τ is given by   ab ac τ = . c a −ba  −   10 With Z0 = i we have 2 0 1  − + 0 z 00 p = z C and p− = wC. 00 ∈ w0 ∈      

The Cartan involution θ is given by conjugating by Z0 and the holomorphic

extension to SL(2, C)is ab a b − . cd7→ cd   −  0 z Thus τθ(X)= XT. Identify p+ and k with C by z respectively − C 7→ 00   w 0 w and K C∗. A simple calculation now shows that 0 w 7→ C '  −  ab 1b/d 1/d 0 10 = cd 01 0 d c/d 1       + ab ab b if d =0.ThusP K P−= d=0 ,andζ = .Wehave 6 C cd 6 cd d      ab b= z C z <1 and = z C z =1 . Furthermore k = D { ∈ || | } S { ∈ || | } cd 1/d. Thus we recover the well known facts:  

ab az + b ab 1 z= and k ,z =(cz + d)− . cd· cz + d cd    

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0 1 01 Let Y 0 = i − .ThenX0= 1 .Thus 2 10 −2 10     + rr r r q = i rRand q− = i − r R . rr ∈ r r ∈  −−    −   t t 0 cos 2 sin 2 1 11 Furthermore exp tiY = t t which implies that c = sin cos √2 11 − 2 2  −  01 and c2 = . Hence 10 −  1z + 2 Lemma 2.7. Let G = SU(1, 1).LetZ= p.Thenc−exp Z = 01 ∈   0 1 2 + − and c− exp Z P K P − if and only if z =0.Furthermoreζ(c)=1, 1 z ∈ C 6  1  2 ζ(c− )= 1,andk(c− ,z)=1/z. − m m Lemma 2.8. Let G =SU(1,1).ThenΦm(z)=z and Ψm(z,w)=(z w) . −

In general we choose Ej gCγj , j =1,... ,r, such that Hj =[Ej,E j], ∈ 0 i r − E j = τ(Ej )=σ(Ej), and Y = j (Ej E j). For j =1,... ,r,de- − − 2 =1 − − fine a homomorphism ϕj : sl(2, C) g by → C P 10 01 00 Hj , Ej and E j . 0 17→ 007→ 107→ −  −     The conjugation of sl(2, C) with respect to su(1, 1) is given by 10 10 σ : X X∗ . su(1,1) 7→ − 0 1 0 1  − − Thus ϕj σ = σ ϕj from which it follows that ϕj(su(1, 1)) g.Further- ◦ su(1,1) ◦ ⊂ more ϕj intertwines the involutions θ, τ and τθ on su(1, 1) and on g.Wedenote the corresponding homomorphism SL(2, C) G also by ϕj . As the roots γj are → C strongly orthogonal gj := ϕj (su(1, 1)) commutes with gk if j = k.Usingϕj calcu- 6 lations involving the strongly orthogonal roots and the algebras gj are reduced to calculations in sl(2, C) and SL(2, C). In particular we get E = Ej. Let ξ be a character of KC and assume it is unitary on K.Thendξ iz(k). We write kdξ := ξ(k), k K . The following is well known: P ∈ ∈ C Lemma 2.9. Let (π, V) be a finite dimensional and irreducible representation of

GC with lowest weight µ.Letuobe a non-zero vector of weight µ. Then the KC- module generated by uo is irreducible. If µ iz(k)∗,thenKC acts on C uo by the character k kµ. ∈ 7→ Proof. Let W be the KC-module generated by uo. Then, as KC normalizes p−, p− W V .HereVis the representation space of π.Letn−= α ∆+kCα.Then ⊂ C ∈ k n− n− p− n− p− n− WC VC⊕ .ButVC⊕ =Cuo.ThusWC is one-dimensionalL and the claim⊂ follows. Lemma 2.10. Let σ : G G be the conjugation with respect to G.Letχ: C → C K C∗be a character. If χ K is unitary, then χ(στ(k)) = χ(k). C→ | Proof. Both sides are holomorphic in k and agree on K as τ z(k)= 1. The lemma | − now follows as K is a real form of KC.

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Lemma 2.11. Let Z p+. Then the following holds: ∈ 2 + 1) If exp Z c P K P −,then ∈ C 2 2 mρn Φm(Z)=(πm(k(c− ,Z))u u )=k(c− ,Z)− . 0 | 0 2mρn 2) Φm(k Z)=k Φm(Z) for every k K . · ∈ C 3) Let z1,... ,zr C.Then ∈ r n 2mρn m(1+d(r 1)/2) Φm(Ad(k) zjEj )=k zj − . j=1 j=1 X Y 2 + 2 Proof. (1) Write c− exp Z = pkq with p P , k = k(c− ,Z), and q P −.As + ∈1 ∈ σ(p)=p− we get πm(p)∗u0 = πm(σθ(p)− )u0 = u0.Thus 2 Φm(Z)=(πm(c−exp Z)u u ) 0 | 0 =(πm(k)πm(q)u πm(p)∗u) 0| 0 =(πm(k)u0u0) mρ | =k− n where the last equality follows from Lemma 2.9 . (2) We have k Z =Ad(k)Z. By Lemma 2.9: · 2 1 Φm(k Z)=(πm(c−kexp Zk− )u0 u0) · 2 | 1 =(πm(τ(k))πm(c− exp Z)πm(k− )u u ) 0 | 0 mρn 2 1 = k (πm(c− exp Z)u πm(στ(k)− )u ) 0 | 0 mρn mρ = k (στ(k)) n Φm(Z)

2mρn = k Φm(Z). (3) This follows from (2) and su(1, 1)-reduction.

+ Theorem 2.12. 1) Let g GC and Z, W p be such that g Z and g W are ∈ ∈mρ mρ · · defined. Then Ψm(g Z, g W)=k(g,Z) n k(g,W) n Ψm(Z, W). · · 2) Let zj ,wj C, j =1,... ,r, and let k K .Then ∈ ∈ C r r r 2mρn m(1+d(r 1)/2) Ψm(k zjEj,k wjEj)=k (zj wj ) − . · · − j=1 j=1 j=1 X X Y 1 Proof. (1) By definition exp g Z = g exp Zk(g,Z)− q, q P −, and similarly for W .Thus · ∈ exp(g Z g W )=exp(gW)exp(g Z) · − · −· ·1 1 1 =(gexp(W )k(g,W)− p)− g exp(Z)k(g,Z)− q 1 1 = p− k(g,W)exp( W)exp(Z)k(g,Z)− q − 2 2 2 + for some p P −.NowAd(c− )=θτ.Thusc− P−c =P .Asaboveweget ∈ 2 1 1 Ψm(g Z, g W)=(πm(c−p−k(g,W)exp( W)exp(Z)k(g,Z)− )u u ) · · − 0 | 0 mρn mρn = k(g,Z) k(g,W) Ψm(Z, W) .

(2) We have k j zj Ej k j wj Ej = k j (zj wj )Ej as KC acts by linear transformations.· The claim− follows· now from Lemma− 2.11. P P P

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3. The Causal Compactification of M In this section we show that the map

gH g (E, E) ξ Ψm(ξ)=0 M3 7→ · − ∈{ ∈S1 | 6 }⊂S1 is a causal compactification of . Recall that the causal structure on is deter- mined by the cone C in q generatedM by Ad(H)Z0, cf. [5, 14, 15]. TheM cone C can also be described in the following way,− cf. [5, 14]:

0 + H K Lemma 3.1. Write Y = Y Y − with Y ± q± ∩ .LetC+be the H -invariant + − + ∈ cone in q given by C+ =Ad(H)Y . Then the following hold: 0 + 1) Z = Y + Y −. − 2) C+ is regular and self-dual. + 3) Let C − = θ(C )=Ad(H)Y−.ThenC=C++C . − Proof. (1) is a standard su(1, 1)-reduction. (2) and (3) are from [5], Theorem 2.6.8.

+ + Identify the tangent space of at ξ with (g g)/((h+q−) (h+q )) q q−. S1 0 × × ' × Let D be the image of C+ C in Tξ 1 under this identification. Then D is an × − 0 S H H-invariant regular cone in Tξ ( ). × 0 S1 + Lemma 3.2. D is an (HQ−) (HQ )-invariant cone in Tξ ( ). × 0 S1 + + Proof. Let q =expX Q and p =expY Q−.Let(R, T ) q q−.Then ∈ ∈ ∈ × Ad(p, q)(R, T )=(ead Y R, ead X T ) =(R+[Y,R]+[Y,[Y,R]],T +[X, T]+[X, [X, T]]).

But [Y,R] h and [Y,[Y,R]] q−.ThusR+[Y,R]+[Y,[Y,R]] = R mod (h + ∈ ∈ + q−). Similarly T +[X, T]+[X, [X, T]] = T mod (h + q ). It follows that Ad(p, q) restricted to Tξ ( ) is the identity map. By this the claim follows. 0 S1 As explained in Section 1 it now follows that D defines a G G-invariant causal structure on by translating the cone D by G G. × S1 × Let Y j = i Ad(c)E j , j =1,... ,r.Inthecaseofsu(1, 1) we have ± − ± (3.1)

i 11 + + 0 1 Y1 = q,Y1=θ(Y1)q,and Y = (Y1 Y 1) . −2 11∈ − − ∈ 2 − − −− Lemma 3.3. Let j =1,... ,r.Then + 1) Yj q and Y j = θ(Yj ) q−. 0∈ 1 r − ∈ + r r 2) Y = j (Yj Y j), i.e., Y =2 j Yj and Y − =2 j Y j. 2 =1 − − =1 =1 − 3) Y k C. ± ∈ P P P Proof. (1) By Lemma 1.4 τ Ad(c)=Ad(c) θ.Thusτ(Yj)= iτ(Ad(c)Ej )= ◦ ◦ − 1 iAd(c)θ(Ej )= Yj,asθ(Ej)= Ej. From σ Ad(c)=Ad(c− ) σwe get − − − ◦ ◦ σ(Yj )=iAd(c)θτ(σ(Ej )) = i Ad(c)( Ej )=Yj. This shows that Yj q.Fur- 0 0 − ∈ + thermore [X ,Yj]= Ad(c)[Z ,Ej]= iAd(c)Ej = Yj . Hence Yj q .As − − ∈ θAd(c)=Ad(c)τ we get θ(Yj )= iAd(c)τ(Ej )= iAd(c)(E j )=Y j. (2) This follows by equation (3.1),− Lemma 3.1, and− su(1, 1)-reduction.− −

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0 1 01 (3) Let X = ϕj h p.Thenansu(1, 1)-reduction shows that j − 2 10 ∈ ∩    0 Yk,k=j, Ad(exp tXj )Yk = t 6 e Yk,k=j  and

0 Y k,k=j, Ad(exp tXj )Y k = t− 6 − e− Y k,k=j.  − In particular r r t t ad X0 t ad tX0 lim e− e j (Yk + Y k)=Yj and lim e e j (Yk + Y k)=Y j. t − t − − →∞ k →−∞ k X=1 X=1 The claim follows now from (2) and Lemma 3.1. We will also need the following lemma, cf. [3] or [5]: r Lemma 3.4. Let aq := j=1 R(Yj Y j).Thenaqis a maximal abelian subalgebra of p contained in q p. − − ∩ P + r Lemma 3.5. q =Ad(H K) RYj . ∩ j=1 Proof. Let X q+.ThenX θ(PX) q p. Therefore we can find k K H and ∈ − ∈ ∩r ∈ ∩ xj R such that Ad(k)(X θ(X)) = j xj (Yj Y j) aq.Asθ(X),Y j q− ∈ − =1 − − ∈ − ∈ it follows that Ad(k)X = r x Y . j=1 j j P + r Lemma 3.6. Q E =Ad(PK H) zjEj zj =1,zj = 1 . · ∩ { j=1 ∈S|| | 6 − } r Proof. Let X =Ad(k) rj Yj Pq with rj R.Thenasimplesu(1, 1)- j=1 ∈ ∈ r 1 irj 1 ir calculation shows that exp X E =Ad(k) − Ej .But − = z, z P j=1 1+irj 1+ir C · 1 1 z ∈ has a solution if and only if z = 1. The solution r is then given by − . 6 − P i 1+z r Corollary 3.7. =(K H) zjEj zj =1 . S ∩ j=1 ∈S | | n + r o Proof. By Lemma 3.6 we have QP E (K H ) zjEj zj =1 and · ⊂ ∩ { j=1 ∈S|| | } the second set is closed in . On the other hand Q+ E is dense in as Q+HQ P − is dense in G. S · S

Theorem 3.8. Let m N be such that Ψm is defined. Define Φ: 1 by Φ(gH):=g ξ.Then∈( ,Φ) is a G-equivariant, causal compactificationM→S of · 0 S1 M with Φ( )= ξ Ψm(ξ)=0 . M { ∈S1 | 6 } Proof. The function Φ is by definition G-equivariant. As the causal structure on

and 1 are G-invariant we only have to show that (dΦ)o(C) Dξ0 .Butthis isM obviousS from the definition of D. To show that the image is dense⊂ it suffices to show that it is given as stated. It follows by Theorem 2.12 that the left-hand side is contained in the right-hand side. Assume now that Ψm(Z, W) =0,(Z, W) 1. Let g G be such that g W = E and then choose k K6 H such∈S that ∈ · − r ∈ ∩ k (g Z)= zjEj.Then(kg) ξ =( zjEj, E). By Theorem 2.12 we · · · j=1 − have Ψm((kg) ξ) = 0. By part 2 of that theorem zj = 1, j =1,... ,r.By P· 6 P + 6 − Lemma 3.6 and Lemma 2.4 we can find q Q such that q (kg ξ)=ξ0. Hence 1 ∈ · · ξ =(qkg)− ξ . · 0

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4. The Domain Ξ

+ + The function Ψm is a holomorphic function on p p . By restricting it, we × can view it as a holomorphic function on b b. In this section we recall the o D ×D o construction of the domain Ξ from [6] and show that Ξ = z b b Ψm(z) = 0 . { ∈D ×D | 6 } Let pr : g q be the orthogonal projection. Let CG be a G-invariant regular q → cone in g such that τ(CG)=CG,andC=CG q=prq(CG), cf. [5], Theorem − ∩ 0 4.5.8. The minimal extension is the minimal cone Cmin generated by Ad(G)Z − and the maximal extension is the maximal cone Cmax = Cmin∗ .LetΓ=Γ(CG)be the Olshanskii semigroup Γ = G exp iCG. Then, cf. [5, 6]

1 Lemma 4.1. 1) Γ γ GC γ− b b . o o ⊂{ ∈ o | D ⊂D } 2) Γ = G exp iCG G CG. o ' 1 × 3) Γ γ G γ− b b . ⊂{ ∈ C | D ⊂D} 1 o o 1 o As in [6] we set Ξ = Γ− GC/HC.ThenΞ =(Γ)− G H ( iC ). o M⊂ M' × − In particular Ξ (and Ξ) are independent of the extension CG. The map Φ from

Theorem 3.8 is defined for all g GC/HC such that g ξ0 exists. We have by Lemma 4.1, part 3, ∈ ·

o o 1 (4.1) Ξ =(Γ )− ξ b b. · 0 ⊂D ×D The following has also been proven by M. Chadli, cf. [1].

o o Theorem 4.2. Ξ = z b b Ψm(z)=0. In particular Ξ is open and { ∈D ×D | 6 } dense in b b. D ×D Proof. By Theorem 2.12 we see that the left-hand side is contained in the right hand side. Let z =(Z, W) b b be such that Ψm(z) =0.AsGacts tran- ∈D ×D 6 sitively on b and 0 b, we may assume that W =0.WriteZ=g 0. Then D ∈D r · g = k1ak2, with k1,k2 K and a =exp j rj (Yj Y j) Aq := exp aq.As =1 − ∈ − r ∈ 0 K-stabilizes 0 we can assume that z =(a 0,0) = ( i j=1 tanh(rj )Ej , 0). ad Z + ·P −0 r acts by multiplication by i on p . Hence exp([π/2]Z ) ( i j=1 tanh(rj )Ej , 0) = r P· − ( j=1 tanh(rj )Ej , 0). Theorem 2.12, part 2, implies that rj =0forj=1,... ,r. r P6 Let γ1 =exp(i j Y j ) Γ (by Lemma 3.3). By su(1, 1)-reduction we get =1 − P1 ∈ γ1− ξo =(E,0). By Lemma 2.4 and 0 < tanh(rj ) < 1weseethatγ2 := · r P o 1 r exp( j=1 log(tanh(rj ))Hj ) Γ . Also γ2− (E,0) = ( j=1 tanh(rj )Ej , 0). The Theorem− follows now as ΓΓo ∈Γo. · P ⊂ P Example 4.3 (SU(n, n)). Let G =SU(n, n). Then

+ 0 Z p = Z M(n, C) M(n, C) 00 ∈ '   

and the action of SU(n, n)onp+ is given by

AB 1 Z=(AZ + B)(CZ + D)− . CD·   The maximal compact subgroup is S(U(n) U(n)) and the bounded realization is × SU(n, n)/S(U(n) U(n)) Z M(n, C) In Z∗Z>0 . × '{ ∈ | − }

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The Shilov boundary is U(n)andEcorresponds to the identity matrix. The Cayley type involution τ is given as AB DC CD7→ BA     and AB H = SU(n, n) GL(n, R)+ . BA∈ '    mn By Lemma 2.11 we get Φm(Z)=det(Z) .Thus

SU(n, n)/ GL(n, R)+ (Z, W) U(n) U(n) det(Z W ) =0 '{ ∈ × | − 6 } and Ξo is given by

(Z, W) M(n, C) M(n, C) In Z∗Z>0,In W∗W>0,det(Z W ) =0 . { ∈ × | − − − 6 } 5. L2-spaces and Double Coverings The quasi-invariant and normalized measure µ on is given by S1

(5.1) f(ξ) dµ(ξ)= f(k1 E,k2 E)dk1dk2 . K K/(H K H K) · · ZS1 Z × ∩ × ∩ Let x G. Define k(x) K, h(x) H and q−(x) Q− by x = k(x)h(x)q−(x). Define ∈ ∈ ∈ ∈

2ρ+ + 1 JR(x, k E)=h(xk) := det(Ad(h) q ) = (det Ad(h) q−)− . · | | Then by [13]

(5.2) f((a Z, b W))JR(a, Z)JR(b, W ) dµ(Z, W)= f(ξ)dµ(ξ) · · ZS1 ZS1 + + where µ is as in (5.1). For g G and Z p such that g exp Z P K P − let ∈ C ∈ ∈ C J(g,Z):=k(g,Z)2ρn . This is the holomorphic Jacobian of the action on p+.The connection between J and JR is given by the following lemma. Lemma 5.1. Suppose that Z = k E for some k K. Then the following hold: · ∈S ∈ 1) g Z =Ad(k(gk))E. · ρ ρ 2) k(g,Z)2 n = h(gk)2 + . | | Proof. Write c =expEkop−. Simple calculation shows that 1 1 1 g exp Z = k(gk)expEko− (c− h(gk)c)kok− q 1 for some q P −. The lemma now follows from c− H c = K , q− =Ad(c)p−,and ∈ C C C k2ρn =1forallk K. | | ∈ ρ 1 From Theorem 2.6 we know that J(a, Z)=k(a, Z) n , a Γ− and Z cl( b) is well defined if g is not one of the Lie algebras sp(2n, )or∈ so(2, 2n +1).∈ D In p R general there is a double covering Γ,˜ respectively G˜, of Γ, respectively G, such that o J(a, Z) is defined as a function on Γ˜ cl( b) holomorphic on Γ˜ b,where × D ×D Γ˜o=κ 1(Γo), κ : Γ˜ Γ the canonical projection. The group G˜ acts on and p − b bya ˜ Z = κ(a) Z. The→ cocycle relation D S · · (5.3) J(ab, Z)= J(a, b Z) J(b, Z) · p p p

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still holds for Γ,˜ cf. (1.2). By Lemma 5.1 and (5.2) we have the following theorem: Theorem 5.2. The formula

1 1 1 1 [λ (a, b)f](Z, W):= J(a ,Z) J(b ,W)f(a− Z, b− W) 1 − − · · defines a unitary representationp of G˜ G˜ pon L2( ). × S1 We define also a double covering of the set := ξ p+ p+ Ψ (ξ) =0 by X { ∈ × | 2 6 } ˜ 2 (5.4) = (ξ,z) C z =Ψ2(ξ) . X { ∈X× | } The covering projection and the square-root √Ψ2 of Ψ2 are given by the projection onto the first, respectively second, variable:

(5.5) p(ξ,z):=ξ and Ψ2(ξ,z)=z. p 2 By Theorem 2.12 J(g,Z) J(g,W) Ψ (Z, W) =Ψ(g (Z, W)). Hence 2 2 · we have a well-definedp action ofpG˜ on ˜ givenp by  X (5.6) a ((Z, W),z):=(a (Z, W), J(a, Z) J(a, W )z) . · · Definition 5.3. A function F in ˜ is even,p respectivelyp odd, if F (ξ,z)=F(ξ, z) respectively F (ξ, z)= F(ξ,z). X − − − The pullback of the G-invariant (respectively quasi-invariant) measure on ˜ ˜ M (respectively 1) defines a G-invariant measure on (respectively 1). We will denote this measureS by the same letter as before. ThusM the correspondingS L2-spaces are defined. We will denote the subspace of even, respectively odd, functions by 2 2 Leven, respectively Lodd. The same convention will be used for other function spaces. Notice that all those coverings are trivial (i.e. splits) in case g is not sp(2n, R)orso(2, 2n + 1) because we can take a global square-root of both J and Ψ2. 1 Lemma 5.4. Let κ− (1) = 1, .Then { } 1 1 1 1 J(a− ,Z) J(b− ,W)= J(a− , Z) J(b− , W ) . Proof. Let  bep as in thep lemma. Then pJ(, Z)=p1 for every Z.Theclaim follows now from (5.3). − p It follows that the representation λ factors to G˜ G/˜ (1, 1), (, ) . Restricted 1 × { } to the diagonal ∆(G˜) it defines a representation of ∆(G˜)/ (1, 1), (, ) G.We { }' notice also that Lemma 5.4 and Theorem 2.12 imply that action of G˜ and Γon˜ Ξ˜ factors to an action of G and Γ as  acts trivially. The same argument shows that the covering Ξ˜ Ξ is trivial on every G and Γo-orbit. → 1 ˜ ˜ ˜ Lemma 5.5. 1) Let p− (ξo )= ξo,ξ1 .Let j:= G ξj , j =0,1.Then ˜ { } M · = o 1 (disjoint union) and both o and 1 are isomorphic to asMG-spaces.M∪M M M M 2 2 2) The spaces L ( ˜ )even and L ( ˜ )odd are G-invariant subspaces both of which M 2 M 2 are equivalent to L ( ). The same holds also for the spaces L ( o) and L2( ). We have M M M1 2 2 2 2 2 L ( ˜ ) L ( ˜ )even L ( ˜ )odd L ( o) L ( ) . M ' M ⊕ M ' M ⊕ M1

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1 Proof. 1) Let (m, z) p− ( ). Let m = g o.Aszis a square-root of Ψ (m)we ∈ M · 2 have also fixed a square root zo of Ψ (o). It follows that (m, z)=g (o,zo). That 2 · is obvious as the stabilizer of ξo and ξ is in both cases H. M0 'M1 'M 1 2) If f L2( ˜ ) is continuous, then we can write ∈ M 1 1 f(m, z)= (f(m, z)+f(m, z)) + (f(m, z) f(m, z)) 2 − 2 − − which defines a continuous projection onto the space of even, respectively odd, functions. On the other hand, if f is even or odd, then f is determined by the restriction to o or . This gives the needed isomorphism. M M1 We prove the following lemma in the same way as part 1) in the last lemma: o o 1 ˜o o o Lemma 5.6. Let Ξ := (Γ )− i.ThenΞ =Ξ Ξ . i ·M o∪ 1 The following theorem follows now by the definition of Ψ1 and Ψ2, cf. (2.1). Theorem 5.7. Let λ be the regular representation of G on L2( ). Then the following hold: M 1) The G-invariant measure on Φ( ) is up to a constant given by M ⊂S1 1 f(ξ) Ψ (ξ) − dµ(ξ). | 2 | ZS1 2 1/2 2 2) The map L ( 1) f (f Ψ2 ) L ( ) is a G-equivariant isometry of (L2( ),λ SG)3onto7→(L2(| |),λ).|M ∈ M S1 0| M 3) If g = sp(2n, R) or g = so(2, 2n +1),then 6 6 L2( ) f (fΨ ) L2( ) S1 3 7→ 1 |M ∈ M 2 2 defines a G-equivariant isometry (L ( 1),λ1 G) onto (L ( ),λ). 2 S2 | M 4) Let f L ( ).Then(f p)√Ψ L(˜)odd and ∈ S1 ◦ 2∈ M 2 2 L ( ) f (f p) Ψ L ( ˜ )odd S1 3 7→ ◦ 2 ∈ M is an isometric G-isomorphism. p Remark 5.8. The above theorem shows in particular that, restricted to the diag- 2 onal, the different representations of G =∆(G)onL( 1), i.e. those defined by using the real Jacobian, respectively the complex Jacobian,S are equivalent.

6. The Hardy Spaces For a complex manifold U let (U) be the space of holomorphic functions on U. o O 1 For γ Γandf (Ξ )letγ f(ξ)=f(γ− ξ). The Hardy space 2 is the space of holomorphic∈ functions∈O on Ξo·such that H

sup (γ f) L2( ) < , γ Γo k · |Mk M ∞ ∈ 2 cf.[6].Iff 2 the L -limit β(f) := limγ 1(γ f) exists and defines a G- equivariant isometry∈H into L2( ). The image→ of β is· given|M by the direct sum of the holomorphic discrete series constructedM in [17, 18]. Let π be an irreducible unitary representation of K on the finite dimensional Hilbert space Vπ.Denotebythe

same letter π the holomorphic extension to KC.Letπ∗denote the contragradient representation on V∗ . We will assume that π∗ contains a non-zero H K fixed π C ∩ C vector uo. We normalize uo to have norm 1. Let t be a Cartan subalgebra of k and g containing a.Letµπ it∗be the highest weight of π with respect to the positive ∈

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+ + system ∆k , cf. Lemma 2.3. Then µπ ia∗.ThesetHCKCP is open and dense 1 1 1 + ∈1 + in G .Forx− =h(x− )kH(x− )p (x− ) HKP define C ∈ C C 1 1 Φπ,u(x):= u, π∗(kH (x− )− )uo ,uVπ. h i ∈ Let 1 1 r ρ = mαα = (1 + d(r j)) γj 2 2 − α ∆+ j=1 X∈ X where mα =dimgα. Then the following holds, cf. [6]:

+ Theorem 6.1. Let := µ t∗ α ∆ : µ + ρ, α < 0 . Then we have E { ∈ C |∀ ∈ n

2 5) If u =0,thenΦπ,u L ( ) if and only if µπ . In that case Φπ,u 6 |M ∈ M ∈E2 generates an irreducible G-invariant subspace Eπ in L ( ).Furthermore M Vπ u Φπ,u Eπ is a K-intertwining operator. 3 7→ ∈ 6) If u =0,thenΦπ,u if and only if µπ . In that case β(Φπ,u)= 6 ∈H2 ∈E Φπ,u . |M 7) Im β = µ Eπ. π∈E Lemma 6.2.LLet u be a weight vector of weight µ.Then

r µ(Hj )/2 Φπ,u(E,0) = 2 u, uo .   h i j=1 Y   In particular Φπ,u(E,0) = 0 if µ ia∗. 6∈ r Proof. We recall that exp i j=1 Y j ξ =(E,0). On the other hand su(1, 1)- reduction shows that − − · P r 1 r kH exp Y j =exp (log 2)Hj.  −  2 j=1 j=1 X X   The claim follows now from Theorem 6.1, part 4.

r r Theorem 6.3. Suppose that Z W =Ad(k) xj Ej Ad(K) xjEj − j=1 ∈ { j=1 | j : xj =0 be such that Φπ,u(Z, W) is defined. Choose rj C such that xj = ∀ 6 } 1 P ∈ P exp 2rj.Writeπ(k−)u= νaν(k)uν,whereuν are weight vectors of weight µµ. Then P r r µν (Hj )/2 µν (Hj )/2 Φπ,u(Z, W)= aν(k) x− 2 uν ,uo .  j    h i ν j=1 j=1 X Y Y    

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+ Proof. Let Φν := Φπ,uν . By Theorem 6.1 we know that Φπ,u is P -invariant. As exp rjHj (E,0) = ( xjEj, 0) we get · PΦπ,u(Z, W)=ΦPπ,u(Z W, 0) − r = aν(k)Φν ( xj Ej, 0) ν j=1 X X r = aν(k)Φν (exp( rjHj) (E,0)) · ν j=1 X X = aν(k)Φπ,exp( r r H )u (E,0) − j=1 j j ν ν P X r = aν(k)exp( rj µν (Hj))Φν (E,0) − ν j=1 X X r µν (Hj )/2 = aν(k) x− Φν (E,0)  j  ν j=1 X Y  r  r µν (Hj )/2 µν (Hj )/2 = aν(k) x− 2 uν ,uo .  j    h i ν j=1 j=1 X Y Y     From this the theorem now follows.

Remark 6.4. Notice that the above theorem applies to Φπ,u Ξ. | We want now to relate the Hardy space 2 to the classical Hardy space. The cl H classical Hardy space is the space of holomorphic functions on b b that 2 H ˜ D ×D extend to L -functions on 1. We have a representation of G on the Hardy space given by S

1 1 (6.1) [λHsp(a)f]= J(a ,Z)f(a− Z). − · In particular the boundary value mapp is G-equivariant, resp. G˜-equivariant. We remark that we could have replaced, in all the constructions G,ΓandΞo by the ˜ corresponding coverings to get the corresponding Hardy space 2 and boundary value map β˜ : ˜ Λ2( ˜ ). Note that it is known from [6] and [17]H that the image H2 → M of β˜ is again the sum over the holomorphic discrete series constructed in the same ˜ way as before, using the lifting of the map kH . 2 is a G-space as the action of  on ˜ is trivial and β is an isometric G-map. TheH boundary value map maps even functionsX into even functions and odd functions into odd functions. Then Lemma 5.5 and Lemma 5.6 show that

Lemma 6.5. We have the decomposition of ˜ into equivalent G-spaces H2 ˜ ˜ even ˜ odd H2 ' H2 ⊕ H2 and both ˜ even and ˜ odd are equivalent to . H2 H2 H2 Theorem 6.6. Assume that g = sp(2n, R) or g = so(2, 2n +1). Define I : cl by I(f):=(fΨ )Ξo.ThenI6 is a G-equivariant6 unitary G-isomorphism.H → H2 1 |

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o o Proof. I is injective as Ξ is open and dense in b b and Ψ is non-zero on Ξ . D ×D 1 1 1 1 I(f)(g− (Z, W)) = f(g− (Z, W))Ψ (g− (Z, W)) · · 1 · 1 1 1 = J(g ,Z) J(g ,W)f(g− (Z, W))Ψ (Z, W) − − · 1 = Ip(λ(g)f)(Z,p W) . So what we have to show is that for f the function f/Ψ is bounded. It ∈H2 1 follows then that this function extends to a holomorphic function on b b and D ×D is actually continuous on 1. We will prove this in the following steps. By Theorem 2.12 we haveS r d(r 1) 1+ − Ψ (Z, W)=Ψ (Z W, 0) = k2ρn x 2 1 1 − j j=1 Y r 2rj where Z W =Ad(k) j=1 xjEj .Chooserj Csuch that e = xj .This − ∈1 is possible as xj =0.Letu Vπ and write k− u = aν(k)uν ,whereaν is a 6 P ∈ continuous function of k and uν is a weight vector for the weight µν .Thenby Theorem 6.3 we now get P

r r µν (Hj ) d(r 1) Φ (Z, W) − π,u 2ρn µn(Hj )/2 ( 2 +1+ 2 ) = k− aν(k) uν ,uo 2 x− . Ψ (Z, W)  h i  j 1 ν j=1 j=1 X Y Y   µν (Hj ) d(r 1) We will now show in Lemma 6.8 that actually ( 2 +1+ 2− ) 0. That will then prove the theorem. − ≥

Lemma 6.7. Let µ = µπ be the highest weight of π Kˆ such that π has a K H- + ∈ µ,Hj d(r 1) ∩ fixed vector and µ + ρ, α < 0 for every α ∆ .Thenh i +1+ − 0. h i ∈ n 2 2 ≤ r + Proof. Write µ = j=1 njγj.Asµis ∆k -dominant, n1 n2 nr.By[19] 1 r ≥ ≥ ··· ≥ we have nj Z for every j.Asρ= (1 + d(r j))γj and Hj corresponds to ∈ P 2 j=1 − 2γj/ γj under the isomorphism ia ia∗, the condition µ + ρ, α < 0reads | | ' P h i (6.2) j :2nj+1+d(r j)<0. ∀ − 1 d(r 1) r On the other hand ρ = 1+ − γ . Hence by (6.2) n 2 2 j j=1   X µ, Hj d(r 1) d(r 1) h i +1+ − n +1+ − 2 2 ≤ 1 2 1 1 = (2n +1+d(r 1)) + 2 1 − 2 1 < . 2 d(r j) d(r j) But nj, 2− Z. Hence nj +1+ 2− 0 and the claim follows for the roots ∈ ≤ 1 γj . But the other positive noncompact roots are of the form 2 (γi + γj)andthe claim follows.

Lemma 6.8. Let π be as above. Let µ be a weight of π such that µ ia∗.Then µ,Hj d(r 1) ∈ h i +1+ − 0 for every j. 2 2 ≤

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1 Proof. If µ is a weight, then µπ nij (γi γj ) with nij 0. But then − 2 i

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on the base point (1, 1) in . This gives us − S×S u u z = g 1= ,w=g ( 1) = − · u · − u where u =cosht+isinh t. This we may insert in the above expression for K to obtain u2+u2 1 K =2 = (u u)(u u) (sinh t)2 − − which may also be written as 2 K = . cosh 2t 1 − In these same coordinates the invariant distribution for the holomorphic discrete series is 2(2n +1)Qn(cosh 2t), so since the Cauchy-Szeg¨o kernel is the sum of these, we get Heine’s formula. Observe that the general situation allows the calculation of the Cauchy-Szeg¨o kernel as a product of the formulas for SU(1, 1): Again it suffices to let the anti- holomorphic argument be the base point and the holomorphic argument along the direction of the Cartan subspace. Since the classical Hardy space for b has the kernel D r (1+d(r 1)/2) Kcl(z,z )= (1 zj z ,j)− − 1 − 1 j=1 Y and the classical kernel for the tensor product, i.e. the classical Hardy space we are comparing with, is the product of two such terms as in the rank-one case, we obtain the following formula for the Cauchy-Szeg¨o kernel for the general Cayley type space: r (1+d(r 1)/2) K = (cosh 2tj 1)− − . − j=1 Y Here the coordinates tj are the same as in the SU(1, 1) case, one for each strongly orthogonal root. It would be interesting to calculate the exact normalizations be- tween the generalized Legendre functions in several variables Qλ and the invariant distributions for the holomorphic discrete series to make explicit the formula

K = Qλ λ X where now the left-hand side is known by the above.

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Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803 E-mail address: [email protected] Matematisk Institut, Odense Universitet, Campusvej 55, DK-5230 Odense M, Denmark E-mail address: [email protected]

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