THE SEGAL-BARGMANN TRANSFORM ON COMPACT SYMMETRIC SPACES AND THEIR DIRECT LIMITS 1 GESTUR O´LAFSSON AND KENG WIBOONTON 1 0 2 Abstract. We study the Segal-Bargmann transform, or the heat transform, H for n t a a compact symmetric space M = U/K. We prove that Ht is a unitary isomorphism J Ht : L2(M) t(MC) using representation theory and the restriction principle. We → H 8 thenshowthattheSegal-Bargmanntransformbehavesnicelyunderpropagationofsym- 1 metric spaces. If M =U /K ,ι is a direct family of compact symmetric spaces n n n n,m n { } ] such that2Mm propagates Mn, m ≥ n, then this gives rise to direct families of Hilbert T spaces L (Mn),γn,m and t(MnC),δn,m such that Ht,m γn,m = δn,m Ht,n. We { } {H } ◦ ◦ R also consider similar commutative diagrams for the Kn-invariant case. These lead to 2 . isometricisomorphismsbetweenthe HilbertspaceslimL (Mn) lim (MnC)aswellas h ≃ H t limL2(Mn)Kn lim (MnC)Kn. −→ −→ a −→ ≃−→H m [ 1 v Introduction 3 6 Denote by h (x) = (4πt)−n/2e−kxk2/4t the heat kernel on Rn and denote by dµ (x) = 4 t t 3 h (x)dx the heat kernel measure on Rn. Denote by ∆ the Laplace operator on Rn. The t 1. Segal-Bargmann transform H , also called the heat kernel transform, on L2(Rn) or on t 0 L2(Rn,µ ) is defined by mapping a function f L2(Rn) to the holomorphic extension to 1 t ∈ 1 Cn of f ht = et∆f. The image of L2(Rn,µt) under the Segal-Bargmann transform is ∗ : the Fock space (Cn) of holomorphic functions F : Cn C such that (2πt)−n F(x+ v t F → | i iy) 2e−kx+iyk2/2tdxdy < . Thus (Cn) = L2(R2n,dµ (x)dµ (y)) (Cn) whereas X | ∞ Ft t/2 t/2 ∩ O R the image of L2(Rn) is (Cn) = L2(R2n,dµ (y)) (Cn), also called the Fock space. r t t/2 H ∩O a This idea, in a slightly different form, was first introduced by V. Bargmann in [3]. An infinite dimensional version was considered by I. E. Segal in [34]. A short history of the Segal-Bargmann transforms for Rn can be found in [13] and [14]. For infinite dimensional analysis one is forced to consider the heat kernel transform de- fined on L2(Rn,µ ). The reason is, that the heat kernel measure forms a projective family t of probability measures on Rn and hence L2(Rn,µ ) forms a direct and projective t n∈N { } family of Hilbert spaces. Similarly, (Cn) forms a direct and projective family t n∈N {F } 1991 Mathematics Subject Classification. 22E45,32A25, 44A15. Keywords and phrases. HeatEquation,Segal-BargmannTransform,CompactRiemanniansymmetric spaces, Direct limits of compact symmetric spaces. Both authors were supported by NSF grant DMS-0801010. 1 THE SEGAL-BARGMANN TRANSFORM ON COMPACT SYMMETRIC SPACES 2 of Hilbert spaces and the H : L2(Rn,µ ) (Cn) is direct and defines a unitary t,n t t { → F } isomorphism limL2(Rn,µ ) lim (Cn). t t → F −→ −→ The symmetric spaces of compact and noncompact type form a natural settings for generalizations of the heat kernel transform and the Segal-Bargmann transform. This was first done in [12] where the Segal-Bargmann transforms were extended to the com- pact group case and compact homogeneous spaces U/K. As in the flat case, the Segal- Bargmann transform H on L2(U/K) is given by the holomorphic extension of f h to t t ∗ the complexification U of U. The author showed that the Segal-Bargmann transform is C an unitary isomorphism from L2(U) onto (U ) L2(U ,ν ), where (U ) denotes the C C t C O ∩ O space of holomorphic functions on U and ν is the U-average heat kernel on U . Anal- C t C ogous results for compact symmetric spaces were given by Stenzel in [37]. The image of the Segal-Bargmann transform H is a L2-Hilbert space of holomorphic functions on the t complexification U /K of U/K. In [12] the heat kernel measure on U /U was used to C C C define the Fock space, whereas [37] uses the heat kernel measure on the non-compact dual G/K of U/K. Both measures coincide as can be shown by using the Flensted-Jensen duality [7]. In [20] and [41] the unitarity of the Segal-Bargmann transform was proved using the restriction principle introduced in [30]. Some work has been done on constructing a heat kernel measure on the direct limit of some complex groups. In [11], Gordina constructed the Fock space on SO( ,C), using ∞ the heat kernel measure determined by an inner product on the Lie algebra so( ,C). ∞ Another direction is taken in [17] where the Segal-Bargmann transform on path-groups is considered. In the noncompact case new technical problems arise. In particular, in the compact case, every eigenfunction of the algebra of invariant differential operators as well as the heat kernel itself, extends to a holomorphic function on U /K . This follows from the C C fact, that each irreducible representation of U extends to a holomorphic representation of U with a well understood growth. In the noncompact case this does not hold anymore. C The natural complexification in this case is the Akhiezer-Gindikin domain Ξ G /K , C C ⊂ see [1]. Using results from [21] it was shown in [22] that the image of the Segal-Bargmann transform on G/K can be identified as a Hilbert-space of holomorphic functions on Ξ, but in this case the norm on the Fock space is not given by a density function as in the flat case. Some special cases have also been considered in [15, 16]. A different description that also works for arbitrary positive multiplicity functions was given in [27]. From the point of view of infinite dimensional analysis, the drawback of all of those articles is that only the invariant measure on G/K is considered, so far no description of the image of the space L2(G/K,µ ) under the holomorphic extension of f h exists, t t ∗ except one can describe the space in terms of its reproducing kernel. The first step to consider the limit of noncompact symmetric spaces was done in [35]. There it was shown for a special class of symmetric spaces G /K that L2(G /K ,µ ) n n n n t n { } THE SEGAL-BARGMANN TRANSFORM ON COMPACT SYMMETRIC SPACES 3 forms projective family of Hilbert spaces. But no attempt was made to consider the Segal-Bargmann transform. Our main goal in this article is to use some ideas from the work of J. Wolf, in particular [44], to construct the Segal-Bargmann on limits of special classes of compact symmetric spaces. In [31] the authors introduced the concept of propagation of symmetric spaces. The results of [44] applies to this situation resulting in an isometric embedding γ of n,m L2(U /K ) into L2(U /K ) for m > n. Let M = U /K , and M = U /K . We n n m m n n n nC nC nC show, using the ideas from [44] that we have an isometric embedding δ : (M ) n,m t Cn H → (M ) such that H γ = δ H . This then results in a unitary isomorphism t Cm t,m n,m n,m t,n H ◦ ◦ H : limL2(M ) lim (M ). t,∞ n t nC → H −→ −→ In the K -invariant case, in general j (L2(M )Kn) * L2(M )Km for m > n. So dif- n n,m n m ferent maps have to be considered in this case. In this article, we define an isometric embedding η : L2(M )Kn L2(M )Km and similarly for (M )Kn with the embed- n,m n m t nC → H dings φ such that H η = φ H resulting in a unitary isomorphism of the n,m t,m n,m n,m t,n ◦ ◦ directed limits. This is the result from [41]. The article is organized as follows. In Section 1 we introduce the basic notation used in this article. In Section 2 we discuss needed results from representation theory and Fourier analysis related to symmetric spaces. The Fock space (M) is introduced in Section 3. t H We show that (M) is a reproducing kernel Hilbert space and determine its reproducing t H kernel. We also describe (M) as a sequence space. The Segal-Bargmann transform is t H introduced in Section 4 and we show that it is an unitary U-isomorphism in Theorem 4.3. In Section 5 we recall the notion of propagation of symmetric spaces introduced in [31]. The infinite limit is considered in the last two sections, Section 6 and Section 7. 1. Basic Notations Let M be a symmetric space of the compact type. Thus, there exists a connected compact semisimple Lie group U and a nontrivial involution θ : U U such that Uθ K Uθ → 0 ⊆ ⊆ and M = U/K. Here, Uθ = u U θ(u) = u denotes the subgroup of θ-fixed points, { ∈ | } and the index stands for the connected component containing the identity. 0 To simplify the exposition we assume that U is simply connected. In this case Uθ is connected and hence K = Uθ is connected and U/K is simply connected. The more general case can be treated by following the ideas in Section 2 in [29]. The base point eK M will be denoted by o. Denote the Lie algebra of U by u. θ ∈ defines a Lie algebra involution on u which we will also denote by θ. Then u = k s, ⊕ where k = X u θ(X) = X and s = X u θ(X) = X . Note that k is the Lie { ∈ | } { ∈ | − } THE SEGAL-BARGMANN TRANSFORM ON COMPACT SYMMETRIC SPACES 4 algebra of K, s T (M) via the map X D , K o X ≃ 7→ d D f(o) = f(exp(tX) o) X dt · (cid:12)t=0 (cid:12) and T(M) ≃ U ×Ad|s s. (cid:12)(cid:12) AsU is compact, there is a faithful representation of U, so we can–andwill–assume that U is linear: U U(n) GL(n,C) for some n N. Then u u(n). Define a U-invariant ⊆ ⊂ ∈ ⊆ inner product on u by X,Y = TrXY = TrXY∗. h i − By restriction, this defines a K-invariant inner product on s and hence a U-invariant metric on M. We note that k and s are orthogonal subspaces of u with respect to , . h· ·i The inner product on u determines an inner product on the dual space u∗ in a canonical way. Furthermore, theseinnerproductsextendtotheinner productsonthecorresponding complexifications u and u∗. All these bilinear forms are denoted by the same symbol C C , . h· ·i Let a s be a maximal abelian subspace of s. View a∗ as the space of C-linear maps ⊆ C a C. Then a∗ = λ a∗ λ(a) R and ia∗ = λ a∗ λ(a) iR . C → { ∈ C | ⊆ } { ∈ C | ⊆ } For α a∗, let u = X u ( H a ) [H,X] = α(H)X . If u = 0 then ∈ C C { ∈ C | ∀ ∈ C } Cα 6 { } α ia∗ and u u = 0 . If u = 0 , then α is called a restricted root. Denote by Cα Cα ∈ ∩ { } 6 { } Σ = Σ(u ,a ) ia∗ the set of restricted roots. Then C C ⊂ u = a m u C C C Cα ⊕ ⊕ α∈Σ M where m = z (a) is the centralizer of a in k. k The simply connected group U is contained as a maximal compact subgroup in the simply connected complexLiegroupU GL(n,C)withLiealgebrau = u C. Denote C C R ⊆ ⊗ by θ : U U the holomorphic extension of θ. Let σ : U U be the conjugation on C C C C → → U with respect to U. Thus the derivative of σ is given by X +iY X iY, X,Y u. C 7→ − ∈ σ is the Cartan involution on U with U = Uσ. We will also write g = σ(g) for g U . C C ∈ C Let K = Uθ. Then K has Lie algebra k and K is a maximal compact subgroup of C C C C K . K is connected as U is simply connected and M = U /K is a simply connected C C C C C C complex symmetric space. As σ(K ) = K it follows that σ defines a conjugation on M C C C with (Mσ) = M. Thus M is a totally real submanifold of M . In particular, C o C Lemma 1.1. If F (M ) and F = 0, then F = 0. C M ∈ O | Let g = k + is = uθσ and let G = Uθσ denote the analytic subgroup of U with Lie C C C algebra g. Md = G/K is a symmetric space of the noncompact type and Md = (Mσθ) . C o Hence, Md is also a totally real submanifold of U /K . Md is called the noncompact dual C C of M. THE SEGAL-BARGMANN TRANSFORM ON COMPACT SYMMETRIC SPACES 5 The following is clear (and well known) using the Cartan decomposition of U and G: C Lemma 1.2. Let g U . Then there exists a unique u U and a unique X iu such C ∈ ∈ ∈ that g = uexpX. We have g G if and only if u K and X is. ∈ ∈ ∈ 2. L2 Fourier Analysis In this section, we give a brief overview of the representation theory related to harmonic analysis on M. Since U is assumed to be simply connected, there is a one-to-one correspondence be- tween U, the set of equivalence classes of irreducible unitary representations of U, and the semi-lattice of dominant algebraically integral weights on a Cartan subalgebra containing a. We bdenote this correspondence by µ (π ,V ). (π ,V ) is spherical if µ µ µ µ ↔ VK = v V ( k K) π (k)v = v = 0 . µ { ∈ µ | ∀ ∈ µ } 6 { } There exists an isometric U-intertwining operator V ֒ L2(M) if and only if VK = µ → µ 6 0 . In that case dimVK = 1. The description of the highest weights of the spherical { } µ representations is given by the Cartan-Helgason theorem, see Theorem 4.1, p. 535 in [19]. Fix a positive system Σ+ Σ. Let ⊂ α,µ (2.1) Λ+(U) = µ ia∗ ( α Σ+) h i Z+ = 0,1,... . K ∈ ∀ ∈ α,α ∈ { } (cid:26) (cid:12) h i (cid:27) (cid:12) As both U and K will be fixed fo(cid:12)r the moment we simply write Λ+ for Λ+(U). Λ+ is (cid:12) K contained in the semi-lattice of dominant algebraically integral weights. Theorem 2.1. Let (π ,V ) be an irreducible representation of U with highest weight µ. µ µ Then π is spherical if and only if µ Λ+. µ ∈ Ifnothingelseissaid, thenwewillfromnowonassumethat(π ,V )isspherical. , µ µ µ h· ·i will denote a U-invariant inner product on V . The corresponding norm is denoted by µ . Let d(µ) = dimV . Then µ d(µ) extends to a polynomial function on a∗ of k · kµ µ 7→ C degree dim u . We fix e VK with e = 1. The function g π (g)e is α∈Σ+ C Cα µ ∈ µ k µk 7→ µ µ right K-invariant and defines a V valued function on M. We write π (x)e = π (g)e if µ µ µ µ µ P x = g o, g U. · ∈ For u V let µ ∈ (2.2) πu(x) = u,π (x)e . µ h µ µiµ The representation π extends to a holomorphic representation of U which we will also µ C denote by π . As µ (2.3) π (g)∗ = π (σ(g)−1) µ µ THE SEGAL-BARGMANN TRANSFORM ON COMPACT SYMMETRIC SPACES 6 on U and both sides are anti-holomorphic on U , it follows that (2.3) holds for all g U . C C ∈ We extend πu to a holomorphic function on M by µ C (2.4) πu(z) = u,π (σ(z))e := u,π (σ(g))e = π (g−1)u,e , z = g o. µ h µ µiµ h µ µiµ h µ µiµ · We normalize the invariant measure on compact groups so that the total measure of e the group is one. Then f(m)dm = f(a o)da defines a normalized U-invariant M U · measure on M. The corresponding L2-inner product, respectively norm, is denoted by R R , , respectively . 2 2 h· ·i k · k Recall that by Schur’s orthogonality relations we have 1 (2.5) u,π (g)v π (g)x,y du = δ u,x y,v . µ µ δ δ µ,ν µ µ h i h i d(µ)h i h i ZU In particular, V L2(M), u d(µ)1/2πu is a unitary U-isomorphism onto its image µ → 7→ µ L2(M) L2(M). Furthermore µ ⊂ L2(M) = L2(M) . µ µ∈Λ+ M Furthermore, each function f L2(M) has a holomorphic extension f to M . µ C ∈ Lemma 2.2. Let the notations be as above. e (1) Let µ,δ Λ+, u V , v V , and H ,H a . Then µ δ 1 2 C ∈ ∈ ∈ ∈ δ πu(gexpH )πv(gexpH )dg = µ,δ u,v π (expH )e ,π (expH )e µ 1 δ 2 d(µ)h iµh µ 2 µ µ 1 µiµ ZU δ µ,δ e e = u,v e ,π (exp(H σ(H ))e . µ µ µ 1 2 µ µ d(µ)h i h − i (2) Let L M be compact. Then there exists a constant C > 0 such that C L ⊂ πu(z) eCLkµk u | µ | ≤ k kµ for all z L. ∈ Proof. (1) This follows from Schur’s orthogonality relations (2.5): πu(gexpH )πv(gexpH )dg = u,π (g)π (expH )e π (g)π (expH )e ,v dg µ 1 δ 2 h µ µ 1 µiµh δ δ 2 δ iδ ZU ZU δ µ,δ e e = u,v π (expH )e ,π (expH )e µ µ 2 µ µ 1 µ µ d(µ)h i h i δ µ,δ = u,v e ,π (exp(H σ(H )))e µ µ µ 1 2 µ µ d(µ)h i h − i where we used that π (expH )∗ = π (σ(expH )−1). (2) follows from Lemma 3.9 in µ 2 µ 2 (cid:3) [4]. THE SEGAL-BARGMANN TRANSFORM ON COMPACT SYMMETRIC SPACES 7 For f L2(U) L1(U) let ∈ ⊂ π (f) = f(g)π (g)dg µ µ ZU be the integrated representation. Denote by P = π (k)dk : V VK the orthogonal µ K µ µ → µ projection v v,e e . If f L2(M) = L2(U)K, then π (f) = π (f)P . Define the 7→ h µi µ ∈ R µ µ µ (vector valued) Fourier transform of f L2(M) by ∈ (2.6) f := π (f)e . µ µ µ Denote the left-regular representation of U on L2(M) by L. Thus (L(a)f)(x) = f(a−1 x). b · Then (2.7) L\(a)f = π (a)f . µ µ µ To describe the image of the Fourier transform lebt V := (v ) ( µ Λ+) v V and d(µ) v 2 < µ  µ µ∈Λ+ (cid:12) ∀ ∈ µ ∈ µ k µkµ ∞ µ∈MΛ+,d  (cid:12)(cid:12) µX∈Λ+  (cid:12) (cid:12)   = v : Λ+ (cid:12)Π V ( µ Λ+) v(µ) V and d(µ) v(µ) 2 < .  → µ∈Λ+ µ (cid:12) ∀ ∈ ∈ µ k k ∞  (cid:12)(cid:12) µX∈Λ+  (cid:12) Then µ∈Λ+,dVµ is a Hilbert space (cid:12)(cid:12)with the inner product  L (v ),(w ) = d(µ) v ,w . µ µ µ µ µ h i h i µ∈Λ+ X The group U acts unitarily on V by (L(a)(u )) = (π (a)u ) . µ∈Λ+,d µ µ µ µ µ µ Theorem 2.3. If f L2(M) tLhen (f ) V and : L2(M) V is ∈ µ µ ∈ µ∈bΛ+,d µ → µ∈Λ+,d µ a unitary U-isomorphism with inverse L L b v d(µ)πv(µ). c 7→ µ µ∈Λ+ X In particular if f L2(M) then ∈ b (2.8) f = d(µ) f ,π ( )e = d(µ)πfµ and f 2 = d(µ) f 2 h µ µ · µiµ µ k k2 k µkµ µ∈Λ+ µ∈Λ+ µ∈Λ+ X X X b b where the first sum is taking in L2(M). If f is smooth, then the sum converges in the C∞-topology. Furthermore (1) If f L2(M) then the orthogonal projection of f into L2(M) is given by f = µ µ ∈ d(µ) f ,π ( )e . µ µ µ h · i b THE SEGAL-BARGMANN TRANSFORM ON COMPACT SYMMETRIC SPACES 8 (2) f has a holomorphic continuation f to M which is given by µ µ C (2.9) f (z) = d(µ) f ,π (z¯)e . µ eh µ µ µi (3) If L M is compact, then there exists a constant C > 0 such that C L ⊂ e b sup f (x) d(µ)eCLkµk f . µ 2 | | ≤ k k x∈L e Proof. This is well known but we indicate how the statements follows from the general Plancherel formula for compact groups, see [9], p. 134. We have f(x) = d(µ)Tr(π (x−1)π (f)) and f 2 = d(µ)Tr(π (f)∗π (f)). µ µ k k2 µ µ µ∈Λ+ µ∈Λ+ X X Extending e to an orthonormal basis for V , it follows from π (f) = π (f)P that µ µ µ µ K Tr(π (x−1)π (f)) = π (x−1)π (f)e ,e = f(µ),π (x)e µ µ µ µ µ µ µ µ µ µ h i h i and e Tr(π (f)∗π (f)) = π (f)∗π (f)e ,e = π (f)e ,π (f)e = f(µ) 2 . µ µ h µ µ µ µiµ h µ µ µ µiµ k kµ The L2-part of the theorem follows now from the Plancherel formula for U. The inter- b twining property is a consequence of (2.7). For the last statement see [38]. That the orthogonal projection f f is given by f (x) = d(µ) f ,π (x)e follows µ µ µ µ µ 7→ h i from (2.8). The last part follows from (2.4) and f f . (cid:3) µ µ 2 k k ≤ k k b The spherical function on M associated with µ bis the matrix coefficient (2.10) ψ (g) = πeµ(g) = e ,π (g)e , g U . µ µ h µ µ µiµ ∈ It is independent of the choice of e as long as e = 1. We will view ψ as a K- µ µ µ µ k k biinvariant function on U or as a K-invariant function on M. ψ is the unique element µ in L2(M)K which takes the value one at the base point o. µ If f L2(M)K then f = f ,e e VK. Furthermore, ∈ µ h µ µiµ µ ∈ µ fµ,eµ µ = πµ(bf)eµ,ebµ µ = f(m)ψµ(m)dm = f(a o)ψµ(a−1)da. h i h i · ZM ZU This motibvates the definition of the spherical Fourier transform on L2(M)K by (2.11) f(µ) = f,ψ . µ 2 h i Define the weighted ℓ2-space ℓ2(Λ+) by d b ℓ2d(Λ+) := (aµ)µ∈Λ+ (cid:12) aµ ∈ C and d(µ)|aµ|2 < ∞.  (cid:12)(cid:12) µX∈Λ+  (cid:12) (cid:12)   (cid:12) THE SEGAL-BARGMANN TRANSFORM ON COMPACT SYMMETRIC SPACES 9 Then ℓ2(Λ+) is a Hilbert space. d Theorem 2.4. The spherical Fourier transform is a unitary isomorphism of L2(M)K onto ℓ2(Λ+) with inverse d (a ) d(µ)a ψ µ µ µ µ 7→ µ∈Λ+ X where the sum is taken in L2(M)K. It converges in the C∞-topology if f is smooth. Furthermore, f 2 = d(µ) f(µ) 2. k k2 | | µ∈Λ+ X b (cid:3) Proof. This follows directly from Theorem 2.3. Notethatψ hasa holomorphicextension ψ toM given by ψ (z) = e ,π (σ(z))e . µ µ C µ µ µ µ µ h i Lemma 2.5. Let f L2(M). Then f (z) = d(µ)f ψ (z). µ e µ e ∈ ∗ Proof. We have e f (z) = d(µ) f ,π (σ(z))e µ µ µ µ µ h i = f(g o) π (g)e ,π (σ(z))e dg b µ µ µ µ µ · h i ZU = f(g o) e ,π (σ(g−1z))e dg µ µ µ µ · h i ZU = f ψ (z). µ ∗ (cid:3) e 3. The Fock Space (M ) t C H In this section, we start by recalling some needed and well-known facts on integration on Md = G/K, thenoncompact dualofM. Wethenintroducetheheat kernel hd onMd. For t more details and proofs for the statements involving hd we refer to [21, 22, 27, 28] and the t references therein. We introduce the Fock space (M ). Using the restriction principle t C H introduced in [30] we show that (M ) is isomorphic to L2(M) as a U-representation. In t C H the next section we will show that the Segal-Bargmann transform H : L2(M) (M ) t t C → H is a unitary isomorphism. Let (ia) = H ia ( α Σ+) α(H) > 0 . + { ∈ | ∀ ∈ } The following is a well known decomposition theorem for an involution commuting with a given Cartan involution, see [8] or Proposition 7.1.3. in [36]. THE SEGAL-BARGMANN TRANSFORM ON COMPACT SYMMETRIC SPACES 10 Lemma 3.1. Let z M . Then there exist u U and H ia such that z = uexp(H) o. C ∈ ∈ ∈ · If u exp(H ) o = u exp(H ) o then there exists w W such that H = w H . If we 1 1 2 2 2 1 · · ∈ · choose H (ia) , then H is unique. + ∈ Let m be a U -invariant measure on M . C C Theorem 3.2. We can normalize m such that for f L1(M ) C ∈ f(z)dm(z) = f(uexpH o)J(H)dHdu, · ZMC ZU Z(ia)+ where J(H) = sinh(2 α,H ) . h i α∈Σ+ Y Proof. This follows from the general integration theorem for symmetric space applied to M , see [8] or Proposition 8.1.1 in [36], using that sinh(2x) = 2sinh(x)cosh(x). (cid:3) C Let m be a G-invariant measure on Md. 1 Theorem 3.3. We can normalize m such that for f L1(M) 1 ∈ f(x)dm (x) = f(kexpH o)J (H)dHdk. 1 1 · ZMd ZK Z(ia)+ where J (H) = J(2−1H). 1 Corresponding to the positive system Σ+ there is an Iwazawa decomposition G = KAdN of G, where Ad = exp(ia). Write x G as x = k(x)a(x)n(x). For α Σ let ∈ ∈ m = dim u and let ρ = 2−1 m α ia∗. Let α C C,α α∈Σ+ α ∈ Pϕ (x) = a(gk)iλ−ρdk λ ZK denote the spherical functions on Md with spectral parameter λ, see [19], Theorem 4.3, p. 418, and p. 435. Lemma 3.4. Let µ Λ+. Then ϕ extends to a holomorphic function ϕ on M i(µ+ρ) i(µ+ρ) C ∈ and ψ = ϕ . µ i(µ+ρ) e Prooff. Seeethe proof of Lemma 2.5 in [4] and the fact that ϕλ(g−1) = ϕ−λ(g), see [19], p. (cid:3) 419. Consider the complex-bilinear extension (·, ·)of h·, ·i|(ia)∗×(ia)∗ to a∗C. We write λ·µ = (λ,µ) and λ2 = (λ,λ). The trace form (X,Y) = Tr(XY∗) defines a K-invariant metric on is and hence a Gd- − invariant metric onMd. We consider the Laplace operator ∆d associated with this metric.