MULTIPLICITY-FREE HOMOGENEOUS OPERATORS IN THE COWEN-DOUGLAS CLASS ADAMKORA´NYIANDGADADHARMISRA Abstract. Inarecentpaper,theauthorshaveconstructedalargeclassofoperatorsinthe Cowen-Douglas class Cowen-Douglas classof theunitdisc Dwhich arehomogeneous with respect to the action of the group M¨ob – the M¨obius group consisting of bi-holomorphic 9 automorphismsoftheunitdiscD. Theassociatedrepresentationforeachoftheseoperators 0 0 is multiplicity free. Here we give a different independent construction of all homogeneous 2 operators in theCowen-Douglas class with multiplicity freeassociated representation and n verify that they are exactly the examples constructed previously. a J 7 The homogeneous operators form a class of bounded operators T on a Hilbert space H. ] Theoperator T is saidto behomogeneous if its spectrumiscontained in theclosed unitdisc A andforeveryM¨obiustransformationg theoperatorg(T), definedviatheusualholomorphic F functional calculus, is unitarily equivalent toT. To every homogeneous irreducibleoperator . h T there corresponds an associated unitary representation π of the universal covering group t a G˜ of the M¨obius group G: m [ π(gˆ)∗T π(gˆ)= (pgˆ)(T), gˆ∈ G˜, 1 v where p : G˜ → G is the natural homomorphism. In the paper [6] (see also [3]), it was 4 9 shown that each homogeneous operator T, not necessarily irreducible, in B (D) admits m+1 7 an associated representation. The representations of G˜ are quite well-known, but we are 0 . still far from a complete description of the homogeneous operators. In the recent paper [6], 1 0 the following theorem was proved. 9 0 Theorem 0.1. For any positive real number λ > m/2, m ∈ N and an (m+1) - tuple of : v positive reals µ = (µ ,µ ,...,µ ) with µ = 1, there exists a reproducing kernel K(λ,µ) on i 0 1 m 0 X the unit disc such that the adjoint of the multiplication operator M(λ,µ) on the corresponding ar Hilbert space A(λ,µ)(D) is homogeneous. The operators (M(λ,µ))∗ are in the Cowen-Douglas class B (D), irreducible and mutually inequivalent. m+1 In the paper [6], we have presented the operators M(λ,µ) in as elementary a way as possible, butthispresentation hidesthenaturalways inwhich theseoperators can befound to begin with. Here we will describe another independent construction of the operators M(λ,µ). We will also give an exposition of some of the fundamental background material. Finally, we will prove that if T is an irreducible homogeneous operator in B (D) whose m+1 associated representation is multiplicity free then, up to equivalence, T is the adjoint of of the multiplication operator M(λ,µ) for some λ> m/2 and µ≥ 0. This research was supported in part by a DST- NSFS&TCooperation Programme. 1 2 ADAMKORA´NYIANDGADADHARMISRA 1. Background material Although,weintendtodiscusshomogeneousoperatorsintheCowen-DouglasclassB (D), n the material below is presented in somewhatgreater generality. Here we discuss commuting tuples of operators in the Cowen-Douglas class B (D) for some bounded open connected n set D ⊆ Cm. The unitary equivalence class of a commuting tuple in B (D) is in one to one n correspondence with a certain class of holomorphic Hermitian vector bundles (hHvb) on D [4]. These are distinguished by the property, among others, that the Hermitian structure on the fibre at w ∈ D is induced by a reproducing kernel K. It is shown in [4] that the correspondingoperator can berealized astheadjointof thecommutingtuplemultiplication operator M on the Hilbert space H of holomorphic functions with reproducing kernel K. Start with a Hilbert space H of Cn - valued holomorphic functions on a bounded open connected set D ⊆ Cm. Assume that the Hilbert space H contains the set of vector valued polynomials and that these form a dense subset in H. We also assume that there is a reproducing kernel K for H. We use the notation K (z) := K(z,w). w Recall that a positive definite kernel K : D×D → Cn×n on D defines an inner product on the linear span of {K (·)ξ : w ∈D,ξ ∈ Cn} ⊆ Hol(D,Cn) by the rule w hK (·)ξ,K (·)ηi = hK (u)ξ,ηi, ξ,η ∈ Cn. w u w (On the right hand side h,i denotes the inner product of Cn. We denote by ε ,...,ε 1 n the natural basis of Cn.) The completion of this subspace is then a Hilbert space H of holomorphic functions on D (cf. [1]) in which the set of vectors {K : w ∈ D} is dense. w The kernel K has the reproducing property, that is, hf,K ξi = hf(w),ξi, f ∈H, w ∈ D, ξ ∈ Cm. w Now, for 1≤ i ≤ m, we have M∗K ξ = w¯ K ξ, w ∈ D, where M f (z) = z f(z), f ∈ H i w i w i i and {K ε }n is a basis for ∩m ker(M −w )∗,(cid:0)w ∈ D(cid:1). w i i=1 i=1 i i The joint kernel of the commuting m - tuple M∗ = (M∗,...,M∗), which we assume 1 m to be bounded, then has dimension n. The map σ : w 7→ K ε , w ∈ D¯, 1 ≤ i ≤ n, i w¯ i provides a trivialization of the corresponding bundle E of Cowen - Douglas (cf. [4]). Here D¯ := {z ∈ Cm |z¯∈D}). On the other hand, suppose we start with an abstract Hilbert space H and a m-tuple of commuting operators T = (T ,...,T ) in the Cowen - Douglas class B (D). Then we have 1 m n a holomorphic Hermitian vector bundle E over D with the fibre E = ∩n ker(T −w ) at w i=1 i i w ∈ D. Following [4], one associates to this a reproducingkernel Hilbert space Hˆ consisting of holomorphic functions on D¯ as follows. Take a holomorphic trivialization σ : D → H i with σ (w), 1 ≤ i ≤ n, spanning E . For f ∈ H, define fˆ(w) := hf,σ (w¯)i , w ∈ D¯. Set i w j j H hfˆ,gˆi := hf,gi . The function K ε := σ\(w¯) then serves as the reproducing kernel for Hˆ H w j j the Hilbert space Hˆ. Note that hKw(z)εj,εiiCn = hKwεj,KzεiiHˆ \ \ = hσ (w¯),σ (z¯)i j i Hˆ = hσ (w¯),σ (z¯)i , z,w ∈ D¯. j i H MULTIPLICITY-FREE HOMOGENEOUS OPERATORS IN THE COWEN-DOUGLAS CLASS 3 If one applies this construction to the case where H is a Hilbert space of holomorphic functions on D, possesses a reproducing kernel, say K, and the operator M∗ is in B (D¯) n then using the trivialization σ (w) = K ε , w ∈ D¯ for the bundle E defined on D¯, the i w¯ i reproducing kernel for Hˆ is hKw(z)εj,εiiCn = hKwεj,KzεjiH = hσ (w¯),σ (z¯)i j i H = hK ε ,K ε i , z,w ∈ D. w j z i Hˆ Thus H = Hˆ. LetG beaLie groupacting transitively on thedomain D ⊆ Cd. LetGL(n,C)denotethe set of non-singular n×n matrices over the complex field C. We start with a multiplier J, that is, a smooth family of holomorphic mapsJ : D → Cn×n satisfying the cocycle relation g (1.1) J (z) = J (z)J (h·z), for all g,h ∈G, z ∈ D, gh h g Let Hol(D,Cn) be the linear space consisting of all holomorphic functions on D taking values in Cn. We then obtain a natural (left) action U of the group G on Hol(D,Cn): (1.2) (Ugf)(z)= Jg−1(z)f(g−1 ·z), f ∈Hol(D,Cn), z ∈ D. Let K ⊆ G be the compact subgroup which is the stabilizer of 0. For h,k in K, we have J (0) = J (0)J (0) so that k 7→ J (0)−1 is a representation of K on Cn. kh h k k As in [6], we say that if a reproducing kernel K transforms according to the rule (1.3) J(g,z)K(g(z),g(ω))J(g,ω)∗ = K(z,ω) for all g ∈ G˜; z,ω ∈ D, then K is quasi-invariant. Proposition 1.1 ([6], Proposition 2.1). Suppose H has a reproducing kernel K. Then U defined by (1.2) is a unitary representation if and only if K is quasi-invariant. Let g be an element of G which maps 0 to z, that is g ·0 = z. z z For quasi-invariant K we have (1.4) K(g ·0,g ·0) = (J (0))−1K(0,0)(J (0)∗)−1, z z gz gz which shows that K(z,z) is uniquely determined by K(0,0). For each z in D, the positive definite matrix K(z,z) gives the Hermitian structure of our vector bundle. Given any positive definite matrix K(0,0) such that (1.5) J (0)−1K(0,0) = K(0,0)J (0)∗ for allk ∈ K, k k that is, the inner product hK(0,0)· | ·i is invariant under J (0), (1.4) defines a Hermitian k structure on the homogeneous vector bundle determined by J (z). In fact, K(z,z), for any g z ∈D iswell defined,becauseifg′ isanother element of Gsuchthatg′ ·0 = z theng′ = g k z z z z 4 ADAMKORA´NYIANDGADADHARMISRA for some k ∈ K. Hence K(g′ ·0,g′ ·0) = K(g k·0,g k·0) z z z z = (J (0))−1K(0,0)(J (0)∗)−1 gzk gzk = J (0)J (k·0) −1K(0,0) J (k·0)∗J (0)∗ −1 k gz gz k = (cid:0)(Jgz(0))−1(Jk(0(cid:1)))−1K(0,0(cid:0))(Jk(0)∗)−1(Jgz(0(cid:1))∗)−1 = (J (0))−1K(0,0)(J (0)∗)−1 gz gz = K(g ·0,g ·0) z z Thisgives agood overview of alltheHermitian structuresof ahomogeneous holomorphic vector bundle. But not all such bundles arise from a reproducing kernel. Starting with a positive matrix satisfying (1.5), (1.4) gives us K(z,z), but there is no guarantee (and is false in general) that K(z,z) extends to a positive definite kernel on D×D. It is, however, true that if there is such an extension then it is uniquely determined by K(z,z) (because K(z,w) is holomorphic in z and antiholomorphic in w). This leaves us with the following possiblestrategy for findingthe homogeneous operators in the Cowen - Douglas class. Find all multipliers, (i.e., holomorphic homogeneous vector bundles(hhvb))suchthat thereexists K(0,0) satisfying (1.5)and consider all such K(0,0). ThendeterminewhichoftheK(z,z)obtainedform(1.4)extendstoapositivedefinitekernel on D ×D. Then check if the multiplication operator is well-defined and bounded on the corresponding Hilbert space. Let H be a Hilbert space consisting of Cn - valued holomorphic functions on some do- main D possessing a reproducing kernel K. The sections of the corresponding holomorphic Hermitian vector bundle defined on D have many different realizations. The connection between two of these is given by a n×n invertible matrix valued holomorphic function ϕ on D. For f ∈ H, consider the map Γ : f 7→ f˜, where f˜(z) = ϕ(z)f(z). Let H˜ = {f˜: f ∈ H}. ϕ The requirement that the map Γ is unitary, prescribes a Hilbert space structure for the ϕ function space H˜. The reproducing kernel for H˜ is easily calculated (1.6) K˜(z,w) = ϕ(z)K(z,w)ϕ(w)∗. It is also easy to verify that Γ MΓ∗ is the multiplication operator M : f˜ 7→ zf˜ on the ϕ ϕ Hilbert space H˜. Suppose we have a unitary representation U given by a multiplier J acting on H according to (1.2). Transplanting this action to H˜ under the isometry Γ , it ϕ becomes U˜g−1f˜ (z) = J˜g(z)f˜(g·z), where the new multiplier J˜is gi(cid:0)ven in(cid:1)terms of the original multiplier J by J˜(z) = ϕ(z)J (z)ϕ(g·z)−1. g g Of course, now K˜ transforms according to (1.3), with the aid of J˜. If we want, we can now ensure that, by passing from H to an appropriate H˜, K˜(z,0) ≡ 1. We merely have to set ϕ(z) = K(0,0)1/2K(z,0)−1. Thus the reproducing kernel K˜ is almost unique. The only freedom left is to multiply ϕ(z) by a constant unitary n ×n matrix. Once the kernel is normalized, we have J (z) =J (0), z ∈ D, k ∈K. k k MULTIPLICITY-FREE HOMOGENEOUS OPERATORS IN THE COWEN-DOUGLAS CLASS 5 In fact, I = K(z,0) = J (z)K(k·z,0)J (0)∗ = J (z)J (0)−1 k k k k and the statement follows. Therefore, once the kernel K is normalized, we have Uk−1f (z) = Jk(0)f(k·z), k ∈ K. (cid:0) (cid:1) Given a multiplier J, there is always the following method for constructing a Hilbert space with a quasi-invariant Kernel K transforming according to (1.4). We look for a func- tional Hilbert space possessing this property among the weighted L2 spaces of holomorphic functions on D. The norm on such a space is (1.7) kfk2 = f(z)∗Q(z)f(z)dV(z) ZD with some positive matrix valued function Q(z). Clearly, this Hilbert space possesses a reproducing kernel K. The condition that Ug−1 in (1.2) is unitary is f(g·z)∗J∗(z)Q(z)J (z)f(g·z)dV(z) = f(w)∗Q(w)f(w)dV(w) g g ZD ZD 2 ∂(g·z) = f(g·z)∗Q(g·z)f(g·z) dV(z), ∂(z) ZD (cid:12) (cid:12) (cid:12) (cid:12) that is, (cid:12) (cid:12) (cid:12) (cid:12) −2 ∂(g·z) (1.8) Q(g·z)= J (z)∗Q(z)J (z) , g g ∂(z) (cid:12) (cid:12) (cid:12) (cid:12) which is equation (1.3) with Jg(z) replaced by ∂∂(g(z·z))J(cid:12)(cid:12)g(z)∗−1.(cid:12)(cid:12) Given the multiplier J (z), Q(z) is again determined by Q = Q(0), and (just as in the g case of K(0,0) = A) it must be a positive matrix commuting with all J (0), k ∈ K. (It is k assumed that each J (0) is unitary). k In this way, we can construct many examples of homogeneous operators in B (D) but n not all. Even,notallthethehomogeneousoperatorsinB (D)comefromthisconstruction. There 1 is a homogeneous operator in the class B (D) corresponding to the multiplier J(g,z) = 1 (g′(z))λ, λ ∈ R exactly when λ> 0. The reproducing kernel is K(z,w) = (1−zw¯)−2λ. But such an operator arises from the construction outlined above only if λ ≥ 1/2. Never the less, the homogeneous operators constructed in the manner described above are of interest since they happen to be exactly the subnormal homogeneous operators in this class (cf. [2]). 2. Computation of the multipliers for the unit disc In the case of B (D), it is shown in [6] that the bundle corresponding to a homogeneous n Cowen-Douglas operator admits an action of the covering group G˜ of the group G = M¨ob via unitary bundle maps. This suggests the strategy of first finding all the homogeneous holomorphic Hermitian vector bundles (a problem easily solved by known methods) and then determining which of these correspond to an operator in the Cowen-Douglas class. 6 ADAMKORA´NYIANDGADADHARMISRA We are going to use the method of holomorphic induction. For this, first we describe some basic facts and fix our notation. We follow the notation of [7] which we will use as a reference. 0 1 i 0 The Lie algebra g of G˜ is spanned by X = 1 , X = 1 and Y = 1 2 1 0 0 2 0 −i ! ! 0 −i 1 . The subalgebra k corresponding to K˜ is spanned by X . In the complexified 2 i 0 0 ! C Lie algebra g , we mostly use the complex basis h,x,y given by 1 1 0 h = −iX = 0 2 0 −1 ! 0 1 x = X +iY = 1 0 0 ! 0 0 y = X −iY = 1 1 0 ! We write GC for the (simply connected group) SL(2,C). Let G = SU(1,1) be the 0 z 0 subgroup corresponding to g. The group GC has the closed subgroups KC = : 0 1 z n(cid:16) (cid:17) 1 z 1 0 z ∈ C,z 6= 0 , P+ = : z ∈ C , P− = : z ∈ C ; the corresponding Lie 0 1 z 1 o n(cid:16) (cid:17) o n(cid:16) (cid:17) o c 0 0 c 0 0 algebras kC = : c ∈ C , p+ = : c ∈ C , p− = : c ∈ C are 0 −c 0 0 c 0 n(cid:16) (cid:17) o n(cid:16) (cid:17) o n(cid:16) (cid:17) o a 0 spannedbyh,xandy,respectively. TheproductKCP− = :0 6= a∈ C,b∈ C b 1 a ! is a closed subgroup to be denoted T; its Lie algebra is tn= Ch+ Cy. The product seot P+KCP− = P+T is dense open in GC, contains G, and the product decomposition of each of its elements is unique. (GC/T is the Riemann sphere, gK˜ → gT, (g ∈ G) is the natural embedding of D into it.) Accordingtoholomorphicinduction[5,Chap13]theisomorphismclassesofhomogeneous holomorphic vector bundles are in one to one correspondence with equivalence classes of linear representations ̺ of the pair (t,K˜). Since K˜ is connected, here this means just the representations of t. Such a representation is completely determined by the two linear transformations ̺(h) and ̺(y) which satisfy the bracket relation of h and y, that is, (2.9) [̺(h),̺(y)] = −̺(y). TheG˜-invariantHermitianstructuresonthehomogeneousholomorphicvectorbundle(mak- ing it into a homogeneous holomorphic Hermitian vector bundle), if they exist, are given by ̺(K˜)-invariant inner products on the representation space. An inner product is ̺(K˜)- invariant if and only if ̺(h) is diagonal with real diagonal elements in an appropriate basis. We will be interested only in bundles with a Hermitian structure. So, we will assume without restricting generality, that the representation space of ̺ is Cd and that ̺(h) is a real diagonal matrix. MULTIPLICITY-FREE HOMOGENEOUS OPERATORS IN THE COWEN-DOUGLAS CLASS 7 Furthermore, we will beinterested only in irreduciblehomogeneous holomorphic Hermit- ian vector bundles, this corresponds to ̺ not being the orthogonal direct sum of non-trivial representations. Suppose we have such a ̺; we write V for the eigenspace of ̺(h) with α eigenvalue α. Let −η be the largest eigenvalue of ̺(h) and m be the largest integer such that −η,−(η+1),...,−(η+m) areall eigenvalues. From (2.9)we have ̺(y)V ⊆ V ; this α α−1 and orthogonality of the eigenspaces imply that V = ⊕m V and its orthocomplement j=0 −(η+j) are invariant under ̺. So, V is the whole space, and have proved that the eigenvalues of ̺(h) are −η,...,−(η+m). ¿From this it is clear that ̺ can be written as the tensor product of the one dimensional representation σ given by σ(h) = −η, σ(y) = 0, and the representation ̺0 given by ̺0(h) = ̺(h)+ηI, ̺0(y) = ̺(y). Correspondingly, the bundle for ̺ is the tensor product of a line bundle L and the bundle corresponding to ̺0. η Therepresentation̺0 hasthegreatadvantagethatitliftstoaholomorphicrepresentation of the group T. It follows that the homogeneous holomorphic vector bundle it determines for D,G˜, can be obtained as the restriction to D of the homogeneous holomorphic vector C bundleover G /T obtained by ordinaryinduction in thecomplex analytic category. So, (as 1 z C a convenient choice) take the local holomorphic cross section z 7→ s(z) := of G /T 0 1 (cid:16) (cid:17) a b over D. In thetrivialization given bys(z), the multiplier then appearsfor g = ∈ GC c d as (cid:16) (cid:17) J0(z) = ̺0 s(z)−1g−1s(g·z) g (cid:0) cz+d 0 (cid:1) = ̺0 −c (cz+d)−1 ! −c (2.10) = ̺0 exp y ̺0 exp(2log(cz+d)h) . cz+d (cid:16) (cid:17) (cid:0) (cid:1) (cid:0) (cid:1) The last two equalities are simple computations. For the line bundle L , the multiplier is g′(z)η (we write g′(z) = ∂g(z)). Consequently, η ∂z the multiplier corresponding to the original ̺ is (2.11) J (z) = g′(z) ηJ0(z). g g (cid:0) (cid:1) 3. Conditions imposed by the reproducing kernel We now assume that we have a homogeneous holomorphic vector bundleinduced by ̺ as in the preceding sections and that it has a reproducing kernel. Then we derive conditions abouttheactionofG˜ thatfollowfromthishypothesis. Inthefinalsection, wewillshowthat these conditions are sufficient: they lead directly to the construction of all homogeneous operators the Cowen-Douglas class with multiplicity free representations. Under our hypothesis there is a Hilbert space structure on our sections in which the action of G˜ given by (1.4) is unitary. We will study this representation through its K - types (i.e., its restriction to K˜). We first compute the infinitesimal representation. 8 ADAMKORA´NYIANDGADADHARMISRA For X ∈g, and holomorphic f, we have (U f)(z) := d U f (z) X dt |t=0 exp(tX) (cid:0) (cid:1)∂ ex(cid:0)p(−tX)·(cid:1)z η (3.12) = d J0 (z)f(exp(−tX)·z) . dt |t=0 ∂z exp(−tX) (cid:0) (cid:1) n(cid:16) (cid:17) o (cid:0) (cid:1) C C There is a local action of G , so this formula remains meaningful also for X ∈g . There are three factors to differentiate. For the last one, d f(exp(−tX)·z) = −(Xz)f′(z), dt |t=0 1 t (cid:0) (cid:1) and we see that exp(tx)·z = ·z = z+t gives x·z = 1; by similar computations, 0 1 ! y · z = −z2, h · z = z. For the first factor, we interchange the differentiations and get −η ∂ (X ·z), i.e., 0,2ηz,−η, respectively for x,y and h. ∂z To differentiate the factor in the middle, we use its expression (2.10). First for X = y, we have d d ̺0 exp(−t(tz+1)−1y) = exp(−t(tz+1)−1̺0(y) dt dt (cid:12)t=0 (cid:12)t=0 (3.13) (cid:12)(cid:12) (cid:0) (cid:1) = −̺(cid:12)(cid:12)0(y)(cid:0) (cid:1) (cid:12) (cid:12) and d d ̺0(exp(2log(tz+1)h)) = exp(2log(tz+1)̺0(h)) dt dt (cid:12)t=0 (cid:12)t=0 (3.14) (cid:12)(cid:12) = 2z̺(cid:12)(cid:12)0(h) (cid:12) (cid:12) ¿From these, following the conventions of [7] in defining H,E,F, it follows that d (Ff)(z) := (U f)(z) = J (z)f(exp(ty)·z) −y exp(ty) dt (cid:12)|t=0 (cid:12) (3.15) = −(cid:12)2ηzI +2z̺0(h)−̺0(y) f(z)−z2f′(z). (cid:12) (cid:0) 1 t (cid:1) Similar, simpler computations give, for g = exp(tx) = 0 1 ! (3.16) (Ef)(z) := U f (z) = −f′(z). x et/2 0 (cid:0) (cid:1) Finally, for g = exp(th) = , we have 0 e−t/2 ! e−t/2 0 J (z) = ̺ = exp(−t)̺0(h). exp(th) 0 et/2 ! Hence it is not hard to verify that (3.17) (Hf)(z) := U f (z) = −ηI +̺0(h) f(z)−zf′(z). h Under our hypothesis, we have(cid:0)a rep(cid:1)roduci(cid:0)ng kernel and(cid:1)U is unitary. From our computa- tions above, we can determine how U decomposes into irreducibles. The infinitesimal rep- resentation of U acts on the vector valued polynomials; a good basis for this space is {ε zn : j n ≥ 0}; ε is the jth natural basis vector in Cd. We have H(ε zn) = −(η+j+n)(ε zn), so j j j the lowest K - types of the irreducible summands are spanned by the ε . This space is also j MULTIPLICITY-FREE HOMOGENEOUS OPERATORS IN THE COWEN-DOUGLAS CLASS 9 the kernel of E. So, U is direct sum of discrete series representations (Uη+j, in the notation of [7]), each one appearing as many times as −(η+j) appears on the diagonal of ̺(h). 4. The multiplicity-free case In order to be able to use the computations of [6] without confusion, we introduce the parameter λ = η+ m. 2 From the last remark of the preceding section, it is clear that if U is multiplicity-free then ̺(h) is an (m+1)×(m+1) matrix with eigenvalues −λ+m,−λ+m−1,...,−λ−m. 2 2 2 As ̺(h)ε = −(λ− m +j)ε , (2.9) shows that j 2 j m ̺(h) ̺(y)ε = −(λ+ +j +1)̺(y)ε ,that is,̺(y)ε = constε . j j j j+1 2 So, ̺(y) is a low(cid:0)er tria(cid:1)ngular matrix (with non-zero entries, otherwise we have a reducible bundle). The homogeneous holomorphic vector bundle determines ̺(y) only up to a conju- gacy by a matrix commuting with ̺(h), that is, a diagonal matrix. So, we can choose the realization of our bundle by applying an appropriate conjugation such that ̺(y) = S , the m triangular matrix whose (j,j −1) element is j for 1 ≤ j ≤ m. By standard representation theory of SL(2,R), the vectors (−F)nε are orthogonal and j the irreducible subspaces H(j) for U are span{(−F)nε : n ≥ 0} for 0 ≤ j ≤ m. There is j also precise information about the norms. Using this, we can construct an orthonormal basis for our representation space. For any n≥ 0, we let uj(z) = (−F)nε . n j j To proceed further, we need to find the vectors u (z) explicitly. This is facilitated by the n following Lemma. Lemma 4.1. Let u be a vector with u (z) = u zn−ℓ, 0 ≤ ℓ ≤ m and n≥ 0. We then have ℓ ℓ (−Fu) (z) = (2λ−m+ℓ+n)u zn+1−ℓ+ℓu zn+1−ℓ, 0≤ ℓ ≤ m. ℓ ℓ ℓ−1 Proof. We recall (3.15) that −(Ff)(z) = 2λzf(z) + S f(z) − 2zD f(z) + z2f′(z) for m m f ∈ H(n),whereD = −̺0(h)isthediagonaloperatorwithdiagonal{−m,−m+1,..., m} m 2 2 2 and S is the forward weighted shift with weights 1,2,...,m. Therefore we have m (−Fu) (z) = 2λu +ℓu −(m−2ℓ)u +(n−ℓ)u zn+1−ℓ ℓ ℓ ℓ−1 ℓ ℓ completing the proof. (cid:0) (cid:1) (cid:3) Lemma 4.2. For 0 ≤ j ≤ m and 0≤ ℓ ≤ m, we have 0 if 0≤ ℓ ≤ j −1 j u (z) = n,ℓ ( nk (j +1)k(2λ−m+2j +k)n−kzn−k if j ≤ℓ ≤ m, k = ℓ−j, where uj (z) is the(cid:0)s(cid:1)calar valued function at the position ℓ of the Cm+1 - valued function n,ℓ uj(z) := (−F)nε . n j Proof. The proof is by induction on n. The vectors uj are in H(n) for 0 ≤ j ≤ m. For a n j fixed but arbitrary positive integer j, 0 ≤ j ≤ m, we see that u (z) is 0 if n < ℓ−j. We n,ℓ j j have to verify that (−Fu )(z) = u (z). From the previous Lemma, we have n n+1 (−Fuj) (z) = (2λ−m+ℓ+n+j)uj zn+j+1−ℓ+ℓuj zn+j+1−ℓ, n ℓ n,ℓ n,ℓ−1 10 ADAMKORA´NYIANDGADADHARMISRA where (−Fuj) (z) is the scalar function at the position ℓ of the Cm+1 - valued function n ℓ j (−Fu )(z). To complete the proof, we note (using k = ℓ−j) that n (−Fuj) (z) n j+k = n (j +1) (2λ−m+2j +k) (2λ−m+2j +k+n)+ k k n−k (cid:0)(cid:0)k−n(cid:1)1 (j +1)k(2λ−m+2j +k−1)n−k zn+1−k = (cid:0)(j +(cid:1)1)k(2λ−m+2j +k)n−k (cid:1) n (2λ−m+2j +k+n)+ n (2λ−m+2j +k−1) zn+1−k k k−1 = (cid:0)(j(cid:0)+(cid:1)1)k(2λ−m+2j +k)n−k(cid:0) (cid:1) (cid:1) ( n + n (2λ−m+2j +k−1)+(n+1) n zn+1−k k k−1 k = (cid:0)(j(cid:0)+(cid:1)1)k((cid:0)2λ−(cid:1) m+2j +k)n−k (cid:0) (cid:1)(cid:1) n+1 (2λ−m+2j +k−1)+ n+1 (n−k+1) zn+1−k k k = (cid:0)(j(cid:0)+1(cid:1))k(2λ−m+2j +k)n−k (cid:0)n+k1 (cid:1)(2λ−m+2(cid:1)j +n) zn+1−k = (j +1)k n+k1 (2λ−m+2j +(cid:0)k(cid:0))n+1(cid:1)−k zn+1−k (cid:1) j = u (cid:0)((cid:0)z) (cid:1) (cid:1) n+1,j+k for a fixed but arbitrary j, 0≤ j ≤ m and k, 0≤ k ≤ m−j. This completes the proof. (cid:3) On H(j), we have the representation Uλj acting (0 ≤ j ≤ m), where λ = λ− m +j. j 2 Its lowest K - type is spanned by ε (= uj) and Hε = λ ε . By [7, Prop 6.14] we have j 0 j j j k(−F)kε k2 = σjk(−F)k−1ε k2 with j k j j σ = (2λ +k−1)k k j for all k ≥ 1. (Here we used that the constant q in [7, equation (6.33)] equals λ (1−λ ) by j j [7, Theorem 6.2].) We write n σj = σj n k k=1 Y which can be written in a compact form (4.18) σj = ((2λ ) (1) ), n j n n where (x) = (x+1)···(x+n−1). We stipulate that the binomial co-efficient n as well n k as (x) are both zero if n < k. n−k (cid:0) (cid:1) 1 Thepositivity ofthenormalizingconstants σj 2 (n ≥ j)isequivalenttotheexistence n−j of an inner product for which the set of vectors ej defined by the formula: (cid:0) n−(cid:1)j ej = (σj)−12uj (z), n ≥ j, 0 ≤ j ≤ m n−j n n−j forms an orthonormal set. Of course, the positivity condition is fulfilled if and only if 2λ > m. j Inthisway,forfixedj,eache hasthesamenormforalln ≥ j. Hencetheonlypossible n−j j choice for an orthonormal system is {µ e : n ≥ j} for some positive real numbers µ > 0 j n−j j