Invariant Inner Product in Spaces of Holomorphic Functions on Bounded Symmetric Domains

We provide new integral formulas for the invariant inner product on spaces of holomorphic functions on bounded symmetric domains of tube type. Our main concern in this work is to provide concrete formulas for the invariant inner products and hermitian forms on spaces of holomorphic functions on Cartan domains D of tube type. As will be explained below, the group Aut(D) of all holomorphic automorphisms of D acts transitively. Aut(D) acts projectively on function spaces on D via f 7 ! U () (')f := (f ') (J') =p ; ' 2 Aut(D); 2 C, but these actions are not irreducible in general. The inner products we consider are those obtained from the holomorphic discrete series by analytic continuation. The associated Hilbert spaces generalize the weighted Bergman spaces, the Hardy and the Dirichlet space. By \concrete" formulas we mean Besov-type formulas, namely integral formulas involving the functions and some of their derivatives. Possible applications include the study of operators (Toeplitz, Hankel) acting on function spaces and the theory of invariant Banach spaces of analytic functions (where the pairing between an invariant space and its invariant dual is computed via the corresponding invariant inner product). Our problem is closely related to nding concrete realizations (by means of integral formulas) of the analytic continuation of the Riesz distribution. Ri], Go], FK2], Chapter VII. 214 Arazy and Upmeier In principle, the analytic continuation is obtained from the integral formulas associated with the weighted Bergman spaces (i.e. the holomorphic discrete series) by \partial integration with respect to the radial variables". This program has been successful in the case of rank 1 (i.e. when D is the open unit ball of C d , see A3]). The case of rank r > 1 is more diicult, and concrete formulas are known only in special cases, see A2], Y4], Y1], Y2]. This paper consists of two main parts. In the rst part (Sections 2, 3, and 4) we develop in full generality the techniques of A2], Y4], and obtain integral formulas for the invariant inner products associated with the so-called Wallach set and pole set. In the second part (section 5) we introduce new techniques (integration on boundary orbits), to obtain new integral formulas for the invariant inner products in the important special cases of Cartan domains of type I and IV. This approach has the potential for further generalizations and applications, including the innnite dimensional …


Introduction
Our main concern in this work is to provide concrete formulas for the invariant inner products and hermitian forms on spaces of holomorphic functions on Cartan domains D of tube type.As will be explained below, the group Aut(D) of all holomorphic automorphisms of D acts transitively.Aut(D) acts projectively on function spaces on D via f 7 !U ( ) (')f := (f ') (J') =p ; ' 2 Aut(D); 2 C, but these actions are not irreducible in general.The inner products we consider are those obtained from the holomorphic discrete series by analytic continuation.The associated Hilbert spaces generalize the weighted Bergman spaces, the Hardy and the Dirichlet space.By \concrete" formulas we mean Besov-type formulas, namely integral formulas involving the functions and some of their derivatives.Possible applications include the study of operators (Toeplitz, Hankel) acting on function spaces and the theory of invariant Banach spaces of analytic functions (where the pairing between an invariant space and its invariant dual is computed via the corresponding invariant inner product).
Our problem is closely related to nding concrete realizations (by means of integral formulas) of the analytic continuation of the Riesz distribution.Ri], Go], FK2], Chapter VII.
In principle, the analytic continuation is obtained from the integral formulas associated with the weighted Bergman spaces (i.e. the holomorphic discrete series) by \partial integration with respect to the radial variables".This program has been successful in the case of rank 1 (i.e. when D is the open unit ball of C d , see A3]).
The case of rank r > 1 is more di cult, and concrete formulas are known only in special cases, see A2], Y4], Y1], Y2].
This paper consists of two main parts.In the rst part (Sections 2, 3, and 4) we develop in full generality the techniques of A2], Y4], and obtain integral formulas for the invariant inner products associated with the so-called Wallach set and pole set.In the second part (section 5) we introduce new techniques (integration on boundary orbits), to obtain new integral formulas for the invariant inner products in the important special cases of Cartan domains of type I and IV.This approach has the potential for further generalizations and applications, including the in nite dimensional setup.
The paper is organized as follows.Section 1 provides background information on Cartan domains, the associated symmetric cones and Siegel domains and the Jordan theoretic approach to the study of bounded symmetric domains Lo], FK2], U2].We also explain some general facts concerning invariant Hilbert spaces of analytic functions on Cartan domains and the connection to the Riesz distribution.Section 2 is devoted to the study of invariant di erential operators on symmetric cones.We study the \shifting operators" introduced by Z. Yan and their derivatives with respect to the \spectral parameter".Section 3 is devoted to our generalization of Yan's shifting method, to nd explicit integral formulas for the invariant inner products obtained by analytic continuation of the holomorphic discrete series.In section 4 we study the expansion of Yan's operators, and obtain applications to concrete integral formulas for the invariant inner products.Some of these results were obtained independently by Z. Yan, J. Faraut and A. Kor anyi,FK2], Y4].We include these results and our proofs, in order to make the paper self contained, and also because in most cases our results go beyond the results in FK2], Y4].
In section 5 we propose a new type of integral formulas for the invariant inner products.These formulas involve integration on boundary orbits and the application of the localized versions of the radial derivative associated with the boundary components of Cartan domains.We are able to establish the desired formulas in the important special cases of type I and IV.The techniques established in this section can be used in the study of the remaining cases.
Finally, in the short section 6 we use the quasi-invariant measures on the boundary orbits of the associated symmetric cone in order to obtain integral formulas for some of the invariant inner products in the context of the unbounded realization of the Cartan domains (tube domains).These results are essentially implicitly contained in VR], where the authors use the Lie-theoretic and Fourier-analytic approach to analysis on symmetric Siegel domains.We use the Jordan-theoretic approach which yields simpler formulation of the results and simpler proofs.which admits the structure of a JB -triple, namely there exists a continuous mapping Z Z Z 3 (x; y; z) !fx; y; zg 2 Z (called the Jordan triple product) which is bilinear and symmetric in x and z, conjugate-linear in y, and so that the operators D(x; x) : Z !Z de ned for every x 2 Z by D(x; x)z := fx; x; zg are hermitian, have positive spectrum, satisfy the "C -axiom" kD(x; x)k = kxk 2 , and the operators (x) := iD(x; x) are triple derivations, i.e. the Jordan triple identity holds (x)fy; z; wg = f (x)y; z; wg + fy; (x)z; wg + fy; z; (x)wg; 8y; z; w 2 Z: The norm k k is called the spectral norm.We put also D(x; y)z := fx; y; zg.An element v 2 Z is called a tripotent if fv; v; vg = v.Every tripotent v 2 Z gives rise to a direct-sum Peirce decomposition Z = Z 1 (v) + Z1 2 (v) + Z 0 (v); where Z (v) := fz 2 Z; D(v; v)z = zg; = 1; 1 2 ; 0: The associated Peirce projections are de ned for z 2 Z (v), = 1; 1 2 ; 0, by P (v)(z 1 + z1 2 + z 0 ) = z ; = 1; 1 2 ; 0: In this paper we are interested in the important special case where Z contains a unitary tripotent e, for which Z = Z 1 (e).In this case Z has the structure of a JB -algebra with respect to the binary product x y := fx; e; yg and the involution z := fe; z; eg, and e is the unit of Z.The binary Jordan product is commutative, but in general non-associative.The triple product is expressed in terms of the binary product and the involution via fx; y; zg = (x y ) z+(z y ) x (x z) y .In this case the open unit ball D of Z is a Cartan domain of tube-type.This terminology is related to the unbounded realization of D, to be explained later.
Let X := fx 2 Z; x = xg be the real part of Z.It is a formally-real (or euclidean) Jordan algebra.Every x 2 X has a spectral decomposition x = P r j=1 j e j , where fe j g r j=1 is a frame of pairwise orthogonal minimal idempotents in X, and f j g r j=1 are real numbers called the eigenvalues of x.The trace and determinant (or, \norm") are de ned in X via tr(x respectively, and they are polynomials on X.The maximal number r of pairwise orthogonal minimal idempotents in X is called the rank of X.The positive-de nite inner product in X, hx; yi = tr(x y); x; y 2 X, satis es hx y; zi = hx; y zi; x; y; z 2 X: Equivalently, the multiplication operators L(x)y := x y; x; y 2 X, are self-adjoint.
The trace and determinant polynomials, as well as the multiplication operators, have unique extensions to the complexi cation X C := X + iX = Z.Let := fx 2 ; x 2 X; N(x) 6 = 0g: Then is a symmetric, open convex cone, i.e. is self polar and homogeneous with respect to the group GL( ) of linear automorphisms of .We denote the connected component of the identity in GL( ) by G( ).De ne P(x) := 2L(x) 2 L(x 2 ); x 2 X; (1.1) then P(x) 2 G( ) for every x 2 , and x = P(x 1=2 )e.Thus G( ) is transitive on .The map x !P(x) from X into End(X) is called the quadratic representation because of the identity P(P(x)y) = P(x)P(y)P(x); 8x; y 2 X: (1. 2) The domain T( ) := X + i , called the tube over .It is an irreducible symmetric domain which is biholomorphically equivalent to D by means of the Cayley transform c : D !T( ), de ned by c(z) := i e + z e z ; z 2 Z: This explains why D is called a tube-type Cartan domain.Let e 1 ; e 2 ; : : : ; e r be a xed frame of minimal, pairwise orthogonal idempotents satisfying e 1 + e 2 + : : : + e r = e, where e is the unit of Z.Let Z = X 1 i j r Z i;j be the associated joint Peirce decomposition, namely Z i;j := Z1 2 (e i ) \ Z1 2 (e j ) for 1 i < j r and Z i;i := Z 1 (e i ) for 1 i r.The characteristic multiplicity is the common dimension a = dim(Z i;j ); 1 i < j r, and d = r + r(r 1)a=2.The number p := (r 1)a + 2 is called the genus of D. It is known that Det(P(x)) = N(x) p ; 8x 2 X; where \Det" is the usual determinant polynomial in End(Z).From this and (1.2) it follows that N(P(x)y) = N(x) 2 N(y) 8x; y 2 X: (1.3) Let u j := e 1 + e 2 + : : : + e j and let Z j := P 1 i k j Z i;k be the JB -subalgebra of Z whose unit is u j .Let N j be the determinant polynomials of the Z j ; 1 j r; they are called the principal minors associated with the frame fe j g r j=1 .Notice that Z r = Z and N r = N.For an r-tuple of integers m = (m 1 ; m 2 ; : : : ; m r ) write m 0 if m 1 m 2 : : : m r 0. Such r-tuples m are called signatures (or, \partitions").The conical polynomial associated with the signature m is N m (z) := N 1 (z) m1 m2 N 2 (z) m2 m3 N 3 (z) m3 m4 : : : N r (z) mr ; z 2 Z: Notice that N m ( P r j=1 t j e j ) = Q r j=0 t mj j , thus the conical polynomials are natural generalizations of the monomials.Let Aut(D) be the group of all biholomorphic automorphisms of D, and let G be its connected component of the identity.Let K := fg 2 G; g(0) = 0g = G \ GL(Z) be the maximal compact subgroup of G.
For any signature m let P m := spanfN m k; k 2 Kg.Clearly, P m P `, where `= jmj = P r j=1 m j and P `is the space of homogeneous polynomials of degree `.By de nition, P m are invariant under the composition with members of K. Let hf; gi F := @ f (g ] )(0) = 1 d Z Z f(z)g(z)e jzj 2 dm(z) (1.4) be the Fock-Fischer inner product on the space P of polynomials, where g ] (z) := g(z ), @ f = f( @ @z ), jzj is the unique K-invariant Euclidean norm on Z normalized so that je 1 j = 1, and dm(z) is the corresponding Lebesgue volume measure.(Thus h1; 1i F = 1).The following result (Peter-Weyl decomposition) is proved in Sc], see also U1].Here the group K acts on functions on D via (k)f := f k 1 ; k 2 K. Notice that P `, `= 0; 1; 2; : : : and P are invariant under this action.Theorem 1.1 (i) The spaces fP m g m 0 , are K-invariant and irreducible.The representations of K on the spaces P m are mutually inequivalent, the P m 's are mutually orthogonal with respect to h ; i F , and P = P m 0 P m .
(ii) If H is a Hilbert space of analytic functions on D with a K-invariant inner product in which the polynomials are dense, then H is the orthogonal direct sum H = P m 0 P m .Namely, every f 2 H is expanded in the norm convergent series f = P m 0 f m , with f m 2 P m , and the spaces P m are mutually orthogonal in H.Moreover, there exist positive numbers fc m g m 0 so that for every f; g 2 H with expansions f = P m 0 f m and g = P m 0 g m we have hf; gi H = X m 0 c m hf m ; g m i F : For every signature m let K m (z; w) be the reproducing kernel of P m with respect to (1.4).Clearly, the reproducing kernel of the Fock-Fischer space F (the completion of P with respect to h ; i F ) is F(z; w) := X m K m (z; w) = e hz;wi : An important property of the norm polynomial N is its transformation rule under the group K where : K !T := f 2 C; j j = 1g is a character.In fact, (k) = N(k(e)) = Det(k) 2=p 8k 2 K. Notice that (1.5) implies that P (m;m;:::;m) = C N m for m = 0; 1; 2; : : :.The subgroup L of K de ned via (1.6) plays an important role in the theory.For example, (m;m;:::;m) = N m by (1.5).The L-invariant real polynomial on X h(x) = h(x; x) := N(e x 2 ) admits a unique K-invariant, hermitian extension h(z; w) to all of Z. Thus, h(k(z); k(w)) = h(z; w) for all z; w 2 Z and k 2 K, h(z; w) is holomorphic in z and anti-holomorphic in w, and h(z; w) = h(w; z), FK1].The transformation rule of h(z; w) under Aut(D) is h('(z); '(w)) = J'(z) 1 p h(z; w) J'(w) where J'(z) := Det(' 0 (z)) is the complex Jacobian of ', and p is the genus of D.
We recall two important formulas for integration in polar coordinates FK2], Chapters VI and IX.The rst formula uses the fact that K = Z, namely the fact that every z 2 Z can be written (not uniquely) in the form z = k(x), where x 2 and k 2 K.This is the rst (or \conical") type of polar decomposition of x, and it generalizes the polar decomposition of matrices.This leads to the formula which holds for every f 2 L 1 (Z; m).Next, x a frame e 1 ; : : : ; e r , and de ne R := span R fe j g r j=1 and R + := f r X j=1 t j e j ; t 1 > t 2 > : : : > t r > 0g and R r + := ft = (t 1 ; : : : t r ); t 1 > t 2 > : : : > t r > 0g: Then Z = K R, namely every z 2 Z has a representation z = k( P r j=1 t j e j ) for some (again, not unique) P r j=1 t j e j 2 R and k 2 K.This representation is the second type of polar decomposition of z.Moreover, m(Z n K R + ) = 0, namely up to a subset of measure zero, every z 2 Z has a representation z = k( P r j=1 t 1=2 j e j ) with t 1 > t 2 > : : : > t r > 0. This leads to the formula (t i t j ) a dt 1 dt 2 : : : dt r ; (1.12) which holds for every f 2 L 1 (Z; m).The constant c 0 will be determined as a byproduct of our work in section 5 below.For convenience, we can write (1.12) in the form (1.16) The two series converge absolutely, (1.15) converges uniformly on compact subsets of ( ; x) 2 C ( \ (e )), and (1.16) converges uniformly on compact subsets of ( ; z; w) 2 C D D.
In particular, it follows that for xed z; w 2 D, the function !h(z; w) is analytic in all of C (this can be proved also by showing that h(z; w) 6 = 0 for z; w 2 D).
The Wallach set of D, denoted by W(D), is the set of all 2 C for which the function (z; w) !h(z; w) is non-negative de nite in D D, namely X i;j a i a j h(z i ; z j ) 0 for all nite sequences fa j g C and fz j g D. For 2 W(D) let H be the completion of the linear span of the functions fh( ; w) ; w 2 Dg with respect to the inner product h ; i determined by hh( ; w) ; h( ; z) i = h(z; w) ; z; w 2 D: Since h(z; w) is continuous in D D, it is the reproducing kernel of H .The transformation rule (1.8) implies that h ; i is K-invariant, namely hf k; g ki = hf; gi for all f; g 2 H and k 2 K. Thus, by Theorems 1.1 and 1.2, for every f; g 2 H with Peter-Weyl expansions f = P m 0 f m , g = P m 0 g m , we have hf; gi = X m 0 hf m ; g m i F ( ) m : (1.17) This formula de nes 7 !hf; gi as a meromorphic function in all of C, whose poles are contained in the pole set P(D) of , see (1.10) and (1.16).Of course, for 2 C n W(D) (1.17) is not an inner product, but merely a sesqui-linear form; it is hermitian precisely when 2 R.
Using (1.16) and (1.17) one obtains a complete description of the Wallach set W(D) and the Hilbert spaces H for 2 W(D).Theorem 1.3 (i) The Wallach set is given by W(D) = W d (D) W c (D) where W d (D) := f j = (j 1) a 2 ; 1 j rg is the discrete part, and W c (D) := ( r ; 1) is the continuous part.(iii) For 1 j r, let S 0 ( j ) := fm 0; m j = m j+1 = : : : = m r = 0g.Then H j = P m2S0( j) P m and h(z; w) j = X m2S0( j) ( j ) m K m (z; w); z; w 2 D: For 2 C, ' 2 G and a functions f on D, we de ne Then, U ( ) (id D ) = I and for '; 2 G we have where c ('; ) is a unimodular scalar which transforms as a cocycle (projective representation of G).In particular, U ( ) (' 1 ) = U ( ) (') 1 .
Theorem 1.4 Let 2 C and let 0 j q( ).
(iv) For f; g 2 M ( ) j with Peter-Weyl expansions f = P m f m and g = P m g m , we have hf; The forms h ; i ;j are hermitian if and only if 2 R.
(vi) The quotient M ( ) j =M ( ) j 1 is unitarizable (namely, h ; i ;j is either positive definite or negative de nite on M ( ) j =M ( ) j 1 ) if and only if either: 2 W(D) and j = 0, or: 2 P(D), j = q( ), and r 2 N. The sequence (1.20) is called the composition series of P ( ) .Definition 1.1 H ;j = H ;j (D) is the completion of M ( ) j =M ( )   j 1 with respect to h ; i ;j .
The main objective of this work is to provide natural integral formulas for the U ( ) -invariant hermitian forms h ; i ;j , with special emphasis on the case where the forms are de nite, namely the case where H ;j is a U ( ) -invariant Hilbert space.These integral formulas provide a characterization of the membership in the spaces H ;j in terms of niteness of some weighted L 2 -norms of the functions or of some of their derivatives.We discuss now some examples which motivate our study.
The weighted Bergman spaces: It is known FK1] that for 2 R the integral c( (The same argument yields the invariance of the measure d 0 (z) := h(z; z) p dm(z)).
From this it follows that the operators U ( ) (') are isometries of L2 (D; ) which leave L 2 a (D; ) invariant.It is easy to verify that point evaluations are continuous linear functionals on L 2 a (D; ) and that the reproducing kernel of L 2 a (D; ) is h(z; w) .(For w = 0 this is trivial, and the general case follows by invariance.)It follows that H = L 2 a (D; ).
The Hardy space: The Shilov boundary S of a general Cartan domain D is the set of all maximal tripotents in Z. S is invariant and irreducible under both of G and K. Let be the unique K-invariant probability measure on S, de ned via The Hardy space H 2 (S) is the space of all analytic functions f on D for which is nite.The polynomials are dense in H 2 (S) and every f 2 H 2 (S) has radial limits f( ) := lim !1 f( ) at -almost every 2 S.Moreover, for f 2 H 2 (S), kfk H 2 (S) = k fk L 2 (S; ) .This identi es H 2 (S) as the closed subspace of L 2 (S; ) consisting of those functions f 2 L 2 (S; is nite.Here dA(z) :=1 dx dy.Clearly, B 2 is a Hilbert space modulo constant functions, and kf 'k B2 = kfk B2 for every f 2 B 2 and ' 2 Aut(D).Thus, B 2 is U (0) -invariant.The composition series corresponding to = 1 = 0 is C1 = M (0) 0 M (0)  generalizing (1.24) and (1.25) for the norms in H ;q( ) for 2 W d (D), in the context of a Cartan domain of tube type (in A1] these formulas are extended to all 2 P(D)).Formula (1.24) says that f 2 B 2 = H 0;1 if and only if f 0 2 H 2 .Namely, di erentiation \shifts" the space corresponding to = 0 to the one corresponding to = 2.This shifting technique is developed in Y3] in order to get integral formulas for the inner products in certain spaces H with p 1. The general idea is to obtain such integral formulas via \partial integration in the radial directions", see Ri], Go], and Finally, we describe the relationship between the invariant inner product and the Riesz distribution.The Riesz distribution was introduced in Ri] for the Lorentz cone, i.e. the symmetric cone associated with the Cartan domain of type IV (the \Lie ball").It was studied in Go] for the cone of symmetric, positive de nite real matrices (associated with the Cartan domain of type III) and for a general symmetric cone in FK2], chapter VII.Let be the symmetric cone associated with the Cartan domain of tube type D. For 2 C with < > (r 1) a 2 let R be the linear functional on the Schwartz space S(X) of X de ned via R 1 is the fundamental solution for the \wave operator" @ N := N( @ @x ).These formulas permit analytic continuation of 7 !R to an entire meromorphic function.
It is very interesting to nd the explicit description of the action of R where where the convolution of functions u and v on is u(y) v(x y) dy: Also, the inner product h ; i , > p 1, in the context of the tube domain where (f g) )(y) := R X f(x + iy) g(x + iy) dx; y 2 .
In view of these formulas the problem of obtaining an explicit description of the analytic continuation of the maps 7 !hf; gi is equivalent to the problem of determining the analytic continuation of the maps 7 !R d r (u).
2 G( )-invariant differential operators Let be the symmetric cone associated with the Cartan domain of tube type D, i.e. the interior of the cone of squares in the Euclidean Jordan algebra X.In this section we study G( )-invariant di erential operators that will be used later for the invariant inner products.The ring Di ( ) G( ) of G( )-invariant di erential operators is a (commutative) polynomial ring C X 1 ; X 2 ; : : : ; X r ], He], FK2].By FK2], Proposition IX.1.1, is a set of uniqueness for analytic functions on Z (namely, if an analytic function on Z vanishes identically on , it vanishes identically on Z).
Similarly, \ D = \ (e ) is a set of uniqueness for analytic functions on D.
Thus, if f; g and q are polynomials on Z so that @ f (g)(x) = f( d dx )g(x) = q(x) for every x 2 , then @ f (g)(z) = f( @ @z )g(z) = q(z) for every z 2 Z.We begin with the following known result FK2], Proposition VII.1.6.
Lemma 2.1 For every s = (s 1 ; s 2 ; : : : ; s r ) 2 C r and `2 N, we have N `( d dx ) N s (x) = s (`) N s `(x); 8x 2 ; (s) N( d dx ) N s (x 1 ) = ( 1) r (s + 1) N s+1 (x 1 ): Let N j be the norm polynomial of the JB -subalgebra V j := P r j+1 j k r Z i;k , where Z i;k are the Peirce subspaces of Z associated with the xed frame fe j g r j=1 .For every s = (s 1 ; : : : ; s r ) 2 C r let N s (x) := N 1 (x) s1 s2 N 2 (x) s2 s3 : : : N r (x) sr ; x 2 ; and s := (s r ; s r 1 ; s r 2 ; : : : ; s 1 ): Then we have N s (x 1 ) = N s (x) for x 2 , FK2],Proposition VII.1.5.Definition 2.1 For `2 N and 2 C let D `( ) be the operator on C 1 ( ) de ned by D `( ) = N d r (x) N `( d dx ) N `+ d r (x): In the special case of the Cartan domain of type II the operators D 1 ( ) have been considered by Selberg (see T], p.208).The operators D `( ) were studied in full generality in Y3], see also FK2], Chapter XIV.Notice that by Lemma 2.1 we have D `( ) N s = (s + + `) (s + ) N s : (2.2) In section 4 below we will extend D `( ) to a polynomial di erential operator on Z, i.e.D `( ) = Q `; (z; @ @z ) for some polynomial Q `; .Lemma 2.2 The operator D `( ) is K-invariant, i.e.D `( )(f k) = (D `( )f) k 8f 2 C 1 ( ); 8k 2 K: Proof: We have N(kz) = (k)N(z) for every z 2 Z.Since the operator @ N = N( @ @z ) is the adjoint of the operator of multiplication by N with respect to the inner product Using (2.2) and the fact that \ D = \ (e ) is a set of uniqueness for analytic functions on D, we obtain the following result.Notice that the assumption that is not in P(D) is used in the above proof to ensure that ( ) m 6 = 0 for every m 0. This is due to the fact that the zero set of the polynomial ( ) m is Z(( ) m ) = r j=1 f j k; k = 0; 1; : : :; m j 1g; (2.6) while P(D) = r j=1 ( j N) = m 0 Z(( ) m ).Similarly, for each m 0 the zero set of the polynomial de ned by (2.3) is given by Z( `;m ( )) = r j=1 f j k; m j k m j + ` 1g: (2.7) The multiplicities of the zeros are equal to the number of their appearances on the right hand side of (2.7).
Proof: By the L-invariance of D `( ) (see Lemma 2.2) it is enough to verify the NA-invariance of D `( ) for functions f of the form f = N s `for some s 2 C r and `2 L. Let 2 NA, and decompose ` uniquely as ` = 0 `0 with 0 2 NA and `0 2 L.Then, using (2.2) and (2.9), we get Corollary 2.5 The operators D `( ) are G( )-invariant. (m j + 1) x y ; 8x; y 2 R (an easy consequence of (1.9) and Stirling's formula) ensures that S ; is continuous on H(D).For 2 P(D) we de ne operators S ; ;j , 0 j q( ), on the space of analytic functions on D of the form f = P m2Sj( ) f m by S ; ;j f := lim ! ( where ( ) m;j are de ned by (1.21).Again, S ; ;j is continuous in the topology of H(D).Also, S ; ;0 = S ; : Proposition 3.1 Let ; > (r 1) a 2 .Then hf; gi = hS ; f; gi for every f; g 2 H .
Proof: By (1.17) the operator S 1 2 ; : H ! H de ned by In a similar way one proves the following result.Proposition 3.2 Let > (r 1) a 2 and let 2 P(D).Then for every 0 j q( ) and all f; g 2 H ;j , hf; gi ;j = hS ; ;j f; gi : (3.1) The operators S ; ;j allow the computation of the invariant hermitian forms hf; gi ;j by \shifting" the point to the point .This is the \shifting method".One typically chooses either = d r or > p 1, so the forms hf; gi ;j can be computed by the integral-type inner products of H 2 (D) or L 2 a (D; ).In order for the shifting method to be useful, one has to identify the operators S ; ;j as di erential or pseudodi erential operators.Essentially, this is our aim in the rest of the paper.Yan's paper Y3] deals with the case where `:= is a su ciently large natural number.The following result is a minor generalization of a result of Y3].Our main result in this section is a generalization of both Theorem 3.1 and the other results of Y3] to the case of invariant hermitian forms associated with the pole set P(D) = r j=1 ( j N).Since W(D) P(D), this covers cases not studied in A1].
Theorem 3.3 Let ; `; q, and j be as in Theorem 3.2.Then H ;j is unitarizable if and only if either (a) j = q and r 2 N, or (b) j = 0 and 2 W d (D) = f j g r j=1 .
For the proof of Theorems 3.2 and 3.3 we consider separately the cases j = 0, j = q, and 1 j q 1. Case 1: j = 0. Since 2 P(D), there is a smallest k 2 f1; 2; : : :; rg and a unique s 2 N so that = k s.We claim that S 0 ( ) = fm 0; m k sg.Indeed, because m n m k s.This establishes the claim.Notice that since + ` r + 1, we have ( + `)m > 0 for any m 0. Also, deg (( ) (`;`;:::;`) ) = q by Lemma 2.3.It follows that for m 2 S 0 ( ), deg ( `;m ) = q, and q `;m ( ) = 1 q! ( @ @ ) q `;m ( ) j = = 1 q! ( @ @ ) q ( + `)m ( ) m ( ) (`;`;:::;`) j = = ( + `)m ( ) m 1 q! ( @ @ ) q ( ) (`;`;:::;`) j = = ( + `)m In this case ( ) m assumes both positive and negative values as m ranges over S 0 ( ).Indeed, if m and n are de ned by m i = n i = 0 for 1 i k 1 and k < i r, and m k = 0, n k = s 1, then ( ) m and ( ) n have di erent signs.Thus h ; i ;0 is not de nite (positive or negative), and thus H ;0 is not unitarizable.This proves Theorem 3.3 in case j = 0. Case 2: j = q.In this case = q j = 0. Also, Lemma 2.3 yields deg ( `;m ) = 0 if m 2 S q ( ) and deg ( `;m ) 1 if m 2 S q 1 ( ).It follows that for f; g 2 H ;q , hD `( )f; gi +`= where is the non-zero constant de ned in (3.4).It follows that hD `( )f; gi +`= X m2Sq( ) hf m ; g m i F ( ) m;q = hf; gi ;q : This proves Theorem 3.2 in case j = q.To prove Theorem 3.3 in this case, assume rst that = r s for some s 2 N. We claim now that S q ( ) n S q 1 ( ) = fm 0; m r s + 1g: This proves Theorem 3.2 in case 1 j q 1.To prove Theorem 3.3 in this case we need to show that as m varies in S j ( ) n S j 1 ( ), ( ) m;j assumes both positive and negative values.Notice rst that there exists a unique pair (k; s) of integers with 1 k < s r so that k and s are positive integers and m 2 S j ( ) n S j 1 ( ) () m k k + 1 and m s s : In fact, s = k + 1 if the characteristic multiplicity a is even, and s , and m k+1 = 0, n k+1 = 1.Then m; n 2 S j ( ) n S j 1 ( ) and ( ) n;j = ( ) m;j ( s ).Thus ( ) n;j and ( ) m;j have di erent signs, and so H ;j is not unitarizable.This proves Theorem 3.3 in case 1 j q 1.
A special case of Theorem 3.2 is the following essentially known result.
Corollary 3.2 Let 2 P(D) be so that s = s( ) := d r 2 N. Then (i) H ;q is unitarizable, and hf; gi ;q = Z S N s ( )(@ s N f)( ) g( ) d ( ); 8f; g 2 H ;q : Thus, an analytic function f on D belongs to H ;q if and only if (N s @ s N ) 1=2 f 2 H 2 (S).
In the last statement (D `( )) 1=2 is the positive square root of the positive operator D `( ), see Corollary 2.1 Indeed, part (i) follows from Theorem 3.2 with j = q, = q j = 0, `= s and D s ( ) = N s @ s N .In this case H +s = Hd r is the Hardy space H 2 (S) on the Shilov boundary S. Corollary 3.2 (i) for 2 W d (D) was proved in A2].The proof of part (ii) is similar.
The case where 2 P(D) and s := d r 2 N (i.e. the highest quotient of the composition series of U ( ) -invariant spaces is unitarizable) is of particular interest.
Theorem 3.4 Let 2 P(D) and assume that s := d r 2 N.Then, for each ' 2 Aut(D) and f 2 H(D) Namely, the operator @ s N intertwines the actions U ( ) and U (p ) of Aut(D).Moreover, hf; gi ;q = c 1 h@ s N f; @ s N gi p ; 8f; g 2 H ;q ; (3.9) Documenta Mathematica 2 (1997) 213{261 where c 1 1 := ( d r ) (s;s;:::;s) r Y j=1 0 Y s 1 u=0 ( + u j ); (3.10) and the product Q 0 s 1 u=0 ranges over all non-zero terms.In particular, if < 1, then hf; gi ;q = c 1 c(p ) Z D (@ s N f)(z) (@ s N g)(z) h(z; z) dm(z); 8f; g 2 H ;q : (3.11) Proof: (3.8) is proved in A1], Theorem 6.4.For the proof of (3.9) and (3.11) we de ne an inner product on the polynomials modulo M ( ) q 1 by f; g] := h@ s N f; @ s N gi p ; f; g 2 H ;q : We claim that ; ] is U ( ) -invariant.Indeed, using (3.8) we see that for every ' 2 Aut(D) and polynomials f and g, Since polynomials are dense in H ;q , the fact that its inner product is the unique U ( ) -invariant inner product (see AF], A1]) implies that hf; gi ;q = c 1 f; g]; 8f; g 2 H ;q : The value (3.10) of c 1 is found by taking f = g = N s , and using the facts that hN s ; N s i F = ( d r ) (s;s;:::;s) , N s ; N s ] = (@ s N N s ) 2 = hN s ; N s i 2 F , and hN s ; N s i ;q = lim ! ( ) q hN s ; N s i F ( ) (s;s::: : Example: In the special case where = 0 and s := d r 2 N, H 0;q is the generalized Dirichlet space, and formula (3.11) is the generalized Dirichlet inner product hf; gi 0;q = c 1 c(p ) Z D (@ s N f)(z) (@ s N g)(z) dm(z); 8f; g 2 H 0;q : 4 The expansion of the operators D `( ) Yan's operators D `( ) = N d r @ Ǹ N +` d r and their derivatives play an important role in the previous section.In this section we obtain an expansion of D `( ) in powers of .This expansion will exhibit D `( ) as a polynomial in z, @ @z , and , showing that D `( ) is a di erential operator (with parameters and `) in the ordinary sense.It also facilitates the computation of the derivatives D `( ) = 1 !( @ @ ) D `( ) j = ; needed in formulas (3.3), (3.5) and (3.6) for the forms hf; gi ;j .Another consequence will be that for any r distinct complex numbers 1 ; : : : ; r the operators D 1 ( 1 ); : : : ; D 1 ( r ) are algebraically independent generators of the ring of invariant di erential operators on the cone , a result obtained independently also by Kor anyi and Yan (see FK2], Chapter XIV).We shall work in the framework of the cone , but all the results will be valid for Z, because is a set of uniqueness for analytic functions on Z.
Consider a general Cartan domain of tube-type D C d with rank r.Let be the associated symmetric cone in the Euclidean Jordan algebra X and x a frame fe 1 ; : : : ; e r g of pairwise orthogonal primitive idempotents in X, whose sum is the unit element e.For 1 r, let := 1 be the spherical polynomial associated with the signature 1 := (1; 1; : : : ; 1; 0; 0; : : :; 0), where there are \1"'s and r \0"'s.Put also 0 (z) 1.Let f g r =0 be the di erential operators on de ned via where for b 2 X, P(b) is de ned via (1.1).Recall that P(b) 2 G( ) for every b 2 , and that = fP(b)e; b 2 g since P(a 1 2 )e = a.Moreover, the L-invariance of the 's and the \polar decomposition" for imply that ( )f(a) := ( d dx )(f( (x))) jx=e ; a 2 (4.5) for every 2 G( ) for which (e) = a.This implies that the operators f g r =0 are G( )-invariant, namely We remark that (4.4) and (4.5) are equivalent to e hx;yi jx=a = ( (y)) e ha;yi = (P (a 1 2 )y) e ha;yi ; a; y 2 ; where 2 G( ) GL(X) satis es (e) = a, is the adjoint of with respect to the inner product h ; i on X, and di erentiates the coordinate x.Notice also that the operators can be written as = c m K m (x; @ @x ); where m = (1; 1; : : : ; 1; 0; : : :; 0) ( \ones" and r zeros), and c m is an appropriate constant.
For = 0; 1, r it is easy to compute .Clearly, 0 = I.Since N is L-invariant, r = N.Using (4.6) and (1.3), we nd that r = N @ N : Also, 1 (x) = 1 r tr(x) = 1 r hx; ei.Indeed, using N 1 (x) = hx; e 1 i and the fact that L is transitive on the frames, we get r hx; ei: Using the fact that tr(P(a 12 )y) = hP(a 1 2 )y; ei = hy; P(a 1 2 )ei = hy; ai; 8a; y 2 , we nd that 1 = 1 r R; where Rf(x) := @ @t f(tx) jt=1 is the radial derivative.Our main result in this section is the expansion of D 1 ( ) = N d r @ N N 1+ d r .
This result was obtained independently by A. Kor anyi, see FK2], Proposition XIV.1.5.(4.7) Proof: For x 2 , the function !N(x) is entire in .Hence both sides of (4.7) are entire in , and it is therefore enough to prove (4.7) for with < < 0. Let = r .Since < > r , we get for every x 2 where d (t) := N(t) d r dt is the G( )-invariant measure on .Fix a; y 2 and put f y (x) := e hx;yi .Then e he;P(a 1 2 )ti N(P(a 1 Comparing this with (4.6), we conclude that Using the relations = r and d r = 1 + r , we obtain (4.7).
Remark: The \binomial formula" (4.8) yields that for every = 1; 2; : : : ; r and every x 2 X, ( where = ( 1 ; 2 ; : : : ; r ) is the sequence of eigenvalues of x, and S r; is the elementary symmetric polynomial of degree in r variables.
Documenta Mathematica 2 (1997) 213{261 Corollary 4.2 The operators f k g r k=1 are algebraically independent generators of the ring Di ( ) G( ) of G( )-invariant di erential operators on .
Proof: Comparing the two expansions (4.11) and (4.14) of D `( ), we see that m 2 C 1 ; 2 ; : : : ; r ] for every signature m 0. Since f m g m 0 is a basis for the space of spherical polynomials, the one-to-one correspondence between spherical polynomials and the elements of Di ( ) G( ) (see FK2], Chapter XIV) implies that f m g m 0 is a basis of Di ( ) G( ) .Thus Di ( ) G( ) = C 1 ; 2 ; : : : ; r ].Since the minimal number of algebraic generators of Di ( ) G( ) is r = rank( ) He], it follows that 1 ; 2 ; : : : ; r are algebraically independent.
In order to describe the facial structure of a Cartan domain of tube-type D C d Lo], A1], let S `be the compact, real analytic manifold of tripotents in Z of rank `= 1; 2; : : : ; r.The group K acts transitively and irreducibly on S `.Let `be the unique K-invariant probability measure on S `given by Z (5.1) where v `is any xed element of S `.For any tripotent v let Z = Z 1 (v)+Z1 2 (v)+Z 0 (v) be the corresponding Peirce decomposition.Then D v := D\Z 0 (v) is a Cartan domain of tube-type, which is the open unit ball of the JB -algebra Z 0 (v).If v 2 S `then the rank of D v is r v := r `, its characteristic multiplicity is a v := a if ` r 2 and a v = 0 if `= r 1, and the genus is p v = p `a.The set v + D v is a face of the closure D of D. For any function f on D let f v be the function on D v de ned by f v (z) := f(v + z); z 2 D v : (5.2) The fundamental polynomial \h" of Z 0 (v) is de ned by h v (z; w) := h(z; w); z; w 2 Z 0 (v): (5.3)For `= 1; 2; : : :; r, @ `D := v2S `(v + D v ) is an orbit of G: @ `D = G(v `).If v 2 S r is a maximal tripotent, then D v = Z 0 (v) = f0g.Hence @ r D = S r = S is the Shilov boundary.In particular, S is a G-orbit.where v 0 is any tripotent orthogonal to v and K v := fk 2 K; k(Z (v)) = Z (v)g; = 0; 1=2; 1, so that K v (v 0 ) = S v .The combination of v; and `yields K-invariant probability measures `; on @ `D, 1 ` r 1, > p `a 1, via Z @ `D fd `; := Next, consider the \sphere bundle" B `, 1 ` r, whose base is S `and the ber at each v 2 S `is v + S v (where S v := @ r `Dv is the Shilov boundary of D v ).The group K acts on B `naturally, and this action is transitive.The combination of the measures v , v 2 S `and `yields K-invariant probability measures `on B `via For v 2 S `, consider the symmetric cone v in Z 0 (v), and let (v)   1 ; (v) 2 ; : : : ; (v)   r `be the canonical generators of the ring Di ( v ) G( v) as in section 4. We also denote (v) 0 = I; (v) := ( (v) 1 ; (v) 2 ; : : : ; (v) r `); and j = (j 1) a 2 ; 0 j r: Conjecture: For every 2 j r and every > j 1 there exists a positive function p j; 2 C 1 ( 0; 1) j 1 ), so that the inner product h ; i j = h ; i j ;0 is given by hf; gi j = Z Sr j+1 hp j; ( (v) )f v ; g v i H (Dv) d r j+1 (v): (5.5) Moreover, if = j 1 + 1 = dim(D v )=rank(D v ), then p j := p j; is a polynomial with positive coe cients.
|large The computation of hf; gi p 1 by integration on @ 1 D We conclude this section with the derivation of a formula for hf; gi p 1 via integration on @ 1 D. (5.30) where the measures v;p 1 are de ned by (5.4).
In this section we obtain integral formulas for the invariant inner products in the spaces H (T ( )).Using the isometry V ( ) : H (D) !H (T ( )) one obtains integral formulas for the inner products in the spaces H (D). Our results are essentially implicitly contained in VR], where the authors determine the Wallach set for Siegel domains of type II, using Lie and Fourier theoretical methods.The Jordantheoretical formalism allows us to formulate our results in a simpler way, avoiding the Lie-theoretical details.Since the Fourier-theoretical arguments in our proofs are contained in VR], we omit all proofs.
Then the following conditions are equivalent: (i) f 2 H (T ( )); (ii) The boundary values f(x) := lim 3y!0 f(x + iy) exist almost everywhere on X, and the Fourier transform f of f(x) is supported in and belongs to L 2 ( ; ).
Moreover, the map f 7 !f is an isometry of H (T ( )) onto L 2 a ( ; ).Proposition 6.1 yields the following result.
By complexi cation, GL( ) is realized as a subgroup of Aut(T( )) which normalizes the translations x (z) := z + x, i.e. ' x ' 1 = '(x) ; 8x 2 X; 8' 2 GL( ): Let G Aut(T( )) be the semi-direct product of X and GL( ).It acts transitively on T( ).Let N G be the semi-direct product of X and N .Then the Iwasawa decomposition of Aut(T( )) 0 is KAN.For k = d r + k a

A
Cartan domain D C d is an irreducible bounded symmetric domain in its Harish-Chandra realization.Thus D is the open unit ball of a Banach space Z = (C d ; k k) spherical (i.e., L-invariant) polynomial in P m satisfying m (e) = 1.
conical polynomials N m .In what follows use the following notation:

(
ii) For 2 W c (D) the polynomials are dense in H and H = P m 0 P m as in Theorem 1.1; ) 1 := R D h(z; z) p dm(z) is nite if and only if > p 1, and in this case cThe weighted Bergman space L 2 a (D; ) consists of all analytic functions in L 2 (D; ).Using (1.8) one obtains the transformation rule of under composition with ' 2 G: d ('(z)) = jJ'(z)j 2 p d (z): ) which extend analytically to D by means of the Poisson integral.Again, the point evaluations f 7 !f(z); z 2 D, are continuous linear functionals on H 2 (S).The corresponding reproducing kernel is called the Szeg o kernel and is given (as a function on S) by S z ( ) = S( ; z) := h( ; z) d=r .See Hu], FK1].This non-trivial fact implies that H d=r = H 2 (S).The transformation rule of the measure under the automorphisms ' 2 G is d ('( )) = jJ'( )j d ( ): Hence, U (d=r) (')f = (f ') (J') 1=2 , ' 2 G, are isometries of L 2 (S; ) which leave H 2 (S) invariant.The Dirichlet space: The classical Dirichlet space B 2 consists of those analytic functions f on the open unit disk D C for which the Dirichlet integral kfk 2

3
Integral formulas via the shifting methodIn this section we develop general shifting techniques (introduced in Y3], for the case of integer shifts).The simplest case where this technique is applied is the case of the Dirichlet space D = H 0;1 over the unit disk D (see Section 2).For any 2 C and 2 C n P(D) we de ne an operator S ; on H( 2 = hS ; f; fi : Now the result follows by polarization. let `2 N. Then for all f; g 2 H hf; gi = ( ; `)hD `( )f; gi +`; short proof for the sake of completeness.Proof: Let f; g 2 H with expansions f = P m 0 f m and g = ( )f)(z) g(z) h(z; z) +` p dm(z):Documenta Mathematica 2 (1997) 213{261 3.2 in case j = 0.If 2 W d (D), i.e. = k and s = 0, then ( ) m > 0 for every m 2 S 0 ( ), namely 0 = m k = m k+1 = = m r .If 2 P(D) n W d (D), then = k s with 1 s.
(zjw) = 2z w is the normalized inner product.Since r = 2 and a = d 2, the Wallach set is W(D) = W d (D) W c (D); W d (D) = f0; can replace in (4.12) and (4.13) P(a 1 2 ) by any 2 G( ) satisfying (e) = a.Hence the operators m are G( )-invariant, namely m (f ) = ( m f) ; 8 2 G( ): Theorem 4.2 For every 2 C and `2 D be a Cartan domain of tube type and rank r 2 in C d , Let D be the Cartan domain of rank r = 2 in C d (the Lie ball), The only tripotent of rank 0 is 0 2 Z, and D = D 0 is also a G-orbit.Thus the decomposition of D into G-orbits is is the Lebesgue measure on D v and c v; is the normalization factor.Similarly, one de nes a probability measure v on the Shilov boundary S v of D v of integration and using the transitivity of K on the frames, w) g v (w) d v;p 1 (w) d 1 (v);

6
Integral formulas in the context of the associated Siegel domain In what follows we shall use the fact FK2] that D is holomorphically equivalent to the tube domain T( ) := X + i via the Cayley transform c : D !T( ), de ned by c(z) := i(e + z)(e z) 1 .For