DUNKL OPERATORS: THEORY AND APPLICATIONS

Transcription

1 DUNKL OPERATORS: THEORY AND APPLICATIONS MARGIT RÖSLER Abstract. These lecture notes are intended as an introduction to the theory of rational Dunl operators and the associated special functions, with an emphasis on positivity and asymptotics. We start with an outline of the general concepts: Dunl operators, the intertwining operator, the Dunl ernel and the Dunl transform. We point out the connection with integrable particle systems of Calogero-Moser- Sutherland type, and discuss some systems of orthogonal polynomials associated with them. A major part is devoted to positivity results for the intertwining operator and the Dunl ernel, the Dunl-type heat semigroup, and related probabilistic aspects. The notes conclude with recent results on the asymptotics of the Dunl ernel. Contents 1. Introduction 2 2. Dunl operators and the Dunl transform Root systems and reflection groups Dunl operators A formula of Macdonald and its analog in Dunl theory Dunl s intertwining operator The Dunl ernel The Dunl transform CMS models and generalized Hermite polynomials Quantum Calogero-Moser-Sutherland models Spectral analysis of abstract CMS Hamiltonians Generalized Hermite polynomials Positivity results Positivity of Dunl s intertwining operator Heat ernels and heat semigroups Notation 31 References 31 1

2 2 MARGIT RÖSLER 1. Introduction While the theory of special functions in one variable has a long and rich history, the growing interest in special functions of several variables is comparatively recent. During the last years, there has in particular been a rapid development in the area of special functions with reflection symmetries and the harmonic analysis related with root systems. The motivation for this subject comes to some extent from the theory of Riemannian symmetric spaces, whose spherical functions can be written as multivariable special functions depending on certain discrete sets of parameters. A ey tool in the study of special functions with reflection symmetries are Dunl operators. Generally speaing, these are commuting differential-difference operators, associated to a finite reflection group on a Euclidean space. The first class of such operators, now often called rational Dunl operators, were introduced by C.F. Dunl in the late nineteen-eighties. In a series of papers ([D1-5]), he built up the framewor for a theory of special functions and integral transforms in several variables related with reflection groups. Since then, various other classes of Dunl operators have become important, in the first place the trigonometric Dunl operators of Hecman, Opdam and the Cheredni operators. These will not be discussed in our notes; for an overview, we refer to [He2]. An important motivation to study Dunl operators originates in their relevance for the analysis of quantum many body systems of Calogero-Moser- Sutherland type. These describe algebraically integrable systems in one dimension and have gained considerable interest in mathematical physics, especially in conformal field theory. A good bibliography is contained in [DV]. The aim of these lecture notes is an introduction to rational Dunl theory, with an emphasis on the author s results in this area. Rational Dunl operators bear a rich analytic structure which is not only due to their commutativity, but also to the existence of an intertwining operator between Dunl operators and usual partial derivatives. We shall first give an overview of the general concepts, including an account on the relevance of Dunl operators in the study of Calogero-Moser-Sutherland models. We also discuss some of the special functions related with them. A major topic will be positivity results; these concern the intertwining operator as well as the ernel of the Dunl transform, and lead to a variety of positive semigroups in the Dunl setting with possible probabilistic interpretations. We mae this explicit at hand of the most important example: the Dunl-type heat semigroup, which is generated by the analog of the Laplacian in the Dunl setting. The last section presents recent results on the asymptotics of the Dunl ernel and the short-time behavior of heat ernels associated with root systems. 2. Dunl operators and the Dunl transform This section gives an introduction to the theory of rational Dunl operators, which we call Dunl operators for short, and to the Dunl transform. References are [D1-5], [DJO], [dj1] and [O1]; for a bacground on reflection groups and root systems the reader is referred to [Hu] and [GB]. We do not intend to give a complete exposition,

3 DUNKL OPERATORS: THEORY AND APPLICATIONS 3 but rather focus on aspects which will be important for the topics of these lecture notes Root systems and reflection groups. The basic ingredient in the theory of Dunl operators are root systems and finite reflection groups, acting on some Euclidean space (a,.,. ) of finite dimension N. It will be no restriction to assume that that a = R N with the standard Euclidean inner product x, y = N j=1 x jy j. For α R N \{0}, we denote by σ α the orthogonal reflection in the hyperplane α perpendicular to α, i.e. σ α (x) = x 2 α, x α 2 α, where x := x, x. Each reflection σ α is orthogonal with respect to the standard inner product. Definition 2.1. A finite subset R R N \ {0} is called a root system, if σ α (R) = R for all α R. The dimension of span R R is called the ran of R. There are two possible additional requirements: R is called reduced, if α R implies 2α / R. crystallographic if R has full ran N and 2 α, β Z for all α, β R. β, β The group W = W (R) O(N, R) which is generated by the reflections {σ α, α R} is called the reflection group (or Coxeter group) associated with R. The dimension of span R R is called the ran of R. If R is crystallographic, then span Z R forms a lattice in R N (called the root-lattice) which is stabilized by the action of the associated reflection group. In rational Dunl theory, one usually wors with reduced root systems which are not necessarily crystallographic. On the other hand, the root systems occuring in Lie theory and in geometric contexts associated with Riemannian symmetric spaces are always crystallographic, and this requirement is also fundamental in the theory of trigonometric Dunl operators. Lemma 2.2. (1) If α is in R, then also α is in R. (2) For any root system R in R N, the reflection group W = W (R) is finite. (3) The set of reflections contained in W is exactly {σ α, α R}. (4) wσ α w 1 = σ wα for all w W and α R. Proof. (1) This follows since σ α (α) = α. (2) As R is fixed under the action of W, the assignment ϕ(w)(α) := wα defines a homomorphism ϕ : W S(R) of W into the symmetric group S(R) of R. This homomorphism is easily checed to be injective. Thus W is naturally identified with a subgroup of S(R), which is finite. Part (3) is slightly more involved. An elegant proof can be found in Section 4.2 of [DX]. Part (4) is straight forward.

4 4 MARGIT RÖSLER Properities (3) and (4) imply in particular that there is a bijective correspondence between the conjugacy classes of reflections in W and the orbits in R under the natural action of W. We shall need some more concepts: Each root system can be written as a disjoint union R = R + ( R + ), where R + and R + are separated by a hyperplane {x R N : β, x = 0} with β / R. Such a set R + is called a positive subsystem. The set of reflecting hyperplanes { α, α R} divides R N into connected open components, called the Weyl chambers of R. It can be shown that the topological closure C of any chamber C is a fundamental domain for W, i.e. C is naturally homeomorphic with the space (R N ) G of all W -orbits in R N, endowed with the quotient topology (see Section 1.12 of [Hu]). W permutes the reflecting hyperplanes as well as the chambers. Examples 2.3. (1) I 2 (n), n 3: Root systems of the dihedral groups. Define D n to be the dihedral group of order 2n, consisting of the orthogonal transformations in the Euclidean plane R 2 which preserve a regular n-sided polygon centered at the origin. It is generated by the reflection at the x-axis and the reflection at the line through the origin which meets the x-axis at the angle π/n. Root system I 2 (n) is crystallographic only for n = 2, 3, 4, 6. (2) A N 1. Let S N denote the symmetric group in N elements. It acts faithfully on R N by permuting the standard basis vectors e 1,..., e N. Each transposition (ij) acts as a reflection σ ij sending e i e j to its negative. Since S N is generated by transpositions, it is a finite reflection group. The root system of S N is called A N 1 and is given by A N 1 = {±(e i e j ), 1 i < j N}. This root system is crystallographic. Its span is the orthogonal complement of the vector e e N, and thus the ran is N 1. (3) B N. Here W is the reflection group in R N generated by the transpositions σ ij as above, as well as the sign changes σ i : e i e i, i = 1,..., N. The group of sign changes is isomorphic to Z N 2, intersects S N trivially and is normalized by S N, so W is isomorphic with the semidirect product S N Z N 2. The corresponding root system is called B N ; it is given by B N = {±e i, 1 i N} {±(e i ± e j ), 1 i < j N}. B N is crystallographic and has ran N. (4) BC N. This is the root system in R N given by BC N BC N = {±e i, ±2e i, 1 i N} {±(e i ± e j ), 1 i < j N}. is crystallographic, but not reduced. A root system R is called irreducible, if it cannot be written as the orthogonal disjoint union R = R 1 R 2 of two root systems R 1, R 2. Any root system can be uniquely written as an orthogonal disjoint union of irreducible root systems. There exists a classification of all irreducible, reduced root systems in terms of Coxeter graphs. There are 5 infinite series: A n, B n, C n, D n (which are crystallographic), as well as the ran 2 root systems I 2 (n) corresponding to the dihedral groups. Apart from those, there

5 DUNKL OPERATORS: THEORY AND APPLICATIONS 5 is a finite numer of exceptional root systems. The root systems BC n (n 1) are the only irreducible crystallographic root systems which are not reduced. We mention that the root system of a complex semisimple Lie algebra is always crystallographic and reduced, and it is irreducible exactly if the Lie algebra is simple. For further details on root systems, the reader is referred to [Hu] and [Kn] Dunl operators. Let R be a reduced (not necessarily crystallographic) root system in R N and W the associated reflection group. The Dunl operators attached with R are modifications of the usual partial derivatives by divided difference operators made up by reflections. The divided difference parts are coupled by parameters, which are given in terms of a multiplicity function: Definition 2.4. A function : R C on the root system R is called a multiplicity function, if it is invariant under the natural action of W on R. The set of multiplicity functions forms a C-vector space whose dimension is equal to the number of W -orbits in R. Throughout these notes, we shall require that the multiplicity is non-negative, that is (α) 0 for all α R. We write 0 for short. Parts of the theory extend to a larger range of multiplicities (depending on R), but the condition 0 is essential for positivity results and probability theory. Definition 2.5. Let : R C be a multiplicity function on R. Then for ξ R N, the Dunl operator T ξ := T ξ () is defined on C 1 (R N ) by T ξ f(x) := ξ f(x) + (α) α, ξ f(x) f(σ αx). α, x α R + Here ξ denotes the directional derivative corresponding to ξ, and R + is a fixed positive subsystem of R. For the i-th standard basis vector ξ = e i R N we use the abbreviation T i = T ei. By the W -invariance of, the definition of the Dunl operators does not depend on the special choice of R +. Also, the length of the roots is irrelevant in the formula for T ξ. This is the basic reason for the convention which requires reduced root systems: a Dunl operator with summation over a non-reduced root system can be replaced by a counterpart with summation about an associated reduced counterpart, teh multiplicities being modified accordingly. Note further that the dependence of T ξ on ξ is linear. In case = 0, the T ξ () reduce to the corresponding directional derivatives. The operators T ξ were introduced and first studied by C.F. Dunl ([D1-5]). They enjoy regularity properties similar to usual partial derivatives on various spaces of functions. We shall use the following notations: Notation Z + := {0, 1, 2,...}. 2. Π := C[R N ] is the C-algebra of polynomial functions on R N. It has a natural grading Π = n 0 P n,

6 6 MARGIT RÖSLER where P n is the subspace of homogeneous polynomials of (total) degree n. 3. S (R N ) denotes the Schwartz space of rapidly decreasing functions on R N, S (R N ) := {f C (R N ) : x β α f,r N < for all α, β Z N +}. It is a Fréchet space with the usual locally convex topology. The Dunl operators T ξ have the following regularity properties: Lemma 2.7. (1) If f C m (R N ) with m 1, then T ξ f C m 1 (R N ). (2) T ξ leaves C c (R N ) and S (R N ) invariant. (3) T ξ is homogeneous of degree 1 on Π, that is, T ξ p P n 1 for p P n. Proof. All statements follow from the representation f(x) f(σ α x) α, x = 1 0 α f ( x t α, x α ) dt for f C 1 (R N ), α R (recall our normalization α, α = 2). (1) and (3) are immediate; the proof of (2) (for S (R N )) is also straightforward but more technical; it can be found in [dj1]. Due to the G-invariance of, the Dunl operators T ξ are G-equivariant: In fact, consider the natural action of O(N, R) on functions f : R N C, given by Then an easy calculation shows: h f(x) := f(h 1 x), Exercise 2.8. g T ξ g 1 = T gξ for all g G. Moreover, there holds a product rule: h O(N, R). Exercise 2.9. If f, g C 1 (R N ) and at least one of them is G-invariant, then T ξ (fg) = T ξ (f) g + f T ξ (g). (2.1) The most striing property of the Dunl operators, which is the foundation for rich analytic structures related with them, is the following Theorem For fixed, the associated T ξ = T ξ (), ξ R N commute. This result was obtained in [D2] by a clever direct argumentation. An alternative proof, relying on Koszul complex ideas, is given in [DJO]. As a consequence of Theorem 2.10 there exists an algebra homomorphism Φ : Π End C (Π) which is defined by For p Π we write Φ : x i T i, 1 id. p(t ) := Φ (p). The classical case = 0 will be distinguished by the notation Φ 0 (p) =: p( ). Of particular importance is the -Laplacian, which is defined by := p(t ) with p(x) = x 2.

7 Theorem = + 2 DUNKL OPERATORS: THEORY AND APPLICATIONS 7 α R + (α)δ α with δ α f(x) = f(x), α α, x here and denote the usual Laplacian and gradient respectively. f(x) f(σ αx) α, x 2 ; (2.2) This representation is obtained by a direct calculation (recall again our convention α, α = 2 for all α R) by use of the following Lemma: Lemma [D2] For α R, define Then ρ α f(x) := f(x) f(σ αx) α, x It is not difficult to chec that (f C 1 (R N )). α,β R + (α)(β) α, β ρ α ρ β = 0. = for any orthonormal basis {ξ 1,..., ξ N } of R N, see [D2] for the proof. Together with the G-equivariance of the Dunl operators, this immediately implies that is G- invariant, i.e. g = g (g G). Examples (1) The ran-one case. In case N = 1, the only choice of R is R = {± 2}, which is the root system of type A 1. The corresponding reflection group is G = {id, σ} acting on R by σ(x) = x. The Dunl operator T := T 1 associated with the multiplicity parameter C is given by T f(x) = f f(x) f( x) (x) +. x Its square T 2, when restricted to the even subspace C 2 (R) e := {f C 2 (R) : f(x) = f( x)} is given by a singular Sturm-Liouville operator: N i=1 T 2 ξ i T 2 C 2 (R) ef = f + 2 x f. (2) Dunl operators of type A N 1. Suppose G = S N with root system of type A N 1. (In contrast to the above example, G now acts on R N ). As all transpositions are conjugate in S N, the vector space of multiplicity functions is one-dimensional. The Dunl operators associated with the multiplicity parameter C are given by T S i and the -Laplacian is S = i + = + 2 j i 1 i<j N 1 σ ij x i x j (i = 1,..., N), 1 x i x j [ ( i j ) 1 σ ij x i x j ].

8 8 MARGIT RÖSLER (3) Dunl operators of type B N. Suppose R is a root system of type B N, corresponding to G = S N Z N 2. There are two conjugacy classes of reflections in G, leading to multiplicity functions of the form = ( 0, 1 ) with i C. The associated Dunl operators are given by T B i where τ ij := σ ij σ i σ j. = i σ i x i + 0 j i [ 1 σij + 1 τ ] ij x i x j x i + x j (i = 1,..., N), 2.3. A formula of Macdonald and its analog in Dunl theory. In the classical theory of spherical harmonics (see for instance [Hel]) the following bilinear pairing on Π, sometimes called Fischer product, plays an important role: [p, q] 0 := ( p( )q ) (0), p, q Π. This pairing is closely related to the scalar product in L 2 (R N, e x 2 /2 dx); in fact, in his short note [M2] Macdonald observed the following identity: [p, q] 0 = (2π) N/2 R N e /2 p(x) e /2 q(x) e x 2 /2 dx. Here e /2 is well-defined as a linear operator on Π by means of the terminating series e /2 ( 1) n p = 2 n n! n p. n=0 Both the Fischer product as well as Macdonald s identity have a useful generalization in the Dunl setting. In the following, we shall always restrict to the case 0. Definition For p, q Π define [p, q] := ( p(t )q ) (0). This bilinear form was introduced in [D4]. We collect some of its basic properties: Lemma (1) If p P n and q P m with n m, then [p, q] = 0. (2) [x i p, q] = [p, T i q] (p, q Π, i = 1,..., N). (3) [g p, g q] = [p, q] (p, q Π, g G). Proof. (1) follows from the homogeneity of the Dunl operators, (2) is clear from the definition, and (3) follows from Exercise 2.8. Let w denote the weight function on R N defined by w (x) = α, x 2(α). (2.3) α R + It is G-invariant and homogeneous of degree 2γ, with the index γ := γ() := α R + (α). (2.4)

9 DUNKL OPERATORS: THEORY AND APPLICATIONS 9 Notice that by G-invariance of, we have ( α) = (α) for all α R. Hence this definition does again not depend on the special choice of R +. Further, we define the constant c := e x 2 /2 w (x)dx, R N a a so-called Macdonald-Mehta-Selberg integral. There exists a closed form for it which was conjectured and proved by Macdonald [M1] for the infinite series of root systems. An extension to arbitrary crystallographic reflection groups is due to Opdam [O1], and there are computer-assisted proofs for some non-crystallographic root systems. As far as we now, a general proof for arbitrary root systems has not yet been found. We shall need the following anti-symmetry of the Dunl operators: Proposition [D5] Let 0. Then for every f S (R N ) and g Cb 1(RN ), T ξ f(x)g(x)w (x)dx = R N f(x)t ξ g(x)w (x)dx. R N Proof. A short calculation. In order to have the appearing integrals well defined, one has to assume 1 first, and then extend the result to general 0 by analytic continuation. Proposition For all p, q Π, [p, q] = c 1 e /2 p(x) e /2 q(x) e x 2 /2 w (x)dx. (2.5) R N This result is due to Dunl ([D4]). As the Dunl Laplacian is homogeneous of degree 2, the operator e /2 is well-defined and bijective on Π, and it preserves the degree. We give here a direct proof which is partly taen from an unpublished part of M. de Jeu s thesis ([dj2], Chap. 3.3). It involves the following commutator results in End C (Π), where as usual, [A, B] = AB BA for A, B End C (Π). Lemma For i = 1,..., N, (1) [ x i, /2 ] = T i ; (2) [ x i, e /2 ] = T i e /2. Proof. (1) follows by direct calculation, c.f. [D2]. Induction then yields that [ xi, ( /2) n] = n T i ( /2 ) n 1 for n 1, and this implies (2). Proof of Proposition Let i {1,..., N}, and denote the right side of (2.5) by (p, q). Then by the anti-symmetry of T i in L 2 (R N, w ), the product rule for T i and

10 10 MARGIT RÖSLER the above Lemma, (p, T i q) = c 1 e /2 p (T i e /2 q ) e x 2 /2 w dx R N ( = c 1 T i e x 2 /2 e /2 p ) (e /2 q ) w dx R N = c 1 e /2 (x i p) (e /2 q ) e x 2 /2 w dx = (x i p, q). R N But the form [.,. ] has the same property by Lemma 2.15(2). It is now easily checed that the assertion is true if p or q is constant, and then, by induction on max(deg p, deg q), for all homogeneous p, q. This suffices by the linearity of both forms. Corollary Let again 0. Then the pairing [.,. ] on Π is symmetric and non-degenerate, i.e. [p, q] = 0 for all q Π implies that p = 0. Exercise Chec the details in the proofs of Prop and Cor Dunl s intertwining operator. It was first shown in [D4] that for non-negative multiplicity functions, the associated commutative algebra of Dunl operators is intertwined with the algebra of usual partial differential operators by a unique linear and homogeneous isomorphism on polynomials. A thorough analysis in [DJO] subsequently revealed that for general, such an intertwining operator exists if and only if the common ernel of the T ξ, considered as linear operators on Π, contains no singular polynomials besides the constants. More precisely, the following characterization holds: Theorem [DJO] Let K reg := { K : ξ R Ker T N ξ () = C 1 }. Then the following statements are equivalent: (1) K reg ; (2) There exists a unique linear isomorphism ( intertwining operator ) V of Π such that V (P n ) = P n, V P0 = id and T ξ V = V ξ for all ξ R N. The proof of this result is by induction on the degree of homogeneity and requires only linear algebra. The intertwining operator V commutes with the G-action: Exercise g 1 V g = V (g G). Hint: Use the G-equivariance of the T ξ and the defining properties of V. Proposition { K : 0} K reg. Proof. Suppose that p n 1 P n satisfies T ξ ()p = 0 for all ξ R N. Then [q, p] = 0 for all q n 1 P n, and hence also [q, p] = 0 for all q Π. Thus p = 0, by the non-degeneracy of [.,. ].

11 DUNKL OPERATORS: THEORY AND APPLICATIONS 11 The complete singular parameter set K \ K reg is explicitly determined in [DJO]. It is an open subset of K which is invariant under complex conjugation, and contains { K : Re 0}. Later in these lectures, we will in fact restrict our attention to non-negative multiplicity functions. These are of particular interest concerning our subsequent positivity results, which could not be expected for non-positive multiplicities. Though the intertwining operator plays an important role in Dunl s theory, an explicit closed form for it is nown so far only in some special cases. Among these are 1. The ran-one case. Here K reg = C \ { 1/2 n, n Z + }. The associated intertwining operator is given explicitly by ( 1 ( 1 2) 2) V ( x 2n ) = ( n) n x 2n ; V ( x 2n+1 ) = ( ) n+1 n+1 x 2n+1, where (a) n = Γ(a + n)/γ(a) is the Pochhammer-symbol. For Re > 0, this amounts to the following integral representation (see [D4], Th. 5.1): V p(x) = Γ( + 1/2) Γ(1/2) Γ() 1 2. The case G = S 3. This was studied in [D6]. Here 1 p(xt) (1 t) 1 (1 + t) dt. (2.6) K reg = C \ { 1/2 n, 1/3 n, 2/3 n, n Z + }. In order to bring V into action in a further development of the theory, it is important to extend it to larger function spaces. For this we shall always assume that 0. In a first step, V is extended to a bounded linear operator on suitably normed algebras of homogeneous series on a ball. This concept goes bac to [D4]. Definition For r > 0, let B r := {x R N : x r} denote the closed ball of radius r, and let A r be the closure of Π with respect to the norm p Ar := p n,br for p = p n, p n P n. n=0 Clearly A r is a commutative Banach- -algebra under the pointwise operations and with complex conjugation as involution. Each f A r has a unique representation f = n=0 f n with f n P n, and is continuous on the ball B r and real-analytic in its interior. The topology of A r is stronger than the topology induced by the uniform norm on B r. Notice also that A r is not closed with respect to.,br and that A r A s with. Ar. As for s r. n=0 Theorem V p,br p,br for each p P n. The proof of this result is given in [D4] and can also be found in [DX]. It uses the van der Corput-Schaae inequality which states that for each real-valued p P n, sup { p(x), y : x, y B 1 } n p,b1.

12 12 MARGIT RÖSLER Notice that here the converse inequality is trivially satisfied, because p(x), x = np(x) for p P n. The following is now immediate: Corollary V f Ar f Ar for every f Π, and V extends uniquely to a bounded linear operator on A r via V f := V f n for f = f n. n=0 Formula (2.6) shows in particular that in the ran-one case with > 0, the operator V is positivity-preserving on polynomials. It was conjectured by Dunl in [D4] that for arbitrary reflection groups and non-negative multiplicity functions, the linear functional f V f(x) on A r should be positive. We shall see in Section 4.1. that this is in fact true. As a consequence, we shall obtain the existence of a positive integral representation generalizing (2.6), which in turn allows to extend V to larger function spaces. This positivity result also has important consequences for the structure of the Dunl ernel, which generalizes the usual exponential function in the Dunl setting. We shall introduce it in the following section. Exercise The symmetric spectrum S (A) of a (unital) commutative Banach- -algebra A is defined as the set of all non-zero algebra homomorphisms ϕ : A C satisfying the -condition ϕ(a ) = ϕ(a) for all a A. It is a compact Hausdorff space with the wea-*-topology (sometimes called the Gelfand topology). Prove that the symmetric spectrum of the algebra A r is given by n=0 S (A r ) = {ϕ x : x B r }, where ϕ x is the evaluation homomorphism ϕ x (f) := f(x). Show also that the mapping x ϕ x is a homeomorphism from B r onto S (A r ) The Dunl ernel. Throughout this section we assume that 0. Moreover, we denote by.,. not only the Euclidean scalar product on R N, but also its bilinear extension to C N C N. For fixed y C N, the exponential function x e x,y belongs to each of the algebras A r, r > 0. This justifies the following Definition [D4] For y C N, define E (x, y) := V ( e.,y )(x), x R N. The function E is called the Dunl-ernel, or - exponential ernel, associated with G and. It can alternatively be characterized as the solution of a joint eigenvalue problem for the associated Dunl operators. Proposition Let 0 and y C N. Then f = E (., y) is the unique solution of the system T ξ f = ξ, y f for all ξ R N (2.7) which is real-analytic on R N and satisfies f(0) = 1.

13 DUNKL OPERATORS: THEORY AND APPLICATIONS 13 Proof. E (. y) is real-analytic on R N by our construction. Define E (n) (x, y) := 1 n! V., y n (x), x R N, n = 0, 1, 2,.... Then E (x, y) = n=0 E(n) (x, y), and the series converges uniformly and absolutely with respect to x. The homogeneity of V immediately implies E (0, y) = 1. Further, by the intertwining property, T ξ E (n) (., y) = 1 n! V ξ., y n = ξ, y E (n 1) (., y) (2.8) for all n 1. This shows that E (., y) solves (2.7). To prove uniqueness, suppose that f is a real-analytic solution of (2.7) with f(0) = 1. Then T ξ can be applied termwise to the homogeneous expansion f = n=0 f n, f n P n, and comparison of homogeneous parts shows that f 0 = 1, T ξ f n = ξ, y f n 1 for n 1. As { K : 0} K reg, it follows by induction that all f n are uniquely determined. While this construction has been carried out only for 0, there is a more general result by Opdam which assures the existence of a general exponential ernel with properties according to the above lemma for arbitrary regular multiplicity parameters. The following is a weaened version of [O1], Prop. 6.7; it in particular implies that E has a holomorphic extension to C N C N : Theorem For each K reg and y C N, the system T ξ f = ξ, y f (ξ R N ) has a unique solution x E (x, y) which is real-analytic on R N and satisfies f(0) = 1. Moreover, the mapping (x,, y) E (x, y) extends to a meromorphic function on C N K C N with pole set C N (K \ K reg ) C N We collect some further properties of the Dunl ernel E. Proposition Let 0, x, y C N, λ C and g G. (1) E (x, y) = E (y, x) (2) E (λx, y) = E (x, λy) and E (gx, gy) = E (x, y). (3) E (x, y) = E (x, y). Proof. (1) This is shown in [D4]. (2) is easily obtained from the definition of E together with the homogeneity and equivariance properties of V. For (3), notice that f := E (., y), which is again real-analytic on R N, satisfies T ξ f = ξ, y f, f(0) = 1. By the uniqueness part of the above Proposition, E (x, y) = E (x, y) for all real x. Now both x E (x, y) and x E (x, y) are holomorphic on C N and agree on R N. Hence they coincide. Just as with the intertwining operator, the ernel E is explicitly nown for some particular cases only. An important example is again the ran-one situation:

14 14 MARGIT RÖSLER Example In the ran-one case with Re > 0, the integral representation (2.6) for V implies that for all x, y C, E (x, y) = Γ( + 1/2) Γ(1/2) Γ() This can also be written as 1 1 e txy (1 t) 1 (1 + t) dt = e xy 1F 1 (, 2 + 1, 2xy). E (x, y) = j 1/2 (ixy) + xy j +1/2(ixy), where for α 1/2, j α is the normalized spherical Bessel function j α (z) = 2 α Γ(α + 1) Jα(z) ( 1) n (z/2) 2n = Γ(α + 1) z α n! Γ(n + α + 1). (2.9) This motivates the following Definition [O1] The -Bessel function (or generalized Bessel function) is defined by J (x, y) := 1 E (gx, y) (x, y C N ). (2.10) G g G Thans to Prop J is G-invariant in both arguments and therefore naturally considered on Weyl chambers of G (or their complexifications). In the ran-one case, we have J (x, y) = j 1/2 (ixy). It is a well-nown fact from classical analysis that for fixed y C, the function f(x) = j 1/2 (ixy) is the unique analytic solution of the differential equation f + 2 x f = y 2 f which is even and normalized by f(0) = 1. In order to see how this can be generalized to the multivariable case, consider the algebra of G-invariant polynomials on R N, Π G = {p Π : g p = p n=0 for all g G}. If p Π G, then it follows from the equivariance of the Dunl operators (Exercise 2.8) that p(t ) commutes with the G-action; a detailed argument for this is given in [He1]. Thus p(t ) leaves Π G invariant, and we obtain in particular that for fixed y C N, the -Bessel function J (., y) is a solution to the following Bessel-system: p(t )f = p(y)f for all p Π G, f(0) = 1. According to [O1], it is in fact the only G-invariant and analytic solution. We mention that there exists a group theoretic context in which, for a certain parameters, generalized Bessel functions occur in a natural way: namely as the spherical functions of a Euclidean type symmetric space, associated with a so-called Cartan motion group. We refer to [O1] for this connection and to [Hel] for the necessary bacground in semisimple Lie theory.

15 DUNKL OPERATORS: THEORY AND APPLICATIONS 15 The Dunl ernel is of particular interest as it gives rise to an associated integral transform on R N which generalizes the Euclidean Fourier transform in a natural way. This transform will be discussed in the following section. Its definition and essential properties rely on suitable growth estimates for E. In our case 0, the best ones to be expected are available: Proposition [R3] For all x R N, y C N and all multi-indices α Z N +, α y E (x, y) x α max g G ere gx,y. In particular, E ( ix, y) 1 for all x, y R N. This result will be obtained later from a positive integral representation of Bochnertype for E, c.f. Cor M. de Jeu had slightly weaer bounds in [dj1], differing by an additional factor G. We conclude this section with two important reproducing properties for the Dunl ernel. Notice that the above estimate on E assures the convergence of the involved integrals. Proposition Let 0. Then (1) e /2 p (x) E (x, y) e x 2 /2 w (x)dx = c e y,y /2 p(y) (p Π, y C N ). R N (2) E (x, y) E (x, z) e x 2 /2 w (x)dx = c e ( y,y + z,z )/2 E (y, z) (y, z C N ). R N Proof. (c.f. [D5].) We shall use the Macdonald-type formula (2.5) for the pairing [.,. ]. First, we prove that In fact, if p P n, then [E (n) (x,. ),. ] = p(x) for all p P n, x R N. (2.11) p(x) = ( x, y n /n!)p(y) and V x p(x) = E (n) (x, y)p(y). Here the uppercase index in V x denotes the relevant variable. Application of V y to both sides yields V x p(x) = E(n) (x, T y )V y p(y). As V is bijective on P n, this implies (2.11). For fixed y, let L n (x) := n j=0 E(j) (x, y). If n is larger than the degree of p, it follows from (2.11) that [L n, p] = p(y). Thus in view of the Macdonald formula, c 1 R N e /2 L n (x)e /2 p(x)e x 2 /2 w (x)dx = p(y). On the other hand, it is easily checed that lim n e /2 L n (x) = e y,y /2 E (x, y). This gives (1). Identity (2) then follows from (1), again by homogeneous expansion of E.

16 16 MARGIT RÖSLER 2.6. The Dunl transform. The Dunl transform was introduced in [D5] for nonnegative multiplicity functions and further studied in [dj1] in the more general case Re 0. In these notes, we again restrict ourselves to 0. Definition The Dunl transform associated with G and 0 is given by. : L 1 (R N, w ) C b (R N ); f (ξ) := c 1 f(x)e ( iξ, x) w (x)dx (ξ R N ). R N The inverse transform is defined by f (ξ) = f ( ξ). Notice that f C b (R N ) results from our bounds on E. The Dunl transform shares many properties with the classical Fourier transform. Here are the most basic ones: Lemma Let f S (R N ). Then for j = 1,..., N, (1) f C (R N ) and T j ( f ) = (ix j f). (2) (T j f) (ξ) = iξ j f (ξ). (3) The Dunl transform leaves S (R N ) invariant. Proof. (1) is obvious from (2.7), and (2) follows from the anti-symmetry relation (Prop. 2.16) for the Dunl operators. For (3), notice that it suffices to prove that α ξ (ξβ f (ξ)) is bounded for arbitrary multi-indices α, β. By the previous Lemma, we have ξ β f (ξ) = ĝ (ξ) for some g S (R N ). Using the growth bounds of Proposition 2.34 yields the assertion. Exercise (1) C c (R N ) and S (R N ) are dense in L p (R N, w ), p = 1, 2. (2) Conclude the Lemma of Riemann-Lebesgue for the Dunl transform: f L 1 (R N, w ) = f C 0 (R N ). Here C 0 (R N ) denotes the space of continuous functions on R N which vanish at infinity. The following are the main results for the Dunl transform; we omit the proofs but refer the reader to [D5] and [dj1]: Theorem (1) The Dunl transform f f is a homeomorphism of S (R N ) with period 4. (2) (Plancherel theorem) The Dunl transform has a unique extension to an isometric isomorphism of L 2 (R N, w ). We denote this isomorphism again by f f. (3) (L 1 -inversion) For all f L 1 (R N, w ) with f L 1 (R N, w ), f = ( f ) a.e.

17 DUNKL OPERATORS: THEORY AND APPLICATIONS CMS models and generalized Hermite polynomials 3.1. Quantum Calogero-Moser-Sutherland models. Quantum Calogero-Moser- Sutherland (CMS) models describe quantum mechanical systems of N identical particles on a circle or line which interact pairwise through long range potentials of inverse square type. They are exactly solvable and have gained considerable interest in theoretical physics during the last years. Among the broad literature in this area, we refer to [DV], [LV], [K], [BHKV], [BF1]-[BF3], [Pa], [Pe], [UW], [D7]. CMS models have in particular attracted some attention in conformal field theory, and they are being used to test the ideas of fractional statistics ([Ha], [Hal]). While explicit spectral resolutions of such models were already obtained by Calogero and Sutherland ([Ca], [Su]), a new aspect in the understanding of their algebraic structure and quantum integrability was much later initiated by [Po] and [He1]. The Hamiltonian under consideration is hereby modified by certain exchange operators, which allow to write it in a decoupled form. These exchange modifications can be expressed in terms of Dunl operators of type A N 1. The Hamiltonian of the linear CMS model with harmonic confinement in L 2 (R N ) is given by H C = + g 1 i<j N 1 (x i x j ) 2 + ω2 x 2 ; (3.1) here ω > 0 is a frequency parameter and g 1/2 is a coupling constant. In case ω = 0, (3.1) describes the free Calogero model. On the other hand, if g = 0, then H C coincides with the Hamiltonian of the N -dimensional isotropic harmonic oscillator, H 0 = + ω 2 x 2. The spectral decomposition of this operator in L 2 (R N ) is well-nown: The spectrum is discrete, σ(h 0 ) = {(2n + N)ω, n Z + }, and the classical multivariable Hermite functions (tensor products of one-variable Hermite functions, c.f. Examples 3.5), form a complete set of eigenfunctions. The study of the Hamiltonian H C was initiated by Calogero ([Ca]); he computed its spectrum and determined the structure of the bosonic eigenfunctions and scattering states in the confined and free case, respectively. Perelomov [Pe] observed that (3.1) is completely quantum integrable, i.e. there exist N commuting, algebraically independent symmetric linear operators in L 2 (R N ) including H C. We mention that the complete integrability of the classical Hamiltonian systems associated with (3.1) goes bac to Moser [Mo]. There exist generalizations of the classical Calogero-Moser-Sutherland models in the context of abstract root systems, see for instance [OP1], [OP2]. In particular, if R is an arbitrary root system on R N and is a nonnegative multiplicity function on it, then the corresponding abstract Calogero Hamiltonian with harmonic confinement is given by with the formal expression F H = F + ω 2 x 2 = 2 1 (α)((α) 1) α, x. 2 α R +

18 18 MARGIT RÖSLER If R is of type A N 1, then H just coincides with H C. For both the classical and the quantum case, partial results on the integrability of this model are due to Olshanetsy and Perelomov [OP1], [OP2]. A new aspect in the understanding of the algebraic structure and the quantum integrability of CMS systems was initiated by Polychronaos [Po] and Hecman [He1]. The underlying idea is to construct quantum integrals for CMS models from differential-reflection operators. Polychronaos introduced them in terms of an exchange-operator formalism for (3.1). He thus obtained a complete set of commuting observables for (3.1) in an elegant way. In [He1] it was observed in general that the complete algebra of quantum integrals for free, abstract Calogero models is intimately connected with the corresponding algebra of Dunl operators. Let us briefly describe this connection: Consider the following modification of F, involving reflection terms: F = 2 (α) α, x ((α) σ α). (3.2) 2 α R + In order to avoid singularities in the reflecting hyperplanes, it is suitable to carry out a gauge transform by w 1/2. A short calculation, using again results from [D2], gives w 1/2 F w 1/2 =, c.f. [R4]. Here is the Dunl Laplacian associated with G and. Now consider the algebra of Π G of G-invariant polynomials on R N. By a classical theorem of Chevalley (see e.g. [Hu]), it is generated by N homogeneous, algebraically independent elements. For p Π G we denote by Res (p(t )) the restriction of the Dunl operator p(t ) to Π G (Recall that p(t ) leaves Π G invariant!). Then A := { Res p(t ) : p Π G } is a commutative algebra of differential operators on Π G containing the operator Res ( ) = w 1/2 F w 1/2, and A has N algebraically independent generators, called quantum integrals for the free Hamiltonian F Spectral analysis of abstract CMS Hamiltonians. This section is devoted to a spectral analysis of abstract linear CMS operators with harmonic confinement. We follow the expositions in [R2], [R5]. To simplify formulas, we fix ω = 1/2; corresponding results for general ω can always be obtained by rescaling. We again wor with the gauge-transformed version with reflection terms, H := w 1/2 ( F x 2 ) w 1/2 = x 2. Due to the anti-symmetry of the first order Dunl operators (Prop. 2.16), this operator is symmetric and densely defined in L 2 (R N, w ) with domain D(H ) := S (R N ). Notice that in case = 0, H is just the Hamiltonian of the N -dimensional isotropic harmonic oscillator. We further consider the Hilbert space L 2 (R N, m ), where m is the probability measure dm := c 1 /2 e x 2 w (x)dx (3.3)

19 and the operator DUNKL OPERATORS: THEORY AND APPLICATIONS 19 J := + N x i i in L 2 (R N, m ), with domain D(J ) := Π. It can be shown by standard methods that Π is dense in L 2 (R N, m ). We do not carry this out; a proof can be found in [R3] or in [dj3], where a comprehensive treatment of density questions in several variables is given. The next theorem contains a complete description of the spectral properties of H and J and generalizes the already mentioned well-nown facts for the classical harmonic oscillator Hamilonian. For the proof, we shall employ the sl(2)-commutation relations of the operators i=1 E := 1 2 x 2, F := 1 2 and H := N x i i + (γ + N/2) on Π (with the index γ = γ() as defined in (2.4)) which can be found in [He1]. They are [ H, E ] = 2E, [ H, F ] = 2F, [ E, F ] = H. (3.4) Notice that the first two relations are immediate consequences of the fact that the Euler operator N ρ := x i i (3.5) satisfies ρ(p) = np for each homogeneous p P n. We start with the following Lemma 3.1. On D(J ) = Π, i=1 i=1 J = e x 2 /4 (H (γ + N/2)) e x 2 /4. In particular, J is symmetric in L 2 (R N, m ). Proof. From (3.4) it is easily verified by induction that [, E n] = 2nE n 1 H + 2n(n 1)E n 1 for all n N, and therefore [, e E/2 ] = e E/2 H Ee E/2. Thus on Π, H e E/2 = e E/ Ee E/2 = e E/2 + e E/2 H = e E/2 (J + γ + N/2). Theorem 3.2. The spaces L 2 (R N, m ) and L 2 (R N, w ) admit orthogonal Hilbert space decompositions into eigenspaces of the operators J and H respectively. More precisely, define V n := {e /2 p : p P n } Π, W n := {e x 2 /4 q(x), q V n } S (R N ).

20 20 MARGIT RÖSLER Then V n is the eigenspace of J corresponding to the eigenvalue n, W n is the eigenspace of H corresponding to the eigenvalue n + γ + N/2, and L 2 (R N, m ) = n Z + V n, L 2 (R N, w ) = n Z + W n. Remar 3.3. A densely defined linear operator (A, D(A)) in a Hilbert space H is called essentially self-adjoint, if it satisfies (i) A is symmetric, i.e. Ax, y = x, Ay for all x D(A); (ii) The closure A of A is selfadjoint. In fact, every symmetric operator A in H has a unique closure A (because A A, and the adjoint A is closed). If H has a countable orthonormal basis {v n, n Z + } D(A) consisting of eigenvectors of A corresponding to eigenvalues λ n R, then it is straightforward that A is essentially self-adjoint, and that the spectrum of the selfadjoint operator A is given by σ(a) = {λ n, n Z + }. (See for instance Lemma of [Da]). In our situation, the operator H is densely defined and symmetric in L 2 (R N, w ) (the first order Dunl operators being anti-symmetric), and the same holds for J in L 2 (R N, m ). The above theorem implies that H and J are essentially self-adjoint and that σ(h ) = {n + γ + N/2, n Z + }, σ(j ) = Z +. Proof of Theorem 3.2. Equation (3.4) and induction yield the commuting relations [ ρ, n ] = 2n n for all n Z +, and hence [ ρ, e /2 ] = e /2. If q Π is arbitrary and p := e /2 q, it follows that ρ(q) = (ρe /2 )(p) = e /2 ρ(p) + e /2 p = e /2 ρ(p) + q. Hence for a C there are equivalent: ( + ρ)(q) = aq ρ(p) = ap a = n Z + and p P n. Thus each function from V n is an eigenfunction of J corresponding to the eigenvalue n, and V n V m for n m by the symmetry of J. This proves the statements for J because Π = V n is dense in L 2 (R N, m ). The statements for H are then immediate by the previous Lemma Generalized Hermite polynomials. The eigenvalues of the CMS Hamiltonians H and J are highly degenerate if N > 1. In this section, we construct natural orthogonal bases for them. They are made up by generalizations of the classical N - variable Hermite polynomials and Hermite functions to the Dunl setting. We follow [R2], but change our normalization by a factor 2. The starting point for our construction is the Macdonald-type identity: if p, q Π, then [p, q] = e /2 p(x)e /2 q(x)dm (x), (3.6) R N

21 DUNKL OPERATORS: THEORY AND APPLICATIONS 21 with the probability measure m defined according to (3.3). Notice that [.,. ] is a scalar product on the R- vector space Π R of polynomials with real coefficients. Let {ϕ ν, ν Z N +} be an orthonormal basis of Π R with respect to the scalar product [.,.] such that ϕ ν P ν. As homogeneous polynomials of different degrees are orthogonal, the ϕ ν with fixed ν = n can for example be constructed by Gram-Schmidt orthogonalization within P n Π R from an arbitrary ordered real-coefficient basis. If = 0, the canonical choice of the basis {ϕ ν } is just ϕ ν (x) := (ν!) 1/2 x ν. Definition 3.4. The generalized Hermite polynomials {H ν, ν Z N +} associated with the basis {ϕ ν } on R N are given by H ν (x) := e /2 ϕ ν (x). (3.7) Moreover, we define the generalized Hermite functions on R N h ν (x) := e x 2 /4 H ν (x), ν Z N +. (3.8) H ν is a polynomial of degree ν satisfying H ν ( x) = ( 1) ν H ν (x) for all x R N. By virtue of (3.6), the H ν, ν Z N + form an orthonormal basis of L 2 (R N, m ). Examples 3.5. (1) Classical multivariable Hermite polynomials. Let = 0, and choose the standard orthonormal system ϕ ν (x) = (ν!) 1/2 x ν, with respect to [.,. ] 0. The associated Hermite polynomials are given by H ν (x) = 1 N e 2 i /2 (x ν i i ) = 2 ν /2 ν! ν! i=1 N i=1 by Ĥ νi (x i / 2), (3.9) where the Ĥn, n Z + are the classical Hermite polynomials on R defined by Ĥ n (x) = ( 1) n e x2 dn dx n e x2 (2) The one-dimensional case. Up to sign changes, there exists only one orthonormal basis with respect to [.,. ]. The associated Hermite polynomials are given, up to multiplicative constants, by the generalized Hermite polynomials Hn(x/ 2) on R. These polynomials can be found in [Chi] and were further studied in [Ros] in connection with a Bose-lie oscillator calculus. The Hn are orthogonal with respect to x 2 e x 2 and can be written as { H2n(x) = ( 1) n 2 2n n! Ln 1/2 (x 2 ), H2n+1(x) = ( 1) n 2 2n+1 n! xl +1/2 n (x 2 ); here the L α n are the usual Laguerre polynomials of index α 1/2, given by L α n(x) = 1 ( n! x α e x dn x n+α e ). x dx n (3) The A N 1 -case. There exists a natural orthogonal system {ϕ ν }, made up by the so-called non-symmetric Jac polynomials. For a multiplicity parameter > 0, the associated non-symmetric Jac polynomials E ν, ν Z N +, as introduced in [O2] (see also [KS]), are uniquely defined by the following conditions:.

22 22 MARGIT RÖSLER (i) E ν (x) = x ν + µ< P ν c ν, µ x µ with c ν,µ R; (ii) For all µ < P ν, ( E ν (x), x µ) = 0 Here < P is a dominance order defined within multi-indices of equal total length (see [O2]), and the inner product (.,.) on Π Π R is given by (f, g) := f(z)g(z) z i z j 2 dz T N i<j with T = {z C : z = 1} and dz being the Haar measure on T N. If f and g have different total degrees, then (f, g) = 0. The set {E ν, ν = n} forms a vector space basis of P n Π R. It can be shown (by use of A N 1 -type Cheredni operators) that the Jac polynomials E ν are also orthogonal with respect to the Dunl pairing [.,. ] ; for details see [R2]. The corresponding generalized Hermite polynomials and their symmetric counterparts have been studied in [La1], [La2] and in [BF1] - [BF3]. As an immediate consequence of Theorem 3.2 we obtain analogues of the classical second order differential equations for generalized Hermite polynomials and Hermite functions: Corollary 3.6. (i) ( N ) + x i i Hν = ν H ν. i=1 (ii) ( x 2) h ν = ( ν + γ + N/2)h ν. Various further useful properties of the classical Hermite polynomials and Hermite functions have extensions to our general setting. We conclude this section with a list of them. The proofs can be found in [R2]. For further results on generalized Hermite polynomials, one can also see for instance [vd]. Theorem 3.7. Let {H ν } be the Hermite polynomials and Hermite functions associated with the basis {ϕ ν } on R N and let x, y R N. Then (1) H ν (x) = ( 1) ν e x 2 /2 ϕ ν (T )e x 2 /2 (Rodrigues-Formula) (2) e y 2 /2 E (x, y) = H ν (x)ϕ ν (y) (Generating relation) ν Z N + ν Z N + (3) (Mehler formula) For all 0 < r < 1, { } H ν (x)h ν (y)r ν 1 = exp r2 ( x 2 + y 2 ) ( rx ) E (1 r 2 ) γ+n/2 2(1 r 2 ) 1 r, y. 2 The sums on the left are absolutely convergent in both cases. The Dunl ernel E in (2) and (3) replaces the usual exponential function. It comes in via the following relation with the (arbitrary!) basis {ϕ ν }: E (x, y) = ϕ ν (x)ϕ ν (y) (x, y R N ). ν Z N +

23 DUNKL OPERATORS: THEORY AND APPLICATIONS 23 Proposition 3.8. The generalized Hermite functions {h ν, ν Z N +} are a basis of eigenfunctions of the Dunl transform on L 2 (R N, w ) with h ν = ( i) ν h ν. 4. Positivity results 4.1. Positivity of Dunl s intertwining operator. In this section it is always assumed that 0. The reference is [R3]. We shall say that a linear operator A on Π is positive, if A leaves the positive cone Π + := {p Π : p(x) 0 for all x R N } invariant. The following theorem is the central result of this section: Theorem 4.1. V is positive on Π. Once this is nown, more detailed information about V can be obtained by its extension to the algebras A r, which were introduced in Definition This leads to Theorem 4.2. For each x R N there exists a unique probability measure µ x on the Borel-σ-algebra of R N such that V f(x) = f(ξ) dµ x(ξ) for all f A x. (4.1) R N The representing measures µ x are compactly supported with supp µ x co {gx, g G}, the convex hull of the orbit of x under G. Moreover, they satisfy µ rx(b) = µ x(r 1 B), µ gx(b) = µ x(g 1 (B)) (4.2) for each r > 0, g G and each Borel set B R N. The proof of Theorem 4.1 affords several steps, the crucial one being a reduction from the N -dimensional to a one-dimensional problem. We shall give an outline, but beforehand we turn to the proof of Theorem 4.2. Proof of Theorem 4.2. Fix x R N and put r = x. Then the mapping Φ x : f V f(x) is a bounded linear functional on A r, and Theorem 4.1 implies that it is positive on the dense subalgebra Π of A r, i.e. Φ x ( p 2 ) 0. Consequently, Φ x is a positive functional on the full Banach- -algebra A r. There exists a representation theorem of Bochner for positive functionals on commutative Banach- -algebras (see for instance Theorem 21.2 of [FD]). It implies in our case that there exists a unique measure ν x M + b ( S(A r )) such that Φ x (f) = f(ϕ) dν x (ϕ) for all f A r, S (A r) with f the Gelfand transform of f. Keeping Exercise 2.27 in mind, one obtains representing measures µ x supported in the ball B r ; the sharper statement on the support is obtained by results of [dj1]. The remaining statements are easy.