E K Narayanan

A Pasquale
email: angela.pasquale@univ-lorraine.fr

HYPERGEOMETRIC FUNCTIONS OF TYPE BC AND STANDARD MULTIPLICITIES

Keywords: 2010 Mathematics Subject Classification. Primary, 33C67; secondary, 43A32, 43A90

We study the Heckman-Opdam hypergeometric functions associated to a root system of type BC and a multiplicity function which is allowed to assume some non-positive values (a standard multiplicity function). For such functions, we obtain positivity properties and sharp estimates which imply a characterization of the bounded hypergeometric functions. As an application, our results extend known properties of Harish-Chandra's spherical functions on Riemannian symmetric spaces of the non-compact type G/K to spherical functions over homogeneous vector bundles on G/K which are associated to certain small K-types.

Introduction

Let G be a connected non-compact real semisimple Lie group with finite center, K a maximal compact subgroup of G, and X = G/K the corresponding Riemannian symmetric space of the non-compact type. Harish-Chandra's theory of spherical functions on X has been extended into two different directions.

The first direction is Heckman-Opdam's theory of hypergeometric functions on root systems. It originated from the fact that, by restriction to a maximally flat subspace of X, the K-invariant functions on X become Weyl-group-invariant functions on the corresponding Cartan subspace a. The analogue of Harish-Chandra's spherical functions are the hypergeometric functions associated with root systems, also known as Heckman-Opdam's hypergeometric functions. Heckman and Opdam introduced them in the late 1980s and, in the middle 1990s, Cherednik contributed in simplifying their original definition; see [START_REF] Heckman | Harmonic analysis and special functions on symmetric spaces[END_REF][START_REF]Harmonic analysis for certain representations of graded Hecke algebras[END_REF], as well as [START_REF] Anker | An introduction to Dunkl theory and its analytic aspects[END_REF] for a recent overview of the theory. In the context of special functions associated with root systems, the symmetric space X is replaced by a triple (a, Σ, m), where a is a Euclidean real vector space (playing the role of the Cartan subspace of X), Σ is a root system in the real dual space a * of a, and m : Σ → C is a multiplicity function, i.e. a function on Σ which is invariant with respect to the Weyl group W of Σ. If a is a Cartan subspace of the symmetric space X, Σ is the corresponding system of (restricted) roots and the multiplicity function m is given by the dimensions of the root spaces, then Heckman-Opdam's hypergeometric functions associated with (a, Σ, m) are precisely the restrictions to a of Harish-Chandra's spherical functions. Here and in the following, we adopt the convention of identifying the Cartan subspace a with its diffeomorphic image A = exp(a) inside G.

Notice that, even if the multiplicity function in a triple (a, Σ, m) might be generally complex-valued, most of the properties of Heckman-Opdam's hypergeometric functions and also the harmonic analysis associated with them, are known assuming that the multiplicity functions have values in [0, +∞). This is of course the most natural setting to include Harish-Chandra's spherical harmonic analysis on Riemannian symmetric spaces.

The second direction extending Harish-Chandra's theory of spherical functions concerns the so-called τ -spherical functions, where τ is a unitary irreducible representation of K. These functions already appeared in the work of Godement [START_REF] Godement | A theory of spherical functions[END_REF] and Harish-Chandra [START_REF] Harish-Chandra | Representations of semisimple Lie groups on a Banch space[END_REF][START_REF]Representations of semisimple Lie groups[END_REF], and they agree with Harish-Chandra's spherical functions on X when τ is the trivial representation. They have been studied either in the context of the representation theory of G or in relation to the harmonic analysis on homogeneous vector bundles over X. Among the references related to the present paper, we mention [START_REF] Harish-Chandra | Differential equations and semisimple Lie groups[END_REF][START_REF] Takahashi | Fonctions sphériques dans les groupes Sp(n, 1). Théorie du potentiel et analyse harmonique[END_REF][START_REF]The Plancherel formula for spherical functions with a one-dimensional K-type on a simply connected simple Lie group of Hermitian type[END_REF][START_REF] Heckman | Harmonic analysis and special functions on symmetric spaces[END_REF][START_REF] Olbrich | Die Poisson-Transformation für homogene Vektorbündel[END_REF][START_REF] Camporesi | The Helgason Fourier transform for homogeneous vector bundles over Riemannian symmetric spaces[END_REF][START_REF] Pedon | Analyse harmonique des formes diffŕentielles sur l'espace hyperbolique réel. I. Transformation de Poisson et fonctions sphériques[END_REF][START_REF] Van Dijk | Harmonic analysis on vector bundles over Sp(1, n)/Sp(1) × Sp(n)[END_REF][START_REF] Oda | Spherical functions for small K-types. Geometric and harmonic analysis on homogeneous spaces[END_REF][START_REF] Ricci | Spherical analysis on homogeneous vector bundles[END_REF][START_REF] Honda | N : Inversion formula for the hypergeometric Fourier transform associated with a root system of type BC[END_REF]. Several new features appear when τ is non-trivial. For instance, the algebra D(G/K; τ ) of invariant differential operators acting on the sections of the homogeneous vector bundle, might not be commutative. Let V τ be the space of τ and let L 1 (G//K; τ ) denote the convolution algebra of End(V τ )-valued functions f on G satisfying f (k 1 gk 2 ) = τ (k -1

2 )f (g)τ (k -1

2 ) for all g ∈ G and k 1 , k 2 ∈ K. It is a classical result that D(G/K; τ ) is commutative if and only if L 1 (G//K; τ ) is commutative. In this case, (G, K, τ ) is said to be a Gelfand triple. A convenient criterion to check whether (G, K, τ ) is a Gelfand triple is due to Deitmar [START_REF] Deitmar | Invariant operators on higher K-types[END_REF]. Namely, D(G/K; τ ) is commutative if and only if the restriction of τ to M , the centralizer in K of the Cartan subspace a, is multiplicity free. This happens for instance when τ | M is irreducible, i.e. τ is a small K-type. For example, a 1-dimensional representation τ is necessarily a small K-type. For Gelfand triples, the theory of τ -spherical functions can be set up exactly as in the case of τ trivial. For instance, the τ -spherical functions can be equivalently defined either as joint eigenfunctions of D(G/K; τ ) or as characters of L 1 (G//K; τ ). Nevertheless, they are much more difficult to handle than in the trivial case, and many of their properties are still not known.

The present paper is situated at a crossing of the two directions of extensions of Harish-Chandra's theory of spherical functions on X mentioned above. It finds its motivations in Shimeno's paper [START_REF]The Plancherel formula for spherical functions with a one-dimensional K-type on a simply connected simple Lie group of Hermitian type[END_REF], in Heckman's chapter [START_REF] Heckman | Harmonic analysis and special functions on symmetric spaces[END_REF]Chapter 5] and, more generally, in the recent paper [START_REF] Oda | Spherical functions for small K-types. Geometric and harmonic analysis on homogeneous spaces[END_REF] by Oda and Shimeno on τ -spherical functions corresponding to small K-types. When τ is a small K-type, a τ -spherical function is uniquely determined by its restriction to a Cartan subspace a of G. By Schur's lemma, this restriction is of the form ϕ • id where id is the identity on V τ and ϕ is function on a which is Weyl-group invariant. The main theorem of [START_REF] Oda | Spherical functions for small K-types. Geometric and harmonic analysis on homogeneous spaces[END_REF] proves that, up to multiplication by an explicit nonvanishing smooth cosh-like factor, the function ϕ is a Heckman-Opdam's hypergeometric function. It corresponds to a triple (a, Σ(τ ), m(τ )) in which Σ(τ ) is possibly not the root system associated with the symmetric space X and m(τ ) need not be positive. This motivates a systematic study of Heckman-Opdam's hypergeometric functions corresponding to multiplicity functions which may assume negative values.

In this paper we take up this line of investigations in the case of the so-called standard multiplicities. More precisely, let Σ be a root system in a * of type BC r , where r is the dimension of a, and let m = (m s , m m , m l ) be a multiplicity function on Σ. Set

M 1 = {(m s , m m , m l ) ∈ M : m m > 0, m s > 0, m s + 2m l > 0} .
In [14, Definition 5.5.1], the elements of M 1 were called the standard multiplicities. Standard multiplicities have been introduced by Heckman and their definition is linked to the regularity of Harish-Chandra's c-function; see [14, Lemma 5.5.2] and (26) below.

All positive multiplicity functions are standard, but standard multiplicites also allow negative values for the long roots. Because of this, the arguments leading to the known properties of the hypergeometric functions corresponding to positive multiplicity functions do not apply to this case. By suitable modifications of their proofs, we show in this paper that various results, including positivity properties and estimates for hypergeometric functions, extend to standard multiplicities (see Proposition 2.5, Theorem 2.10 and Theorem 2.11).

Let a *

C denote the complex dual vector space of a and consider the Heckman-Opdam's (symmetric and non-symmetric) hypergeometric functions F λ (m, x) and G λ (m, x) with λ ∈ a * C ; see subsection 1.2 for the definitions. Under the condition that m ∈ M 1 we prove:

(1) F λ and G λ are real and strictly positive on a for all λ ∈ a * , (2)

|F λ | ≤ F Re λ and |G λ | ≤ G Re λ on a for all λ ∈ a * C , (3) max{|F λ (x)|, |G λ (x)|} ≤ |W |e max w∈W (wλ)(x) for all λ ∈ a *
C and x ∈ a, (4) F λ+µ (x) ≤ F µ (x)e max w∈W (wλ)(x) and G λ+µ (x) ≤ G µ (x)e max w∈W (wλ)(x) for all λ ∈ a * , µ ∈ (a * ) + and x ∈ a, (5) asymptotics on a for F λ when λ ∈ a * C is fixed but not necessarily regular, (6) sharp estimates on a for F λ when λ ∈ a * is fixed but not necessarily regular.

The estimates (but not the asymptotics) pass by continuity to the boundary of M 1 as well. The above properties extend to M 1 the corresponding properties proved for nonnegative multiplicities by Opdam [START_REF]Harmonic analysis for certain representations of graded Hecke algebras[END_REF], Ho and Ólafsson [START_REF] Ho | Paley-Wiener theorem for line bundles over compact symmetric spaces and new estimates for the Heckman-Opdam hypergeometric functions[END_REF], Schapira [START_REF] Schapira | Etude analytique et probabiliste de laplaciens associés à des systèmes de racines : laplacien hypergéométrique de Heckman-Opdam et laplacien combinatoire sur les immeubles affines[END_REF], Rösler, Koornwinder and Voit [START_REF] Rösler | Limit transition between hypergeometric functions of type BC and type A[END_REF], and the authors and Pusti [START_REF] Narayanan | Asymptotics of Harish-Chandra expansions, bounded hypergeometric functions associated with root systems, and applications[END_REF].

The set of real-valued multiplicities on which both function F λ and G λ are naturally defined is

M 0 = {(m s , m m , m l ) ∈ M : m m ≥ 0, m s + m l ≥ 0} .
Clearly, M 1 ⊂ M 0 . In section 2 we introduce and discuss other subsets of M. For instance, the estimates (3) hold in fact for a larger set of multiplicities than M 1 ; see Lemma 2.4.

Let ρ(m) ∈ a * be the half-sum of the positive roots in Σ, counted with their multiplicities; see [START_REF] Branson | The Paley-Wiener theorem and the local Huygens' principle for compact symmetric spaces: the even multiplicity case[END_REF]. Moreover, let C(ρ(m)) denote the convex hull of the finite set {wρ(m) : w ∈ W }. Suppose first that the triple (a, Σ, m) is associated with a symmetric space X and consider Harish-Chandra's parametrization of the spherical functions by the elements of a * C . Let ϕ λ denote the spherical function corresponding to λ ∈ a * C . The spherical functions which are bounded can be identified with the elements of the spectrum of the (commutative) convolution algebra of L 1 K-invariant functions on X. They have been determined by the celebrated theorem of Helgason and Johnson [START_REF] Helgason | The bounded spherical functions on symmetric spaces[END_REF]. It states that the spherical function ϕ λ on X is bounded if and only if λ belongs to the tube domain in a * C over C(ρ). It is a fundamental result in the L 1 spherical harmonic analysis on X. For instance, it implies that the spherical transform of a K-invariant L 1 function on X is holomorphic in the interior of the tube domain over C(ρ). In particular, the spherical transform of an L 1 function cannot have compact support, unlike what happens for the classical Fourier transform.

The theorem of Helgason and Johnson has been recently extended to Heckman-Opdam's hypergeometric functions in [START_REF] Narayanan | Asymptotics of Harish-Chandra expansions, bounded hypergeometric functions associated with root systems, and applications[END_REF] under the assumption that the multiplicity function m is positive. On the boundary of M 1 there are nonzero multiplicities m for which ρ(m) = 0. However, ρ(m) = 0 for m ∈ M 1 . It is therefore natural to ask if Helgason-Johnson's characterization of boundedness by means of the tube domain over C(ρ) holds for Heckman-Opdam's hypergeometric functions associated with arbitrary standard multiplicity functions m ∈ M 1 and not only to the positive ones. The answer to this question is positive and is given by Theorem 2.11.

Let M + denote the set of nonnegative multiplicity functions on a fixed root system of type BC. In section 3 we consider the 2-parameter deformations of m

= (m s , m m , m l ) ∈ M + of the form m( , ) = (m s + 2 , m m + 2 , m l -2 ) ,
where , ∈ R. The corresponding non-symmetric and symmetric ( , )-hypergeometric functions are respectively defined for λ ∈ a * C and x ∈ a by

G , ,λ (m; x) = u(x) -v(x) -G λ (m( , ); x) , F , ,λ (m; x) = u(x) -v(x) -F λ (m( , ); x) ,
where u(x) and v(x) are suitable products of hyperbolic cosine functions depending on the roots; see [START_REF]Harmonic analysis on homogeneous vector bundles on hyperbolic spaces[END_REF] and [START_REF]A formula for the hypergeometric function of type BC n[END_REF].

Given m = (m s , m m , m l ) ∈ M + , then m( , ) ∈ M 1 if and only if -ms 2 < < ms 2 + m l and > -mm 2 .
For these values, the results proved in section 2 for G λ (m( , )) and F λ (m( , )) extend to corresponding results for the ( , )-hypergeometric functions, but some care is needed to take the factors u -and v -into account. In particular, Theorem 3.7 characterizes the F , ,λ 's which are bounded. This theorem as well as other properties extends to all 's for which -(m l -1)/2 < (m s + m l + 1)/2 thanks to the symmetry relation F , ,λ = F -+m l -1, ,λ proved in (44).

The final section of this paper, section 4, is devoted to geometric examples based on [START_REF] Heckman | Harmonic analysis and special functions on symmetric spaces[END_REF][START_REF]The Plancherel formula for spherical functions with a one-dimensional K-type on a simply connected simple Lie group of Hermitian type[END_REF] and [START_REF] Oda | Spherical functions for small K-types. Geometric and harmonic analysis on homogeneous spaces[END_REF]. Namely, we prove that the τ -spherical functions on G/K associated with a small K-type τ and for which the root system of G/K is of type BC can be described as symmetric ( , )-hypergeometric functions on the root system Σ(τ ) of [START_REF] Oda | Spherical functions for small K-types. Geometric and harmonic analysis on homogeneous spaces[END_REF] and specific choices of a multiplicity function m ∈ M + and of the deformation parameters ( , ). Our general results from section 3 apply therefore to these cases.

Notation and preliminaries

In this section we collect the basic notation and some preliminary results. We refer to [START_REF] Heckman | Harmonic analysis and special functions on symmetric spaces[END_REF] and [START_REF] Narayanan | Asymptotics of Harish-Chandra expansions, bounded hypergeometric functions associated with root systems, and applications[END_REF] for a more extended exposition.

If F (m) is a function on a space X that depends on a parameter m, we will denote its value at x ∈ X by F (m; x) rather than F (m)(x). Given two nonnegative functions f and g on a same domain D, we write f g if there exist positive constants C 1 and C 2 so that C 1 ≤ f (x) g(x) ≤ C 2 for all x ∈ D. 1.1. Root systems. Let a be an r-dimensional real Euclidean vector space with inner product •, • , and let a * be the dual space of a. For λ ∈ a * let x λ ∈ a be determined by the condition that λ(x) = x, x λ for all x ∈ a. The assignment λ, µ := x λ , x µ defines an inner product in a * . Let a C and a * C respectively denote the complexifications of a and a * . The C-bilinear extension to a C and a * C of the inner products on a * and a will also be indicated by •, • . We denote by | • | = •, • 1/2 the associated norm. We shall often employ the notation

λ α := λ, α α, α (1)
for elements λ, α ∈ a * C with |α| = 0. Let Σ be a root system in a * with set of positive roots of the form

Σ + = Σ + s Σ + m Σ + l , where Σ + s = β j 2 : 1 ≤ j ≤ r , Σ + m = β j ± β i 2 : 1 ≤ i < j ≤ r , Σ + l = {β j : 1 ≤ j ≤ r} .
The positive Weyl chamber a + consists of the elements x ∈ a for which α(x) > 0 for all α ∈ Σ + ; its closure is a + = {x ∈ a : α(x) ≥ 0 for all α ∈ Σ + }. Dually, the positive Weyl chamber (a * ) + consists of the elements λ ∈ a * for which λ, α > 0 for all α ∈ Σ + . Its closure is denoted (a * ) + . We assume that the elements of Σ + l form an orthogonal basis of a * and that they all have the same norm p. We denote by W the Weyl group of Σ. It acts on the roots by permutations and sign changes. For a ∈ {s, m, l} set Σ a = Σ + a (-Σ + a ). A multiplicity function on Σ is a W -invariant function on Σ. It is therefore given by a triple m = (m s , m m , m l ) of complex numbers so that m a is the (constant) value of m on Σ a for a ∈ {s, m, l}.

It will be convenient to refer to a root system Σ as above as of type BC r even if some values of m are zero. This means that root systems of type C r will be considered as being of type BC r with m s = 0 and m m = 0. Likewise, the direct products of rank-one root systems (BC 1 ) r and (A 1 ) r will be considered of type BC r with m m = 0 and with m m = m s = 0, respectively.

The dimension r of a is called the (real) rank of the triple (a, Σ, m). Finally, we set

ρ(m) = 1 2 α∈Σ + m α α = 1 2 r j=1 m s 2 + m l + (j -1)m m β j ∈ a * . (2)
Example 1.1 (Geometric multiplicities). For special values of the multiplicities m = (m s , m m , m l ), triples (a, Σ, m) as above appear as restricted root systems of a large family of irreducible symmetric spaces G/K of the non-compact type. Among these spaces, a remarkable class consists of the noncompact Hermitian symmetric spaces; see [START_REF] Helgason | Differential geometry, Lie groups, and symmetric spaces[END_REF] or [START_REF]The Plancherel formula for spherical functions with a one-dimensional K-type on a simply connected simple Lie group of Hermitian type[END_REF]. For these spaces m l = 1. Their root systems and multiplicity functions are listed in Table 1. The literature on Hermitian symmetric spaces refers to the situations where Σ = C r and Σ = BC r as to the Case I and the Case II, respectively. In the following, the triples m = (m l , m m , m s ) appearing as root multiplicities of Riemannian symmetric spaces of the non-compact type and root system of type BC will be called geometric multiplicities.

Other examples of geometric multiplicities will be considered in Section 4.

G K Σ (m s , m m , m l )
SU(p, q) S ( U (p) × U (q)) p = q: C p p < q: BC p p = q: (0, 2, 1) p < q: (2(q -p), Notice that in this paper we adopt the notation commonly used in the theory of symmetric spaces. It differs from the notation in the work of Heckman and Opdam in the following ways. The root system R and the multiplicity function k used by Heckman and Opdam are related to our Σ and m by the relations R = {2α : α ∈ Σ} and k 2α = m α /2 for α ∈ Σ. 1.2. Cherednik operators and hypergeometric functions. In this subsection, we review some basic notions on the hypergeometric functions associated with root systems. This theory has been developed by Heckman, Opdam and Cherednik. We refer the reader to [START_REF] Heckman | Harmonic analysis and special functions on symmetric spaces[END_REF], [START_REF]Harmonic analysis for certain representations of graded Hecke algebras[END_REF], and [START_REF]Lecture notes on Dunkl operators for real and complex reflection groups[END_REF] for more details and further references.

Let S(a C ) denote the symmetric algebra over a C considered as the space of polynomial functions on a * C , and let S(a C ) W be the subalgebra of W -invariant elements. Every p ∈ S(a C ) defines a constant-coefficient differential operators p(∂) on A C and on a C such that ξ(∂) = ∂ ξ is the directional derivative in the direction of ξ for all ξ ∈ a. The algebra of the differential operators p(∂) with p ∈ S(a C ) will also be indicated by S(a C ).

Set a reg = {x ∈ a : α(x) = 0 for all α ∈ Σ}. Let R denote the algebra of functions on a reg generated by 1 and

g α = (1 -e -2α ) -1 (α ∈ Σ + ) . (3)
Notice that for ξ ∈ a and α ∈ Σ + we have

∂ ξ g α = 2α(ξ) g 2 α -g α g -α = 1 -g α . ( 4
)
Hence R is stable under the actions of S(a C ) and of the Weyl group. Let D R = R ⊗ S(a C ) be the algebra of differential operators on a reg with coefficients in R. The Weyl group W acts on D R according to

w ϕ ⊗ p(∂)) := wϕ ⊗ (wp)(∂).
We indicate by D W R the subalgebra of W -invariant elements of D R . The space D R ⊗ C[W ] can be naturally endowed with the structure of an associative algebra.

The differential component of a differential-reflection operator

P ∈ D R ⊗ C[W ] is the differential operator β(P ) ∈ D R such that P f = β(P )f . ( 5
) for all f ∈ C[A C ] W . If P = w∈W P w ⊗ w with P w ∈ D R , then β(P ) = w∈W P w . For ξ ∈ a the Cherednik operator (or Dunkl-Cherednik operator) T ξ (m) ∈ D R ⊗ C[W ] is defined by T ξ (m) := ∂ ξ -ρ(m)(ξ) + α∈Σ + m α α(ξ)(1 -e -2α ) -1 ⊗ (1 -r α ) (6)
The Cherednik operators commute with each other (cf. [START_REF]Harmonic analysis for certain representations of graded Hecke algebras[END_REF], Section 2). Therefore the map ξ → T ξ (m) on a extends uniquely to an algebra homomorphism 2 , where {ξ j } r j=1 is any orthonormal basis of a, is called the Heckman-Opdam Laplacian. Explicitly,

p → T p (m) of S(a C ) into D R ⊗ C[W ]. If p ∈ S(a C ) W , then D p (m) := β T p (m) turns out to be in D W R . Let p L ∈ S(a C ) W be defined by p L (λ) := λ, λ for λ ∈ a * C . Then T p L (m) = r j=1 T ξ j (m)
T p L (m) = L(m) + ρ(m), ρ(m) - α∈Σ + m α α, α sinh 2 α ⊗ (1 -r α ) , (7)
where

L(m) := L a + α∈Σ + m α coth α ∂ xα , (8)
L a is the Laplace operator on a, and ∂ xα is the directional derivative in the direction of the element x α ∈ a corresponding to α in the identification of a and a * under •, • , as at the beginning of subsection 1.1. See [31, (1)] and [32, §2.6] for the computation of [START_REF] Erdélyi | Higher transcendental functions[END_REF]. In ( 7) and ( 8) we have set

sinh α = e α -e -α 2 = e α 2 (1 -e -2α ) , (9)
coth α = 1 + e -2α 1 -e -2α = 2 1 -e -2α -1 . (10)
Moreover,

D p L (m) = L(m) + ρ(m), ρ(m) . Set D(m) := {D p (m) : p ∈ S(a C ) W }. The map γ(m) : D(m) → S(a C ) W defined by γ(m) D p (m) (λ) := p(λ) (11)
is called the Harish-Chandra homomorphism. It defines an algebra isomorphism of D(m) onto S(a C ) W (see [14, Theorem 1.3.12 and Remark 1.3.14]). From Chevalley's theorem it therefore follows that D(m) is generated by r(= dim a) elements. The next lemma will play a decisive role for us.

Lemma 1.2. For every multiplicity function m, the algebra D(m) is the centralizer of L(m) in D W R . Proof. This is [START_REF] Heckman | Harmonic analysis and special functions on symmetric spaces[END_REF]Theorem 1.3.12]. See also [START_REF] Heckman | Harmonic analysis and special functions on symmetric spaces[END_REF]Remark 1.3.14] for its extension to complex-valued multiplicities.

Let λ ∈ a * C be fixed. The Heckman-Opdam hypergeometric function with spectral parameter λ is the unique W -invariant analytic solution F λ (m) of the system of differential equations

D p (m)f = p(λ)f (p ∈ S(a C ) W ), (12)
which satisfies f (0) = 1. The non-symmetric hypergeometric function with spectral parameter λ is the unique analytic solution G λ (m) of the system of differential equations

T ξ (m)g = λ(ξ)g (ξ ∈ a), (13)
which satisfies g(0) = 1. These functions are linked by the relation

F λ (m; x) = 1 |W | w∈W G λ (m; w -1 x) (x ∈ a) , (14)
where |W | denotes the cardinality of W . The existence of the hypergeometric functions requires suitable assumptions on m. We refer the reader to Appendix A for additional information.

For geometric multiplicities, D(m) coincides with the algebra of radial components of the G-invariant differential operators on G/K. Moreover, F λ (m) agrees with the restriction to A ≡ a of Harish-Chandra's spherical function on G/K with spectral parameter λ. A geometric intepretation of the functions G λ (m) has been recently given in [START_REF] Oda | Differential-difference operators and radial part formulas for non-invariant elements[END_REF].

Hypergeometric functions associated with standard multiplicities

In this section we look at the hypergeometric functions associated with standard multiplicites. Basic estimates and a characterization of the bounded hypergeometric functions (for these sets of multiplicity functions) are established.

Let M denote the set of real-valued multiplicity functions m = (m s , m m , m l ) on a root system Σ of type BC r . We will consider the following subsets of M:

M + = {m ∈ M : m α ≥ 0 for every α ∈ Σ} , ( 15
) M 0 = {(m s , m m , m l ) ∈ M : m m ≥ 0, m s + m l ≥ 0} , ( 16
) M 1 = {(m s , m m , m l ) ∈ M : m m > 0, m s > 0, m s + 2m l > 0} , ( 17
) M 2 = {(m s , m m , m l ) ∈ M : m m ≥ 0, m l ≥ 0, m s + m l ≥ 0} , ( 18
) M 3 = {(m s , m m , m l ) ∈ M : m m ≥ 0, m l ≤ 0, m s + 2m l ≥ 0} . ( 19
)
So M + consists of the non-negative multiplicity functions and M 1 = (M + ∪ M 3 ) 0 , the interior of M + ∪ M 3 . For real-valued multiplicities, M 0 is the natural set for which both hypergeometric functions G λ (m) and F λ (m) are defined for all λ ∈ a * C ; see Appendix A. Recall that the elements of M 1 are called standard multiplicity functions (see [START_REF] Heckman | Harmonic analysis and special functions on symmetric spaces[END_REF]Definition 5.5.1]). These sets of multiplicities are represented in Figure 1. (a) For all λ ∈ a * the functions G λ (m) and F λ (m) are real and strictly positive.

m s m l m s + 2m l = 0 m s + m l = 0 M + M 2 M 1 M 3
(b) For all λ ∈ a * C |G λ (m)| ≤ G Re λ (m) , |F λ (m)| ≤ F Re λ (m) . ( 20
) (c) For all λ ∈ a * and all x ∈ a G λ (m; x) ≤ G 0 (m; x)e maxw(wλ)(x) F λ (m; x) ≤ F 0 (m; x)e maxw(wλ)(x) . ( 21
)
More generally, for all λ ∈ a * , µ ∈ (a * ) + and all x ∈ a

G λ+µ (m; x) ≤ G µ (m; x)e maxw(wλ)(x) F λ+µ (m; x) ≤ F µ (m; x)e maxw(wλ)(x) . ( 22
)
(In the above estimates, max w denotes the maximum over all w ∈ W .)

Proposition 2.1 lists the sharpest estimates known so far for Heckman-Opdams hypergeometric functions. The following lemma, even if quite elementary, is the key result allowing us to modify their proofs and to extend these estimates to multiplicities m ∈ M 3 .. It will play a similar role also for extending the asymptotics and the boundedness characterization..

Lemma 2.2. Let m = (m s , m m , m l ) ∈ M 0 and β ∈ Σ + l .
Then the following inequalities hold for all x ∈ a:

(a) m s 2 + m l 1 1 + e -β(x) ≥ 0 if m ∈ M + ∪ M 3 ; (b) m s 2 + m l 1 + e -2β(x) (1 + e -β(x) ) 2 ≥ 0 if m ∈ M 2 ∪ M 3 . Proof. If m ∈ M + ∪ M 3 , then m s ≥ 0 and m l ≥ -1 2 m s . Hence, for every 0 ≤ C ≤ 1, we have ms 2 + Cm l ≥ (1 -C) ms 2 ≥ 0.
We therefore obtain both (a) and (b) for these m's by observing that for every t = -β(x) ∈ R we have

0 ≤ 1 1 + e t ≤ 1 and 0 ≤ 1 + e 2t (1 + e t ) 2 ≤ 1 . Suppose now that m ∈ M 2 . Since m l ≥ 0, m s + m l ≥ 0 and 1+|z| 2 |1+z| 2 ≥ 1 2
for all z ∈ C, we immediately obtain that for all t = -β(x) ∈ R we have

m s 2 + m l 1 + e 2t (1 + e t ) 2 ≥ 1 2 (m s + m l ) ≥ 0 .
This proves the lemma.

Remark 2.3. Since lim t→+∞ m s 2 + m l 1 1 + e t = m s 2 and 1 2 m s + m l 1 + e 2t (1 + e t ) 2 t=0 = 1 2 m s + m l , it is clear that the inequality (a) of Lemma 2.2 cannot extend to m ∈ M 0 \ M + ∪ M 3 and (b) cannot extend to m ∈ M 0 \ M 2 ∪ M 3 .
To use the methods from [30, §3], we will also need an extension of the real case of Opdam's estimates, [26, Proposition 6.1 [START_REF] Anker | An introduction to Dunkl theory and its analytic aspects[END_REF]]. We will do this for m ∈ M 2 ∪ M 3 . Notice that the complex variable version of Opdam's estimates, [26, Proof. Regrouping the terms corresponding to β and β/2 and setting z = x ∈ a, we can rewrite the first term on the right-hand side of the first displayed equation in [26, p. 101] as

w∈W,α∈Σ + m m α α(ξ)(1 -e -4α(x) ) (1 -e -2α(x) ) 2 |ϕ w -ϕ rαw | 2 e -2µ(x) + w∈W,β∈Σ + l m β/2 β 2 (ξ)(1 -e -2β(x) ) (1 -e -β(x) ) 2 + m β β(ξ)(1 -e -4β(x) ) |1 -e -2β(x) ) 2 |ϕ w -ϕ r β w | 2 e -2µ(x)
in which α(ξ)(1 -e -4α(x) ) is positive for all α ∈ Σ + when ξ ∈ a is chosen in the same Weyl chamber of x. (In [26, p. 101], there is an additional multiplicative constant -1 2 . Our argument takes into account this sign change.) The coefficient of

|ϕ w -ϕ r β w | 2 e -2µ(x) for β ∈ Σ + m is then clearly positive. For β ∈ Σ + l , this coefficient is equal to β(ξ)(1 -e -2β(x) ) (1 -e -β(x) ) 2 m s 2 + m l 1 + e -2β(x) (1 + e -β(x) ) 2 ,
which is positive by Lemma 2.2(b). Therefore, the same proof as in [26, Proposition 6.1] allows us to obtain the required inequalities.

The following proposition extends the positivity properties and basic estimates from [31, §3.1] and [30, §3] to the multiplicity functions in M 3 . Suppose G λ (m) is not positive and let x ∈ a be a zero of G λ (m) of minimal norm. As in [START_REF] Schapira | Etude analytique et probabiliste de laplaciens associés à des systèmes de racines : laplacien hypergéométrique de Heckman-Opdam et laplacien combinatoire sur les immeubles affines[END_REF]Lemma 3.1], one has to distinguish whether x is regular or singular. If x is regular, take ξ in the same chamber of x. Evaluation at x of the equation

T ξ (m)G λ (m) = λ(ξ)G λ (m) (24)
yields

∂ ξ G λ (m; x) = α∈Σ + m α α(ξ) 1 -e -2α(x) G λ (m; r α x) -G λ (m; x) + ρ(m) + λ (ξ)G λ (m; x)
in which G λ (m; x) vanishes. In the sum over Σ + , the coefficient of G λ (m; r α x) -G λ (m; x) is always non-negative for α ∈ Σ + m . Moreover, grouping together those corresponding to β/2 and β ∈ Σ + l , we obtain as coefficient of G λ (m;

r β x) -G λ (m; x) m β/2 β 2 (ξ) 1 -e -β(x) + m β β(ξ) 1 -e -2β(x) = β(ξ) 1 -e -β(x) m s 2 + m l 1 1 + e -β(x) , (25)
which is positive by Lemma 2.2(a).

If x is singular, let I = {α ∈ Σ + : α(x) = 0} and let ξ be in the same face of x so that α(ξ) = 0 for all α ∈ I. In this case, (24) evaluated at x gives

∂ ξ G λ (m; x) = - α∈I m α α(ξ) α, α ∂ xα G λ (m; x) + α∈Σ + m \I m m α(ξ) 1 -e -2α(x) G λ (m; r α x) -G λ (m; x) + β∈Σ + l \I m β/2 β 2 (ξ) 1 -e -β(x) + m β β(ξ) 1 -e -2β(x) G λ (m; r β x) -G λ (m; x) + ρ(m) + λ (ξ)G λ (m; x) ,
in which the first sum and G λ (m; x) vanish, and the sum over Σ + l \ I has coefficient as in [START_REF] Opdam | Root systems and hypergeometric functions[END_REF].

In both cases, one can argue as in [31, Lemma 3.1] by replacing the multiplicities k α ≥ 0 in that proof with ms 2 + m l 1 1+e -β(x) and using Lemma 2.2(a). Having that G Re λ (m) is real and positive, to prove (b) for G λ (m), one can make the same grouping and substitution of multiplicities, as done above for Finally, [START_REF] Oda | Differential-difference operators and radial part formulas for non-invariant elements[END_REF] has been proven for multiplicities m α ≥ 0 in [30, Theorem 3.3] using the original versions of (a), (b), (c) and ( 21) together with a clever application of the Phragmén-Lindelöf principle that does not involve the root multiplicities. With Opdam's estimate for m ∈ M 3 , as in Lemma 2.4, the original proof from [START_REF] Rösler | Limit transition between hypergeometric functions of type BC and type A[END_REF] extends to the case of m ∈ M 3 and yields [START_REF] Oda | Differential-difference operators and radial part formulas for non-invariant elements[END_REF]. 2.2. Asymptotics. We now investigate the asymptotic behavior of the hypergeometric functions F λ (m) for m ∈ M 1 . For this we follow [START_REF] Narayanan | Asymptotics of Harish-Chandra expansions, bounded hypergeometric functions associated with root systems, and applications[END_REF]. Before we state our results, it is useful to recall the methods adopted in [START_REF] Narayanan | Asymptotics of Harish-Chandra expansions, bounded hypergeometric functions associated with root systems, and applications[END_REF]. Let m be an arbitrary non-negative multiplicity function. One of the main results in [START_REF] Narayanan | Asymptotics of Harish-Chandra expansions, bounded hypergeometric functions associated with root systems, and applications[END_REF] is Theorem 2.11, where a series expansion away from the walls was obtained for F λ (m) for all λ ∈ a * C (even when λ is non-generic). Recall from [14, §4.2] (or from [21, §1.2] in symmetric space notation) that, for generic λ, the function F λ (m) is given on a + by w∈W c(m; wλ)Φ wλ (m), where c(m; λ) is Harish-Chandra's c-function (see (68) in the appendix) and Φ λ (m; x) admits the series expansion Φ λ (m;

∂ ξ G λ (m; x), inside the formula for ∂ ξ |Q λ | 2 (x)
x) = e (λ-ρ(m))(x) µ∈2Λ Γ µ (m; λ)e -µ(x) (x ∈ a + ).
The coefficients Γ µ (m, λ) are determined from the recursion relations

µ, µ -2λ Γ µ (m, λ) = 2 α∈Σ + m α n∈N,µ-2nα∈Λ Γ µ-2nα (m, λ) µ + ρ(m) -2nα -λ, α ,
with the initial condition that Γ 0 (m, λ) = 1. The above defines Γ µ (m; λ) as meromorphic functions on a * C . For a non-generic point λ = λ 0 a series expansion for F λ (m) was obtained in [START_REF] Narayanan | Asymptotics of Harish-Chandra expansions, bounded hypergeometric functions associated with root systems, and applications[END_REF] following the steps given below:

Step I: List the possible singularities of the c-function and the coefficients Γ µ (m; λ) at λ = λ 0 .

Step II: Identify a polynomial p so that

λ → p(λ) w∈W c(m; wλ)e (λ-ρ(m))(x) µ∈2Λ Γ µ (m; λ)e -µ(x)
is holomorphic in a neighborhood of λ 0 .

Step III: Write F λ 0 (m) = a ∂(π)(pF λ (m))| λ=λ 0 where ∂(π) is the differential operator corresponding to the highest degree homogenous term in p and a, is a non-zero constant. This gives the series expansion of F λ 0 (m).

A careful examination shows that the same proofs go through even with the assumption that the multiplicity function belongs to M 1 . Indeed, the possible singularities of the c-function and the Γ µ (m) are contained in the same set of hyperplanes as listed in [

+ s Γ (λ 0 ) α 2 + m s 4 + 1 2 Γ (λ 0 ) α 2 + m s 4 + m l 2 α∈Σ + m Γ (λ 0 ) α 2 + m m 4 + 1 2 Γ (λ 0 ) α 2 + m m 4
(26) (where the second factor in the product does not appear if the rank r is one) is nonsingular. It is clear that the expression in ( 26) is nonsingular for

λ 0 ∈ a * C with Re λ 0 ∈ (a * ) + if m ∈ M 1 .
Notice that the nonvanishing of b 0 (m; λ 0 ) identifies the main term in the expansion of F λ 0 (m; x) as b 0 (m;λ 0 ) π 0 (ρ 0 (m)) π 0 (x)e (λ 0 -ρ(m))(x) , where ρ 0 (m) is defined as in [21, (58)]. Notice also that c(m; λ 0 ), and hence b 0 (m; λ 0 ), can vanish for m ∈ M 0 \ M 1 .

It follows from the above that [21, Theorem 2.11] and, as a consequence, [21, Theorem 3.1] continue to hold true for a multiplicity function m ∈ M 1 . We state it below and refer to [START_REF] Narayanan | Asymptotics of Harish-Chandra expansions, bounded hypergeometric functions associated with root systems, and applications[END_REF] for any unexplained notation.

Theorem 2.6. Suppose m ∈ M 1 . Let λ 0 ∈ a * C with Re λ 0 ∈ (a * ) + , and let x 0 ∈ a + be fixed. Then, there are constants C 1 > 0, C 2 > 0 and b > 0 (depending on m, λ 0 and x 0 ) so that for all x ∈ x 0 + a + :

F λ 0 (m; x)e -(Re λ 0 -ρ(m))(x) π 0 (x) - b 0 (m; λ 0 ) π 0 (ρ 0 (m)) e i Im λ 0 (x) + w∈W Re λ 0 \W λ 0 b w (m; λ 0 )π w,λ 0 (x) c 0 π 0 (x) e iw Im λ 0 (x) ≤ C 1 (1 + β(x)) -1 + C 2 (1 + β(x)) |Σ + λ 0 | e -bβ(x) , ( 27
)
where β(x) is the minimum of α(x) over the simple roots α ∈ Σ + and the term C 1 (1 + β(x)) -1 on the right-hand side of ( 27) does not occur if α, λ 0 = 0 for all α ∈ Σ.

Notice that, for fixed x 0 ∈ a + , we have β(x) |x| as x → ∞ in x 0 + a + , where |x| is the Euclidean norm on a.

Likewise, one can extend to m ∈ M 1 the following corollary, which restates Theorem 2.6 in the special case where λ 0 ∈ (a * ) + .

Corollary 2.7. Suppose m ∈ M 1 . Let λ 0 ∈ (a * ) + , and let x 0 ∈ a + be fixed. Then there are constants C 1 > 0, C 2 > 0 and b > 0 (depending on m, λ 0 and x 0 ) so that for all x ∈ x 0 + a + :

F λ 0 (m; x) - b 0 (m; λ 0 ) π 0 (m; ρ 0 ) π 0 (x)e (λ 0 -ρ(m))(x) ≤ ≤ C 1 (1 + β(x)) -1 + C 2 (1 + β(x)) |Σ + λ 0 | e -bβ(x) π 0 (x)e (λ 0 -ρ(m))(x) . ( 28
)
The term C 1 (1 + β(x)) -1 on the right-hand side of (28) does not occur if α, λ 0 = 0 for all α ∈ Σ.

It might be useful to observe that for m ∈ M 0 the only obstruction to [START_REF]Lecture notes on Dunkl operators for real and complex reflection groups[END_REF] is that b 0 (m; λ 0 ) = 0. For arbitrary values of λ 0 ∈ a * C with Re λ 0 ∈ (a * ) + , by [START_REF]Harmonic analysis for certain representations of graded Hecke algebras[END_REF], this happens if and only if

m s > -2 , m s + 2m l > 0 and, if r > 1 m m > 0 .
Corollary 2.8. Let m ∈ M 0 . Then Theorem 2.6 holds for every λ 0 ∈ a * C with Re λ 0 ∈ (a * ) + such that (26) is nonsingular.

Sharp estimates.

In this subsection we assume that m ∈ M 3 . Let M 0 3 denote the interior of M 3 . Since M 0 3 ⊂ M 1 , the results of both subsections 2.1 and 2.2 are available on M 0 3 . Let λ ∈ a * . Using the nonsymmetric hypergeometric functions G λ (m), their relation to the hypergeometric function, and the system of differential-reflection equations they satisfy, Schapira proved in [START_REF] Schapira | Etude analytique et probabiliste de laplaciens associés à des systèmes de racines : laplacien hypergéométrique de Heckman-Opdam et laplacien combinatoire sur les immeubles affines[END_REF] the following local Harnack principle for the hypergeometric function

F λ (m): for all x ∈ a + ∇F λ (m; x) = - 1 |W | w∈W w -1 (ρ(m) -λ)G λ (m; wx) , (29)
the gradient being taken with respect to the space variable x ∈ a. It holds for every multiplicity function m for which both G λ (m) and F λ (m) are defined. See [START_REF] Schapira | Etude analytique et probabiliste de laplaciens associés à des systèmes de racines : laplacien hypergéométrique de Heckman-Opdam et laplacien combinatoire sur les immeubles affines[END_REF]Lemma 3.4]. Since ∂ ξ F = ∇F, ξ and since G λ (m) and F λ (m) are real and non-negative for m ∈ M 3 , one obtains as in loc. cit. that for all ξ ∈ a

∂ ξ e K ξ ξ,• |ξ| 2 F λ (m; •) ≥ 0 ,
where K ξ = max w∈W (ρ(m) -λ)(wξ). See Appendix B for a proof. This in turn yields the following subadditivity property, which is implicit in [START_REF] Schapira | Etude analytique et probabiliste de laplaciens associés à des systèmes de racines : laplacien hypergéométrique de Heckman-Opdam et laplacien combinatoire sur les immeubles affines[END_REF] for m ∈ M + and in fact holds also for m ∈ M 0 3 and, by continuity, on M 3 . Lemma 2.9. Suppose m ∈ M 3 . Let λ ∈ a * . Then for all x, x 1 ∈ a we have F λ (m; x + x 1 )e min w∈W (ρ(m)-λ)(wx 1 ) ≤ F λ (m; x) ≤ F λ (m; x + x 1 )e max w∈W (ρ(m)-λ)(wx 1 ) . [START_REF] Rösler | Limit transition between hypergeometric functions of type BC and type A[END_REF] In particular, if λ ∈ (a * ) + and x 1 ∈ a + , then

F λ (m; x + x 1 )e -(ρ(m)-λ)(x 1 ) ≤ F λ (m; x) ≤ F λ (m; x + x 1 )e (ρ(m)-λ)(x 1 ) (31)
for all x ∈ a.

Together with Corollary 2.7, the above lemma yields the following global estimates of F λ (m; x). Theorem 2.10. Let m ∈ M 0 3 ⊂ M 1 and λ 0 ∈ (a * ) + . Then for all x ∈ a + we have F λ 0 (m; x)

α∈Σ 0 λ 0 (1 + α(x)) e (λ 0 -ρ(m))(x) , (32)
where Σ 0 λ 0 = {α ∈ Σ + s ∪ Σ + m : α, λ 0 = 0}. Proof. Same as in [START_REF] Narayanan | Asymptotics of Harish-Chandra expansions, bounded hypergeometric functions associated with root systems, and applications[END_REF]Theorem 3.4].

We end this section with the following characterization of the bounded hypergeometric functions corresponding to multiplicity functions m ∈ M 1 . Proof. The same arguments as in [21, Theorem 4.2] work using the results in this section.

Applications and developments

In this section we consider two-parameter deformations of the multiplicities in M + and study a class of hypergeometric functions associated with them. As we shall see in Section 4, for specific values of m ∈ M + and of the deformation parameters ( , ), these hypergeometric functions turn out to agree with the τ -spherical functions on the homogeneous vector bundles over G/K, when τ is a small K-type and G/K has root system of type BC. The general properties proved in this section will provide for most of the BC cases in [START_REF] Oda | Spherical functions for small K-types. Geometric and harmonic analysis on homogeneous spaces[END_REF] symmetry properties, estimates, aymptotics and a characterization of the τ -spherical functions which are bounded.

3.1.

A two-parameter deformation of a multiplicity function. Let (a, Σ, m) be a triple as in subsection 1.1, with m = (m s , m m , m l ). For any two parameters , we define a deformation m( , ) of m as follows:

m α ( , ) =      m s + 2 if α ∈ Σ s m m + 2 if α ∈ Σ m m l -2 if α ∈ Σ l . (33)
We shall suppose in the following that m ∈ M 0 , , ∈ R and ≥ -m m /2. When this does not cause any confusion, we shall shorten the notations m( , 0) and m(0, ) and write m( ) and m( ), respectively. Since m s ( , ) + m l ( , ) = m s + m l , the above assumptions ensure that m( , ) ∈ M 0 . The two deformations, in and , are independent. So, m( , ) = m( )( ) = m( )( ). Since we are assuming that ≥ -m m /2, the additional parameter does not increase the range of possible multiplicities of the middle roots. Its relevance will appear in the definitions the of ( , )-hypergeometric functions in (48) and (49).

The following theorem shows that every element of M + ∪ M 3 is of the form m( ) for some m ∈ M + . For a fixed m = (m s , m m , m l ) we shall use the notation

min (m) = - m s 2 and max (m) = m s 2 + m l . (34)
We simply write min and max when this does not cause any ambiguity. Observe that satisfies -ms 2 = -

m 0 s +m 0 l 2 ≤ ≤ m 0 s +m 0 l 2 = ms 2 + m l because m 0 s ≥ 0 and m 0 s + 2m 0 l ≥ 0.
The inequalities for are strict if m 0 ∈ M 1 since in this case m 0 s > 0. Conversely, suppose that m 0 = m( ) for m ∈ M + and as in the statement. Then

-m s ≤ 2 ≤ m s + 2m l and m s + m l = m 0 s + m 0 l . Hence m 0 s = m s + 2 ≤ 2(m s + m l ) = 2(m 0 s + m 0 l ), i.e. m 0 s + 2m 0 l ≥ 0. Moreover, m 0 s = m s + 2 ≥ m s -m s = 0. Thus m 0 ∈ M + ∪ M 3 .
All the inequalities are strict if ∈ min (m), max (m) . Hence, in this case, m 0 ∈ M 1 . Lemma 3.1 is pictured in Figure 2 below. Recall that in this paper we consider the root system C r as a root system BC r with m s = 0. The diagonal segments belong to lines m s + m l = constant. If m 0 ∈ M + ∪ M 3 belongs to such a line, the corresponding element m ∈ M + in the first part of the proof, is the intersection of this line with the m s -axis. The specific segments drawn are those passing through geometric multiplicities (m s , m m , m l = 1): the segments contain the values of (m s ( ), 8 -4 5 SU(p, q) q > p 2(q -p) p -q q -p + 1

m l ( )) = (m s + 2 , 1 -2 ) with ∈ min , max ]. m s m l m s + 2 m l = 0 m l = 1 Σ = C r S U ( p , p + 1 )
u(x) = r j=1 cosh β j (x) 2 , (35)
v(x) = 1≤i<j≤r cosh β j (x) -β i (x) 2 cosh β j (x) + β i (x) 2 . ( 36
)
For ξ ∈ a and , ∈ R we define the Cherednik operator T , ,ξ by

T , ,ξ (m) = u -v -• T ξ (m( , )) • u v . (37)
A simple computation (using that u and v commute with 1 -r α for α ∈ Σ) shows that

T , ,ξ (m) = T ξ (m( , )) + u -1 ∂ ξ (u) + v -1 ∂ ξ (v) ,
where

u -1 ∂ ξ (u) = 1 2 r j=1 β j (ξ) tanh β j 2 (38)
and

v -1 ∂ ξ (v) = 1 2 1≤i<j≤r (β j (ξ) -β i (ξ)) tanh β j -β i 2 + (β j (ξ) + β i (ξ)) tanh β j + β i 2 .
(39) Let R 0 be the algebra of functions on a reg generated by 1 and (1 ± e -α ) -1 with α ∈ Σ + ; see [12, pp. 63-64]

. So R is a subalgebra of R 0 . In general, T , ,ξ ∈ D R 0 ⊗ C[W ],
where

D R 0 = R 0 ⊗ S(a C ). However, if = 0, then T ,ξ = T ,0,ξ ∈ D R ⊗ C[W ]
. By construction, {T , ,ξ (m) : ξ ∈ a} is a commutative family of differential-reflection operators. So, the map ξ → T , ,ξ (m) extends uniquely to an algebra homomorphism p → T , ,p (m) from

S(a C ) to D R 0 ⊗ C[W ] such that T , ,p (m) = u -v -• T p (m( , )) • v u . In particular, one can define the ( , )-Heckman-Opdam Laplacian T , ,p L (m) = r j=1 T , ,ξ j (m) 2 = u -v -• T p L (m( )) • u v , ( 40
)
where {ξ j } r j=1 is any orthonormal basis of a and p L is defined by p L (λ) := λ, λ for λ ∈ a * C . Next, we compute the differential part of the ( , )-Heckman-Opdam Laplacian in a closed form, allowing us to deduce some symmetry properties that will be useful in later sections. For an arbitrary root system Σ on a * and a multiplicity function m, let us set

f Σ (m) = α∈Σ + m α (2 -m α -2m 2α ) α, α (e α -e -α ) 2 . ( 41
)
We start by recalling the following lemma.

Lemma 3.2. Let Σ be an arbitrary root system on a * and m a multiplicity function. If δ Σ (m)

1 2 = α∈Σ + (e α -e -α ) mα 2 , then δ Σ (m) 1 2 • (L Σ (m) + ρ Σ (m), ρ Σ (m) ) • δ Σ (m) -1 2 = L a + f Σ (m)
, where L Σ (m) is the Heckman-Opdam Laplacian associated to (Σ, m) and ρ Σ (m) is the half sum of positive roots.

Proof. See [START_REF] Heckman | Harmonic analysis and special functions on symmetric spaces[END_REF]Theorem 2.1.1] From now onwards, Σ will denote a root system of type BC, as in Section 1. Recall the operator β from [START_REF] Deitmar | Invariant operators on higher K-types[END_REF]. The same definition extends β to an operator from

D R 0 ⊗ C[W ] to D R 0 . If p ∈ S(a C ), then β T , ,p (m) = u -v -• β T p (m( , )) • u v ∈ D R 0 . Set D , ,p (m) = β T , ,p (m) for p ∈ S(a C ) W . Furthermore, set D , (m) = {D , ,p (m) : p ∈ S(a C ) W } . ( 42
)
Lemma 3.3. Consider the root system Σ = 2Σ and the multiplicity function m = (m s + m l , m m + 2 , 0). Then we have the identity

u -ms 2 v -mm 2 • D , ,p L (m) • u ms 2 v mm 2 = L Σ ( m) + ρ Σ ( m), ρ Σ ( m) + f Σ (m( , )) -f Σ ( m).
Proof. We start with δ Σ (m( , ))

1 2 • L Σ (m( , )) + ρ(Σ, m( , )), ρ(Σ, m( , )) • δ Σ (m( , )) -1 2 (43)
The above, by Lemma 3.2 equals

L a + α>0 m α ( , ) 2 -m α ( , ) -2m 2α ( , ) α, α (e α -e -α ) 2 = L a + f Σ (m( , )).
Replacing L a with δ Σ ( m)

1 2 • L Σ ( m) + ρ Σ ( m), ρ Σ ( m) • δ Σ ( m) -1 2 -f Σ ( m) we get the result as δ Σ (m( , )) 1 2 δ Σ ( m) -1 2 = 2 M u -( ms 2 + ) v -( mm 2 + )
, where M is a constant depending on m and r. (44)

Proof. A simple computation shows that

f Σ (m( , )) = f Σ (m( )) + p 2 4 ( + 1 -m l ) r j=1 1 cosh 2 β j 2 .
The stated equality follows then from Lemma 3.3.

Proposition 3.5. In the above notation, D , (m

) = u -v -• D(m( , )) • u v is a commu- tative subalgebra of D W R 0 of rank r. It is the centralizer of D , ,p L in D W R 0 . As a consequence, D , (m) = D -+m l -1, (m). In particular, D , (m) = D -, (m) if m l = 1.
If = 0, then R 0 can be replaced by R. Furthermore, if m = (m s , m m , m l = 1) is geometric and ∈ Z, = 0, then D ,0 (m) = δ (D(G/K; τ )).

Proof. Because of ( 38), [START_REF] Warner | Harmonic analysis on semi-simple Lie groups, II. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen[END_REF] and since u and v are W -invariant, one can easily check that the conjugation by u v maps D W R 0 into itself and D W R into itself if = 0. Now the proof follows from the more general result in [START_REF] Oda | Spherical functions for small K-types. Geometric and harmonic analysis on homogeneous spaces[END_REF]Lemma 5.2], showing that the centralizers of

D p L (m( , )) in D W
R 0 and D W R agree, together with (44). Here, δ (D(G/K, τ ) denotes the τ -radial components of the commutative algebra D(G/K; τ ) (see subsection 4.1).

( , )-Hypergeometric functions. Let λ ∈ a *

C be fixed. The ( , )-Heckman-Opdam hypergeometric function with spectral parameter λ is the unique W -invariant analytic solution F , ,λ (m) of the system of differential equations

D , ,p (m)f = p(λ)f (p ∈ S(a C ) W ), (45)
which satisfies f (0) = 1.

The non-symmetric ( , )-hypergeometric function with spectral parameter λ is the unique analytic solution G , ,λ (m) of the system of differential equations

T , ,ξ (m)g = λ(ξ)g (ξ ∈ a), (46)
which satisfies g(0) = 1.

The symmetric and non-symmetric ( , )-Heckman-Opdam hypergeometric functions are therefore (suitably normalized) joint eigenfunctions of the commutative algebras D , (m) and {T , ,p : p ∈ S(a C )}, respectively. Notice that the equality for D , (m) in Proposition 3.5 yields

F , ,λ (m) = F -+m l -1, ,λ (m) (λ ∈ a * C ) . ( 47
)
On the other hand, an analogous symmetry relation is generally not true for G , ,λ (m) as can be seen from rank one examples. We omit the details. By definition,

F , ,λ (m; x) is W -invariant in x ∈ a and in λ ∈ a * C . Furthermore, since u v • D , ,p (m) • u -v -= D p (m( , )) and u v • T , ,ξ (m) • u -v -= T ξ (m( , )), one obtains for all λ ∈ a * C : F , ,λ (m) = u -v -F λ (m( , )) , (48)
G , ,λ (m) = u -v -G λ (m( , )) . ( 49
)
As in the case = = 0,

F , ,λ (m; x) = 1 |W | w∈W G , ,λ (m; w -1 x) (x ∈ a) . ( 50
)
As a consequnce of (48) and of the corresponding properties of the Heckman-Opdam hypergeometric functions (see e. 48) and (49), we can obtain from the results of section 2 some estimates and asymptotic properties for the ( , )-hypergeometric functions. The factors u -and v -require some additional care for the asymptotics.

By Lemma 3.1, we have m( , ) ∈ M + ∪M 3 for m ∈ M + and ∈ min (m), max (m) , ≥ -mm 2 . This implies that Proposition 2.5, Theorem 2.6 and Corollary 2.7 hold true for F λ (m( , )) for all m ∈ M + and ∈ min (m), max (m) . In turn, (48) yields analogous statements for the ( , )-hypergeometric functions. Recall that min = -ms 2 and max = ms 2 + m l . (We are omitting from the notation the dependence on m of min and max ). So min ≤ 0 < max .

Recalling also that F , ,λ = F -+l-1, ,λ , we see that we can extend the inequalities for

F , ,λ to ∈] -ms 2 -1, ms 2 + m l [ (or to ∈ [-ms 2 -1, ms 2 + m l ]
where the extension by continuity in the multiplicity parameter is possible; see Theorem A.1 in the appendix). For a fixed multiplicity m = (m s , m m , m l ), the symmetries of F , ,λ in the -parameter can be pictured as symmetries of the deformations m( ) around the point m( m l -1

2 ), as in Figure 3.

Similarly, the inequalities of Lemma 2.4 extend to any ∈ R because either the condition ≤ max or -+m l -1 ≤ max is always satisfied. In the first case, m( , ) ∈ M 2 ∪M 3 ; in the other, m(-

+ m l + 1, ) ∈ M 2 ∪ M 3 .
We leave to the reader the simple task of modifying the statements of Theorem 2.6 and Corollary 2.7 in this case, and we collect the estimates obtained for G , ,λ (m) and F , ,λ (m) in the corollary below.

Corollary 3.6. Let m = (m s , m m , m l ) ∈ M + be a nonnegative multiplicity function on a root system of type BC r , and let , ∈ R, ≥ -mm 2 . Then the following properties hold. (a) For all ≤ max (m), λ ∈ a * C and x ∈ a,

|G , ,λ (m; x)| ≤ |W | u -(x)v -(x)e maxw(wλ)(x) (51)
and for all ∈ R, λ ∈ a * C and x ∈ a,

|F , ,λ (m; x)| = |F -+m l -1,λ (m; x)| ≤ |W | u -(x)v -(x)e maxw(wλ)(x) . (52)
m s m l m s + 2 m l = 0 m l = 1 m s + m l = c o n s t M + M 3 m(l max ) m(l) m( m l -1 2 ) m = m(0) m(-l + m l -1) m(l min ) m(l min -1) -2 l min = -ms 2 l max = ms 2 + m l Figure 3. Symmetries of m( ) around m( m l -1 2 ) if m l ≥ 1 (b) Suppose ∈ [ min -1, max ]. For all λ ∈ a * the function F , ,λ (m) is real and strictly positive. For all λ ∈ a * C |F , ,λ (m)| ≤ F , ,Re λ (m) . (53)
Moreover, for all λ ∈ a * and all x ∈ a F , ,λ (m; x) ≤ F , ,0 (m; x)e maxw(wλ)(x) .

More generally, for all λ ∈ a * , µ ∈ (a * ) + and all x ∈ a F , ,λ+µ (m; x) ≤ F , ,µ (m; x)e maxw(wλ)(x) .

The same properties hold for G , ,λ (m) provided ∈ [ min , max ].

(c) Suppose ∈ [ min -1, max ], ≥ 0, λ ∈ (a * ) + and x 1 ∈ a + . Further assume that m l ≥ 1. Then for all x ∈ a we have

F , ,λ (m; x + x 1 )e -(λ+ρ(m(2 )))(x 1 ) ≤ F , ,λ (m; x) ≤ F , ,λ (m; x + x 1 )e (λ+ρ(m(2 )))(x 1 ) . ( 56
) (d) Suppose ∈] min , max [, > 0 if m m = 0, ≥ 0 if m m > 0 and λ ∈ (a * ) + .
Then for all x ∈ a + we have

F , ,λ (m; x) α∈Σ 0 λ (1 + α(x)) e (λ-ρ(m(2 )))(x) , (57)
where

Σ 0 λ = {α ∈ Σ + s ∪ Σ + m : α, λ = 0}. The asymptotics (57) extend to ∈ ] min -1, max [ if m s + m l ≥ 1.
Proof. We prove (c) and (d). For (c), first assume that ∈ [ min , max ]. since F , ,λ = F -+m l -1, ,λ we may assume that ≥ m l -1 2 ≥ 0. To keep track of the exponents appearing in our formulas, it will be convenient to introduce the following notation,

1≤i<j≤r (β j ± β i ) = 1≤i<j≤r (β j + β i ) + 1≤i<j≤r (β j -β i ) = r j=1 2(j -1)β j . (58)
For a ≥ 0 and b ≥ 0, cosh a cosh b ≤ cosh(a + b) ≤ cosh a e b .

Hence, if ≥ 0, ≥ 0 and x, x 1 are as above,

u (x)u (x 1 ) ≤ u (x + x 1 ) ≤ u (x)e 2 r j=1 β j (x 1 ) , v (x)v (x 1 ) ≤ v (x + x 1 ) ≤ v (x)e 2 1≤i<j≤r (β j ±β i )(x 1 ) . (59)
Notice that

ρ(m(2 )) = ρ(m( , )) + 2 r j=1 β j + 2 1≤i<j≤r (β j ± β i ) . (60)
By [START_REF] Schapira | Etude analytique et probabiliste de laplaciens associés à des systèmes de racines : laplacien hypergéométrique de Heckman-Opdam et laplacien combinatoire sur les immeubles affines[END_REF], which applies since m( ,

) ∈ M 3 , F λ (m( , ); x + x 1 )e -(λ+ρ(m( , )))(x 1 ) ≤ F λ (m( , ); x) ≤ F λ (m( , ); x + x 1 )e (λ+ρ(m( , )))(x 1 ) .
Multiplying each term of the inequality by u(x) -v(x) -and using the definition of F , ,λ , we obtain

u (x + x 1 )v (x + x 1 ) u (x)v (x) F , ,λ (m; x + x 1 )e -(λ+ρ(m( , )))(x 1 ) ≤ F , ,λ (m; x) ≤ u (x + x 1 )v (x + x 1 ) u (x)v (x) F , ,λ (m; x + x 1 )e (λ+ρ(m( , )))(x 1 ) .
The right hand side inequality in (c) follows from the second inequalities in (59) and (60). Again, using (60) (and (58)), the expression on the left hand side inequality of (d) becomes

u (x + x 1 )v (x + x 1 ) u (x 1 )v (x 1 ) e 2 r j=1 β j (x 1 ) e 2 1≤i<j≤r (β j ±β i )(x 1 ) F , ,λ (m; x + x 1 )e -(λ+ρ(m(2 )))(x 1 ) .
This implies (c) as

u (x + x 1 )v (x + x 1 ) u (x 1 )v (x 1 ) e 2 r j=1 β j (x 1 ) e 2 1≤i<j≤r (β j ±β i )(x 1 ) ≥ 1.
For ∈ [ min -1, min [, notice that min ≤ -+ m l -1 ≤ max as m l ≥ 1 and use the equality F , ,λ = F -+m l -1, ,λ . Next we prove (d). For a ≥ 0 we have a e a . Hence

u -(x) e -2 r j=1 β j (x) v -(x) e - 2 1≤i<j≤r (β j ±β i )(x) . (61)
Since m( , ) ∈ M 0 3 , we have by Theorem 2.10,

F λ (m( , ); x) α∈Σ 0 λ (1 + α(x)) e (λ-ρ(m( , )))(x) .
Then (57) follows by multiplying both sides of this asymptotics by u -(x)v -(x), together with (61) and (60).

If ∈] min -1, min [, then -+ m l -1 ∈]min + m l -1, max [. The lower bound satisfies min + m l -1 ≥ min if and only if m s + m l ≥ 1. In this case, the above asymptotics hold for -l + m l -1 as well. They lead to (57) using F , ,λ (m) = F -+m l -1, ,λ (m).

3.5. Bounded ( , )-hypergeometric functions. In this section we address the problem of characterizing the ( , )-hypergeometric functions which are bounded. Recall that we are considering a multiplicity function m = (m s , m m , m l ) ∈ M + and its deformation m( , ) as in [START_REF] Shimeno | Eigenspaces of invariant differential operators on a homogeneous line bundle on a Riemannian symmetric space[END_REF]. To apply the estimates from subsection 3.4, we will assume that ∈ [ min , max ]. The asymptotics from Theorem 2.6 hold under the stronger assumption that ∈] min , max [.

Recall also that

ρ(m( , )) = 1 2 α∈Σ + m α ( , )α = ρ(m) - 2 r j=1 β j + 2 1≤i<j≤r (β j ± β i )
where 1≤i<j≤r (β j ± β i ) is given by (58) and so,

ρ(m(2 )) = ρ(m( , )) + 2 r j=1 β j + 2 1≤i<j≤r (β j ± β i ).
Notice that, if we suppose ∈ [0, max ], ≥ -ms 2 then m( , ) ∈ M + ∪ M 3 by Lemma 3.1. Hence ρ(m( , )) ∈ (a * ) + . Notice also that ρ(m(2 )) ≥ ρ(m) if ≥ 0. Proof. First note that m( , ) ∈ M 1 . Since F , ,λ = F -+m l -1, ,λ we may assume that 2))) + ia * and consider the holomorphic function λ → F , ,λ (m; x) (for a fixed x). Since

0 ≤ m l -1 2 ≤ < max . We show that F , ,λ is bounded if λ ∈ C(ρ(m(2 ))) + ia * . Let λ ∈ C(ρ(m(
|F , ,λ (m; x)| ≤ F , ,Re λ (m; x) = u -(x)v -(x)F Re λ (m( , ); x),
we can argue as in the proof of [START_REF] Narayanan | Asymptotics of Harish-Chandra expansions, bounded hypergeometric functions associated with root systems, and applications[END_REF]Theorem 4.2] to obtain that the maximum of

|F , ,λ (m; x)| is attained at {wρ(m(2 )) : w ∈ W }. That is |F ,λ (m; x)| ≤ u -(x)v -(x)F ρ(m(2 )) (m( , ); x).
As noticed before, ρ(m( , )) ∈ (a * ) + . Applying [START_REF] Oda | Differential-difference operators and radial part formulas for non-invariant elements[END_REF], we obtain

β j (x) + 2 1≤i<j≤r (β j ± β i )(x) = 2 r j=1 β j (x 0 ) + 2 1≤i<j≤r (β j ± β i )(x 0 )
Since u and v are W -invariant, from the above we get

|F , ,λ (m; x)| ≤ u -(x 0 )v -(x 0 )e 2 r j=1 β j (x 0 ) e 2 1≤i<j≤r (β j ±β i )(x 0 ) ≤ 2 M
where M is a constant depending on , and r.

To prove the other way, we proceed as in [START_REF] Narayanan | Asymptotics of Harish-Chandra expansions, bounded hypergeometric functions associated with root systems, and applications[END_REF] (see pages 251-252). Let λ 0 be such that Re λ 0 ∈ (a * ) + \ C(ρ(m(2 ))). Let x 1 ∈ a + be such that (Re λ 0 -ρ(m(2 )))(x 1 ) > 0. If F , ,λ 0 (m) is bounded we have lim t→∞ F , ,λ 0 (m; tx 1 )e -t(Re λ 0 -ρ(m(2 )))(x 1 ) t -d = 0. Since, F , λ 0 (m; x) = u -(x)v -(x)F λ 0 (m( , ); x) and cosh a behaves like e a for large positive a, we have

F λ 0 (m( , ); tx 1 )e -t(Re λ 0 -ρ(m( , )))(x 1 ) t -d → 0 t → ∞.
So, by Theorem 2.6 (which is applicable as m( , ) ∈ M 1 ), the proof can be completed as in Theorem 4.2 of [START_REF] Narayanan | Asymptotics of Harish-Chandra expansions, bounded hypergeometric functions associated with root systems, and applications[END_REF].

Remark 3.8. The proof of Theorem 3.7 shows that the ( , )-hypergeometric function

F , ,λ (m) is bounded for λ ∈ C(ρ(m(2 ))) + ia * , provided ∈ [ min -1, max ] and ≥ 0.
For general root systems of type BC r , we cannot prove that when = max the function F , ,λ (m) is not bounded for λ / ∈ C(ρ(m(2 ))) + ia * . This is because the function b 0 appearing in Theorem 2.6 might vanish. However the above theorem continues to hold for = max provided r = 1. Indeed, for = max we have m s ( ) + 2m l ( ) = 0 and, by [START_REF]Harmonic analysis for certain representations of graded Hecke algebras[END_REF], the function b 0 (m( max ); λ 0 ) vanishes for λ 0 ∈ a * C with Re λ 0 ∈ (a * ) + , if and only if there is j ∈ {1, . . . , r} such that (λ 0 ) β j = 0. Outside these λ 0 , the asymptotics from Theorem 2.6 still hold. In the rank-one case, there is only one λ 0 for which the asymptotics do not hold, namely λ 0 = 0 which is not outside C(ρ(m)) + ia * . So the above result holds under the assumption ∈ [ min , max ], in the rank one case. Remark 3.9. It is natural to expect that F , ,λ is bounded by one if λ ∈ C(ρ(m(2 )))+ia * , though we are not able to prove this. However, this holds true for the geometric case as can be seen from Corollary 4.2 and from the integral formula (66) together with the case-by-case computation that ρ G/K = ρ(m(2 )); see subsection 4.2.

4. Some geometric cases 4.1. Spherical functions on line bundles over Hermitian symmetric spaces. In this subsection we shall assume that = 0 and consequently suppress the index from the notation.

Let m = (m s , m m , m l = 1) be a geometric multiplicity function corresponding to a non-compact Hermitian symmetric space G/K, as in Example 1.1. Let τ be a fixed one dimensional unitary representations of K. So ∈ Z (or ∈ R by passing to universal covering spaces). Let E denote the homogeneous line bundle on G/K associated with τ . The smooth τ -spherical sections of E can be identified with the space C ∞ (G//K; τ ) of functions on G satisfying f (k

1 gk 2 ) = τ -(k 1 k 2 )f (g) for all g ∈ G and k 1 , k 2 ∈ K.
Recall our convention of identifying the Cartan subspace a with its diffeomorphic image A = exp(a) ⊂ G. So every element ϕ ∈ C ∞ (G//K; τ ) is uniquely determined by its Weylgroup invariant restriction ϕ| a to A = a. Recall also the notation D(G/K; τ ) for the (commutative) algebra of the G-invariant differential operators on E . Then the τ -spherical functions on G/K are the (suitably normalized) joint eigenfunctions of D(G/K; τ ) belonging to C ∞ (G//K; τ ). For a fixed , they are parametrized by λ ∈ a * C (modulo the Weyl group). We denote by ϕ ,λ the τ -spherical function of spectral parameter λ.

Many authors have studied the τ -spherical functions by relating them to hypergeometric functions on root systems. See e.g. [14, §5.2-5.5], [33, §6], [34, §3 and 5], [START_REF] Ho | Paley-Wiener theorem for line bundles over compact symmetric spaces and new estimates for the Heckman-Opdam hypergeometric functions[END_REF]. Indeed, D (m) = δ (D (G/K)), where δ denotes the τ -radial component in the sense of Harish-Chandra [START_REF] Harish-Chandra | Differential equations and semisimple Lie groups[END_REF]; see [START_REF] Oda | Spherical functions for small K-types. Geometric and harmonic analysis on homogeneous spaces[END_REF]Section 3.6]. More specifically,

D ,p L = δ (Ω) -τ -(Ω m ) , (62)
where Ω and -Ω m are the Casimir operator on g C and m C , respectively. See [START_REF]The Plancherel formula for spherical functions with a one-dimensional K-type on a simply connected simple Lie group of Hermitian type[END_REF]Lemma 2.4] and [23, Section 3.5]. Hence

ϕ ,λ | a = F ,λ (m) = u -F λ (m( )) . (63)
Since m l = 1, the classical symmetry ϕ ,λ = ϕ -,λ is a special case of the symmetry F ,λ = F -+m l -1,λ from (47).

In the usual parametrization, the trivial representation of K corresponds to = 0. In this case, C ∞ (G//K; τ 0 ) is the space of the smooth K-invariant functions on G/K, D 0 (G/K) coincides with the algebra D(G/K) of G-invariant differential operators on G/K, and the τ 0 -spherical functions ϕ 0,λ are precisely Harish-Chandra's spherical functions ϕ λ on G/K.

The following proposition summarizes the basic properties of the τ -spherical functions. As in the K-biinvariant case, they can be explicitly given by an integral formula, which extends to arbitrary real 's the classical integral formula by Harish-Chandra's for the spherical function ϕ λ = ϕ 0,λ on G/K. e (λ-ρ)(H(gk)) τ (kκ(gk) -1 ) dk, g ∈ G.

(64)

Then the set of functions ϕ ,λ , λ ∈ a * C exhausts the class of (elementary) τ -spherical functions on G. Two such functions ϕ ,λ and ϕ ,µ are equal if and only if µ = wλ for some w ∈ W. Moreover, for a fixed g

∈ G, ϕ ,λ (g) is holomorphic in (λ, ) ∈ a * C × C. Furthermore, ϕ ,λ = ϕ -,λ . Proof. See [33, Proposition 6.1].
The integral formula together with the properties of Harish-Chandra's spherical functions ϕ λ ( = 0 case) automatically implies several of the properties of the τ -spherical functions we are studying in this paper. Nevertheless, others, such as their positivity or the full characterization of the bounded spherical functions, do not seem to follow from (64). We collect all properties in the following corollary. Recall that max = ms 2 +1 because m l = 1. Corollary 4.2. Let ∈ R, then we have:

(1) ϕ ,λ is real valued for λ ∈ a * .

( Proof. The first property follows from the equality ϕ ,λ = ϕ -,λ because τ + τ -is real valued. For part [START_REF] Branson | The Paley-Wiener theorem and the local Huygens' principle for compact symmetric spaces: the even multiplicity case[END_REF], by the symmetry in , we can suppose that ≥ 0. So m( ) ∈ M 1 .

The positivity of ϕ ,λ | a follows then from Propositions 2.1,(a) and 2.5. Part ( 3) is a consequence of the integral formula, and the fact that τ is a unitary character. Part (4) follows from (3) and the theorem by Helgason and Johnson characterizing the parameters λ for which the Harish-Chandra's spherical functions ϕ λ are bounded; see [START_REF] Helgason | The bounded spherical functions on symmetric spaces[END_REF]. Finally, for (5), we can suppose that ≥ 0 so that m( ) ∈ M 1 . Then ( 5) is an immediate consequence of (3) together with the second part of Theorem 3.7.

4.2. τ -spherical functions for other small K-types τ . We start by briefly recalling the main result from [START_REF] Oda | Spherical functions for small K-types. Geometric and harmonic analysis on homogeneous spaces[END_REF], which expresses spherical functions associated to small K-types as hypergeometric functions multiplied with an explicit function involving hyperbolic sines and cosines. We refer to [START_REF] Oda | Spherical functions for small K-types. Geometric and harmonic analysis on homogeneous spaces[END_REF] for the proofs of the properties mentioned here and for further information.

Let G be a non-compact connected real semisimple Lie group with finite center. Let G = KAN be an Iwasawa decomposition and let M be the centralizer of A in K. Recall that a K-type (τ, V τ ) (i.e. an irreducible representation of K) is called small if it is irreducible as an M -module. We denote by ϕ τ λ the τ -spherical function on G with spectral parameter λ ∈ a * . Moreover, with a slight abuse of notation, we denote by ϕ τ λ | a the Winvariant scalar function such that the restriction of ϕ τ λ to A ≡ a is ϕ τ λ | a id, where id denotes the identity operator on V τ . Theorem 4.3. (See [START_REF] Oda | Spherical functions for small K-types. Geometric and harmonic analysis on homogeneous spaces[END_REF]Theorem 1.6]) Suppose (τ, V ) is a small K-type of a non-compact real simple Lie group G with finite center. If G is a simply-connected split Lie group G 2 of type G 2 , we further suppose that τ is not the small type π 2 specified in Theorem 2.2 of [START_REF] Oda | Spherical functions for small K-types. Geometric and harmonic analysis on homogeneous spaces[END_REF]. Then there exists a root system Σ(τ ) in a * and a multiplicity function m(τ

) on Σ(τ ) such that ϕ τ λ | a = δ -1 2 G/K δ Σ(τ ) (m(τ )) 1 2 F λ (Σ(τ ), m(τ )) for all λ ∈ a *
C . In the above, F λ (Σ(τ ), m(τ )) stands for the Heckman-Opdam hypergeometric function associated with the triple (a, Σ(τ ), m(τ )). Here, we recall that we use the symmetric space notation and hence our Σ(τ ) and m(τ ) are related to Σ τ and k τ of [START_REF] Oda | Spherical functions for small K-types. Geometric and harmonic analysis on homogeneous spaces[END_REF] by Σ τ = 2Σ(τ ) and k τ 2α = mα(τ ) 2 . For any root system (Σ, m), the function δ Σ (m) is defined as:

δ Σ (m) = α∈Σ + sinh α |α| mα (65)
and δ G/K corresponds to the root system for G/K.

The classification of the small K-types τ and the concrete choices (which are one or two) of pairs (Σ(τ ), m(τ )) occurring in Theorem 4.3 can be found in [START_REF] Oda | Spherical functions for small K-types. Geometric and harmonic analysis on homogeneous spaces[END_REF]Section2]. Clearly, the trivial K-type is always small. If G/K is Hermitian, a K-type τ is small if and only if it is a one-dimensional unitary character τ , as in subsection 4.1. All other small K-types have dimension bigger than one.

As in the scalar and the Hermitian cases, the τ -spherical functions corresponding to a small K-type can be represented by the integral formula

ϕ τ λ (g) = K e (λ-ρ G/K )(H(gk)) τ (kκ(gk)) -1 dk , (66)
see [23, (3.7)], where ρ G/K is computed from the root system for G/K. Formula (66) is a special instance of [39, 9.1.5, p. 300] and [3, (3.9)].

Based on the case-by-case analysis of [23, Section 2], we now show that every τ -spherical function for a symmetric space G/K with restricted root system of type BC and for which dim τ ≥ 2 is a ( , )-hypergeometric function for suitable choices of a multiplicity function m = (m s , m m , m l ) ∈ M + , defined on a root subsystem of Σ(τ ), and of values and . It follows, in particular, that m is in general different from both the original root multiplicity function of G/K and m(τ ).

This identification, together with (66), allows us to prove an analogue of Corollary 4.2, in which one has to replace ϕ ,λ with ϕ τ λ | a and the condition | | ≤ max by the conditions min (m) -1 ≤ max (m) and ≥ -mm 2 . Notice that the condition on is equivalent to the more symmetric condition -(m l -1)/2 ≤ (m s + m l + 1)/2. The precise statement depends on the specific values of min (m) = -m s /2 and max (m) = m s /2 + m l determined below. We leave to the reader the task of writing down the precise statement in each case. In fact, the symmetry (47) shows the multiplicity functions m( , ) and m(-+ m l -1, ) are different if = m l -1 2 but yield to the same τ -spherical function. As remarked in [START_REF] Oda | Spherical functions for small K-types. Geometric and harmonic analysis on homogeneous spaces[END_REF]Section 2], the classification of K-types can be given at the level of Lie algebras of non-compact type since a small K-type of G always lifts to that of a finite cover of G. We keep the convention that a root multiplicity 0 is equivalent to the fact that the corresponding root does not belong to the fixed root system. Since m(τ n ) -= m(-+ m l -1), where -+ m l -1 = -n + 1, the fact that the two multiplicity functions lead to the same τ n -spherical function is a special instance of the symmetry (47). 4.2.2. The case of g = so(2r, 1), r ≥ 2. Let G = Spin(2r, 1) be the double cover of SO(2r, 1) and K = Spin(2r). Then G is simply connected. The non-trivial small Ktypes are the irreducible representations τ ± s with highest weight (s/2, . . . , s/2, ±s/2) in standard notation, where s ∈ N. The case s = 1 corresponds to the positive and negative spin representations and the corresponding τ -spherical analysis was studied in [START_REF] Camporesi | Harmonic analysis for spinors on real hyperbolic spaces[END_REF].

The root system of G/K, say {±α}, is of type A 1 and m α = 2r -1. According to [23, Section 2.4], Σ(τ ± s ) = {±α/2, ±α}, m(τ ± s ) = (-2s, 2r + 2s -1) and

ϕ τ ± s λ | a = cosh s α 2 F λ (Σ(τ ± s ), m(τ ± s )) .
Then m(τ ± s ) = m( s ) for m = (0, 2r -1) ∈ M + on Σ(τ ± s ) and s = -s. Notice that s,min = 0 and s,max = 2r -1. The symmetry (47) gives (with = 0) F -s,λ (m) = F s+2r-2,λ (m). For s = 1, the symmetric of s = -1 is 2r -1 and m(2r -1) ∈ M 3 (but / ∈ M 1 ). Observe also that ρ(m) = ρ G/K = (r -1/2)α. 4.2.3. The case of g = so(p, q), p > q ≥ 3. Let G be the double cover of Spin(p, q) (which is simply connected) and K = Spin(p) × Spin(q). The root systems of G/K is {±β j ; 1 ≤ j ≤ q} ∪ {±β j ± β i ; 1 ≤ i < j ≤ q}, with root multiplicities m β j = p -q and m β j ±β i = 1. According to [23, Section 2.5], the small K-types are of two forms:

(i) τ = 1 ⊗ σ, where σ is the spin representation of Spin(q) if q is odd, and either of the two spin representations of Spin(q) if q is even. (ii) τ = σ ⊗ 1, where σ is either of the two spin representations of Spin(p) if p is even and q is odd.

In case (i), Σ(τ ) = {±β j ; 1 ≤ j ≤ q} ∪ { ±β j ±βi 2

; 1 ≤ i < j ≤ q} with m(τ ) = (0, 1, p -q). In case (ii), Σ(τ ) = {± β j 2 ; 1 ≤ j ≤ q} ∪ { ±β j ±βi 2

; 1 ≤ i < j ≤ q} ∪ {±β j ; 1 ≤ j ≤ q} with m(τ ) = (2(p -q), 1, -(p -q)). Furthermore, ϕ τ λ | a equals

1≤i≤j≤q cosh β j + β i 2 cosh β j -β i 2 -1/2
F λ (Σ(τ ), m(τ )), in case (i), and q j=1 cosh β j 2

-(p-q) 1≤i≤j≤q cosh β j + β i 2 cosh β j -β i 2 -1/2
F λ (Σ(τ ), m(τ )), in case (ii) .

The two cases come from the multiplicity on Σ(τ ) equal to m = (0, 0, p -q) ∈ M + and ( , ) = (0, 1/2) in case (i), so that m( , ) = (0, 1, p -q) ; ( , ) = (p -q, 1/2) in case (ii), so that m( , ) = (2(p -q), 1, -(p -q)) .

Since x → F λ (x) and ξ → K ξ are W -invariant, the last inequality extends to all x ∈ a. Hence, for every x, ξ ∈ a and every λ ∈ a * we have

∂ ξ e K ξ ξ,• |ξ| 2 F λ (m; •) = e K ξ ξ,• |ξ| 2 ∂ ξ F λ + K ξ F λ ≥ 0 .
The previous argument is essentially the one leading to [START_REF] Godement | A theory of spherical functions[END_REF] in [START_REF]Contributions to the hypergeometric function theory of Heckman and Opdam: sharp estimates, Schwartz space, heat kernel[END_REF]Lemma 3.3].

We pass from (72) to the inequality (30) in Lemma 2.9 by using the following arguments, which are repeatedly used in [START_REF]Contributions to the hypergeometric function theory of Heckman and Opdam: sharp estimates, Schwartz space, heat kernel[END_REF]. This is why we say that Lemma 2.9 is implicit in [START_REF]Contributions to the hypergeometric function theory of Heckman and Opdam: sharp estimates, Schwartz space, heat kernel[END_REF].

Let λ ∈ a * and x ∈ a be fixed. Set

f (t) = e K ξ ξ,x+tξ |ξ| 2
F λ (m; x + tξ) .

By Lemma 3.2 in [START_REF]Contributions to the hypergeometric function theory of Heckman and Opdam: sharp estimates, Schwartz space, heat kernel[END_REF] and since

f d (t) = d dt t 0 f (t) = ∂ ξ e K ξ ξ,• |ξ| 2 F λ (m; •) (x) ≥ 0 ,
as required.

Figure 1 . 2 . 1 .Proposition 2 . 1 .

 12121 Figure 1. Sets of BC multiplicities

  Proposition 6.1 (2)], has been recently extended by Ho and Ólafsson [18, Appendix A] to m ∈ M 2 and to a domain in a C which is much larger than the original one considered by Opdam. For m ∈ M 2 the inequality (b) of Lemma 2.2 was noticed in the proof of [18, Proposition 5.A, p. 25]. Lemma 2.4. Suppose that m ∈ M 2 ∪ M 3 . Then for all λ ∈ a * C and x ∈ a |G λ (m; x)| ≤ |W |e maxw Re(wλ)(x) , |F λ (m; x)| ≤ |W |e maxw Re(wλ)(x) . (23)

Proposition 2 . 5 .

 25 Proposition 2.1 holds for m ∈ M 3 . Proof. The proofs of (a), (b) and (21) follow the same steps as the proofs of [31, Lemma 3.1 and Proposition 3.1]. So we shall just indicate what has to be modified in the proofs when considering m ∈ M 3 instead of m ∈ M + .

  appearing in the proof of [31, Proposition 3.1 (a)]. A third application of the same grouping in the formula of ∂ ξ R λ (x) in the proof of [31, Proposition 3.1 (b)] yields (21).

  21, Lemma 2.3]. Hence the same polynomials and differential operators can be used in Step II and III above. The crucial detail to be checked is the computation of the constant b 0 (m; λ 0 ) appearing in [21, Lemma 2.6]. The explicit expression in terms of Harish-Chandra's cfunction shows that, for root systems of type BC r \ C r , the function b 0 (m; λ 0 ) is non-zero if and only if

  α∈Σ

Theorem 2 . 11 .

 211 Let m ∈ M 1 . Then F λ (m) is bounded if and only if λ ∈ C(ρ(m)) + ia * ,where C(ρ(m)) is the convex hull of the set {wρ(m) : w ∈ W }.

Lemma 3 . 1 .

 31 Let m 0 = (m 0 s , m 0 m , m 0 l ) ∈ M 0 . Then m 0 ∈ M + ∪ M 3 if and only if there are m = (m s , m m , m l ) ∈ M + and ∈ R so that m 0 = m( ) and ∈ min (m), max (m) . Moreover, m 0 ∈ M 1 = (M + ∪ M 3 ) 0 if and only if m = (m s , m m , m l ) ∈ M + is as above and ∈ min (m), max (m) . Proof. If m 0 ∈ M + ∪ M 3 , then m 0 = m( ) for m = (m s , m m , m l ) = (m 0 s + m 0 l , m 0 m , 0) ∈ M + and = -

Figure 2 .

 2 Figure 2. (m s ( ), m l ( )) for geometric (m s , m l = 1) and ∈ [ min , max ]

Corollary 3 . 4 .

 34 In the above notation, for all l ∈ R and ≥ -mm 2 , D , ,p L (m) = D -+m l -1, ,p L .

3 . 4 .

 34 g.[START_REF] Heckman | Harmonic analysis and special functions on symmetric spaces[END_REF] Theorem 4.4.2],[START_REF]Harmonic analysis for certain representations of graded Hecke algebras[END_REF] Theorem 3.15]), the F , ,λ (m; x) and G , ,λ (m; x) are entire functions of λ ∈ a * C , analytic functions in x ∈ a and mero-morphic functions of m = (m s , m m , m l ) ∈ C 3 . Let M C,0 = {m = (m s , m m , m l ) ∈ C 3 : Re(m s +m l ) ≥ 0}. Observe that m ∈ M C,0 if and only if m( ) ∈ M C,0 . One can show (see Appendix A below) that for fixed (λ, x) ∈ a * C × a, the functions F λ (m; x) and G λ (m; x) are holomorphic in a neighborhood of M C,0 . It follows that F , ,λ (m; x) and G , ,λ (m; x) are holomorphic near each m = (m s , m m , m l ) ∈ M 0 . Estimates and asymptotics for the ( , )-hypergeometric functions. Let m = (m s , m m , m l ) ∈ M + . Using (

Theorem 3 . 7 .

 37 Assume that m l ≥ 1 and ≥ 0 if m m > 0, > 0 if m m = 0. Fix ∈ R with ∈] min -1, max [.Then, the ( , )-hypergeometric function F , ,λ (m) is bounded if and only if λ ∈ C(ρ(m(2 ))) + ia * , where C(ρ(m(2 ))) is the convex hull of the set {wρ(m(2 )) : w ∈ W }.

F

  ρ(m(2 )) (m( , ); x) = F ρ(m( , ))+ 2 r j=1 β j + 2 1≤i<j≤r (β j ±β i ) (m( , ); x) ≤ F ρ(m( , )) (m( , ); x)e maxw w 2 r j=1 β j (x)+ 2 1≤i<j≤r (β j ±β i )(x) .Now, F ρ(m( , )) (m( , ); x) = 1; see e.g. [21, Lemma 4.1] (the proof extends to every multiplicity function for which the symmetric and non-symmetric hypergeometric functions are defined). Choose x 0 in the W -orbit of x so that max

Proposition 4 . 1 .

 41 For ∈ R and λ ∈ a * C set ϕ ,λ (g) = K/Z(G)

( 4 )

 4 ) ϕ ,λ | a is positive for λ ∈ a * and | | ≤ max . (3) |ϕ ,λ | ≤ ϕ Re λ for λ ∈ a * C . |ϕ ,λ (m)| ≤ 1 for λ ∈ C(ρ(m)) + ia * , where C(ρ(m)) is the convex hull of the set {wρ(m) : w ∈ W }. (5) Let | | < max . The τ -spherical function ϕ ,λ (m) on G/K is bounded if and only if λ ∈ C(ρ(m))+ia * , where C(ρ(m)) is the convex hull of the set {wρ(m) : w ∈ W }. Moreover, |ϕ ,λ (m; x)| ≤ 1 for all λ ∈ C(ρ(m)) + ia * and x ∈ a.

4. 2 . 1 .

 21 The case of g = sp(p, 1), p ≥ 2. Let G = Sp(p, 1) (which is simply connected) and K = Sp(p) × Sp(1). The small K-types are precisely the representations of the form τ n = 1⊗τ n , the tensor product of the trivial representation of Sp(p) and the n-dimensional irreducible representation of Sp(1) = SU[START_REF] Branson | The Paley-Wiener theorem and the local Huygens' principle for compact symmetric spaces: the even multiplicity case[END_REF] with n ∈ N. The original root system is given by Σ = {±α, ±2α} (of type BC 1 ), with multiplicities m ±α = 4(p -1) and m ±2α = 3. For the small type τ n , we have Σ(τ n ) = Σ and m(τ n ) ± = (4p -2 ± 2n, 1 ∓ 2n); see[START_REF] Oda | Spherical functions for small K-types. Geometric and harmonic analysis on homogeneous spaces[END_REF] Section 2.3]. The associated spherical function is given byϕ τn λ | a = (cosh α) -1∓n F λ (Σ, m(τ n ) ± ) .(67)Notice that m(τ n ) + = m( n ) for m = (4(p -1), 3) ∈ M + and n = n + 1. Moreover, min = -2(p -1) and max = 2p + 1. So m(τ n ) + ∈ M + ∪ M 3 provided n ≤ 2p and m(τ n ) + ∈ M 1 if n < 2p. Notice that ρ(m) = ρ G/K = (2p + 1)α.

|ξ| 2 e|ξ| 2 F

 22 we conclude that f (t) is increasing on [0, +∞[ and hencee K ξ t F λ (m; x + tξ) = f (t) ≥ f (0) = e λ (m; x) i.e. e K ξ t F λ (m; x + tξ) ≥ F λ (m; x)(73) for all t ≥ 0. Choose first ξ = x 1 ∈ a and t = 1. Then (73) gives e max w∈W (ρ(m)-λ)(wx 1 ) F λ (m; x + x 1 ) ≤ F λ (m; x) . (74) Replace x by x + x 1 and choose ξ = -x 1 and t = 1. Then (73) gives e -min w∈W (ρ(m)-λ)(wx 1 )F λ (m; x) ≥ F λ (m; x + x 1 ) ,

Table 1 .

 1 Root multiplicities of irreducible non-compact Hermitian symmetric spaces

	2, 1)

The authors thank both the referees for a careful reading of the manuscript and pointing out some mistakes. The authors gratefully acknowledge financial support by the Indo-French Centre for the Promotion of Advanced Research (IFCPAR/CEFIPRA), Project No. 5001-1: "Hypergeometric functions: harmonic analysis and representation theory". The first named author thanks SERB, India for the financial support through MATRICS grant MTR/2018/00051.

Appendix A. The Heckman-Opdam hypergeometric functions as functions of their parameters 32 Appendix B. Some computations 33 References Notice that min (m) = 0 and max (m) = m l = p -q ≥ 1. Hence, in both cases, m( , ) ∈ M + ∪ M 3 and m( , ) / ∈ M 1 . The symmetry (47) with + m l -1 = -+ p -q -1 yields the equalities

Since m l = p -q ≥ 1, we see for instance that Theorem 3.7 applies to case (i). Notice also that

Appendix A. The Heckman-Opdam hypergeometric functions as functions of their parameters

Let a be an r-dimensional Euclidian real vector space, with an inner product •, • , and let Σ be a root system Σ in a * of Weyl group W . Let M C denote the set of complex-valued multiplicity functions m = {m α } on Σ. (Hence M C ≡ C d where d is the number of Weyl group orbits in Σ.)

In this appendix we collect the regularity properties of the (symmetric and nonsymmetric) hypergeometric functions

Most of the results are known, but scattered in the literature. Recall the Gindikin-Karpelevich formula for Harish-Chandra's c-function:

where c(m; λ) is given in terms of the positive indivisible roots α ∈ Σ + i by

and Γ is the classical gamma function. Set (see [25, p. 196])

is not singular}

, where r α is the reflection across ker(α) and χ δ is the character of δ. Let d δ be the lowest embedding degree of δ in C[a C ], and set (see [START_REF]Harmonic analysis for certain representations of graded Hecke algebras[END_REF]Definition 3.13])

The regularity properties of the (symmetric and nonsymmetric) functions are summarized in the following theorem.

Theorem A.1. The hypergeometric function

For a + iΩ replaced by a W -invariant tubular domain V of a in a C , these results are due to Opdam, see [START_REF] Opdam | Root systems and hypergeometric functions[END_REF]Theorem 2.8] and [START_REF]Harmonic analysis for certain representations of graded Hecke algebras[END_REF]Theorem 3.15]. See also [START_REF] Heckman | Harmonic analysis and special functions on symmetric spaces[END_REF]Theorem 4.4.2]. The remark that the maximal tubular domain V is a + iΩ was made by Jacques Faraut, see [2, Remark 3.17].

Moreover, M C,0 is stable under deformations by ∈ C as in [START_REF] Shimeno | Eigenspaces of invariant differential operators on a homogeneous line bundle on a Riemannian symmetric space[END_REF]:

Proof. The inclusion M C,0 ⊂ M F,reg was observed in [START_REF] Heckman | Harmonic analysis and special functions on symmetric spaces[END_REF]Remark 4.4.3] and follows from the properties of the Gamma function. Notice that Re(ρ(m) α ) ≥ Re( mα 2 + m 2α ) for α ∈ Σ + i , with simplifications in the factor of c(m; ρ(m)) corresponding to α in case of equality with mα 2 + m 2α real. For the inclusion M C,0 ⊂ M G,reg , observe that ε δ (m) =

Appendix B. Some computations

In this appendix, we prove two inequalities stated in subsection 2.3. We begin showing that for every λ ∈ a * and ξ ∈ a

Firstly, as in the proof of Lemma 3.4 in [START_REF]Contributions to the hypergeometric function theory of Heckman and Opdam: sharp estimates, Schwartz space, heat kernel[END_REF], for every x ∈ a + , ξ ∈ a and λ ∈ a * we have

(where ξ is fixed and acts on the x-variable). Here G λ (wx) ≥ 0 for m ∈ M 3 (as in [START_REF]Contributions to the hypergeometric function theory of Heckman and Opdam: sharp estimates, Schwartz space, heat kernel[END_REF] for m ∈ M + ). Set K ξ = max w∈W (ρ(m) -λ)(wξ). So K ξ = -min w∈W (λ -ρ(m))(wξ) and