Inversion formula for the hypergeometric Fourier transform associated with a root system of type BC
aa r X i v : . [ m a t h . R T ] J u l INVERSION FORMULA FOR THE HYPERGEOMETRICFOURIER TRANSFORM ASSOCIATED WITH A ROOTSYSTEM OF TYPE BC TATSUO
HONDA , HIROSHI
ODA , AND NOBUKAZU
SHIMENOAbstract.
We give the inversion formula and the Plancherel formulafor the hypergeometric Fourier transform associated with a root sys-tem of type BC , when the multiplicity parameters are not necessarilynonnegative. Introduction
Heckman and Opdam [12, 9, 10, 11, 23, 25, 13] have developed a the-ory of hypergeometric functions associated with root systems. When themultiplicity function k takes some particular values, the Heckman-Opdamhypergeometric function coincides with the restriction to a Cartan subspace a of the zonal spherical function on a Riemannian symmetric space of thenoncompact type.In this group case, associated harmonic analysis have been developed byHarish-Chandra, Gindikin-Karpelevich, Helgason, Gangolli, Rosenberg, andother researchers (cf. [8, 14]). In particular, an explicit inversion formula,the Paley-Wiener theorem, and Plancherel theorem for the spherical Fouriertransform have been established.Opdam [23] generalized those results in the group case to the hypergeo-metric Fourier transform when the multiplicity function takes arbitrary non-negative values. There are only the continuous spectra and the Plancherelmeasure is c p λ, k q ´ c p´ λ, k q ´ dµ p λ q on ?´ a ˚ , where c p λ, k q is Harish-Chandra’s c -function and µ is a normalized Lebesgue measure on ?´ a ˚ .Moreover, Opdam [24] studied the case of negative multiplicity functionswhen the root system is reduced. In this case, there are spectra with sup-ports of lower dimensions in the Plancherel measure, in addition to the mostcontinuous spectra described above.If the root system is of type BC , then the associated Heckman-Opdamhypergeometric function is the Jacobi function, which was introduced and Mathematics Subject Classification.
Key words and phrases.
Heckman-Opdam hypergeometric function, hypergeometricFourier transform, Jacobi function, spherical function.The second author was supported by JSPS KAKENHI Grant Number 18K03346.
HONDA , HIROSHI
ODA , AND NOBUKAZU
SHIMENO studied by Flensted-Jensen and Koornwinder [17, 7, 18]. The Jacobi func-tion is an even eigenfunction of a second order ordinary differential oper-ator and can be written by the Gauss hypergeometric function. In thegroup case, the second order differential operator is the radial part of theLaplace-Beltrami operator on a rank one symmetric space and the Jacobifunction is the restriction of the zonal spherical function to a Cartan sub-space a » R . Let k s and k ℓ denote the multiplicity parameters for the shortroots ˘ β and the long roots ˘ β , respectively. Flensted-Jensen [7, Appen-dix 1] proved the inversion formula for the hypergeometric Fourier transformunder the condition k s , k ℓ P R and k s ` k ℓ ą ´ . Moreover, if k satisfies k s ` k ℓ ` ´ ˇˇ k ℓ ´ ˇˇ ě
0, then there are only the continuous spectra. On thecontrary, there are finite number of discrete spectra in addition to the con-tinuous spectra. These discrete spectra and the corresponding Plancherelmeasure are obtained by residues of c p λ, k q ´ c p´ λ, k q ´ . Flensted-Jensen[7] applied his results on the Jacobi transform to harmonic analysis of spher-ical functions on the universal covering group of SU p n, q associated witha one-dimensional K -type. The Jacobi transform also can be applied tothe harmonic analysis on some homogeneous vector bundles over hyperbolicspaces (cf. [6, 30]). In these group cases, discrete spectra correspond torelative discrete series representations.The Heckman-Opdam hypergeometric function is a real analytic jointeigenfunction of commuting family of differential operators. In group case,the commuting family of differential operators are radial parts of invari-ant differential operators. The inversion formula for the hypergeometricFourier transform gives an expansion of an arbitrary function in terms ofthe Heckman-Opdam hypergeometric functions. Heckman-Opdam theorygives a generalization of Euclidean Fourier analysis (the case of k ” BC with an arbitrary rank when the multiplicity function is not necessarilynonnegative. Except the case of type BC as we mentioned above and agroup case studied by the third author [29], the study of this case haveremained open for many years. We have decided to study this case, becauseof its application to harmonic analysis of spherical functions associated withcertain K -types on connected semisimple Lie groups of finite center (cf. [20,final comment]).For the root system of type BC r with r ě
2, there are three multiplicityparameters k s , k m and k ℓ corresponding to the short, medium, and longroots respectively. We prove the inversion formula, Paley-Wiener theorem, NVERSION FORMULA FOR THE HYPERGEOMETRIC FOURIER TRANSFORM 3 and Plancherel theorem for the hypergeometric Fourier transform underthe condition k s , k m , k ℓ P R and k s ` k ℓ ą ´ , k m ě
0. The condi-tion on k makes the spectral problem well-posed and covers known groupcases. We give the Plancherel measure explicitly in terms of residues of c p λ, k q ´ c p´ λ, k q ´ . In particular, we classify the square integrable hy-pergeometric functions and give their L -norms explicitly. Our method ofcalculus of residues of c p λ, k q ´ c p´ λ, k q ´ follows closely to that of [29],where the third author studied spherical functions associated with a one-dimensional K -type on an irreducible Hermitian symmetric space and ob-tained the inversion formula for the spherical transform.This paper is organized as follows. In Section 1, we review the Heckman-Opdam hypergeometric function. In Section 2, we define the hypergeometricFourier transform associated with a root system of type BC and give ba-sic results (Theorem 2.2 and Theorem 2.3). In Section 3 we define the set(3.11) of parameters of the square integrable hypergeometric functions forthe root system of type BC and give tempered hypergeometric functions(Corollary 4.8). In Section 4 we prove the inversion formula (Theorem 4.1),Paley-Wiener theorem (Theorem 4.6), Plancherel theorem (Theorem 4.7),and give the classification and explicit formulas for the L -norms of squareintegrable hypergeometric functions (Corollary 4.8). Explicit formulate ofthe residues of c p λ, k q ´ c p´ λ, k q ´ that contribute to the Plancherel mea-sure are given by (4.1) and Proposition 4.5.We thank Professor Toshio Oshima for helpful discussions on the subjectmatter of this paper.1. Heckman-Opdam hypergeometric functions
In this section, we give basic notations for a root system of type BC andreview on the Heckman-Opdam hypergeometric function associated with aroot system of type BC . We refer original papers by Heckman and Opdam[9, 12, 23] and survey articles [9, 11, 25, 1, 13] for details.Let N denote the set of non-negative integers. Let r denote a positiveinteger and a a r -dimensional Euclidean space with an inner product x ¨ , ¨ y .We often identify a and a ˚ by using the inner product. We use the samenotation x ¨ , ¨ y for the inner product on a ˚ and the complex symmetricbilinear form on a ˚ C ˆ a ˚ C . For x P a define || x || “ a x x, x y . Let R Ă a ˚ be aroot system of type BC r . It is of the form(1.1) R “ t˘ β i , ˘ β i , ˘ p β p ˘ β q q ; 1 ď i ď r, ď q ă p ď r u , where t β , β , . . . , β r u forms an orthogonal basis of a ˚ with || β p || “ || β q || p ď q ă p ď r q . Moreover, we will assume || β || “
2, so t β , . . . , β r u forms an TATSUO
HONDA , HIROSHI
ODA , AND NOBUKAZU
SHIMENO orthonormal basis of a ˚ . Let R ` denote the positive system of R defined by(1.2) R ` “ t β i , β i , p β p ˘ β q q ; 1 ď i ď r, ď q ă p ď r u . Let α , . . . , α r denote the positive roots defined by(1.3) α i “ p β r ` ´ i ´ β r ´ i q p ď i ď r ´ q , α r “ β . Then(1.4) B “ t α , α , . . . , α r u is the set of the simple roots in R ` . We call ˘ β i , ˘ p β p ˘ β q q p p ă q q , and ˘ β i , short, medium, and long roots, respectively. Let R ` s , R ` m , and R ` ℓ denote the subset of R ` consisting of the short, medium, and long roots,respectively. Let W denote the Weyl group for R . It is the semidirectproduct of Z r and S r . The group Z r and S r act on a ˚ as sign changes andpermutations of t β , . . . , β r u , respectively.Let k be a complex-valued W -invariant function on R , which is calleda multiplicity function and K C the space of the multiplicity functions. Let k s , k m , and k ℓ be the value of k for the short, medium, and long rootsrespectively. We identify k P K C with the 3-tuple p k s , k m , k ℓ q P C , so K C » C . Let ρ p k q “ ÿ α P R ` k α α. For α P R let α _ denote the corresponding coroot α _ “ α {x α, α y . Any λ P a ˚ C can be written in the form(1.5) λ “ r ÿ i “ λ i β i with λ i “ x λ, β _ i y p ď i ď r q . We identify λ P a ˚ C with the r -tuple p λ , . . . , λ r q P C r , so a ˚ C » C r . Inparticular,(1.6) ρ p k q “ p k s ` k ℓ , k s ` k ℓ ` k m , . . . , k s ` k ℓ ` p r ´ q k m q . For k P K C let L p k q denote the differential operator on a defined by(1.7) L p k q “ r ÿ i “ B β i { ` ÿ α P R ` k α coth α B α . Here B α denote the directional derivative corresponding to α . There exists analgebra D p k q of W -invariant differential operators on a with the properties: L p k q P D p k q and there exists an algebra isomorphism γ k : D p k q „ÝÑ S p a C q W ,where S p a C q W denotes the set of the W -invariant elements in the symmetricalgebra S p a C q . For λ P a ˚ C we consider the system of differential equations(1.8) Df “ γ k p D qp λ q f for any D P D p k q . In particular, it contains the equation(1.9) L p k q f “ px λ, λ y ´ x ρ p k q , ρ p k qyq f. Let a ` “ t x P a ; α p x q ą α P R ` u . NVERSION FORMULA FOR THE HYPERGEOMETRIC FOURIER TRANSFORM 5
Let Q ` denote the subset of a ˚ spanned by B over N . There exists a seriessolution Φ p λ, k q of (1.8) of the form(1.10) Φ p λ, k ; x q “ ÿ κ P Q ` a κ p λ, k q e p λ ´ ρ p k q´ κ qp x q p x P a ` q , with a p λ, k q “
1. For generic λ all the coefficients a κ p λ, k q p κ P Q ` q are determined uniquely by (1.9) and Φ p λ, k q converges on a ` . Moreover, Φ p λ, k q satisfies (1.8). We call Φ p λ, k q the Harish-Chandra series. As afunction of the spectral parameter λ P a ˚ C , Φ p λ, k q is meromorphic withsimple poles along hyperplanes of the form(1.11) x λ, α _ y “ j for some α P R ` , j “ , , , . . . . See [10, Proposition 4.2.5]. Moreover, if(1.12) x λ, α _ y R Z for all α P R , then t Φ p wλ, k q ; w P W u forms a basis of the solution space of (1.8) on a ` .The condition (1.12) can be written explicitly as(1.13) 2 λ i R Z p ď i ď r q , λ p ˘ λ q R Z p ď p “ q ď r q . Lemma 1.1.
Let U Ă K C » C be a bounded set, ε ą , and ν P a ˚ . Thenthere exists a polynomial p p λ, k q , constant C , and n P N such that | p p λ q Φ p λ, k ; x q| ď C p ` || λ ||q n e Re p λ ´ ρ p k qqp x q , whenever k P U and Re λ P ν ´ a ˚` and x P a satisfies α p x q ą ε for any α P B .Proof. The lemma follows by extending the estimates for the coefficients a κ p λ, k q due to Gangolli. See [14, Ch IV Lemma 5.6] and [8, Theorem 4.5.4].See also [2, Ch I Lemma 5.1] and [29, Corollary 3.11]. (cid:3) Let Γ p ¨ q denote the Gamma function. Define the meromorphic functions˜ c α p λ, k q p α P R q , ˜ c p λ, k q , and c p λ, k q on a ˚ C ˆ K C by˜ c α p λ, k q “ Γ ` x λ, α _ y ` k α ˘ Γ ` x λ, α _ y ` k α ` k α ˘ , (1.14) ˜ c p λ, k q “ ź α P R ` ˜ c α p λ, k q , (1.15)and(1.16) c p λ, k q “ ˜ c p λ, k q ˜ c p ρ p k q , k q with the convention k α “ α R R . We call c p λ, k q Harish-Chandra’s c -function. By [10, (3.4.6)] and [11, Proposition 5.1],˜ c p λ, k q “ ź ď q ă p ď r Γ ` p λ p ´ λ q q ˘ Γ ` p λ p ` λ q q ˘ Γ ` p λ p ´ λ q ` k m q ˘ Γ ` p λ p ` λ q ` k m q ˘ (1.17) ˆ r ź i “ ´ k s Γ ` λ i ˘ Γ ` p λ i ` q ˘ Γ ` p λ i ` k s ` q ˘ Γ ` p λ i ` k s ` k ℓ q ˘ TATSUO
HONDA , HIROSHI
ODA , AND NOBUKAZU
SHIMENO and(1.18) ˜ c p ρ p k q , k q “ r ź i “ Γ p k s ` p i ´ q k m ` k ℓ q Γ p k m q Γ p p k s ` p i ´ q k m ` k ℓ qq Γ p i k m q . Note(1.19) c p λ, q “ r r ! “ | W | for k s “ k m “ k ℓ “ . Let K reg , K , and K ` denote the subsets of K C » C given by K reg “ t k P K C ; ˜ c p ρ p k q , k q “ u , (1.20) K “ t k P K C ; k s , k m , k ℓ P R u , (1.21) K ` “ t k P K ; k α ě α P R u . (1.22) Lemma 1.2.
Let U Ă K reg be a bounded set and ν P a ˚ . Then there existsa polynomial p p λ, k q , constant C , and n P N such that | p p λ, k q c p´ λ, k q| ´ ď C p ` || λ ||q n whenever k P U and Re λ P ν ´ a ˚` .Proof. The estimate follows from (1.14), (1.15), (1.16), and Stirling’s for-mula as in the group case. See [14, Ch IV, Proposition 7.2], [8, Proposi-tio 4.7.15], [29, Lemma 3.9], and [17, Lemma 2.2]. (cid:3)
Define a ˚` “ t λ P a ˚ ; x λ, α y ą α P R ` u» tp λ , λ , . . . , λ r q P R r ; 0 ă λ ă λ ă ¨ ¨ ¨ ă λ r u . For λ P a ˚ C satisfying (1.12) and k P K reg , define(1.23) F p λ, k ; x q “ ÿ w P W c p wλ, k q Φ p wλ, k ; x q . Each terms c p wλ, k q , Φ p wλ, k q on the right hand side of (1.23) is meromor-phic function on a ˚ C ˆ K C and indeed regular for λ and k satisfying (1.12) and k P K reg . A deep theorem due to Heckman and Opdam says that F p λ, k ; x q continued to an analytic function on a ˚ C ˆ K reg ˆ a and F p λ, k ; 0 q “
1. Itsatisfies F p wλ, k ; x q “ F p λ, k ; x q for all w P W, (1.24) F p λ, k ; wx q “ F p λ, k ; x q for all w P W, (1.25) F p wλ, k ; x q “ F p ¯ λ, ¯ k ; x q . (1.26)Here ¯ denotes the complex conjugate. F p λ, k ; x q is the unique W -invariantreal analytic solution of the hypergeometric system (1.8) on a satisfying F p λ, k ; 0 q “
1. We call F p λ, k ; x q the Heckman-Opdam hypergeometricfunction associated with the root system R . Remark . (1) The Heckman-Opdam hypergeometric function is a gener-alization of the zonal spherical function on a Riemannian symmetric space NVERSION FORMULA FOR THE HYPERGEOMETRIC FOURIER TRANSFORM 7 of the noncompact type. Let G be a connected noncompact semisimple Liegroup of finite center with the Cartan decomposition G “ K exp a K . Let Σ Ă a ˚ be the restricted root system for p g , a q and m α the dimension of theroot space corresponding to α P Σ . Set R “ Σ, k α “ m α { p α P R q . Then L p k q is the radial part of the Laplace-Beltrami operator on G { K , c p λ, k q is Harish-Chandra’s c -function, F p λ, k q is the restriction to a of thezonal spherical function on G { K . (Here R and Σ are not necessarily of type BC .)The zonal spherical function is a bi- K -invariant function on G . Moregenerally, elementary spherical functions associated with some K -types areexpressed by the Heckman-Opdam hypergeometric function. The case ofone-dimensional K -types when G is of Hermitian type is studied by [10,Section 5] and [29]. More generally, the cases of small K -types are studiedby [20].We call above cases “the group case”.(2) When the root system R is of type BC , the Heckman-Opdam hyper-geometric function is the Jacobi function studied by Flensted-Jensen andKoornwinder: F p λ, k ; x q “ φ p α , β q?´ λ p z q : “ F ` p λ ` ρ p k qq , p´ λ ` ρ p k qq ; α ` ´p sinh z q ˘ , with z “ β p x q , α “ k s ` k ℓ ´ , β “ k ℓ ´ , and ρ p k q “ α ` β `
1, where F denote the Gauss hypergeometric function (cf. [17, 7]).(4) The hypergeometric system (1.8) has regular singular points at infinityin a ` and is holonomic of rank | W | . The leading exponents at infinityof (1.8) are of the form wλ ´ ρ p k q p w P W q . If λ P a ˚ C satisfies (1.12),then Φ p wλ, k q p w P W q are solutions of (1.8) with the leading exponents wλ ´ ρ p k q p w P W q . If λ P a ˚ C does not satisfy (1.12), then there are still | W | linearly independent real analytic solutions on a ` with leading exponents wλ ´ ρ p k q (counting the multiplicity on the wall in a ˚ ). They may havepolynomial term (or logarithmic term by taking exp x as a coordinate) asin the case of the Gauss hypergeometric differential equation. For general λ P a ˚ C , the asymptotic expansion (1.23) becomes a convergent expansion on a ` of the form(1.27) F p λ, k ; x q “ ÿ µ P W λ ÿ κ P Q ` p µ,κ p λ, k ; x q e p µ ´ ρ p k q´ κ qp x q , where p µ,κ are polynomials in x . See [12, 9], [10, §4.2], and [26, 27, 28].In order to study the hypergeometric Fourier transform in the next sec-tion, the following result is important. TATSUO
HONDA , HIROSHI
ODA , AND NOBUKAZU
SHIMENO
Theorem 1.4.
Let k P K reg . Let D Ă a be a compact set and p P S p a C q .Then there exist a constant C and n P N such that |Bp p q F p λ, k ; x q| ď C p ` || λ ||q n e max w P W Re wλ p x q for all λ P a ˚ C and x P D . Here Bp p q denote the constant coefficient differen-tial operator on a corresponding to p .Proof. This is [24, Theorem 2.5]. It was shown by [23, Corollary 6.2] thatthe estimate is true for k P K ` . It is still true by using the hypergeometricshift operators. (cid:3) Remark . Ho and Ólafsson [16, Appendix] proved an estimate as in theabove theorem for k P K with k s ` k ℓ ě , k s ě
0, and k m ě k P K satisfying more general conditions k s ` k ℓ ą ´ and k m ě Hypergeometric Fourier transform
Let dx and dλ denote the Lebesgue measures on the Euclidean spaces a and a ˚ , respectively. Then(2.1) dλ “ dλ dλ ¨ ¨ ¨ dλ r , in terms of the coordinates (1.5). (Recall that we assume || β || “
2. Weshould add the factor p|| β ||{ q r on the right hand side of (2.1), if we considera general inner product on a ˚ .) Let dµ p λ q denote the measure on ?´ a ˚ given by(2.2) dµ p λ q “ p π q ´ r d p Im λ q “ p π ?´ q ´ r dλ. The measures dx on a and dµ p λ q on ?´ a ˚ are normalized so that theinversion formula for the Euclidean Fourier transform˜ f p λ q “ ż a f p x q e ´ λ p x q dx p f P C c p a q q is given by f p x q “ ż ?´ a ˚ ˜ f p λ q e λ p x q dµ p λ q . For k P K let δ k denote the weight function on a given by δ k “ ź α P R ` ˇˇ e α ´ e ´ α ˇˇ k α “ ź α P R ` s ˇˇ e α ´ e ´ α ˇˇ k s ` k ℓ ` e α ` e ´ α ˘ k ℓ ź β P R ` m ˇˇ e β ´ e ´ β ˇˇ k m Necessary and sufficient conditions for the locally integrability of δ k are givenby [3, Section 2] and [11, Proposition 5.1]. For k P K , δ k is locally integrableif and only if(2.3) k s ` k ℓ ą ´ ` max t , ´p r ´ q k m u and k m ą ´ r . (Since e t ´ e ´ t “ t ` o p t q , δ k is locally integrable if and only if the Selbergintegral S r p k s ` k ℓ ` , , k m q converges.) Moreover, by (1.18), (2.3) holds NVERSION FORMULA FOR THE HYPERGEOMETRIC FOURIER TRANSFORM 9 if and only if k is in the connected component of K reg X K containing K ` .Thus the problem of giving the spectral decomposition associated with thehypergeometric function F p λ, k q make sense for k satisfying (2.3).Assume k P K satisfies (2.3). The hypergeometric Fourier transform F k of f P C p a q W is defined by F k f p λ q “ ż a ` f p x q F p λ, k ; x q δ k p x q dx (2.4) “ | W | ż a f p x q F p λ, k ; x q δ k p x q dx. (Note F p λ, k ; ´ x q “ F p´ λ, k ; x q “ F p λ, k ; x q , since ´ W P W .)Given a W -invariant convex compact neighborhood C of the origin in a ,consider the function H C p λ q “ max x P C λ p x q for λ P a ˚ . If the support of f P C c p a q is contained in C , then its hypergeometric Fourier transform φ “ F k f of f P C c p a q W is a holomorphic function on a ˚ C that satisfies thecondition(2.5) sup λ P a ˚ C p ` || λ ||q n e ´ H C p´ Re λ q | φ p λ q| ă `8 for any n P N . The above estimate (2.5) for φ “ F k f follows from Theorem 1.4 (see [24,Theorem 4.1]).Let PW p a C q denote the space of all holomorphic functions φ on a ˚ C suchthat there exists a W -invariant convex compact neighborhood C of the ori-gin in a satisfying (2.5). It coincide with the Paley-Wiener space in theEuclidean Fourier analysis (the case of k “ Remark . The above estimate (2.5) for φ “ F k f with k R K ` also can beproved by using the hypergeometric shift operator to reduce to the case of F k ` l with k ` l P K ` . Such argument are used in [15] in the setting of theJacobi polynomial.By (1.17), we can choose η P ´ a ˚` so that c p´ λ, k q ´ is holomorphic inthe region t λ P a ˚ C ; Re λ P η ´ a ˚` u . Here a ˚` denote the closure of a ˚` . Theorem 2.2 (Inversion formula, first form) . Assume k P K satisfies (2.3) .For f P C p a q W and x P a ` , (2.6) f p x q “ ż η `?´ a ˚ F k f p λ q Φ p λ, k ; x q c p´ λ, k q ´ dµ p λ q . Proof.
From Lemma 1.1, Lemma 1.2, and (2.5), the integral on the righthand side of (2.6) does not depend on the choice of η and is holomorphicin k P K reg . If k P K ` , then (2.6) was proved by [23, Theorem 9.13]. Thus(2.6) follows by analytic continuation. (cid:3) Theorem 2.3. If k P K satisfies (2.7) k s ě ´ , k m ě , k s ` k ℓ ě , HONDA , HIROSHI
ODA , AND NOBUKAZU
SHIMENO then for f P C c p a q W , (2.8) f p x q “ | W | ż ?´ a ˚ F k f p λ q F p λ, k ; x q | c p λ, k q| ´ dµ p λ q p x P a q . Proof. If k P K satisfies (2.7), then we can choose η “
0. Thus, by changesof variables and (1.23), we have (2.8). (cid:3) Tempered hypergeometric functions
In this section, we study tempered hypergeometric functions that con-tribute to the Plancherel theorem, which we will study in the next section.Though combinatorial features are different, our argument and result aresimilar to those of [24], where tempered hypergeometric functions are deter-mined for some negative multiplicity functions on reduced root systems.If k P K satisfies (2.7), then there are only continuous spectra in theinversion formula (2.8) for the hypergeometric Fourier transform. In thiscase, F p λ, k q with λ P ?´ a ˚ exhaust tempered hypergeometric functions.If k P K satisfies (2.3) but (2.7), then c p´ λ, k q ´ has singularities and wemust take account of residues during shifts of integral domains in the righthand side of (2.6) as η Ñ
0. The most continuous part of the spectraldecomposition is given by the right hand side of (2.8). Moreover, there arespectra whose supports have dimensions lower than r .In this paper, we exclude the case of k m ă k s ` k ℓ ą ´ , k m ě . Under (3.1) residue calculus can be handle explicitly by using the samemethod as in the proof of [29, Theorem 6.7], where the inversion formulafor the spherical transform associated with a one-dimensional K -type on asemisimple Lie group of Hermitian type is given. The case of(3.2) k s ` k ℓ ą ´ , k m “ BC and easy to analyze. If k m ă
0, then we mightneed a different kind of combinatorial argument as in [24], which we do notgo further in this paper.It is convenient to adopt the parameter α and β for the Jacobi function(cf. [7, 17]), which is the case of r “
1. We substitute(3.3) α “ k s ` k ℓ ´ , β “ k ℓ ´ k s and k ℓ . Then the condition (3.1) holds ifand only if(3.4) α ą ´ , k m ě ρ p k q is written as(3.5) ρ p k q “ p α ` β ` , α ` β ` k m ` , . . . , α ` β ` p r ´ q k m ` q . NVERSION FORMULA FOR THE HYPERGEOMETRIC FOURIER TRANSFORM 11
For 1 ď i ď r let ˜ c i p λ, k q denote the product of the factors of (1.17) forthe roots β i and β i .˜ c i p λ, k q : “ ˜ c β i p λ, k q ˜ c β i p λ, k q (3.6) “ ´ α ` β Γ ` λ i ˘ Γ ` p λ i ` q ˘ Γ ` p λ i ` α ´ β ` q ˘ Γ ` p λ i ` α ` β ` q ˘ . Let Θ be a subset of B , x Θ y the subset of R consisting of the linearcombinations of elements in Θ , and W Θ the subgroups of W generated bysimple reflections with respect to elements of Θ . Define a Θ “ t x P a ; α p x q “ α P Θ u , a p Θ q “ t x P a ; x x, y y “ y P a Θ u . Then x Θ y is a root system in a p Θ q ˚ with a positive system x Θ y ` “ x Θ yX R ` ,the set Θ of simple roots, and the Weyl group W Θ . Let W p Θ q denote theset of w P W satisfying wx “ x for any x P a p Θ q . For λ P a ˚ C we write λ “ λ a p Θ q ` λ a Θ with λ a p Θ q P a p Θ q ˚ C and λ a Θ P a ˚ Θ, C . Thus ρ p k q a p Θ q is “ ρ p k q ”for the root system x Θ y . Let ˜ c Θ p λ, k q and ˜ c Θ p λ, k q denote the meromorphicfunctions on a ˚ C ˆ K C given by(3.7) ˜ c Θ p λ, k q “ ź α P x Θ y ` ˜ c α p λ, k q , ˜ c Θ p λ, k q “ ź α P R ` z x Θ y ` ˜ c α p λ, k q . Set ˜ c H p λ, k q “
1. Thus ˜ c p λ, k q “ ˜ c Θ p λ, k q ˜ c Θ p λ, k q . Moreover, define(3.8) c Θ p λ, k q “ ˜ c Θ p λ, k q ˜ c Θ p ρ p k q , k q , c Θ p λ, k q “ ˜ c Θ p λ, k q ˜ c Θ p ρ p k q , k q , so c p λ, k q “ c Θ p λ, k q c Θ p λ, k q and c Θ p λ, k q is the c -function for the rootsystem x Θ y . Note the estimate given in Lemma 1.2 also holds for the partial c -function c Θ p λ, k q .For 0 ď i ď r let Θ i denote the subset of B given by(3.9) Θ i “ t α j ; r ´ i ` ď j ď r u . Then x Θ i y Ă a p Θ i q ˚ is a root system of type BC i . (If k m “
0, it reduces tothe direct sum of i copies of BC .) Its Weyl group is W Θ i » Z i ¸ S i . Wehave a p Θ i q ˚ “ Span R t β j ; 1 ď j ď i u , a ˚ Θ i “ Span R t β j ; i ` ď j ď r u , and the complex spans give their complexifications. The stabilizer W p Θ i q of a p Θ i q in W acts on a Θ i and W p Θ i q » Z r ´ i ¸ S r ´ i . Define the subset W Θ i of W by W Θ i “ tp ε, σ q P Z r ¸ S r ; σ p q ă ¨ ¨ ¨ ă σ p i q , (3.10) ε p β j q “ β j p @ j P t σ p q , . . . , σ p i qu qu . Then W Θ i is a complete set of representatives for W Θ i z W . HONDA , HIROSHI
ODA , AND NOBUKAZU
SHIMENO
For 1 ď i ď r let D k p Θ i q denote the subset of a p Θ i q ˚ » R i given by D k p Θ i q “ tp λ , . . . , λ i q P R i ; λ ` | β | ´ α ´ P N , λ i ă , (3.11) λ j ` ´ λ j ´ k m P N p ď j ď i ´ qu . Note D k p Θ i q “ H if and only if α ´ | β | ` p i ´ q k m ` ă
0. We set D k pHq “ H for convenience.For 1 ď i ď r define(3.12) F Θ i p λ, k q “ ÿ s P W Θi c Θ i p sλ, k q Φ p sλ, k q and let F H p λ, k q “ Φ p λ, k q . Note F B p λ, k q “ F p λ, k q . An argument similarto the proof of [10, Theorem 4.3.14] shows that the singularities of F Θ i p λ, k q are at most simple poles along hyperplanes of the form(3.13) x λ, α _ y “ j for some α P R ` z x Θ i y and j “ , , . . . . Such generalization for any Θ is given by [22, Theorem 8, Corollary 9]. By(1.23),(3.14) F p λ, k q “ ÿ w P W Θi c Θ i p wλ, k q F Θ i p wλ, k q . Proposition 3.1.
Assume k P K satisfies (3.1) . If λ P a ˚ C satisfies λ a p Θ i q P D k p Θ i q and does not lie on any of the hyperplanes (3.13) , then (3.15) F Θ i p λ, k q “ c Θ i p λ, k q Φ p λ, k q . In particular, if λ P D k p B q , then (3.16) F p λ, k q “ c p λ, k q Φ p λ, k q . Proof. If i “
0, then (3.15) is obvious. So assume i ě λ a p Θ i q P D k p Θ i q .First we assume that k P K reg and λ a Θi P a ˚ Θ i , C are generic so that k m R N and λ “ λ a p Θ i q ` λ a Θi satisfies (1.12). (We can do this by (3.11).) We willshow that all terms except s “ W in the right hand side of (3.12) vanish.If s P W Θ i satisfies s p β q “ ´ β l for some 1 ď l ď i , then c Θ i p sλ, k q “ i ě s p β q “ β l for some 1 ď l ď i .If l ě
2, then c Θ i p sλ, k q contains a factor Γ ` p λ j ´ λ j ` q ˘ Γ ` p λ j ´ λ j ` ` k m q ˘ for some 1 ď j ď i ´
1, which vanishes because p λ j ´ λ j ` ` k m q in thedenominator become a non-positive integer by (3.11). Thus the terms ofthe right hand side of (3.12) vanish unless s p β q “ β . We can show byinduction on j that (3.12) vanish unless s p β j q “ β j for any 1 ď j ď i . Wemay drop our assumption k and λ are generic by analytic continuation. (cid:3) Theorem 3.2.
Assume k P K satisfies (3.1) . If λ P λ P D k p Θ i q ` ?´ a ˚ Θ i satisfies (3.17) x λ, α _ y “ for all α P R ` z x Θ i y NVERSION FORMULA FOR THE HYPERGEOMETRIC FOURIER TRANSFORM 13 then (3.18) F p λ, k q “ ÿ w P W Θi c p wλ, k q Φ p wλ, k q and each term on the right hand side of (3.18) does not vanish.Proof. For w P W Θ i , it follows from Proposition 3.1 that F Θ i p wλ, k q “ c Θ i p wλ, k q Φ p wλ, k q . (Note that the above formula is true for w P W Θ i , but it does not hold forgeneral w P W .) Thus the theorem follows from (3.14). (cid:3) We say that F p λ, k q is tempered if there exist C ě d P N such that(3.19) | F p λ, k ; x q| ď C p ` || x ||q d e ´ ρ p k qp x q for all x P a ` . Corollary 3.3.
Assume k P K satisfies (3.1) . For λ P D k p Θ i q ` ?´ a ˚ Θ i , F p λ, k q is tempered. Moreover, F p λ, k q is a real-valued square integrablefunction for any λ P D k p B q .Proof. By Theorem 3.2, we have a convergent expansion on a ` of the form(3.20) F p λ, k ; x q “ ÿ µ P W Θi λ ÿ κ P Q ` p µ,κ p λ, k ; x q e p µ ´ ρ p k q´ κ qp x q , where p µ,κ are polynomials in x . The leading exponents of F p λ, k ; x q for λ P D k p Θ i q`?´ a ˚ Θ i on a ` are wλ ´ ρ p k q p w P W Θ i q . For w “ p ε, σ q P W Θ i we have x Re wλ, β j y “ x wλ a p Θ i q , β j y ă j P t σ p q , . . . , σ p i qu , x Re wλ, β j y “ j R t σ p q , . . . , σ p i qu . Our results follows from the criterion of Casselman and Miličić ([4, Corol-lary 7.3, Theorem 7.5]. For λ P D k p B q , F p λ, k q is real-valued by (3.16). (cid:3) In the next section, we will show that F p λ, k q p λ P D k p B qq exhaust squareintegrable hypergeometric functions after establishing the Plancherel theo-rem (see Corollary 4.8).4. Inversion and Plancherel formula
In this section, assume that k P K satisfies (3.1). We use the notation α , β given by (3.3). Recall the sets D k p Θ i q Ă a p Θ i q ˚ p ď i ď r q defined by(3.11). For 1 ď i ď r and λ P D k p Θ i q define d Θ i p λ, k q “ ˜ c Θ i p ρ p k q , k q (4.1) ˆ i ź j “ ´ α ´ β ´ λ j π Γ ` p λ j ` α ` | β | ` q ˘ Γ ` p´ λ j ` α ` | β | ` q ˘ Γ ` p λ j ´ α ` | β | ` q ˘ Γ ` p´ λ j ´ α ` | β | ` q ˘ ˆ ź ď q ă p ď i p λ q ´ λ p q Γ ` p λ p ´ λ q ` k m q ˘ Γ ` p´ λ q ´ λ p ` k m q ˘ Γ ` p λ p ´ λ q ´ k m ` q ˘ Γ ` p´ λ q ´ λ p ´ k m ` q ˘ . HONDA , HIROSHI
ODA , AND NOBUKAZU
SHIMENO
Note that d Θ i p λ, k q is a positive number. (If r “ k m “
0, the factorsof (4.1) in the third line should be understood to be one.) The meaning of d Θ i p ξ, k q will be clear in Theorem 4.1, Proposition 4.5, and Corollary 4.8.Let dλ a Θi “ dλ i ` ¨ ¨ ¨ dλ r denote the Euclidean measure on a ˚ Θ i and µ Θ i the measure on ?´ a ˚ Θ i givenby(4.2) dµ Θ i p λ a Θi q “ p π q ´ r ` i d p Im λ a ˚ Θi q “ p π ?´ q ´ r ` i dλ a Θi , which coincides with (2.2) for i “ ď i ď r , let ν k ,Θ i denote the measure on D k p Θ i q ` ?´ a Θ i definedby ż D k p Θ i q`?´ a Θi ψ p λ q dν k ,Θ i p λ q (4.3) “ | W p Θ i q| ÿ λ a p Θi q P D k p Θ i q d Θ i p λ a p Θ i q , k q ż ?´ a ˚ Θi ψ p λ q | c Θ i p λ, k q| ´ dµ Θ i p λ a Θi q . Here we write λ “ λ a p Θ i q ` λ a Θi with λ a p Θ i q P D k p Θ i q and λ a Θi P ?´ a ˚ Θ i .In particular, ż ?´ a ˚ ψ p λ q dν k , H p λ q “ | W | ż ?´ a ˚ ψ p λ q dµ p λ q| c p λ, k q| and ż D k p B q ψ p λ q dν k , B p λ q “ ÿ λ P D k p B q d B p λ, k q ψ p ξ q . For φ P PW p a C q W let J k ,Θ i φ p x q p ď i ď r q denote the functions definedby(4.4) J k ,Θ i φ p x q “ ż D k p Θ i q`?´ a ˚ Θi φ p λ q F p λ, k ; x q dν k ,Θ i p λ q . By (2.5), Theorem 1.4, and Lemma 1.1 (for the root system x Θ i y ), theintegral on the right hand side of (4.4) converges and defines an element of C p a q W .Now we state the main result of this paper: Theorem 4.1 (Inversion formula, second form) . Assume k P K satisfies (3.1) . Then for f P C c p a q W , (4.5) f p x q “ r ÿ i “ J k ,Θ i F k f p x q p x P a q . For the proof of Theorem 4.1 we make some preparations. Let Ξ denotethe subset of B given by(4.6) Ξ “ t α j ; 1 ď j ď r ´ u . NVERSION FORMULA FOR THE HYPERGEOMETRIC FOURIER TRANSFORM 15
Then x Ξ y is a root system of type A r ´ and W Ξ “ S r . We define(4.7) F Ξ p λ, k q “ ÿ s P W Ξ c Ξ p sλ, k q Φ p sλ, k q . If r “
1, then Ξ “ H . In this case we set F Ξ p λ, k q “ Φ p λ, k q , c Ξ p λ, k q “ W Ξ “ t W u . By the definition, F Ξ p sλ, k q “ F Ξ p λ, k q for any s P W Ξ .Note ˜ c Ξ p λ, k q “ ź ď q ă p ď r Γ ` p λ p ´ λ q q ˘ Γ ` p λ p ´ λ q ` k m q ˘ , (4.8) ˜ c Ξ p ρ p k q , k q “ r ź j “ Γ p k m q Γ p j k m q , (4.9) ˜ c Ξ p λ, k q “ ź ď q ă p ď r Γ ` p λ p ` λ q q ˘ Γ ` p λ p ` λ q ` k m q ˘ r ź j “ ˜ c j p λ, k q . (4.10)Then an argument similar to the proof of [10, Theorem 4.3.14] shows thatthe singularities of F Ξ p λ, k q are at most simple poles along hyperplanes ofthe form(4.11) x λ, α _ y “ j for some α P R ` z x Ξ y and j “ , , . . . . Such generalization for any Θ is given by [22, Theorem 8, Corollary 9]. Remark . If k m “
0, then c Ξ p λ, k q “ { r !, so F Ξ p λ, k q “ | W Ξ | ÿ s P W Ξ Φ p sλ, k q . Since Φ p λ, k q is the product of the Harish-Chandra series associated withthe root systems t˘ β i , ˘ β i u p ď i ď r q in this case, the above statementon regularity is obvious.We may choose the subgroup W Ξ : “ Z r of W as a complete set of repre-sentatives for W Ξ z W . By (1.23),(4.12) F p λ, k q “ ÿ w P W Ξ c Ξ p wλ, k q F Ξ p wλ, k q . For 0 ď i ď r ´ W Ξi “ W Ξ X W Θ i . Then W Ξi » Z r ´ i , which actas changes of signs for β i ` , . . . , β r . Put W Ξr “ t W u . Proposition 4.3.
Assume k P K satisfies (3.1) and let ď i ď r . If λ P a ˚ C satisfies (4.13) λ P D k p Θ i q ` ?´ a ˚ Θ i with x λ, α _ y “ for any α P R z x Ξ y , then (4.14) F p λ, k q “ ÿ w P W Ξi c Ξ p wλ, k q F Ξ p wλ, k q and each terms on the right hand side of (4.14) does not vanish. In partic-ular, for λ P D k p B q , (4.15) F p λ, k q “ c Ξ p λ, k q F Ξ p λ, k q . HONDA , HIROSHI
ODA , AND NOBUKAZU
SHIMENO
Proof.
We can prove this proposition in a similar way as Proposition 3.1.For i “ i ě
1. First we assume that k and λ a Θi P ?´ a ˚ Θ i are generic so that k m R N and λ satisfies (1.12).Then the expansion (4.12) holds. By (3.6) and (3.11), c p wλ, k q “ λ satisfying (4.13) and w P W Ξ “ Z r with wβ “ ´ β . By (4.10), c Ξ p wλ, k q “ λ and w . Thus(4.16) F p λ, k q “ ÿ w P W Ξ c Ξ p wλ, k q F Ξ p wλ, k q and c Ξ p wλ, k q “ w P W Ξ1 and λ satisfying (4.13) with i “
1. Thisproves the case of i “ r ě i ě
2. If λ satisfies (4.13),then (4.16) holds by the above argument. If w P W Ξ satisfies wβ “ ´ β ,then by (4.10), c Ξ p wλ, k q contains a factor Γ ` p λ ´ λ q ˘ Γ ` p λ ´ λ ` k m q ˘ , which vanishes for λ satisfying (4.13) because p λ ´ λ ` k m q in the de-nominator become a non-positive integer. By (4.10), c Ξ p wλ, k q “ λ satisfying (4.13) and w P W Ξ with wβ “ ´ β . Thus(4.17) F p λ, k q “ ÿ w P W Ξ c Ξ p wλ, k q F Ξ p wλ, k q and c Ξ p wλ, k q “ w P W Ξ2 and λ satisfying (4.13) with i “
2. Thisproves the case of i “ ď i ď r . Since c Ξ p wλ, k q and F Ξ p wλ, k q are regular for any λ satisfying (4.13) and w P W Ξ i ,we may drop our assumption k and λ are generic by analytic continuation. (cid:3) Remark . Corollary 3.3 follows also from Proposition 4.3.For 2 ď i ď r let W Ξ,i denote the subset of W Ξ “ S r given by(4.18) W Ξ,i “ t s P W Ξ ; s p q ă s p q ă ¨ ¨ ¨ ă s p i qu and set W Ξ, “ W Ξ , “ W Ξ . Proof of Theorem 4.1.
Choose η P ´ a ˚` with η ă α ´ | β | ` η j “ η ´ p j ´ q ε p ď j ď r q for sufficiently small ε ą
0. Then c p´ λ, k q ´ isholomorphic in the region t λ P a ˚ C ; Re λ P η ´ a ˚` u and Theorem 2.2 holds.For φ P PW p a C q W define(4.19) J k φ p x q “ ż η `?´ a ˚ φ p λ q Φ p λ, k ; x q c p´ λ, k q ´ dµ p λ q p x P a ` q . By Lemma 1.1, Lemma 1.2, and (2.5), the integral on the right hand sideof (4.19) does not depend on the choice of η . By Theorem 2.2, we have f p x q “ J k F k f p x q for any x P a ` . NVERSION FORMULA FOR THE HYPERGEOMETRIC FOURIER TRANSFORM 17
By (1.23), (4.4) for i “ J k , H φ p x q “ ż ?´ a ˚ φ p λ q Φ p λ, k ; x q c p´ λ, k q ´ dµ p λ q . for any x P a ` . Note that the integrand of (4.20) does not have singularitieson ?´ a ˚ .The singularities of the function λ ÞÑ c p´ λ, k q ´ in the region t λ P a ˚ C ; Re λ P ´ a ˚` u are precisely across the hyperplanes λ j “ ξ for 1 ď j ď r and ξ P D k p Θ q “ t λ P R ; λ ă , λ ´ α ` | β | ` P N u all oforder one. Note D k p Θ q “ H if and only if α ´ | β | ` ě
0. Since φ p λ q Φ p λ, k ; x q is regular for Re λ P ´ a ˚` there are no other singularitiesof φ p λ q Φ p λ, k ; x q c p´ λ, k q ´ that should be considered.For ξ P D k , let ˇ ξ “ p ξ, ξ, . . . , ξ q P a ˚ Θ . By Cauchy’s residue theorem, J k φ p x q ´ J k , H φ p x q is the sum of ´ p π ?´ q r ´ r ÿ j “ ż ˇ ξ `?´ R r ´ p φ p λ q Φ p λ, k ; x qq| λ j “ ξ (4.21) ˆ Res λ j “ ξ p c p´ λ, k q ´ q dλ t , ,...,r uzt j u over ξ P D k p Θ q . Here dλ t , ,...,r uzt j u “ ś i “ j dλ i . Note that the integrandsas functions of p λ i q i P t , ,...,r uzt j u does not have singularities on ˇ ξ ` ?´ R r ´ ,because singularities of ˜ c i p´ λ, k q ´ along λ i “ ξ p i “ j q is canceled byzeroes along the wall λ i “ λ j coming from ˜ c α p´ λ, k q ´ for either α “ p β i ´ β j q or p β j ´ β i q in R ` .By changes of variables, (4.21) becomes ´ p π ?´ q r ´ p r ´ q ! ż ˇ ξ `?´ a ˚ Θ p φ p λ q F Ξ p λ, k ; x qq| λ “ ξ (4.22) ˆ Res λ “ ξ p c Ξ p λ, k q ´ c p´ λ, k q ´ q dλ a Θ , because (1.17) and (4.8) yield c Ξ p wλ, k q ´ c p´ wλ, k q ´ “ c Ξ p λ, k q ´ c p´ λ, k q ´ for any w P W Ξ . Since λ ÞÑ φ p λ q F Ξ p λ, k ; x q is a W Ξ -invariant regular function on(4.23) t λ P C r ; Re λ j ď p ď j ď r qu and λ ÞÑ c Ξ p λ, k q ´ c p´ λ, k q ´ is W Ξ -invariant, it suffices to consider thesingularities of λ ÞÑ c Ξ p λ, k q ´ c p´ λ, k q ´ in the subset(4.24) t λ P C r ; Re λ ď Re λ ď ¨ ¨ ¨ ď Re λ r ď u of (4.23). Note that c Ξ p λ, k q ´ has no singularities in this set. Thus we willstudy the singularities of c p´ λ, k q ´ in (4.24).We assert that for any 2 ď i ď r ´ ξ P D k p Θ i q , the singularities ofRes λ i “ ξ i ¨ ¨ ¨ Res λ “ ξ Res λ “ ξ c p´ λ, k q ´ as a function of λ i ` for Re λ i ` ě ξ i are λ i ` with p ξ , . . . , ξ i , λ i ` q P D k p Θ i ` q and all of them are simple poles. We provethis assertion by induction on i . We have already seen that the assertion HONDA , HIROSHI
ODA , AND NOBUKAZU
SHIMENO holds for i “
0. Recall c p´ λ, k q ´ “ c Θ i p´ λ, k q ´ c Θ i p´ λ, k q ´ . By (3.9) and (3.7), we haveRes λ i “ ξ i ¨ ¨ ¨ Res λ “ ξ Res λ “ ξ c p´ λ, k q ´ “ p c Θ i p´ λ, k q ˇˇ λ p Θ i q“ ξ q ´ Res λ i “ ξ i ¨ ¨ ¨ Res λ “ ξ Res λ “ ξ c Θ i p´ λ, k q ´ and Res λ i “ ξ i ¨ ¨ ¨ Res λ “ ξ Res λ “ ξ c Θ i p´ λ, k q ´ is a nonzero constant thatdepends only on k and ξ “ p ξ , . . . , ξ i q P D k p Θ i q . Thus, we should examinesingularities of c Θ i p´ λ, k q ´ “ C r ź j “ i ` Γ ` p´ λ j ` α ´ β ` q ˘ Γ ` p´ λ j ` α ` β ` q ˘ Γ ` p´ λ j ` q ˘ Γ ` ´ λ j ˘ ˆ ź ď q ă p ď rp ą i Γ ` p´ λ q ´ λ p ` k m q ˘ Γ ` p λ q ´ λ p ` k m q ˘ Γ ` p´ λ q ´ λ p q ˘ Γ ` p λ q ´ λ p q ˘ with p λ , . . . , λ i q “ p ξ , . . . , ξ i q P D k p Θ i q as a function of λ i ` in(4.25) ξ i ď Re λ i ` ď ¨ ¨ ¨ ď Re λ r ď . Here C is a non-zero constant. Thus possible singularities come from sin-gularities of(4.26) Γ ` p´ λ i ` ` α ´ | β | ` q ˘ i ź q “ Γ ` p ξ q ´ λ i ` ` k m q ˘ Γ ` p ξ q ´ λ i ` q ˘ as a function of λ i ` for Re λ i ` ě ξ i .If k m P N , then Γ ` p ξ q ´ λ i ` ` k m q ˘ Γ ` p ξ q ´ λ i ` q ˘ q “ ´ k m p ξ q ´ λ i ` q ¨ ¨ ¨ p ξ q ´ λ i ` ` k m ´ q . (The right hand side is understood to be 1 if k m “ c Θ i p´ λ, k q ´ with p λ , . . . , λ i q “ p ξ , . . . , ξ i q P D k p Θ i q as a function of λ i ` in (4.25) are singularities of Γ ` p´ λ i ` ` α ´ | β | ` q ˘ p ξ i ´ λ i ` q ¨ ¨ ¨ p ξ i ´ λ i ` ` k m ´ q for Re λ i ` ě ξ i . Thus there are simple poles for λ i ` with p ξ i ´ λ i ` ` k m q P ´ N , which prove our assertion.If k m R N , then we can write (4.26) as a product of(4.27) Γ ` p´ λ i ` ` α ´ | β | ` q ˘ Γ ` p ξ ´ λ i ` q ˘ i ´ ź q “ Γ ` p ξ q ´ λ i ` ` k m q ˘ Γ ` p ξ q ` ´ λ i ` q ˘ and(4.28) Γ ` p ξ i ´ λ i ` ` k m q ˘ . Since λ satisfies p λ , . . . , λ i q “ p ξ , . . . , ξ i q P D k p Θ i q and (4.25), (4.27) isregular for Re λ i ` ě ξ i , hence the singularities come from (4.28). Thusthere are simple poles for λ i ` with p ξ i ´ λ i ` ` k m q P ´ N , which proveour assertion. NVERSION FORMULA FOR THE HYPERGEOMETRIC FOURIER TRANSFORM 19
For 1 ď i ď r and ξ P D k p Θ i q let I Θ i p ξ, λ a Θi , k q “ p´ q i p π ?´ q r ´ i p r ´ i q ! ˆ ` φ p λ q F Ξ p λ, k ; x q c Ξ p λ, k q ´ ˘ˇˇ λ p Θ i q“ ξ ˆ Res λ i “ ξ i ¨ ¨ ¨ Res λ “ ξ Res λ “ ξ c p´ λ, k q ´ and(4.29) φ _ Θ i p x q “ ÿ ξ P D k p Θ i q ż ?´ a ˚ Θi I Θ i p ξ, λ a Θi , k q dλ a Θi . We prove by induction on i that J k φ p x q ´ J k , H φ p x q ´ ř i ´ j “ φ _ Θ j p x q equals(4.30) ÿ ξ P D k p Θ i q ż ˇ ξ i `?´ a ˚ Θi I Θ i p ξ, λ a Θi , k q dλ a Θi . Here ˇ ξ i “ p ξ i , . . . , ξ i q P a ˚ Θ i . We already proved the case of i “
1. We firstchange the integral domain ˇ ξ i `?´ a ˚ Θ i of (4.30) to p ξ i , ξ i ` ε, . . . , ξ i `p r ´ j ´ q ε q ` ?´ a ˚ Θ i , next to ?´ a ˚ Θ i of (4.29) and pick up residues. As we haveseen above, the singularities of I Θ i p ξ, λ a Θi , k q that concern are λ j “ ξ i ` such that p ξ , . . . , ξ i , ξ i ` q P D k p Θ i ` q and i ` ď j ď r . By Lemma 1.1,Lemma 1.2, and (2.5), I Θ i p ξ, λ a Θi , k q as a function of λ a Θi P a ˚ Θ i , C with (4.25)behaves suitably as | Im λ a Θi | goes to infinity. By Cauchy’s residue theoremand changes of variables, the difference of (4.30) and (4.29) is (4.30) with i replaced by i `
1. Thus we have proved that J k φ p x q ´ J k , H φ p x q “ r ÿ i “ φ _ Θ i p x q . If i “ r and D k p B q “ H , then there remains indeed no integral sign and φ _ B p x q give a point spectra.By Proposition 4.3 and Proposition 4.5, φ _ Θ i “ J k ,Θ i φ holds on a ` for any 1 ď i ď r and the theorem follows. Here we use the fact that c Θ i p λ, k q c Θ i p´ λ, k q “ | c Θ i p λ, k q| for any λ P D k p Θ i q ` ?´ a ˚ Θ i (cf. [29,Lemma 6.6]). (cid:3) Proposition 4.5.
Assume k P K satisfies (3.1) and let ď i ď r . For ξ P D k p Θ i q , (4.31) p´ q i Res λ i “ ξ i ¨ ¨ ¨ Res λ “ ξ p c Θ i p λ, k q ´ c Θ i p´ λ, k q ´ q “ d Θ i p ξ, k q . Proof.
Note that in each step the residue is taken for a single Gamma func-tion in c α p´ λ, k q ´ for some α P R ` . We recall the following formulas Γ p z ` q “ zΓ p z q , (4.32) Γ p z q Γ p ´ z q “ π sin πz , (4.33) Res z “ m Γ ´ ´ z ¯ “ p´ q m ` m ! “ p´ q m ` Γ p m ` q p m P N q (4.34) HONDA , HIROSHI
ODA , AND NOBUKAZU
SHIMENO for the Gamma function.We prove (4.31) by induction on i . First we consider the case of i “ ξ “ α ´ | β | ` ` m P D k p Θ q p m P N , ξ ă q . By (3.6) and (4.34),Res λ “ ξ ˜ c p´ λ, k q ´ “ Res λ “ ξ α ´ β ´ λ ´ Γ ` p´ λ ` α ` | β | ` q ˘ Γ ` p´ λ ` α ´ | β | ` q ˘ ? π Γ p´ λ q“ p´ q m ` m ! 2 ´ β `| β |´ m ´ Γ p| β | ´ m q? π Γ p´ α ` | β | ´ ´ m q . By (4.32) and (4.33), ´ ˜ c p ξ , k q ´ Res λ “ ξ ˜ c p´ λ, k q ´ (4.35) “ p´ q m m ! 2 α ´ β ´ Γ p| β | ´ m q Γ p α ` m ` q Γ p α ´ | β | ` m ` q π Γ p´ α ` | β | ´ ´ m q Γ p α ´ | β | ` m ` q . “ p´ q m sin p π p´ α ` | β | ´ m qq sin p π p´ α ` | β | ´ m qˆ α ´ β ´ p´ α ` | β | ´ m ´ q Γ p α ` m ` q Γ p| β | ´ m q π m ! Γ p´ α ` | β | ´ m q“ α ´ β ´ p´ α ` | β | ´ m ´ q Γ p α ` m ` q Γ p| β | ´ m q π m ! Γ p´ α ` | β | ´ m q“ ´ α ´ β ´ ξ π Γ ` p ξ ` α ` | β | ` q ˘ Γ ` p´ ξ ` α ` | β | ` q ˘ Γ ` p ξ ´ α ` | β | ` q ˘ Γ ` p´ ξ ´ α ` | β | ` q ˘ . Thus we proved (4.31) for i “ i and let ξ “ p ξ , ξ i ` q P D k p Θ i ` q with ξ P D k p Θ i q . We will prove ´ ˜ c Θ i ` p ρ p k q , k q ´ d Θ i ` p ξ, k q ˜ c Θ i p ρ p k q , k q ´ d Θ i p ξ , k q (4.36) “ Res λ i ` “ ξ i ` ź α P x Θ i ` y ` z x Θ i y ˜ c α pp ξ , λ i ` q , k q ´ ˜ c α pp´ ξ , ´ λ i ` q , k q ´ . As we see in the proof of Theorem 4.1, if k m R N , then the residue on the righthand side of (4.36) is taken for ˜ c p β i ` ´ β i q pp´ ξ , ´ λ i ` q , k q ´ , otherwise for˜ c i ` pp´ ξ , ´ λ i ` q , k q ´ .First assume k m R N . The set x Θ i ` y ` z x Θ i y consists of roots β i ` , β i ` , p β i ` ˘ β j q p ď j ď i q . Since ξ i ` P α ´ | β | ` i k m ` N , the contributionsof β i ` and β i ` to the right hand side of (4.36) is˜ c i ` p ξ, k q ´ ˜ c i ` p´ ξ, k q ´ (4.37) “ ´ α ´ β ´ sin π ξ i ` sin π p ξ i ` ` q sin π p´ ξ i ` ` α ´ | β | ` q sin π p ξ i ` ` α ´ | β | ` qˆ ξ i ` Γ ` p ξ i ` ` α ` | β | ` q ˘ Γ ` p´ ξ i ` ` α ` | β | ` q ˘ Γ ` p ξ i ` ´ α ` | β | ` q ˘ Γ ` p´ ξ i ` ´ α ` | β | ` q ˘ NVERSION FORMULA FOR THE HYPERGEOMETRIC FOURIER TRANSFORM 21 “ α ´ β ´ sin π p α ´ | β | ` i k m ` q sin πi k m sin π p α ´ | β | ` i k m ` qˆ ξ i ` Γ ` p ξ i ` ` α ` | β | ` q ˘ Γ ` p´ ξ i ` ` α ` | β | ` q ˘ Γ ` p ξ i ` ´ α ` | β | ` q ˘ Γ ` p´ ξ i ` ´ α ` | β | ` q ˘ . Since ξ j ` ´ ξ j P k m ` N for 1 ď j ď i , the contributions of p β i ` ´ β j q p ď j ď i ´ q to the right hand side of (4.36) is˜ c p β i ` ´ β j q p ξ, k q ´ ˜ c p β i ` ´ β j q p´ ξ, k q ´ “ sin π p ξ i ` ´ ξ j q sin π p ξ i ` ´ ξ j ´ k m q ξ i ` ´ ξ j Γ ` p ξ i ` ´ ξ j ` k m q ˘ Γ ` p ξ i ` ´ ξ j ´ k m ` q ˘ “ sin π p i ´ j ` q k m sin π p i ´ j q k m ξ i ` ´ ξ j Γ ` p ξ i ` ´ ξ j ` k m q ˘ Γ ` p ξ i ` ´ ξ j ´ k m ` q ˘ , so i ´ ź j “ ˜ c p β i ` ´ β j q p ξ, k q ´ ˜ c p β i ` ´ β j q p´ ξ, k q ´ (4.38) “ sin πi k m sin π k m i ´ ź j “ ξ i ` ´ ξ j Γ ` p ξ i ` ´ ξ j ` k m q ˘ Γ ` p ξ i ` ´ ξ j ´ k m ` q ˘ . Similarly, the contribution of p β i ` ` β j q p ď j ď i q to the right hand sideof (4.36) is i ź j “ ˜ c p β i ` ` β j q p ξ, k q ´ ˜ c p β i ` ` β j q p´ ξ, k q ´ (4.39) “ sin π p α ´ | β | ` i k m ` q sin π p α ´ | β | ` i k m ` qˆ i ź j “ ´ ξ i ` ´ ξ j Γ ` p´ ξ i ` ´ ξ j ` k m q ˘ Γ ` p´ ξ i ` ´ ξ j ´ k m ` q ˘ . The contributions of p β i ` ´ β i q to the right hand side of (4.36) is˜ c p β i ` ´ β i q p ξ, k q ´ Res λ i ` “ ξ i ` ˜ c p β i ` ´ β i q pp´ ξ , ´ λ i ` q , k q ´ (4.40) “ π sin π k m ξ i ` ´ ξ i Γ ` p ξ i ` ´ ξ i ` k m q ˘ Γ ` p ξ i ` ´ ξ i ´ k m ` q ˘ . By (4.37), (4.38), (4.39), and (4.40), the sine functions cancel each other outand (4.36) holds for i ` k m P N . Then the contribution of p β i ` ˘ β j q p ď j ď i q to the right hand side of (4.36) is˜ c p β i ` ˘ β j q p ξ, k q ´ ˜ c p β i ` ˘ β j q p´ ξ, k q ´ (4.41) “ p´ q k m ´ ξ i ` ¯ ξ j Γ ` p ξ i ` ˘ ξ j ` k m q ˘ Γ ` p ξ i ` ˘ ξ j ´ k m ` q ˘ . HONDA , HIROSHI
ODA , AND NOBUKAZU
SHIMENO
By (4.35), the contributions of β i ` and β i ` to the right hand side of(4.36) is ˜ c i ` p ξ, k q ´ Res λ i ` “ ξ i ` ˜ c i ` pp´ ξ , ´ λ i ` q , k q ´ “ ´ α ´ β ´ ξ i ` π (4.42) ˆ Γ ` p ξ i ` ` α ` | β | ` q ˘ Γ ` p´ ξ i ` ` α ` | β | ` q ˘ Γ ` p ξ i ` ´ α ` | β | ` q ˘ Γ ` p´ ξ i ` ´ α ` | β | ` q ˘ . By (4.41) and (4.42), (4.36) holds for i ` (cid:3) Theorem 4.6 (Paley-Wiener theorem) . Assume k P K satisfies (3.1) . Thenthe hypergeometric Fourier transform F k is a bijection of C p a q W onto PW p a C q W .Proof. We already see in Section 2 that the image is contained in PW p a C q W .Thus it remains to prove the surjectivity. Let f p x q “ J k φ p x q (cf. (4.19)).By sending ´ η to infinity in a ` , we can see as in the proof of [14, Ch IV,Theorem 7.3] that f is a compactly supported function on a . In the proofof Theorem 4.1 we see that f p x q “ J k φ p x q “ ř ri “ J k ,Θ i φ p x q . Thus f is a C -function by Theorem 1.4, Lemma 1.2, and (2.5).If k P K ` , then F k f “ φ by [23, Lemma 9.12]. Since F k f “ F k J k φ depends analytically on k , we have F k f “ φ for k P K satisfying (3.1). (cid:3) Let ν k denote the measure on Ů ri “ p D k p Θ i q ` ?´ a ˚ Θ i q given by ż Ů ri “ p D k p Θ i q`?´ a ˚ Θi q ψ p λ q dν k p λ q “ r ÿ i “ ż D k p Θ i q`?´ a ˚ Θi ψ p λ q dν k ,Θ i p λ q . Theorem 4.7 (Plancherel theorem) . Assume k P K satisfies (3.1) . Thenfor f P C c p a q W , (4.43) 1 | W | ż a | f p x q| δ k p x q dx “ ż Ů ri “ p D k p Θ i q`?´ a ˚ Θi q | F k f p λ q| dν k p λ q . Moreover, the hypergeometric Fourier transform F k extends to an isometryof L p a ; | W | δ k p x q dx q onto L p Ů ri “ p D k p Θ i q ` ?´ a ˚ Θ i q ; dν k q .Proof. By Theorem 4.1, Lemma 1.1, Lemma 1.2, and (2.5), and Fubini’stheorem,1 | W | ż a | f p x q| δ k p x q dx “ | W | ż a f p x q p J k F k f qp x q δ k p x q dx “ ż Ů ri “ p D k p Θ i q`?´ a ˚ Θi q | F k f p λ q| dν k p λ q . Thus F k extends to an isometry of L p a ; | W | δ k p x q dx q into L p Ů ri “ p D k p Θ i q`?´ a ˚ Θ i q ; dν k q . The surjectivity of F k can be proved as in the proof of [24,Theorem 5.5]. (cid:3) Corollary 4.8.
We assume that k P K satisfies the condition (3.1) . Thenthe hypergeometric function F p λ, k q is square integrable if and only if λ P NVERSION FORMULA FOR THE HYPERGEOMETRIC FOURIER TRANSFORM 23
W D k p B q . Each square integrable hypergeometric function is of the form (4.44) F p λ, k q “ c p λ, k q Φ p λ, k q for some λ P D k p B q . For any λ P D k p B q , (4.45) 1 | W | ż a F p λ, k ; x q δ k p x q dx “ d B p λ, k q . For any λ, µ P D k p B q with λ “ µ , (4.46) 1 | W | ż a F p λ, k ; x q F p µ, k ; x q δ k p x q dx “ . Proof.
By Corollary 3.3, we already know that F p λ, k q is square integrablefor λ P D k p B q . It remains to prove that they exhaust the square integrablehypergeometric functions.Assume that F p µ, k q is square integrable for some µ P a ˚ C . Then for D P D p k q , γ k p D qp µ q ż a F p µ, k q F p λ, k q δ k p x q dx “ ż a p DF p µ, k qq F p λ, k q δ k p x q dx “ ż a F p µ, k qp DF p λ, k qq δ k p x q dx “ γ k p D qp λ q ż a F p µ, k q F p λ, k q δ k p x q dx. Here we used the fact that each elements of D p k q is symmetric with respectto δ k p x q dx , which can be proved by the construction of D p k q in terms oftrigonometric Dunkl operators (cf. [10], [23]). Thus p p p µ q ´ p p λ qq ż a F p µ, k q F p λ, k q δ k p x q dx “ p P S p a C q W . Therefore, if λ R W µ , then F k p F p µ, k qqp λ q “ | W | ż a F p µ, k q F p λ, k q δ k p x q dx “ . By Theorem 4.7, µ “ wλ for some w P W and λ P D k p B q , and1 | W | ż a F p λ, k ; x q δ k p x q dx “ ˆ | W | ż a F p λ, k ; x q δ k p x q dx ˙ d B p λ, k q . Thus (4.45) follows. (cid:3)
Remark . The square integrable hypergeometric functions are analyticcontinuation of the Jacobi polynomials. This fact was observed by [29,Remark 5.12] for the group case and mentioned without proof in [3, §6].If β ă
0, then λ P D k p B q if and only if λ r ă µ “ λ ´ ρ p k q satisfies µ P N , µ i ` ´ µ i P N p ď i ď r ´ q . Thus, D k p B q can be written as D k p B q “ t µ ` ρ p k q P a ˚ ; µ ` ρ p k q P ´ w ˚ Ξ a ˚` (4.47) and x µ, α _ y P N for all α P R ` u . Here w ˚ Ξ denote the longest element of W Ξ “ S r . HONDA , HIROSHI
ODA , AND NOBUKAZU
SHIMENO If k P K ` and µ P a ˚ satisfies x µ, α _ y P N for all α P R ` , then Φ p µ ` ρ p k q , k q “ P p µ, k q , which is the Jacobi polynomial of the highest weight µ (cf. [12, 9, 10]). Note k P K ` if and only if α ě ´ , β ě ´ , and k m ě F p µ ` ρ p k q , k q “ c p µ ` ρ p k q , k q P p µ, k q forms a one parameterfamily of polynomials with analytic parameter β and if β ă ´ α ´ p r ´ q k m ´ , then D k p B q “ H and F p µ ` ρ p k q , k q p µ ` ρ p k q P D k p B qq is square in-tegrable. On the one hand, for k P K ` the L -norm of the constantmultiple c p µ ` ρ p k q , k q P p µ, k q of the Jacobi polynomial is an integral of c p µ ` ρ p k q , k q P p µ, k q δ k over a compact torus and its explicit formula isgiven by [10, Corollary 3.5.3]. On the other hand, for k satisfying (3.1)and µ ` ρ p k q P D k p B q the L -norm of F p µ ` ρ p k q , k q is an integral of | W | F p µ ` ρ p k q , k q δ k over a and its explicit formula is given by (4.45) and(4.1). Comparing these formulas we can observe that the two norms coincideup to a constant multiple that depends only on k .We can reduce the case of β ą β ă
0. We introduce themultiplicity function ˜ k associated with k , which is defined by˜ k s “ k s ` k ℓ ´ , ˜ k m “ k m , ˜ k ℓ “ ´ k ℓ . In terms of α , β , associated ˜ α , ˜ β are given by˜ α “ α , ˜ β “ ´ β . By [10, Theorem 2.1.1], δ k ˝ p L p k q ` x ρ p k q ,ρ p k qyq ˝ δ ´ k “ δ ˜ k ˝ p L p ˜ k q ` x ρ p ˜ k q , ρ p ˜ k qyq ˝ δ ; ˜ k ´ . By the characterization of the hypergeometric functions, we have(4.48) F p λ, k q “ r ź i “ ´ cosh β i ¯ ´ k ℓ F p λ, ˜ k q . For the elementary spherical functions associated with a one-dimensional K -type, the above formula was given by [10, Theorem 5.2.2] and [29, Propo-sition 2.6, Remark 3.8]. References [1] J. P. Anker,
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