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An elementary approach to the hypergeometric shift operators of Opdam. (English) Zbl 0721.33009

This article is an important development in analysis on root systems. The author and E. M. Opdam [Compos. Math. 64, 329-352, 353-373 (1987; Zbl 0656.17006 and Zbl 0656.17007); ibid. 67, 21-49, 191-209 (1988; Zbl 0669.33007 and Zbl 0669.33008)] have constructed generalized hypergeometric functions and Jacobi polynomials associated to root systems of Weyl groups. Special cases of these functions are related to classical orthogonal polynomials and spherical functions on Riemannian symmetric spaces. The basic objects of concern are Weyl group invariant differential operators and functions on Euclidean space which are periodic modulo the root lattice. The first constructions for the operators were technically difficult and frequently required case-by-case analysis. In this paper, Heckman constructs an algebra of differential- difference operators for any root system, which very naturally lead to the desired differential operators. The reviewer [Trans. Am. Math. Soc. 331, No.1, 167-183 (1989; Zbl 0652.33004)] first introduced differential- difference oprators in the analysis of orthogonality structures with a Coxeter group invariance for functions on the sphere.

Suppose R is a root system on an Euclidean space E (with R + denoting the positive roots). For αR let α x :=2α/<α,α> and denote the associated reflection r α , where r α (λ):=λ-<α x ,λ>α, λE. By hypothesis, for each α,βR, <α,β> and r α (β)R. Further W=W(R) is a group generated by {r α :αR} and [P] is the group algebra of the free abelian group (the weight lattice) P={λE: <λ,α x > for all αR}. Denote the basis elements of [P] by e λ so that e 0 =1, (e λ ) -1 =e -λ , (λP), and define w(e λ )=e w(λ) for wW. Then [P] W denotes the algebra of W-invariants in [P]. The main problem is to describe orthogonality structures on [P] W and produce explicit orthogonal bases, (this includes group characters, spherical functions, and Jack polynomials). These structures are parametrized by “multiplicity functions” k={k α :αR} (k α is a positive integer with possibly some extra restrictions) where k α =k β when α and β are roots with the same length.

The differential-difference operators are generated by {D ξ :ξE} where

D ξ (e λ ):=<ξ,λ>e λ + αR + k α <ξ,α>

((1-e -α )/(1-e -α ))(e λ -e r α (λ) ), for λP, extended by linearity to [P]. Heckman shows that each D ξ is selfadjoint for the inner product

(f,g) k =CT(fg ¯ αR (e α/2 -e -α/2 ) k α ),f,g[P],

where (constant term) CT( λ a λ e λ )=a 0 , and g ¯:= λ g λ e -λ when g= λ g λ e λ . For d=1,2,3,··. ξE, the operator D ξ,d := ηW ξ D η d is W-invariant and reduce to a differential operator when restricted to [P] W . These operators now allow a neat proof for the orthogonality of the Jacobi polynomials. Further they considerably simplify the construction of E. M. Opdam’s shift operators [Invent. Math. 98, 1-18 (1989; Zbl 0696.33006)], with which he proved Macdonald’s constant-term (q=1) conjectures and also determined the L 2 -norms of the Jacobi polynomials.


MSC:
33C80Connections of hypergeometric functions with groups and algebras
17B20Simple, semisimple, reductive Lie (super)algebras
22E30Analysis on real and complex Lie groups
Keywords:
root systems
References:
[1][B] Beerends, R.: On the Abel transformation and its inversion. Proefschrift Leiden, 1987
[2][BGA] Bernstein, I.N., Gel’fand, I.M., Gel’fand, S.I.: Schubert cells and the cohomology ofG/P. Russ. Math. Surveys28, 1-26 (1973) · doi:10.1070/RM1973v028n03ABEH001557
[3][D] Debiard, A.: Polynômes de Tchébychev et de Jacobi dans un espace Euclidien de dimensionp. C.R. Acad. Sci. Paris296, 529-532 (1983)
[4][De1] Demazure, M.: Désingularisation des variétés de Schubert généralisés. Ann. Sci. Ec. Norm. Supér7, 53-88 (1974)
[5][De2] Demazure, M.: Une nouvelle formule des caractères. Bull. Soc. Math.98, 163-172 (1974)
[6][Du] Dunkl, C.F.: Differential-difference operators associated to reflection groups. Trans. AMS311, 167-183 (1989) · doi:10.1090/S0002-9947-1989-0951883-8
[7][Ha] Harish-Chandra: Spherical functions on a semisimple Lie group I, Am. J. Math.80, 553-613 (1958), or the Collected Works, Vol. 2, pp. 409-478 · doi:10.2307/2372772
[8][HO] Heckman, G.J., Opdam, E.M.: Root systems and hypergeometric functions I. Comp. Math.64, 329-352 (1987)
[9][He1] Heckman, G.J.: Root systems and hypergeometric functions II. Comp. Math.64, 353-373 (1987)
[10][He2] Heckman, G.J.: Hecke algebras and hypergeometric functions. Invent. Math.100, 403-417 (1990) · Zbl 0723.33007 · doi:10.1007/BF01231193
[11][He3] Heckman, G.J.: A remark on the Dunkl differential-difference operators, Proceedings of the Bowdoin conference on Harmonic analysis on reductive groups 1989
[12][Hel1] Helgason, S.: Differential Geometry, Lie groups and Symmetric Spaces. Academic Press: New York 1978
[13][Hel2] Helgason, S.: Groups and Geometric Analysis. Academic Press: New York 1984
[14][K] Koornwinder, T.H.: Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent differential operators, I?IV, Indag. Math.36, 48-66 and 358-381 (1974)
[15][Ma1] Macdonald, I.G.: Some conjectures for root systems. Siam J. Math. Anal.13, 988-1007 (1982) · Zbl 0498.17006 · doi:10.1137/0513070
[16][Ma2] Macdonald, I.G.: Orthogonal polynomials associated to root systems. Oxford 1988 (Preprint)
[17][Ma3] Macdonald, I.G.: Commuting differential operators and zonal spherical functions, Algebraic Groups Utrecht 1986, LNM vol. 1271, pp. 189-200
[18][Mo] Moser, J.: Three integrable systems connected with isospectral deformation. Adv. Math.16, 197-220 (1975) · Zbl 0303.34019 · doi:10.1016/0001-8708(75)90151-6
[19][O1] Opdam, E.M.: Root systems and hypergeometric functions III. Comp. Math.67, 21-49 (1988)
[20][O2] Opdam, E.M.: Root systems and hypergeometric functions IV. Comp. Math.67, 191-209 (1988)
[21][O3] Opdam, E.M.: Some applications of hypergeometric shift operators. Invent. Math.98, 1-18 (1989) · Zbl 0696.33006 · doi:10.1007/BF01388841
[22][O4] Opdam, E.M.: Generalized hypergeometric functions associated with root systems, Proefschrift Leiden 1988
[23][OP1] Olshanetsky, M.A., Perelomov, A.M.: Completely integrable systems connected with semisimple Lie algebras. Invent Math.37, 93-108 (1976) · Zbl 0342.58017 · doi:10.1007/BF01418964
[24][OP2] Olshanetsky, M.A., Perelomov, A.M.: Classical integrable finite dimensional systems related to Lie algebras. Phys. Reps. 71 (1981), 313-400. · doi:10.1016/0370-1573(81)90023-5
[25][OP3] Olshanetsky, M.A., Perelomov, A.M.: Quantum integrable systems related to Lie algebras, Phys. Reps.94, 313-400 (1983) · doi:10.1016/0370-1573(83)90018-2
[26][R] Ruijsenaars, S.N.M.: Finite-dimensional soliton systems. In: Kupershmidt, B. (ed.). Integrable and superintegrable systems. World Scientific Singapore 1990
[27][Se] Sekiguchi, J.: Zonal spherical functions on some symmetric spaces. Publ. RIMS Kyoto Univ.12, 455-459 (1977) · Zbl 0383.43005 · doi:10.2977/prims/1195196620
[28][Sp] Sprinkhuizen-Kuyper, I.G.: Orthogonal polynomials in two variables. A further analysis of the polynomials orthogonal over a region bounded by two lines and a parabola. Siam J. Math. An. 7 (4), 501-518 (1976) · Zbl 0332.33011 · doi:10.1137/0507041
[29][V] Vretare, L.: Formulas for elementary spherical functions and generalized Jacobi polynomials. Siam J. Math. An.15 (4), 805-833 (1984) · Zbl 0549.43006 · doi:10.1137/0515062