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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 $\alpha \in R$ let ${\alpha }^{x}:=2\alpha /<\alpha ,\alpha >$ and denote the associated reflection ${r}_{\alpha }$, where ${r}_{\alpha }\left(\lambda \right):=\lambda -<{\alpha }^{x},\lambda >\alpha$, $\lambda \in E$. By hypothesis, for each $\alpha ,\beta \in R$, $<\alpha ,\beta >\in ℤ$ and ${r}_{\alpha }\left(\beta \right)\in R$. Further $W=W\left(R\right)$ is a group generated by $\left\{{r}_{\alpha }:\alpha \in R\right\}$ and $ℝ\left[P\right]$ is the group algebra of the free abelian group (the weight lattice) $P=\left\{\lambda \in E:$ $<\lambda ,{\alpha }^{x}>\in ℤ$ for all $\alpha \in R\right\}$. Denote the basis elements of $ℝ\left[P\right]$ by ${e}^{\lambda }$ so that ${e}^{0}=1$, ${\left({e}^{\lambda }\right)}^{-1}={e}^{-\lambda }$, ($\lambda \in P\right)$, and define $w\left({e}^{\lambda }\right)={e}^{w\left(\lambda \right)}$ for $w\in W$. Then $ℝ{\left[P\right]}^{W}$ denotes the algebra of W-invariants in $ℝ\left[P\right]$. The main problem is to describe orthogonality structures on $ℝ{\left[P\right]}^{W}$ and produce explicit orthogonal bases, (this includes group characters, spherical functions, and Jack polynomials). These structures are parametrized by “multiplicity functions” $k=\left\{{k}_{\alpha }:\alpha \in R\right\}$ $\left({k}_{\alpha }$ is a positive integer with possibly some extra restrictions) where ${k}_{\alpha }={k}_{\beta }$ when $\alpha$ and $\beta$ are roots with the same length.

The differential-difference operators are generated by $\left\{{D}_{\xi }:\xi \in E\right\}$ where

${D}_{\xi }\left({e}^{\lambda }\right):=<\xi ,\lambda >{e}^{\lambda }+\sum _{\alpha \in {R}_{+}}{k}_{\alpha }<\xi ,\alpha >$

$\left(\left(1-{e}^{-\alpha }\right)/\left(1-{e}^{-\alpha }\right)\right)\left({e}^{\lambda }-{e}^{{r}_{\alpha }\left(\lambda \right)}\right)$, for $\lambda \in P$, extended by linearity to $ℝ\left[P\right]$. Heckman shows that each ${D}_{\xi }$ is selfadjoint for the inner product

${\left(f,g\right)}_{k}=CT\left(f\overline{g}\prod _{\alpha \in R}{\left({e}^{\alpha /2}-{e}^{-\alpha /2}\right)}^{{k}_{\alpha }}\right),\phantom{\rule{1.em}{0ex}}f,g\in ℝ\left[P\right],$

where (constant term) $CT\left({\sum }_{\lambda }{a}_{\lambda }{e}^{\lambda }\right)={a}_{0}$, and $\overline{g}:={\sum }_{\lambda }{g}_{\lambda }{e}^{-\lambda }$ when $g={\sum }_{\lambda }{g}_{\lambda }{e}^{\lambda }$. For $d=1,2,3,··$. $\xi \in E$, the operator ${D}_{\xi ,d}:={\sum }_{\eta \in {W}_{\xi }}{D}_{\eta }^{d}$ is W-invariant and reduce to a differential operator when restricted to $ℝ{\left[P\right]}^{W}$. 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 $\left(q=1\right)$ conjectures and also determined the ${L}^{2}$-norms of the Jacobi polynomials.

##### MSC:
 33C80 Connections of hypergeometric functions with groups and algebras 17B20 Simple, semisimple, reductive Lie (super)algebras 22E30 Analysis on real and complex Lie groups
root systems
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