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A remark on the Dunkl differential-difference operators. (English) Zbl 0749.33005

Harmonic analysis on reductive groups, Proc. Conf., Brunswick/ME (USA) 1989, Prog. Math. 101, 181-191 (1991).
[For the entire collection see Zbl 0742.00061.]
For each root system \(R\) in an Euclidean space \(E\) of dimension \(n\) there exists a commutative algebra, with \(n\) generators, of differential- difference operators which act as differential operators on the polynomials invariant under the Coxeter group \(W\) generated by \(R\). For \(\xi\neq 0\) in \(E\) the corresponding (first-order) operator is defined by \[ D_ \xi p(\lambda):={d\over dt}p(\lambda+t\xi)\mid_{t=0}+\sum_{\alpha\in R_ +}k_ \alpha\langle\alpha,\;\xi\rangle{p(\lambda)-p(r_ \alpha\lambda)\over\langle\alpha,\lambda\rangle}, \] where \(R_ +\) is the set of positive roots, \(r_ \alpha\) is the reflection along \(\alpha\) (so that \(r_ \alpha\lambda:=\lambda- 2(\langle\alpha,\lambda\rangle/\langle\alpha,\alpha\rangle)\alpha,\;\lambda\in E)\) and \(\langle\alpha,\lambda\rangle\) denotes the inner product in \(E\); the parameters (“multiplicity function”) \(k_ \alpha\) satisfy \(k_ \alpha=k_{w\alpha}\) for all \(w\in W\).
The reviewer [Trans. Am. Math. Soc. 311, No. 1, 167-183 (1989; Zbl 0652.33004)] constructed these operators and proved \(D_ \xi D_ \eta=D_ \eta D_ \xi\) for all \(\xi,\eta\in E\).
The present paper contains an exposition of the proof for commutativity. The author shows that the ring of \(W\)-invariant operators in the algebra generated by \(\{D_ \xi:\xi\in E\}\), when restricted to invariant polynomials, is a commutative algebra of differential operators with \(n\) generators (corresponding to the fundamental invariants of \(W\)). One of these is the Laplacian \(L(k):=\sum^ n_{j=1}D^ 2_{\xi_ j}\), where \(\{\xi_ j:j=1,\ldots,n\}\) is any orthonormal basis for \(E\).
This gives a proof of the complete integrability of the generalized nonperiodic Calogero-Moser system. Also the author uses the operators \(D_ \xi\) to construct the Opdam shift operators [E. M. Opdam, Invent. Math. 98, No. 1, 1-18 (1989; Zbl 0696.33006)] which intertwine the Laplacians \(L(k)\) and \(L(k')\) for contiguous multiplicity functions, that is \(k_ \alpha'=k_ \alpha\) or \(k_ \alpha+1\).

MSC:

33C80 Connections of hypergeometric functions with groups and algebras, and related topics
20F55 Reflection and Coxeter groups (group-theoretic aspects)
13N10 Commutative rings of differential operators and their modules
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