Intertwining Operators Associated to a Family of Differential-Reflection Operators

Abstract

We introduce a family of differential-reflection operators \({\Lambda_{A, \varepsilon}}\) acting on smooth functions defined on \({{\mathbb R}}\). Here, A is a Sturm–Liouville function with additional hypotheses and \({-1 \leq \varepsilon \leq 1}\). For special pairs \({(A, \varepsilon),}\) we recover Dunkl’s, Heckman’s and Cherednik’s operators (in one dimension). The spectral problem for the operators \({\Lambda_{A, \varepsilon}}\) is studied. In particular, we obtain suitable growth estimates for the eigenfunctions of \({\Lambda_{A, \varepsilon}}\). As the operators \({\Lambda_{A, \varepsilon}}\), are a mixture of \({{\rm d}/{\rm d}x}\) and reflection operators, we prove the existence of an intertwining operator \({V_{A, \varepsilon}}\) between \({\Lambda_{A, \varepsilon}}\) and the usual derivative. The positivity of \({V_{A, \varepsilon}}\) is also established.