Orthogonal polynomials and reflection groups (Q2760152)

In this survey article, the author describes his work on orthogonal polynomials of several variables which are invariant under reflection groups. After presenting some motivating classical one-variable and multivariable examples (e.g. Hermite and Laguerre polynomial and their multivariable versions), he introduces a commuting family of first order differential-difference operators \({\mathcal D}_i\) associated with a reduced root system \(R\) and a multiplicity function \(k\) on \(R\). The \({\mathcal D}_i\), commonly known after the author as Dunkl operators, can be considered as a deformation of the usual partial derivatives, which correspond to the limit \(k \rightarrow 0\). Connection with inner product structure are given. In particular, the \(h\)-harmonic polynomials are introduced as analogues of the spherical harmonics corresponding to a certain weight function \(h\) which is invariant under the Weyl group of \(R\). The analogue \(K(x,y)\) of the exponential function \(\exp(<x,y>)\) (with \(x,y \in \mathbb R^d\)) is constructed by means of certain intertwining operators. Integration against \(K(x,-iy)\) yields a generalized Fourier transform which can be considered as the spectral resolution of the operators \({\mathcal D}_i\). The \(h\)-harmonic polynomials can be used to describe an orthogonal basis for \(L^2(\mathbb R^d, h(x)^2 dx)\). NEWLINENEWLINENEWLINEProofs are omitted in this article, but precise references are provided. See e.g. the recent monograph by \textit{C. F. Dunkl} and \textit{Y. Xu} [Orthogonal polynomials of several variables, Encyclopedia of Mathematics and Its Applications. 81. Cambridge: Cambridge University Press (2001; Zbl 0964.33001)].NEWLINENEWLINEFor the entire collection see [Zbl 0969.00053].