Vector-Valued Polynomials and a Matrix Weight Function with B2-Action - Charles F. Dunkl (Symmetry, Integrability and Geometry: Methods and Applications, 2013)
The structure of orthogonal polynomials on $mathbbR^{2}$ with the weight function $vert x_{1}^{2}-x_{2}^{2}vert ^{2k_{0}}vertx_{1}x_{2}vert ^{2k_{1}}e^{-( x_{1}^{2}+x_{2}^{2})/2}$ is based on the Dunkl operators of type $B_{2}$. This refers to the full symmetry group of the square, generated by reflections in the lines $x_{1}=0$ and $x_{1}-x_{2}=0$. The weight function is integrable if $k_{0},k_{1},k_{0} +k_{1}>-frac{1}{2}$. Dunkl operators can be defined for polynomials taking values in a module of the associated reflection group, that is, a vector space on which the group has an irreducible representation. The unique $2$-dimensional representation of the group $B_{2}$ is used here. The specific operators for this group and an analysis of the inner products on the harmonic vector-valued polynomials are presented in this paper. An orthogonal basis for the harmonic polynomials is constructed, and is used to define an exponential-type kernel. In contrast to the ordinary scalar case the inner product structure is positive only when $( k_{0},k_{1})$ satisfy $-frac{1}{2} < k_{0}pm k_{1} < frac{1}{2}$. For vector polynomials $(f_{i})_{i=1}^{2}$, $(g_{i})_{i=1}^{2}$ the inner product has the form $iint_{mathbbR^{2}}f(x) K(x) g(x)^{T}e^{-( x_{1}^{2}+x_{2}^{2})/2}dx_{1}dx_{2}$ where the matrix function $K(x)$ has to satisfy various transformation and boundary conditions. The matrix $K$ is expressed in terms of hypergeometric functions.