Vector-Valued Polynomials and a Matrix Weight Function with $B_2$-Action

The structure of orthogonal polynomials on ${\mathbb{R}}^2$ with the weight function $| x_{1}^{2}-x_{2}^{2}|^{2k_{0}}| x_{1}x_{2}|^{2k_{1}}e^{-(x_{1}^{2}+x_{2}^{2})/2}$ 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 $\iint_{$\mathbb{R}$^{2}}f(x) K(x) g(x)^{T}e^{-(x_{1}^{2}+x_{2}^{2})/2}dx_{1}dx_{2}$wherethematrixfunction$K(x)$ has to satisfy various transformation and boundary conditions. The matrix $K$ is expressed in terms of hypergeometric functions.