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

Abstract

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 ${\iint }_{{\mathbb{R}}^2}f(x)K(x)g(x{)}^T{e}^{-(x_1^2+x_2^2)/2}d{x}_{1}d{x}_2$ where the matrix function $K(x)$ has to satisfy various transformation and boundary conditions. The matrix $K$ is expressed in terms of hypergeometric functions.

This is the expanded version of an invited lecture presented at the Conference on Harmonic Analysis, Convolution Algebras, and Special Functions, TU München, 10 Sept. 2012; v2: the construction of an exponential-type function has been added; v3: published version

References in corpus (1)

• Unitary representations of rational Cherednik algebras