### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 007, 23 pages      arXiv:1210.1177      https://doi.org/10.3842/SIGMA.2013.007

### Vector-Valued Polynomials and a Matrix Weight Function with B2-Action

Charles F. Dunkl
Department of Mathematics, University of Virginia, PO Box 400137, Charlottesville VA 22904-4137, USA

Received October 16, 2012, in final form January 23, 2013; Published online January 30, 2013

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

Key words: matrix Gaussian weight function; harmonic polynomials.

pdf (494 kb)   tex (116 kb)

References

1. Carter R.W., Finite groups of Lie type. Conjugacy classes and complex characters, Wiley Classics Library, John Wiley & Sons Ltd., Chichester, 1993.
2. Dunkl C.F., Differential-difference operators and monodromy representations of Hecke algebras, Pacific J. Math. 159 (1993), 271-298.
3. Dunkl C.F., Monodromy of hypergeometric functions for dihedral groups, Integral Transform. Spec. Funct. 1 (1993), 75-86.
4. Dunkl C.F., Opdam E.M., Dunkl operators for complex reflection groups, Proc. London Math. Soc. 86 (2003), 70-108, math.RT/0108185.
5. Dunkl C.F., Xu Y., Orthogonal polynomials of several variables, Encyclopedia of Mathematics and its Applications, Vol. 81, Cambridge University Press, Cambridge, 2001.
6. Etingof P., Stoica E., Unitary representations of rational Cherednik algebras, Represent. Theory 13 (2009), 349-370, arXiv:0901.4595.
7. Griffeth S., Orthogonal functions generalizing Jack polynomials, Trans. Amer. Math. Soc. 362 (2010), 6131-6157, arXiv:0707.0251.
8. Olver F.W.J., Lozier D.W., Boisvert R.F., Clark C.W. (Editors), NIST handbook of mathematical functions, U.S. Department of Commerce National Institute of Standards and Technology, Washington, DC, 2010.