Harmonic polynomials and peak sets of reflection groups (Q1826003)

For a finite reflection group G, considered as a subgroup of the orthogonal group on \({\mathbb{R}}^ N\), let \(\{v_ j:\) \(1\leq j\leq m\}\) be the set of positive roots. Further choose parameters \(\alpha_ j\), \(1\leq j\leq m\) so that \(\alpha_ i=\alpha_ j\) whenever \(wv_ i=v_ j\) for some \(w\in G\). This paper deals with the peak set E of \[ h(x)=\prod^{m}_{j=1}| <x,v_ j>|^{\alpha_ j}, \] that is the set where h achieves its maximum on the unit sphere: \(\{x: <x,x>=1\}\) in \({\mathbb{R}}^ N\). I. Schur and T. Stieltjes showed that the peak sets for groups of the types \(A_ N\), \(B_ N\), and \(D_ N\) are related to zeros of Hermite and Laguerre polynomials. The peak set for \(F_ 4\) (a one-parameter family) is determined in this paper. By use of differential-difference operators [the author, Trans. Am. Math. Soc. 311, 167-183 (1989; Zbl 0652.33004)] a Laplacian is defined on the space of functions on E. This leads to a structure of harmonic polynomials and orthogonality. Further the co-invariant algebra of G is faithfully represented as an algebra of operators on this function space.