Invariants of class $C^k$ of finite Coxeter groups and their representation in terms of anisotropic spaces

A. O. Gokhman

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 25, Tome 247 (1997), pp. 46-70

Résumé

The article is devoted to the study of representation of $C^k(\mathbb R^n)$-smooth functions $f$ invariant with respect to finite Coxeter groups $W$ in the form $f=F\,\circ\,p$, where $p$ is a base in the algebra of $W$-invariant polynomials. We examine the drop of smoothness of $F$ as compared with $f$ and conclude that this drop has anisotropic nature and that, more precisely, at each point $p_0$ it is described by a vector $\bar\mu(p_0)\in\mathbb R^n$. We examine the cases $W=A_n$, $B_n$, $D_n$, $\mathfrak D_m$; in each case the greatest component $\mu_j$ of $\bar\mu$ is equal to the Coxeter number of the stabilizer $W_{y_0}$ of the point $y_0$, where $p_0=p(y_0)$.