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

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 $p_0=p(y_0)$.