Differential Operators on $G(r,n)$-Invariant Functions

Abstract

We generalize known results on normalized symmetric coordinates and their dual differential operators, established for the symmetric group, to the complex monomial reflection group $G(r,n):=\mu_r\wr S_n$ (with $G(1,n)=S_n$). The central tool, proved in detail, is a \emph{transfer principle}: the substitution $y_i=x_i^r$ identifies the invariant ring of $G(r,n)$ with that of $S_n$ and transports, term by term, the corresponding operators and coordinates into explicit rational objects in the original variables $x_i$. From this we deduce the $G(r,n)$-analogues of the known results for $S_n$ existence and uniqueness of the dual coordinates $U_k$, and a Weyl algebra structure localized at the discriminant of $G(r,n)$ with complete and self-contained proofs for the points that do not follow directly from the transfer (Leibniz rule, main theorem). We then treat the \emph{total diagonal}: its preimage splits into $r^{n-1}$ lines permuted transitively by the group, and there, unlike the transferred operators $\Delta_i$, the \emph{raw} derivatives $\partial_{x_i}$ exhibit a phenomenon specific to $r\ge2$ that we describe completely via a Fa di Bruno-type structure formula. Finally, we give a closed formula for the constants of this structure formula (via Bell polynomials), completely resolve the degeneracy at an isolated point $x_i=0$ (the operator $\Delta_i$ extends holomorphically there, with $\Delta_i\phi=\partial_i^r\phi/r!$), and deduce from this a partial analogue of the description of the tangent space to the GIT quotient $\CC^n/G(r,n)$; the case of several coordinates vanishing simultaneously remains open and is precisely delineated. Full proofs of all new results are given in detail.