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

Ferdinand Kafando, Jean Kaboré and Ibrahim Nonkané

Abstract and Applied Analysis, 2026, vol. 2026, 1-19

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 Gr,n:= μr≀Sn (with G1,n=Sn). Our central tool is a transfer principle, which we prove in detail: the substitution yi=xir identifies the invariant ring of Gr,n with that of Sn, and carries the corresponding operators and coordinates, term by term, into explicit objects in the original variables xi (rational for the operators, polynomial for the coordinates). From this we deduce the Gr,n-analogs of the known results for Sn: a Weyl algebra structure localized at the discriminant of Gr,n, and the existence and uniqueness of the dual coordinates Uk, the latter following readily from the transfer principle itself. The Leibniz rule for the symmetrized operator is the one exception, since it does not follow directly from the transfer; we give a direct, self-contained proof of it instead. For the main theorem we further give a second, independent proof, which exhibits the underlying linear system as triangular. We then turn to the total diagonal, whose preimage splits into rn−1 lines permuted transitively by the group. There, unlike the transferred operators Δi, the raw derivatives ∂xi exhibit a phenomenon specific to r≥2, which we describe in full via a Faà di Bruno-type structure formula, with an explicit closed formula for its constants. We likewise resolve the degeneracy at an isolated point xi=0 completely: the operator Δi extends holomorphically there, with Δiϕ=∂irϕ/r!, from which we deduce a partial analog of the tangent space to the GIT quotient Cn/Gr,n. Only the case of several coordinates vanishing simultaneously remains open, and we delineate it precisely. All proofs are given in detail.MSC2020 Classification: 13A50, 20F55, 16S32, 05E05, 14L24.

Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlaaa:3421318

DOI: 10.1155/aaa/3421318

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