A uniform decomposition theorem for invariant differential operators on imprimitive complex reflection groups G(r,p,n)
Abstract
{We study the module structure of the polynomial ring, localized at the discriminant, over the ring of differential operators on the ring of invariants of the imprimitive complex reflection group G(r,p,n), describing its simple components with explicit generators given by the higher Specht polynomials of Ariki--Terasoma--Yamada. The proof rests on a Jacobian lemma computing the discriminant of G(r,p,n), combined with a double-centralizer argument. As particular cases (r=2, p=2 or p=1) we recover, and considerably shorten, the known decomposition theorems for the real reflection groups W(Dn) and W(Bn); we also treat G(r,r,n) and G(r,1,n) explicitly, with worked examples (D2, D3, B2) and their central idempotents. Finally, applying the Galois descent equivalence of categories of Nonkané to G(r,p,n) for the first time gives a second, generator-free description of the simple summands as twisted invariants.}
Related
- The quaternionic reflection groups with imprimitive complexifications and the finite subgroups of GL(2,C)
- Specht ideals for the imprimitive complex reflection group G(r,1,n): geometric orbit types and a dominance theorem
- Duflo--Serganova functor for chiral differential operators on classical invariant rings
- Langlands Duality and Invariant Differential Operators: the Case SL(2n+1)
- Differential Operators on G(r,n)-Invariant Functions
BibTeX
@misc{kabore2026uniform,
title = {{A uniform decomposition theorem for invariant differential operators on imprimitive complex reflection groups G(r,p,n)}},
author = {Jean Kaboré and Ibrahim Nonkané},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2608.01474},
eprint = {2608.01474},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.01474},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A uniform decomposition theorem for invariant differential operators on imprimitive complex reflection groups G(r,p,n)
AU - Jean Kaboré
AU - Ibrahim Nonkané
PY - 2026
DA - 2026/11/06
PB - arXiv
DO - 10.48550/arxiv.2608.01474
UR - https://arxiv.org/abs/2608.01474
AB - {We study the module structure of the polynomial ring, localized at the discriminant, over the ring of differential operators on the ring of invariants of the imprimitive complex reflection group G(r,p,n), describing its simple components with explicit generators given by the higher Specht polynomials of Ariki--Terasoma--Yamada. The proof rests on a Jacobian lemma computing the discriminant of G(r,p,n), combined with a double-centralizer argument. As particular cases (r=2, p=2 or p=1) we recover, and considerably shorten, the known decomposition theorems for the real reflection groups W(D_n) and W(B_n); we also treat G(r,r,n) and G(r,1,n) explicitly, with worked examples (D_2, D_3, B_2) and their central idempotents. Finally, applying the Galois descent equivalence of categories of Nonkané to G(r,p,n) for the first time gives a second, generator-free description of the simple summands as twisted invariants.}
ER -