A uniform decomposition theorem for invariant differential operators on imprimitive complex reflection groups G(r,p,n)

2026/08/02 by Jean Kaboré, Ibrahim Nonkané
Mathematics · #math.RT #math.AC #math.AG #math.GR #msc:20F55 #msc:13A50 #msc:16S32 #msc:20C15 #msc:05E1

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arxiv created 2026/08/02 · arxiv updated 2026/08/04

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(D_n) and W(B_n); 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.

Citations

• Invariant differential operators and the generalized symmetric group