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 $W(D_n)$ 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 Nonkan\'e to $G(r,p,n)$ for the first time gives a second, generator-free description of the simple summands as twisted invariants.}