Decomposition of modules over invariant differential operators

TL;DR

This paper studies the structure of modules over invariant differential operators related to finite groups, providing new isomorphisms, constructing simple modules, and deriving branching rules with applications to symmetric groups.

Contribution

It introduces a novel isomorphism between module categories, constructs simple modules for generalized symmetric groups, and offers new insights into the representation theory of symmetric groups.

Findings

Established an isomorphism between module categories.

Constructed simple modules as Gelfand models.

Derived branching rules and Young basis construction.

Abstract

Let $G$ be a finite subgroup of the linear group of a finite-dimensional complex vector $V$, $B={\operatorname S}(V)$ be the symmetric algebra, ${\mathcal D}=\mathcal D^G_B$ the ring of $G$-invariant differential operators, and ${\mathcal D}^-$ its subring of negative degree operators. We prove that $M\mapsto M^{ann}= {\operatorname Ann}_{\mathcal D^-}(M)$ defines an isomorphism between the category of ${\mathcal D}$-submodules of $B$ and a category of modules formed as lowest weight spaces. This is applied to a construction of simple ${\mathcal D}$-submodules of $B$ when $G$ is a generalized symmetric group, to show that $B^{ann}$ is a so-called Gelfand model. Using differential algebra and lowest weight methods we also prove branching rules, entailing the main results in the representation theory of the symmetric group, such as a differential construction of the Young basis.