Decomposition of modules over invariant differential operators

Rikard Bögvad, Rolf Källström

Published 2015-06-20Version 1

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, and to prove branching rules. We also get a new construction of Young bases for representations of the symmetric group.