Irreducible non-holonomic modules for rational Cherednik algebras

Abstract

Let $\mathbb{K}$ be an algebraically closed field of characteristic zero and let $A_n(\mathbb{K})$ be the $n$-th Weyl algebra. We prove that for every complex reflection group $G$ and every $n \geq 2$ the ring of invariant differential operators $A_n(\mathbb{K})^G$ has irreducible non-holonomic modules of Gelfand-Kirillov dimension $2n-1$, and we exhibit such modules explicitly. Through the Morita equivalence between $A_n(\mathbb{K})^G$ and the rational Cherednik algebra $H_{\mathfrak c}$ at an integral parameter, we deduce that $H_{\mathfrak c}$ has irreducible non-holonomic modules of Gelfand-Kirillov dimension $2n-1$ for $n \geq 2$. In an appendix, following a proof kindly shared by O. Mathieu, we show that $A_n(\mathbb{K})$ has irreducible modules of every Gelfand-Kirillov dimension in the interval $[n, 2n-1]$.