Towards a classification of finite-dimensional representations of rational Cherednik algebras of type D

Using a combinatorial description due to Jacon and Lecouvey of the wall crossing bijections for cyclotomic rational Cherednik algebras, we show that the irreducible representations $L_c(\lambda^\pm)$ of the rational Cherednik algebra $H_c(D_n, \mathbb{C}^n)$ of type $D$ for symmetric bipartitions $\lambda$ are infinite dimensional for all parameters $c$. In particular, all finite-dimensional irreducible representations of rational Cherednik algebras of type $D$ arise as restrictions of finite-dimensional irreducible representations of rational Cherednik algebras of type $B$.