The $(q,t)$ Catalan character conjecture for rational Cherednik algebras

Let $n$ be the symmetric-group parameter, let $W=S_n$, let $\mathbf e$ be the symmetrizing idempotent, let $L_c(\mathsf{triv})$ be the irreducible spherical lowest-weight module, and let $\mathbb T$ act on the relevant associated graded spaces and on the Hilbert scheme. Write ${\sf Hilb}^n_o(\mathbb C^2)$ for the punctual Hilbert scheme and $\mathcal L$ for its relevant line bundle. For $c=\frac{1}{n}+k$, with $k=0,1,\ldots$, one has the canonical $\mathbb T$-module isomorphism

$${\mathtt{gr}}\bigl(\mathbf e\!\cdot\! L_c(\mathsf{triv})\bigr)\simeq H^0\bigl({\sf Hilb}^n_o(\mathbb C^2),\mathcal L^{\otimes k}\bigr).$$

The $(q,t)$ Catalan character conjecture. In particular, $\chi_{\mathbf e\cdot L(\mathsf{triv})}(q,t)=C_n^{(k)}(q,t)$, the $(q,t)$ Catalan number. This conjecture refines the corresponding dimension formula and connects finite-dimensional rational Cherednik-algebra representations with the geometry of Hilbert schemes; the supplied text gives no resolution status.