The arbitrary-weights character conjecture for homogeneous components of sheaf moduli spaces

Let ${\overline w}\in{\mathbb Z}^r$, let $\alpha,\beta\geq 1$ satisfy $0\leq w_i<\alpha+\beta$ and $\gcd(\alpha,\beta)=1$, and let $a_i=\sharp\{j\mid w_j=i\}$. Let $\alpha'$ be the unique integer with $0\leq\alpha'<\alpha+\beta$ and $\alpha'\alpha\equiv1\pmod{\alpha+\beta}$. Define $\overline a'=(a'_0,\ldots,a'_{\alpha+\beta-1})$ by

$$a'_i=a_{\alpha'i\pmod{\alpha+\beta}},$$

and let $\overline a”\in\mathbb Z^{\alpha+\beta-1}$ be $\overline a'$ without its last coordinate; write $\overline 0=(0,\ldots,0)\in\mathbb Z^{\alpha+\beta-1}$. Here ${\mathcal M}(r,n)^{T^{{\overline w}}_{\alpha,\beta}}$ denotes the corresponding homogeneous component, $h_0$ its zeroth cohomology dimension, and $\chi^{p,p'}_{\overline a,\overline b}$ the associated character. Arbitrary-weights character conjecture.

$$\sum_{n\geq 0}h_0\left({\mathcal M}(r,n)^{T^{{\overline w}}_{\alpha,\beta}}\right)q^n=\left(\prod_{i=1}^{\infty}(1-q^{(\alpha+\beta)i})\right)\chi^{\alpha+\beta,\alpha+\beta+r}_{\overline 0,\overline a”}.$$

This conjecture extends the proposed relation between homogeneous components of sheaf moduli spaces and Virasoro characters from the previously known setting to arbitrary coprime $\alpha,\beta$ and weight vectors $\overline w$.