The local-cohomology interpretation of the character-variety partition function

Let $L\subset\mathrm{Hilb}_n(\mathbb C^2)$ be the Lagrangian cut out by $\sum_{i=1}^n x_i^r=0$ for $r>0$, let $P$ be the Procesi bundle, let $\Omega^*$ be the bundle of differential forms, and let $H_L^n$ denote local cohomology supported on $L$. Let $H_*^{\mathrm{BM}}($Higgs moduli stack$)$ denote Borel–Moore homology of the Higgs moduli stack. Local-cohomology interpretation.

$$H_L^n\bigl(P^{\otimes k}\otimes(\Omega^*)^{\otimes g}\otimes O(g-1)\bigr)^\vee\cong H_*^{\mathrm{BM}}(\text{Higgs moduli stack}).$$

This is proposed as a geometric interpretation of the partition-function identity, replacing global sections by local cohomology; the supplied text gives no resolution status.