Cohomological identification conjectures for BPS complexes of symmetric quotient stacks

Let $G$ act on a smooth variety $X$. For a pair $(\lambda,\alpha)\in\mathscr{Q}_{X,G}$, let $\underline{\mathcal{P}}_{(\lambda,\alpha)}\coloneqq\underline{\mathcal{BPS}}_{(\lambda,\alpha)}$ be the associated cohomologically graded mixed Hodge module. Let $\mathscr{Q}_{X,G}^{\mathrm{st}}\subseteq\mathscr{Q}_{X,G}$ denote the stable pairs, namely those for which there are closed orbits with finite stabilizer for the $G^{\lambda}/G_{(\lambda,\alpha)}$-action on $X^{\lambda}_{\alpha}$. Write $\underline{\mathcal{IC}}(X^{\lambda}_{\alpha}/\!/G^{\lambda})$ for the intersection-cohomology complex and $\mathscr{L}$ for the Lefschetz object.

BPS complex conjectures. The following assertions should hold:

1. $\underline{\mathcal{P}}_{(\lambda,\alpha)}\neq 0$ if and only if $(\lambda,\alpha)\in\mathscr{Q}_{X,G}^{\mathrm{st}}$.
2. For $(\lambda,\alpha)\in\mathscr{Q}_{X,G}$,

$$\underline{\mathcal{P}}_{(\lambda,\alpha)}={\underline{\mathcal{H}}}^{\dim G_{(\lambda,\alpha)}}(\pi_*\underline{\mathbf{Q}}_{X^{\lambda}_{\alpha}/G^{\lambda}}^{\mathrm{vir}})[-\dim G_{(\lambda,\alpha)}].$$

3. There is a natural inclusion

$$\underline{\mathcal{IC}}(X^{\lambda}_{\alpha}/\!/G^{\lambda})\otimes\mathscr{L}^{\dim G_{(\lambda,\alpha)}/2}\hookrightarrow\underline{\mathcal{P}}_{(\lambda,\alpha)},$$

which is an isomorphism.

The conjectures aim to identify precisely when the BPS complexes are nonzero and to describe them in terms of cohomology and intersection cohomology. The first conjecture was proved in the cited follow-up work under an orthogonality assumption for the action of $G$; the general assertions are not established by the supplied text.