Topological mirror symmetry in degree zero for rank two character varieties

Let $C$ be a compact Riemann surface of genus $g$, let $r$ be the rank, and set

$$\Gamma\coloneqq \operatorname{Pic}^0(C)[r]\simeq (\mathbb{Z}/r\mathbb{Z})^{2g}.$$

The group $\Gamma$ acts by tensorisation on $M(C,\operatorname{SL}_r)$, and $\hat{\Gamma}$ denotes its character group. Identify $\Gamma$ with $\hat{\Gamma}$ by Poincaré duality, writing the resulting identification as $w\colon\Gamma\to\hat{\Gamma}$. For $\gamma\in\Gamma$, let $M(C,\operatorname{SL}_r)_\gamma$ be its fixed-point locus, let $F(\gamma)$ be the Fermionic shift, and let $IE$ denote the intersection E-polynomial.

Topological mirror symmetry in degree zero. For every $\kappa\in\hat{\Gamma}$, with $\gamma=w(\kappa)$,

$$IE(M(C,\operatorname{SL}_r))_{\kappa}=IE(M(C,\operatorname{SL}_r)_\gamma/\Gamma;u,v)(uv)^{F(\gamma)}.$$

In particular,

$$IE(M(C,\operatorname{SL}_r))=IE_{\mathrm{st}}(M(C,\operatorname{PGL}_r)).$$

This asserts the degree-zero topological mirror-symmetry relation between the $\operatorname{SL}_r$ and $\operatorname{PGL}_r$ character varieties, matching character-isotypic intersection E-polynomials with the corresponding stringy contributions.