Quantum cohomology, Hitchin systems, and Fourier--Mukai comparison

Let $X$ be the moduli space of stable trace-free rank $r$ Higgs bundles over a smooth projective curve $C$ with fixed determinant $\mathrm{\Lambda }\in \text{Pic}(C)$ and $\gcd (r,\mathrm{deg}\mathrm{\Lambda })=1$. We prove that the ${\text{cc}}^{\ast }$-equivariant quantum product of a divisor on $X$ is given by the Steinberg correspondence of the Lagrangian Steinberg cycle in $X\times X$. The Steinberg cycle is the image of the reduced virtual fundamental cycle of the $2$-pointed Kontsevich stable map space to $X$ under evaluation map. For $\mathrm{\Gamma }={\text{Pic}}^0(C)[r]$, we give a character decomposition for $\mathrm{\Gamma }$-invariant curve classes or finite orbit sums. In rank two and genus two we explicitly determine its fifteen endoscopic lines and the reduced spectral discriminant. %The resulting endoscopic support statement is cohomological, not a determination of the entire Chow cycle. For $\widehat{X}=[X\mathrm{/}\mathrm{\Gamma }]$, which is the moduli space of stable ${\text{PGL}}_r$-Higgs bundles over $C$. We consider the orbifold quantum cohomology theory from a gerbe-twisted theory. The complex K-theory ${\text{KU}}^{\ast }(X)$ and twisted K-theory ${\text{KU}}^{\ast }([X\mathrm{/}\mathrm{\Gamma }],\alpha )$ by the lifting ${\mu }_r$-gerbe $\alpha$ of the universal projective bundle is an isomorphism, and is given by the Fourier-Mukai transformation. We prove that the Fourier-Mukai transformation is compatible with the quantum product by the divisors.