We construct a Hecke structure on equivariant Bredon cohomology with local coefficients and then describe some of its properties. We compare this structure with the Mackey structure defined by T. tom Dieck and with the Illman transfer.

Let X be a space with free loop space ΛX and mod two cohomology R = H*X. We construct functors ${\Omega }_{\lambda }\left(R\right)$ and ℓ(R) together with algebra homomorphisms $e:{\Omega }_{\lambda }\left(R\right)\to H*\left(\Lambda X\right)$ and $\psi :\ell \left(R\right)\to H*\left(E{S}^1{\times }_{S^1}\Lambda X\right)$. When X is 1-connected and R is a symmetric algebra we show that these are isomorphisms.

We describe hypergeometric solutions of the quantum differential equation of the cotangent bundle of a $$𝔤{𝔩}_n$$ partial flag variety. These hypergeometric solutions manifest the Landau-Ginzburg mirror symmetry for the cotangent bundle of a partial flag variety.