Tamely ramified relative Langlands conjecture

Establish the equivalence between the category of (H(F),psi)-equivariant D-modules on the affine flag variety Fl_G and the corresponding category of quasi-coherent sheaves on the fiber product of the dual hyperspherical variety with the GrothendieckSpringer resolution.

Background

The affine-flag or Iwahori extension is presented as the tamely ramified form of the relative Langlands conjecture. Its spectral side involves the GrothendieckSpringer resolution of the dual Lie algebra and is intended to refine the spherical correspondence.

The paper subsequently focuses on a quantum representation-theoretic formulation and proves a restricted Iwahori equivalence under stated hypotheses, but the full tamely ramified relative Langlands conjecture displayed here is formulated as a conjectural equivalence.