  We lift the affine Matsuki correspondence between real and symmetric loop
group orbits in affine Grassmannians to an equivalence of derived categories of
sheaves. In analogy with the finite-dimensional setting, our arguments depend
upon the Morse theory of energy functions obtained from symmetrizations of
coadjoint orbits. The additional fusion structures of the affine setting lead
to further equivalences with Schubert constructible derived categories of
sheaves on real affine Grassmannians.

Representation Theory;Algebraic Geometry