Mirror symmetry and big algebras

First we recall the mirror symmetry identification of the coordinate ring of certain very stable upward flows in the Hitchin system and the Kirillov algebra for the minuscule representation of the Langlands dual group via the equivariant cohomology of the cominuscule flag variety (e.g. complex Grassmannian). In turn we discuss a conjectural extension of this picture to non-very stable upward flows in terms of a big commutative subalgebra of the Kirillov algebra, which also ringifies the equivariant intersection cohomology of the corresponding affine Schubert variety.