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.