consider the orbit stack of the action of an algebraic group g on the tangent variety of some homogeneous variety g/h where h is a nice closed subgroup... for example the case where g/h is the flag variety of g...

so consider a simple lie algebra g, modulo a borel b, as a rep of b ...

hmm, googling on "affine schubert" seems to indicate that that's people talking about schubert calculus for grassmanians of loop groups (or something like that), rather than about relating the action of a simple lie group on the tangent bundle of the flag variety to the action on the square of the flag variety...

let's try considering b2 for example... the tangent space of the "line" grassmanian can be thought of as pseudo-euclidean (2,1)-space, i think... but what about the tangent space of the flag variety??