Webster’s Functors

We start with a warm-up: we explain Nakajima’s construction of the geometric sl2-action on ⊕ v Hmid(T ∗Gr(v, w)). 1.1. Warm-up: Nakajima’s geometric action. The lagrangian subvariety φ−1(0) ⊂ M(v) is easily seen to be the zero section Gr(v, w) ⊂ T ∗Gr(v, w). Indeed, consider the action of C× on M(v) induced from the C×-action on T ∗R by fiberwise dilations. Under the identification ofM(v) with T ∗Gr(v, w), we get the action by fiberwise dilations. Note that φ−1(0) is compact and C×-stable. The only lagrangian subvarierty in T ∗ Gr(v, w) with these properties is Gr(v, w). So we have Hmid(T ∗Gr(v, w)) = Htop(Gr(v, w)) = C. We want to construct operators between Htop(Gr(v, w)) and Htop(Gr(v + 1, w)). Note that we have a correspondence between Gr(v, w),Gr(v + 1, w): the partial flag variety Fl(v, v + 1;w). More precisely, we have morphisms πv : Fl(v, v + 1, w) Gr(v, w), πv+1 : Fl(v, v+1, w) Gr(v+1, w), both are locally trivial fibrations whose fibers are projective spaces Pw−v−1,Pv. So we have maps f := πv+1∗π ∗ v : Htop(Gr(v, w)) → Htop(Gr(v + 1, w)) : πv∗π v+1 =: e. Here π∗ v is the identity map between Htop(Gr(v, w)) → Htop(Fl(v, v+1;w)) and πv∗ is the multiplication by the Euler characteristic of the fiber. The maps π∗ v+1, πv+1∗ are defined similarly. The maps e, f equip the space ⊕ v Htop(Gr(v, w)) with the irreducible w + 1dimensional representation of sl2 (where Htop(Gr(v, w)) has weight w − 2v).