Quiver Grassmannians, quiver varieties and the preprojective algebra. (Q635678)

Let \(\mathfrak g\) be a Kac-Moody algebra and \(Q\) be some orientation of the Dynkin diagram of \(\mathfrak g\). Let \(\widetilde Q\) be the double quiver of \(Q\). The main result of this paper asserts that a certain quiver Grassmannian for \(\widetilde Q\) is homeomorphic to a certain Lagrangian Nakajima quiver variety. The authors also refine this result by finding quiver Grassmannians that are homeomorphic to the Demazure quiver varieties, and others which are homeomorphic to the graded/cyclic quiver varieties. The Demazure quiver Grassmannian allows the authors to describe injective objects in the category of locally nilpotent modules over the preprojective algebra. Motivated by an earlier version of the paper, \textit{I. Shipman} has recently proved [see Math. Res. Lett. 17, No. 5, 969-976 (2010; Zbl 1231.16010)] that Lusztig's bijection between the points of the Lagrangian Nakajima quiver variety and the points of a type of quiver Grassmannian inside a projective object is, in fact, an isomorphism of algebraic varieties. The present paper has an Appendix where the authors explain how this result allows one to conclude that the maps between quiver Grassmannians and Lagrangian Nakajima quiver varieties described in the paper under review are also isomorphisms of algebraic varieties.