0 Quiver Varieties of Type A

We prove a conjecture of Nakajima describing the relation between the geometry of quiver varieties of type A and the geometry of the partial flags varieties and of the nilpotent variety. The kind of quiver varieties we are interested in, have been introduced by Nakajima as a generalization of the description of the moduli space of anti-self-dual connections on ALE spaces constructed by Kroneheimer and Nakajima [3]. They result to have a rich and interesting geometry and they were used by Nakajima to give a geometric construction of the representations of Kac-Moody algebras [5], [6]. A similar construction had already been done in the case of sln by Ginzburg [1] using partial flags varieties. A precise conjecture of Nakajima ([5] or theorem 12 below) describes the relation between the two kind of varieties. I want to thank Corrado De Concini who explained me quiver varieties and pointed out to me this problem and Hiraku Nakajima who pointed out an error in a previous version of this paper and the solution to it. 1. Nakajima’s conjecture We recall some definition and fix some notation on quiver varieties of type An−1 and on partial flags varieties. 1.1. Quiver varieties of type An−1. Let C = 2I − A be the Cartan matrix of type An−1 and consider the “double” graph of type An−1. We will call the vertices and the arrows of this graph according to the following diagram: 1 ar1 66 2 ar2 44 ar1 vv · · · ar2 uu n− 2 arn−2 22 n− 1 arn−2 rr . (1) In particular I = {1, . . . , n−1} is the set of vertices, Ω = {ar1, . . . , arn−2}, Ω = {ar1, . . . , arn−2} and H = Ω ∪ Ω is the set of arrows. We observe that given i, j ∈ I we have that aij = card{h ∈ Ω joining i and j}. Finally if h is an arrow we call h0 its source of h and h1 its target. Notation 1. In this paper v = (v1, . . . , vn−1) and d = (d1, . . . , dn−1) will be vectors of integers. In the case that they are vectors of nonnegative integers Vi and Di will be vector spaces of dimension vi and di for 1