Kac conjecture from Nakajima quiver varieties

We prove a generating function formula for the Betti numbers of Nakajima quiver varieties. We prove that it is a q-deformation of the Weyl-Kac character formula. In particular this implies that the constant term of the polynomial counting the number of absolutely indecomposable representations of a quiver equals the multiplicity of a a certain weight in the corresponding Kac-Moody algebra, which was conjectured by Kac in 1982. Let Γ = (I, E) be a quiver that is an oriented graph on a finite set I = {1, . . . , n} with E ⊂ I × I a finite multiset of oriented (perhaps multiple but no loop) edges. Given two dimension vectors v = (vi) ∈ NI and w = (wi) ∈ N Nakajima [12, 13] constructs, as holomorphic symplectic quotient, a complex variety M(v,w) of dimension 2dv,w, which we call a Nakajima quiver variety. In [14] Nakajima found a combinatorial algorithm to determine the Betti numbers of these varieties. Here we prove and study the following generating function of Betti numbers of Nakajima quiver varieties. Theorem 1 Fix w ∈ N. Denote bi(M(v,w)) := dim ( H(M(v,w)) ) . Then in the notation of (18),(21), (24)