Hiraku Nakajima

Instanton counting on blowup. I. 4-dimensional pure gauge theory

We give a mathematically rigorous proof of Nekrasov’s conjecture: the integration in the equivariant cohomology over the moduli spaces of instantons on ℝ4 gives a deformation of the Seiberg-Witten prepotential for N=2 SUSY Yang-Mills theory. Through a study of moduli spaces on the blowup of ℝ4, we derive a differential equation for the Nekrasov’s partition function. It is a deformation of the equation for the Seiberg-Witten prepotential, found by Losev et al., and further studied by Gorsky et al.

Quiver varieties and finite dimensional representations of quantum affine algebras

Introduction 145 1. Quantum affine algebra 150 2. Quiver variety 155 3. Stratification of M0 163 4. Fixed point subvariety 167 5. Hecke correspondence and induction of quiver varieties 169 6. Equivariant K-theory 174 7. Freeness 178 8. Convolution 185 9. A homomorphism Uq(Lg)→ KGw×C ∗ (Z(w))⊗Z[q,q−1] Q(q) 192 10. Relations (I) 194 11. Relations (II) 202 12. Integral structure 214 13. Standard modules 218 14. Simple modules 224 15. The Ue(g)-module structure 233 Added in proof 236 References 236

–analogs of –characters of Kirillov-Reshetikhin modules of quantum affine algebras

We prove the Kirillov-Reshetikhin conjecture concerning certain finite dimen- sional representations of a quantum affine algebraUq(bg) when b is an untwisted affine Lie algebra of type ADE. We use t-analog of q-characters introduced by the author in an essen- tial way.

McKay correspondence and Hilbert schemes in dimension three

Abstract Let G be a nontrivial finite subgroup of SL n ( C ) . Suppose that the quotient singularity C n /G has a crepant resolution π: X→ C n /G ( i.e. K X = O X ) . There is a slightly imprecise conjecture, called the McKay correspondence, stating that there is a relation between the Grothendieck group (or (co)homology group) of X and the representations (or conjugacy classes) of G with a “certain compatibility” between the intersection product and the tensor product (see e.g. [22]). The purpose of this paper is to give more precise formulation of the conjecture when X can be given as a certain variety associated with the Hilbert scheme of points in C n . We give the proof of this new conjecture for an abelian subgroup G of SL 3 ( C ) .

Crystal bases and two-sided cells of quantum affine algebras

Let

t-Analogs of q-Characters of Quantum Affine Algebras of Type E6, E7, E8

\g

Moduli spaces of anti-self-dual connections on ALE gravitational instantons

be an affine Kac-Moody Lie algebra. Let

Quiver Varieties and Branching

\U^+

t-analogue of the q-characters of finite dimensional representations of quantum affine algebras

be the positive part of the Drinfeld-Jimbo quantum enveloping algebra associated to

Gauge theory on resolutions of simple singularities and simple Lie algebras

\g