Quiver varieties and quantum affine algebras

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Standard modules, introduced by the head investigator via equivariant K-groups of quiver varieties, are proved to be isomorphic to extremal weight modules, introduced by Kashiwara. It was shown by Kashiwara that extremal weight modules have crystal bases. The head investigator proved that they are 'almost orthonormal' with respect to the natural inner product. This result is generalized to arbitrary afiine Lie algebras by a joint work with J. Beck. As an application, we prove the conjecture of Lusztig on cells of quantum affine algebras.On the other hand, the head investigator further studies t-analogs of q-characters. In particular, he gives 1) expressions in terms of Young tableaux for type A and D, and 2) generalization to the case when q is a root of unity. He also proved that the q-characters of a certain class of representations, called Kirillov-Reshetkhin modules, satisfy the recursive system called T-system.Also, the head investigator writes a C program for computing q-characters of finite dimensional representations of quantum affine algebras. He succeeded the calculation except Eg case. For Eg case, he can succeed if he has enough memory (some ten giga bytes) and computer time (about one week). But he did not have enough budget to perform the computation.The head investigator also studies operators on cohomology groups of moduli spaces of vector bundles over K3 surfaces, given by exceptional vector bundles.