Quiver varieties and quantum affine algebras

通过箭形变种的等变K-群引入的标准模与Kashiwara引入的极权模同构。柏原的研究表明，极端重量模块有晶体基座。首席研究人员证明，相对于自然的内积，它们几乎是正交化的。通过与J.Beck的共同工作，将这一结果推广到任意Afiine李代数。作为应用，我们证明了量子仿射代数胞格上的Lusztig猜想。另一方面，首席研究者进一步研究了q-特征标的t-类似。特别地，他给出了1)关于A型和D型的Young画面的表达式，以及2)当Q是单位根的情况下的推广。证明了一类称为Kirillov-Reshetkhin模的表示的Q-特征标满足称为T-系统的递归系统，并编写了计算量子仿射代数有限维表示的Q-特征标的C程序。除情况外，他的计算都成功了。例如，如果他有足够的内存(大约10G字节)和计算机时间(大约一周)，他就可以成功。但他没有足够的预算来进行计算。首席研究员还研究了由例外向量丛给出的K3曲面上向量丛的模空间的上同调群上的算子。

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.