Quiver varieties and quantum affine algebras

证明了由首席研究员通过等变k群引入的标准模与由Kashiwara引入的极值权模同构。Kashiwara证明了极值重量模具有晶体基。首席研究员证明了它们对于自然内积是“几乎正交的”。通过与J. Beck的联合工作，将这一结果推广到任意正则李代数。作为一个应用，我们证明了Lusztig关于量子仿射代数细胞的猜想。另一方面，首席调查员进一步研究了q字符的t类似物。特别地，他给出了1)A和D类型的Young表的表达式，以及2)当q是单位的根时的推广。他还证明了一类称为Kirillov-Reshetkhin模的表示的q-字符满足称为t系统的递归系统。此外，首席研究员编写了一个C程序，用于计算量子仿射代数的有限维表示的q字符。除例外，他计算成功了。例如，如果他有足够的内存（大约10千兆字节）和计算机时间（大约一周），他就可以成功。但是他没有足够的预算来进行计算。首席研究者还研究了K3曲面上向量束模空间上同调群上的算子，这些算子由例外向量束给出。

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.