Representation theory of Lie algebras and quantum groups

1.量子群的旗流形Tanisaki利用量子群的旗流形作为非对易方案，给出了量子群的表示理论。特别是，他考虑了D-模态理论，建立了一个版本的Beillnson-Bernstein对应，证明了Lunts-Roscnberg猜想的一部分。Kashiwara研究了量子仿射代数的有限维表示，给出了基本模的张量积不可约的条件。Nakajima在某些单项式的集合上引入了一种晶体结构，并对标准模是基本模的张量积这一事实给出了一个新的证明。他还证明了Lusztig猜想的细胞结构的量子仿射代数3。与复反射群相关的绿色函数Shoji构造了一种新的Macdonald多项式作为与复反射群相关的Hall-Littlewood多项式的两参数形式4.有限Chevalley群的特征Shoji研究了Lusztig确定特征的程序中剩余的一些标量的确定，并为特殊的lincar群确定了它们。椭圆Lic代数、集合Artin群和Hecke代数Saito用顶点算子代数的方法构造了额外椭圆Lie代数的表示，并计算了它的特征.在p-adic flekis上的球齐性空间中，Kato给出了球函数在某些情况下的维数公式，并建立了对称空间中球函数的一般理论

1. Flag manifolds for quantum groups Tanisaki investigaled represecrtation theory of quantum groups using the flag manifolds of quanlum groups as non-commutative schemes. Especially, he has considered about the D-modyle theory and established a version of Beillnson-Bernstein correspondence proving a part of the conjecture by Lunts-Roscnberg2. Finite dimensional representations of quantum sffine algebras Kashiwara investigated the crystal bases and gave a condition for the tensor product of fundamental modules to be irreducible. Nakajima indroduced a crystal structure on the set of certain monomials and gave a new proof for the fact that the standard modulesare tensor product of fundamental modulcs. He also gave a proof of Lusztig's conjecture about the cell structure of quantum affine algebras3. Green functions assoclated to complex refiection groups Shoji constructed a new type of Macdonald polynomials as a two parameter version of Hall-Littlewood polynomial associated to complex refection groups4. Characters of finite Chevalley groups Shoji investigated about the delermination of some scalars which are the remaining part in Lusztig's program determining the characters, and has decided them for the special lincar groups5. Elliptic Lic algebras and assclated Artin groups and Hecke algebras Saito constructed a representation of exccptional elliptic Lie algebsa using the method of vertex operator algebras and computed its characters. He also get a result about a relation between the elilptic Hecke algebras and the double affino Hecke algebrasSphcrical homogeneous spaces over p-adic flekis Kato gave a dimension formula for spherical functions for some cases and constructed a general theory of spherical functions for symmetric spaces