Representation theory of Lie algebras and quantum groups

1.量子群的标志流形Tanisaki使用量子群的标志流形作为非对易方案研究了量子群的表示理论。特别是，他考虑了D-modyle理论，并建立了Beillnson-Bernstein对应的一个版本，证明了LUNTS-Rvocnberg2猜想的一部分。量子sffine代数的有限维表示Kashiwara研究了其晶基，并给出了基本模的张量积不可约的一个条件。Nakajima在某些单项集上引入了一种晶体结构，并对标准模是基本模的张量积给出了新的证明。他还证明了Lusztig关于量子仿射代数的胞格结构的猜想。与复反射群相关的格林函数，Shoji构造了一种新的Macdonald多项式，作为与复反射群相关的两参数Hall-Littlewood多项式的形式。有限Chvalley群的特征标Shoji研究了Lusztig的特征标程序中剩余的几个标量的判定问题，并对特殊的线性群5进行了判定。椭圆Lic代数、配合Artin群和Hecke代数Saito利用顶点算子代数的方法构造了可取椭圆李代数的一个表示，并计算了它的特征标.他还得到了关于椭圆型Hecke代数与双仿射Hecke代数之间关系的一个结果。p-adadyflekis Kato上的球齐性空间在某些情况下给出了球函数的一个量纲公式，并构造了对称空间的球函数的一般理论

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