Representation theory of Lie algebras and quantum groups

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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