Lie algebras and quantum groups via algebraic analysis

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1. Tanisaki investigated the action of the braid groups on the zero-weight spaces of the integral modules over quantized enveloping algebras. Under a certain condition on the representation the braid group action descends to the Hecke algebra, and a module over the Hecke algebra is obtained. In the case of type A all irreducible modules over the Hecke algebra is derived. Using the Kazhdan-Lusztig bases for the Hecke algebra modules and the global bases for the modules over the quantized enveloping algebras we can show that the construction above makes sense over rings. Hence it is possible to investigate modular representations by this method. Tanisaki gave a new proof of the conjecture by Lascoux-Leclerc-Thibon.2. Tanisaki investigated applications of the geometric Langlands correspondence to representation theory. Especially, he tried to obtain the twining character formula due to Naito etc. using geometric Langalands correspondence.3. Kashiwra investigated level zero fundamental modules over affine quantum algebras and Demazure modules.4. Shoji investigated representations of modified Ariki-Koike algebras introduced by himself. He also obtained interesting results on the corresponding q-Shur algebras.5. Naito investigated crystal bases of the extremal weight modules. Especially, he considered about the action of the diagram automorphisms and obtained results about the elements of the crystal base fixed by this group.6. Satio investigated Macdonald polynomials using Cherednik algebras.7. Kashiwara investigated representations of affine Hecke algebras of type B.8. Shoji investigated generalized Green functions and obtained results about some constant.