Representation theory of Algebraic Groups and Quantum Groups

下面，我们陈述我们关于（1）仿射李代数，（2）Hecke代数，（3）有限Chevalley群，（4）复反射群，和（5）量子群的主要结果。(1)T.Tanisaki（与M.Kashiwara）完全确定了仿射李代数上具有非临界最高权的不可约模的特征标。这是通过减少，使用Jantzen的技巧，问题的情况下合理的权重，这种情况下已经处理以前谷崎和柏原。(2)1992年，K.Uno证明了不可分解Hecke代数模的等价类个数有限的条件。S. Ariki解决了这个Uno猜想肯定在经典的情况下。其余的例外情况似乎也在我们的能力范围之内。(3)T. Shoji给出了一种构造有限经典群的绿色函数的组合方法。这推广了绿色关于有限一般线性群的著名结果.有趣的是，同样的过程对某些复杂的反射群也有意义。(4)Kawanaka为有限复反射群的不可约特征标引入了新的不变量，并在不可约情形下明确地计算了它们。Gyoja和其他人计算了任何有限外尔群的相同不变量，并观察到与Lusztig的双边细胞概念的奇怪关系。(5)J.Murakami（与H.Murakami联合）证明了Kashaev纽结不变量只不过是用对应于量子群U_q（sl_2）的不可约表示的量子R-矩阵定义的有色Jones不变量的特殊化，并利用这一点推广了Kashaev猜想，说明了这个不变量与双曲纽结的补的双曲体积的联系，到一般纽结的情况。

Below, we state our main results on (1) affine Lie algebras, (2) Hecke algebras, (3) finite Chevalley groups, (4) complex reflection groups, and (5) quantum groups.(1) T.Tanisaki (jointly with M.Kashiwara) completely determined the characters of irreducible modules with non-critical highest weight over affine Lie algebras. This was done by reducing, using Jantzen's trick, the problem to the case of rational weights, the case already treated previously by Tanisaki and Kashiwara.(2) K.Uno conjectured, in 1992, the condition under which the number of equivalence Classes of indecomposable Hecke algebra modules is finite. S.Ariki settled this Uno conjecture affirmatively in the classical cases. The remaining exceptional cases also seem to be within our reach.(3) T.Shoji gave a combinatorial method by which one can construct Green functions of finite classical groups. This generalizes the wellknown result of Green for finite general linear groups. It is interesting to note that the same procedure makes sense for certain complex reflection groups.(4) N.Kawanaka introduced new invariants for the irreducible characters of finite complex reflection groups, and calculated them explicitly in the imprimiteve cases. Gyoja and others calculated the same invariants for any finite Weyl groups, and observed a strange relation with Lusztig's notion of two-sided cells.(5) J.Murakami (jointly with H.Murakami) showed the Kashaev knot-invariant is nothing but a specialization of colored Jones invariant defined using quantum R-matrices corresponding to irreducible representations of quantum groups U_q (sl_2), and, using this, generarlzed the Kashaev conjecture, stating a connection of this invariant with hyperbolic volumes of complements of hyperbolic knots, to the case of general knots.