| Co-Investigator(Kenkyū-buntansha) |
MIKI Kei Osaka University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (40212229)
MURAKAMI Jun Waseda University, School of Science and Engineering, Professor, 理工学部, 教授 (90157751)
DATE Etsuro Osaka University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (00107062)
ARIKI Susumu Tokyo University of Mercantile Marine, Faculty of Mercantile Marine Science, Associate Professor, 商船学部, 助教授 (40212641)
TANISAKI Toshiyuki Hiroshima University, Faculty of Science, Professor, 理学部, 教授 (70142916)
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| Research Abstract |
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.
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