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Representation Theory of Algebraic Groups and Quantum Groups
Edited by: Toshiaki Shoji, Nagoya University, Japan, Masaki Kashiwara, RIMS, Kyoto University, Japan, Noriaki Kawanaka, Osaka University, Japan, George Lusztig, Massachusetts Institute of Technology, Cambridge, MA, and Ken-ichi Shinoda, Sophia University, Tokyo, Japan
A publication of the Mathematical Society of Japan.
 Advanced Studies in Pure Mathematics 2004; 512 pp; hardcover Volume: 40 ISBN-10: 4-931469-25-6 ISBN-13: 978-4-931469-25-9 List Price: US$116 Member Price: US$92.80 Order Code: ASPM/40 This book is a collection of research and survey papers written by speakers at the Mathematical Society of Japan's 10th International Conference. It presents a comprehensive overview of developments in representation theory of algebraic groups and quantum groups. Particularly noteworthy are papers containing remarkable results concerning Lusztig's conjecture on cells in affine Weyl groups. The following topics were discussed: cells in affine Weyl groups, tilting modules, tensor categories attached to cells in affine Weyl groups, representations of algebraic groups in positive characteristic, representations of Hecke algebras, Ariki-Koike and cyclotomic $$q$$-Schur algebras, cellular algebras and diagram algebras, Gelfand-Graev representations of finite reductive groups, Green functions associated to complex reflection groups, induction theorem for Springer representations, representations of Lie algebras in positive characteristic, representations of quantum affine algebras, extremal weight modules, crystal bases, tropical Robinson-Schensted-Knuth correspondence and more. The material is suitable for graduate students and research mathematicians interested in representation theory of algebraic groups, Hecke algebras, quantum groups, and combinatorial theory. Volumes in this series are freely available electronically 5 years post-publication. Published for the Mathematical Society of Japan by Kinokuniya, Tokyo, and distributed worldwide, except in Japan, by the AMS. Readership Graduate students and research mathematicians interested in algebra, algebraic geometry, mathematical physics, and combinatorial theory. Table of Contents H. H. Anderson -- Cells in affine Weyl groups and tilting modules S. Ariki and A. Mathas -- Heche algebras with a finite number of indecomposable modules S. Arkihopov -- Algebraic construction of contragradient quasi-Verma modules in positive characteristic R. Bezrukavnikov -- On tensor categories attached to cells in affine Weyl groups D. Gaitsgory -- Appendix: Braiding compatibilities R. Bezrukavnikov and V. Ostrik -- On tensor categories attached to cells in affine Weyl groups II C. W. Curtis and K.-i. Shinoda -- Zeta functions and functional equations associated with the components of the Gelfand-Graev representations of a finite reductive group J. J. Graham and G. I. Lehrer -- Cellular algebras and diagram algebras in representation theory J. C. Jantzen -- Representations of Lie algebras in positive characteristic S.-J. Kang -- Quantum affine algebras and crystal bases G. Lusztig -- An induction theorem for Springer's representations A. Mathas -- The representation theory of the Ariki-Koike and cyclotomic $$q$$-Schur algebras S. Naito and D. Sagaki -- Crystal bases and diagram automorphisms H. Nakajima -- Extremal weight modules of quantum affine algebras M. Noumi and Y. Yamada -- Tropical Robinson-Schensted-Knuth correspondence and birational Weyl group actions T. Shoji -- Green functions attached to limit symbols T. A. Springer -- Cells for a Hecke algebra representation N. Xi -- On the characterization of the set $$\mathcal{D}_1$$ of the affine Weyl group of type $$\tilde{A}_n$$