Introduction to Complex Reflection Groups and Their Braid Groups
Weyl groups are particular cases of complex reflection groups, i.e. finite subgroups of GLr(C) generated by (pseudo)reflections. These are groups whose polynomial ring of invariants is a polynomial algebra. It has recently been discovered that complex ref
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1988
Michel Broué
Introduction to Complex Reflection Groups and Their Braid Groups
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Michel Broué Université Paris Diderot Paris 7 UFR de Mathématiques 175 Rue du Chevaleret 75013 Paris France [email protected]
ISBN: 978-3-642-11174-7 e-ISBN: 978-3-642-11175-4 DOI: 10.1007/978-3-642-11175-4 Springer Heidelberg Dordrecht London New York Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2009943837 Mathematics Subject Classification (2000): 20, 13, 16, 55 c Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. c Anouk Grinberg Cover illustration: Cover design: SPi Publisher Services Printed on acid-free paper springer.com
Preface
Weyl groups are finite groups acting as reflection groups on rational vector spaces. It is well known that these rational reflection groups appear as “skeletons” of many important mathematical objects: algebraic groups, Hecke algebras, Artin–Tits braid groups, etc. By extension of the base field, Weyl groups may be viewed as particular cases of finite complex reflection groups, i.e., finite subgroups of some GLr (C) generated by (pseudo–)reflections. Such groups have been characterized by Shephard–Todd and Chevalley as those finite subgroups of GLr (C) whose ring of invariants in the corresponding symmetric algebra C[X1 , X2 , . . . , Xr ] is a regular graded ring (a polynomial algebra). The irreducible finite complex reflection groups have been classified by Shephard–Todd. It has been recently discovered that complex reflection groups play also a key role in the structure as well as in the representation theory of finite reductive groups i.e., rational points of algebraic connected reductive groups over a finite field – for a survey on that type of questions, see for example [Bro1]. Not only do complex reflection groups appear as “automizers” of peculiar tori (the “cyclotomic Sylow subgroups”), but as much as Weyl groups, they give rise to braid groups and generalized Hecke algebras which govern representation theory of finite reductive groups. In the meantime, it has been understood that many of the known properties of Weyl groups, and more generally of Coxeter finite groups (reflection gro
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