Fourier Transforms of Invariant Functions on Finite Reductive Lie Algebras
The study of Fourier transforms of invariant functions on finite reductive Lie algebras has been initiated by T.A. Springer (1976) in connection with the geometry of nilpotent orbits. In this book the author studies Fourier transforms using Deligne-Luszti
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Emmanuel Letellier
Fourier Transforms of Invariant Functions on Finite Reductive Lie Algebras
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Author Emmanuel Letellier Department of Mathematics Sophia University Kioicho, Chiyodaku Tokyo 102-8554 Japan e-mail: [email protected]
Library of Congress Control Number: 2004115717
Mathematics Subject Classification (2000): 20C33 ISSN 0075-8434 ISBN 3-540-24020-9 Springer Berlin Heidelberg New York DOI: 10.1007/b104209 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm 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 for prosecution under the German Copyright Law. Springer is a part of Springer Science + Business Media http://www.springeronline.com c Springer-Verlag Berlin Heidelberg 2005 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the authors 41/3142/du - 543210 - Printed on acid-free paper
To my parents
Preface
The present work is about the study of the trigonometric sums on finite reductive Lie algebras of Chevalley’s type in the sense of [Spr76]. This subject has been introduced to me by my supervisors Gus Lehrer and Jean Michel in connection with [Leh96][Leh97] while I was starting my PhD under a cotutelle agreement between the university Paris 6 and the university of Sydney. The required background is the standard knowledge of the theory of connected reductive groups and finite groups of Lie type [Spr]. It is a great pleasure to thank my supervisors Gus Lehrer and Jean Michel for their precious advices throughout the elaboration of this work. I am also very grateful to all the others who red the first drafts and suggested improvements, particularly A. Henderson, T. Shoji, J. van Hamel and the editor. Finally I would like to thank G. Lusztig who invented the theory I use in this book. The preparation of this work has been conducted at the following places: “Equipe des groupes finis”(Institut de math´ematiques de Jussieu, Paris), university of Sydney, LAMFA (universit´e de Picardie Jules-Verne), Sophia university (Tokyo). It is a pleasure to thank the previously named institutes for their hospitality. I am grateful to the AEAP (Australian European Award Program), the French ministry of research and to JSPS (Japanese Society for the Promotion of Science) for their generous support.
Tokyo, July 2004
Emmanuel Letellier
Contents
1
Introduction . . . . . . . . . . . . . . . . . .
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