Alternative Pseudodifferential Analysis With an Application to Modul
This volume introduces an entirely new pseudodifferential analysis on the line, the opposition of which to the usual (Weyl-type) analysis can be said to reflect that, in representation theory, between the representations from the discrete and from the (fu
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André Unterberger
Alternative Pseudodifferential Analysis With an Application to Modular Forms
1935
123
Lecture Notes in Mathematics Editors: J.-M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris
1935
André Unterberger
Alternative Pseudodifferential Analysis With an Application to Modular Forms
ABC
Author André Unterberger Mathématiques Université de Reims Moulin de la Housse, BP 1039 51687 Reims Cedex 2 France [email protected]
ISBN: 978-3-540-77910-0 e-ISBN: 978-3-540-77911-7 DOI: 10.1007/978-3-540-77911-7 Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2008921392 Mathematics Subject Classification (2000): 35S99, 22E70, 42A99, 11F11, 11F37, 81S30 c 2008 Springer-Verlag Berlin Heidelberg ° 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. Cover design: WMXDesign GmbH Printed on acid-free paper 987654321 springer.com
To Franc¸ois Treves
Preface
The subject of the present work is pseudodifferential analysis: the motivations lie in harmonic analysis and modular form theory. So far as the last two domains are concerned, nothing more than some minimal familiarity is needed: some knowledge of the metaplectic representation, and of the definition of holomorphic and nonholomorphic modular forms, will help. Even though the symbolic calculus introduced here is entirely new, and does not depend on any technical result concerning pseudodifferential operators, it would not be honest to claim that no previous acquaintance with that field is necessary: the analysis developed here is strikingly different from the usual one, some knowledge of which – in particular, its representation-theoretic aspects – is needed for comparison. Modular form theory is a very appealing subject: some time ago already, we tried to approach it from an angle which, to us, was much more familiar, that of pseudodifferential analysis. It is possible to realize nonholomorphic modular forms as distributions in the plane [35, Sect. 18], the main benefit being that they can then be considered as symbols for a calculus of the usual species, to wit the Weyl calculus. Yes, there are difficulties on the way toward developing the symbolic calculus of associate
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