Progress in Mathematics

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2 Progress in Mathematics Volume 209 Series Editors H. Bass J. Oesterle A. Weinstein

3 Andre Unterberger Automorphic Pseudodiflerential Analysis and Higher Level Weyl Calculi Springer Base! AG

4 Author: A n c Unterberger l n ~ Mathematiques (UMR 6056) Universite de Reims Moulin de la Housse B.P Reims Cedex 2 France 2000 Mathematics Subject Classification 11Fxx, 22E30, 22E45, 35S99, 47G30, 58J40, 81S1O A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at < ISBN ISBN (ebook) DOI / 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, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained Springer Basel AG Originally published by B i r k h Verlag, u s ebasel- r Boston - Berlin in 2003 Softcover reprint of the hardcover 1 st edition 2003 Printed on acid-free paper produced of chlorine-free pulp. TCF 00 ISBN

5 Ferran Sunyer i Balaguer ( ) was a selftaught Catalan mathematician who, in spite of a serious physical disability, was very active in research in classical mathematical analysis, an area in which he acquired international recognition. His heirs created the Fundaci6 Ferran Sunyer i Balaguer inside the Institut d'estudis Catalans to honor the memory of Ferran Sunyer i Balaguer and to promote mathematical research. Each year, the Fundaci6 Ferran Sunyer i Balaguer and the Institut d'estudis Catalans award an international research prize for a mathematical monograph of expository nature. The prize-winning monographs are published in this series. Details about the prize and the Fundaci6 Ferran Sunyer i Balaguer can be found at This book has been awarded the Ferran Sunyer i Balaguer 2002 prize. ~ E S T V D I S A > f-... ~ Z - -MCMVII,. The members of the scientific commitee of the 2002 prize were: Hyman Bass University of Michigan Pilar Bayer Universitat de Barcelona Antonio Cordoba Universidad Autonoma de Madrid Paul Malliavin Universite de Paris VI Alan Weinstein University of California at Berkeley

6 mestvdisa > f- - V'l r- Z - -MCMVlI JC Ferran Sunyer i Balaguer Prize winners: 1992 Alexander Lubotzky Discrete Groups, Expanding Graphs and Invariant Measures, PM Klaus Schmidt Dynamical Systems of Algebraic Origin, PM The scientific committee decided not to award the prize 1995 As of this year, the prizes bear the year in which they are awarded, rather than the previous year in which they were announced 1996 V. Kumar Murty and M. Ram Murty Non-vanishing of L-Functions and Applications, PM A. Bottcher and Y.I. Karlovich Carles on Curves, Muckenhoupt Weights, and Toeplitz Operators, PM Juan J. Morales-Ruiz Differential Galois Theory and Non-integrability of Hamiltonian Systems, PM Patrick Dehornoy Braids and Selj-Distributivity, PM Juan-Pablo Ortega and Tudor Ratiu Hamiltonian Singular Reduction, (to be published) 2001 Martin Golubitsky and Ian Stewart The Symmetry Perspective, PM 200

7 To the memory of Laurent Schwartz

8 Contents o 1 Foreword Introduction 1 Automorphic Distributions and the Weyl Calculus 2 The Weyl calculus, the upper half-plane, and automorphic distributions Eisenstein distributions, Dirac's comb and Bezout's distribution... 4 The structure of automorphic distributions 5 The main formula: a heuristic approach 2 A Higher-level Weyl Calculus of Operators 6 A tamer version of the Weyl calculus: the horocyclic calculus 7 The higher-level metaplectic representations 8 The radial parts of relativistic wave operators 9 The higher-level Weyl calculi Can one compose two automorphic operators? 11 The sharp product of two power-functions: the Weyl case 12 Beyond the symplectic group The Sharp Composition of Automorphic Distributions 13 The Roelcke-Selberg expansion of functions associated with Q ; ~ #, Q ; ~ the 2 : continuous part The Roelcke-Selberg expansion of functions associated with Q ; ~ # l Q ; ~ the 2 : discrete part A proof of the main formula Towards the completion of the multiplication table 4 Further Perspectives 17 Another way to compose Weyl symbols Odd automorphic distributions and modular forms of non-zero weight New perspectives and problems in quantization theory Index of Notation Subject Index. Bibliography

9 o Foreword The present foreword addresses itself to readers with a previous knowledge, or interest, in pseudodifferential analysis, modular form theory or quantization theory. The book itself, however, to start with the introduction which follows, has been written under no such assumption, and everything needed will be recalled in due time. A. Pseudo differential analysis: First and above all, this is the study of a certain class of pseudo differential operators in one variable, namely those whose Weyl symbols are automorphic distributions on ~ 2 i.e.,, distributions invariant under the linear action of the group r = 8L(2,7l,): these symbols are interesting, but singular objects. The spectral theory of the Euler operator in L 2 ( r -\ a ~ Hilbert 2 ) space the very definition of which may involve the Weyl calculus - shows that automorphic distributions are linear superpositions of the following elementary distributions: the Eisenstein distributions (of which a continuous family is needed) and the (exceptional) cusp-distributions. The main object of this study is the construction of a multiplication table - in other words a symbolic calculus - for the associated operators. From the point of view of pseudo differential analysis, one interest of this work lies in that it requires handling, and composing, extremely singular operators. When automorphic distributions are taken as symbols, a composition formula of the familiar type such as (0.1) would be totally inappropriate, since none of the terms on the right-hand side could have in general any signification. This calls for a drastic change of point of view, putting the emphasis, on the phase space ~ too, 2 on spectral-theoretic concepts rather than differential geometry. Also, it is useful to extend the Weyl calculus Op as a more general calculus OpP depending on some integer p 2': 0: besides its role in smoothing up some of the difficulties inherent in the Weyl calculus proper, this extension may have some interest from the point of view of harmonic analysis - it yields a parameter-dependent generalization of the metaplectic representation - and appears in the analysis of certain relativistic wave equations; it is also the natural pseudo differential analysis to use in problems dealing with functions on the real line, flat up to a specified order at zero. Other by-products include an improved version of Cotlar's lemma about sums of "almost-orthogonal" operators (in Section 10), and some understanding and applications, from the point of view of operator theory, of the complement of the symplectic Lie algebra s p ( n in, ~ ) the full linear algebra g l ( 2 n (Section, ~ ) 12). B. Non-holomorphic modular form theory: There is no classical way to turn spaces of automorphic functions on the half-plane into non-commutative (associative) algebras: the main point of the book is that, taking a quantum point of view, such

10 2 o. Foreword a construction is possible, even quite natural. The trick is to associate operators to automorphic functions (relying on some symbolic calculus of operators) and consider the composition of operators. The composition formulas bring to light, in a novel way, much of the structure pertinent to the study of non-holomorphic modular form theory, such as the zeta function, Hecke's theory, L-functions and convolution L-functions. Even though we could have worked throughout on the half-plane, we have found it much better, for several reasons, to transfer automorphic functions to the plane m;2, where they become objects r-invariant under the linear action of some arithmetic group. This forces one to work with distributions rather than functions, but apart from this harmless fact, it has only advantages: one of these is that the algebraic structure on the space of automorphic distributions can be defined solely in terms of the - immensely popular - Weyl calculus of operators (cf. supra). The main result (Sections 5 and 15) expresses the composition of any two Eisenstein distributions as the image of a quite canonical "Bezout distribution" (related to Poincare-Selberg series) under a simple, but interesting, operator: more details can be found in the following introduction. C. Quantization theory: Under this vocable, we mean the definition and study of rules of symbolic calculus associated with the consideration of nice "phase spaces" : such a space could be m;2 (on which there are more possibilities than what is usually believed, including the OpP -calculus referred to in the first part of this foreword), or r\m;2 (the main object of study here), or a homogeneous space, or (tentatively, at least) some of the spaces of interest in algebraic geometry. Writing this monograph has confirmed again our feeling that deformation quantization (the "small parameter" point of view) may not be the more fruitful point of view. Since this position - also based on a fairly wide experimentation with the construction of alternative pseudo differential analyses, over a span of years - lies outside the more popular trends, we have found it useful, in a last, largely self-contained, expository section, to indicate what could be some lines of a program in this direction. It is to be noted, in particular, that concepts relative to the composition of symbols are given a much wider realm than is usually the case, even (cf. Section 17) in the case of the one-dimensional Weyl calculus.