Local Newforms for GSp(4)
Local Newforms for GSp(4) describes a theory of new- and oldforms for representations of GSp(4) over a non-archimedean local field. This theory considers vectors fixed by the paramodular groups, and singles out certain vectors that encode canonical inform
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Brooks Roberts
Ralf Schmidt
Local Newforms for GSp(4)
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Authors Brooks Roberts
Ralf Schmidt
Department of Mathematics University of Idaho Moscow ID 83844-1103 USA e-mail: [email protected]
Department of Mathematics University of Oklahoma Norman OK 73019-0315 USA e-mail: [email protected]
About the diagram. The diagram illustrates natural bases for the new- and oldforms in a generic representation π of GSp(4, F) with trivial central character. The solid dot in the first row is the newform at level Nπ . The solid dots and circles of the k-th row represent vectors in a natural basis for the oldforms at level Nπ + k. Thus, the dimension of the paramodular vectors at level Nπ is 1, the dimension at level Nπ + 1 is 2, the dimension at level Nπ + 2 is 4, and so on. The basis at a particular level is obtained from the newform by application of the commuting level raising operators θ , θ 0 , and the self-dual operator η . The arrows pointing down and to the left correspond to θ , the arrows pointing down and to the right correspond to θ 0 , and the vertical arrows correspond to η . The black dots represent oldforms obtained solely by application of θ ’s and θ 0 ’s. The inner-most circles represent oldforms obtained by a single application of η , the circles immediately around the inner-most circles represent oldforms obtained by two applications of η , etc.
Library of Congress Control Number: 2007929853 Mathematics Subject Classification (2000): 11F46, 11F70, 22E50 ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 ISBN 978-3-540-73323-2 Springer Berlin Heidelberg New York DOI 10.1007/978-3-540-73324-9 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 for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Berlin Heidelberg 2007 ° 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. Typesetting by the authors and SPi using a Springer LATEX macro package Cover design: design & production GmbH, Heidelberg Printed on acid-free paper
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Acknowledgments
This work was funded through NSF grants #0454809 and #0400837, commonly titled “Collaborative Research: Local Newforms for GSp(4)”. Further funding was provided by the RiP program of Mathematisches Forschungsinstitut Oberwolfach in 2005. We thank S
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