Local Theory

  • H. Jacquet
  • R. P. Langlands
Part of the Lecture Notes in Mathematics book series (LNM, volume 114)


Irreducible Representation Invariant Subspace Haar Measure Cartan Subgroup Finite Dimensional Space 
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References for Chapter I

The Weil representation is constructed in

  1. 1.
    Weil, A., Sur certains groupes d'opérateurs unitaires, Acta Math., t. 111, 1964.Google Scholar

One of the first to study representations of groups over non-archimedean local fields was F. Mautner in

  1. 2.
    Mautner, F., Spherical functions over gadic fields, I, Amer. Jour. Math., vol LXXX, 1958.Google Scholar

Absolutely cuspidal representations were first constucted by Gelfand and Graev. References to their work and that of Kirillov will be found in

  1. 3.
    Gelfand, I.M., M.I. Graev, and I.I. Pyatetskii — Shapiro, Representation Theory and Automorphic Functions, W.B. Saunders Co., 1966.Google Scholar

These representations were constructed in terms of the Weil representation by Shalika and by Tanaka.

  1. 4.
    Shalika, J. Representations of the two-by-two unimodular group over local fields, Notes, Institute for Advanced Study.Google Scholar
  2. 5.
    Tanaka, S., On irreducible unitary representations of some special linear groups of the second order, Osaka Jour. Math., 1966.Google Scholar

To classify the representations over an archimedean field we have used a theorem of Harish-Chandra which may be found in

  1. 6.
    Harish-Chandra, Representations of semisimple Lie groups, II, T.A.M.S., vol 76, 1954.Google Scholar

Our discussion of characters owes much to

  1. 7.
    Sally, P.J. and J.A. Shalika, Characters of the discrete series of representations of SL(2) over a local field, P.N.A.S., 1968.Google Scholar

Three standard references to the theory of L — functions are

  1. 8.
    Lang, S., Algebraic numbers, Addison-Wesley, 1964.Google Scholar
  2. 9.
    Tate, J., Fourier analysis in number fields and Hecke's Zeta — functions in Algebraic number theory, Thompson Book Co., 1967.Google Scholar
  3. 10.
    Weil, A., Basic number theory, Springer Verlag, 1967.Google Scholar

In Paragraph 8 we have used a result from

  1. 11.
    Harish-Chandra, Automorphic forms on semisimple Lie groups, Springer-Verlag, 1968.Google Scholar

Tamagawa measures are discussed in

  1. 12.
    Weil, A. Adèles and algebraic groups, Institute for Advanced Study, 1961.Google Scholar

Copyright information

© Springer-Verlag 1970

Authors and Affiliations

  • H. Jacquet
  • R. P. Langlands

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