• Gérard Lion
  • Michèle Vergne
Part of the Progress in Mathematics book series (PM, volume 6)


We have constructed in Part 1 a projective representation R of the symplectic group. As discovered by A. Weil, this representation plays a central role in the transformation properties of the classical Jacobi θ-serie \(\theta (z) = \sum\limits_{n} {{{e}^{{i\pi {{n}^{2}}z}}}}\) and higher dimensional θ-series, when interpreted as suitable coefficients of this representation R. We indicate now the nature of the relation between R and theta series:
  • Let D be the Siegel upper half-plane, i.e.

  • D = {Z. (n × n) complex symmetric matrices, such that Im Z ≫ 0}


Modular Form Congruence Subgroup High Weight Vector Theta Series Lagrangian Subspace 
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Copyright information

© Springer Science+Business Media New York 1980

Authors and Affiliations

  • Gérard Lion
    • 1
  • Michèle Vergne
    • 2
  1. 1.X U.E.R. de Sciences EconomiquesUniversité de ParisNanterreFrance
  2. 2.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

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