The Shale-Weil representation associated to a quadratic form

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


Let (V, B) be a symplectic space. The representation \( \mathop{R}\limits^{\sim } \) of \(\mathop{{Sp}}\limits^{\sim } \) is “almost” irreducible. It is a sum of two irreducible representations corresponding to even and odd functions. We will similarly study the decomposition into irreducible components of the k-tensor products of the representation \(\mathop{R}\limits^{\sim } \). The principle is the following: Let (E, S) be an orthogonal vector space of dimension k, with a quadratic form S. Then the space (V ⊗ E, B ⊗ S) with the bilinear form B ⊗ S is a symplectic space. The representation R of the group Sp(B × S) restrict to Sp(B) × O(S) in RS. When S is positive definite, RS, as a representation of Sp(B), is the k-tensor product of R and the compact group O(k) plays an analogous role for the de-k composition of \(\mathop{ \otimes }\limits^{k} \) that the symmetric group σk for the representation of GL(n, ℂ) \(\mathop{ \otimes }\limits^{k} {{\rm{C}}^{n}} \) let us consider an irreducible representation X of O(k) occurring in RS, then the restriction of the representation RS to the space of vectors of type X under O(k) is jointly irreducible under Sp(B) x O(k). Thus the unitary representation of Sp(B) occurring in RS are in this way naturally parametrized by representations of the compact group O(k).


Irreducible Representation Unitary Representation Harmonic Polynomial Symplectic Space 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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