The cocycle of the Shale-Weil representation and the Maslov index

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


Let ℓ1, ℓ2, ℓ3 be three Lagrangian planes in the symplectic vector space (V, B). We consider the canonical unitary intertwining operator (we leave implicit the choice of e1, e2, e3):
$${{\mathcal{F}}_{{i,j}}}:H({{\ell }_{j}}) \to H({{\ell }_{i}})$$
which intertwines W(ℓj) and W(ℓi) defined in 1.4.8. It is clear that the operator Ƒ1,3 Ƒ3,2 Ƒ2,1 is proportional to the identity operator on H(ℓ1) as this operator intertwines the irreducible representation W(ℓ1) with itself. Hence there is a scalar of modulus one a(ℓ1, ℓ2, ℓ3) such that:
$${{\mathcal{F}}_{{{{\ell }_{1}},{{\ell }_{3}}}}}{{\mathcal{F}}_{{{{\ell }_{3}},{{\ell }_{2}}}}}{{\mathcal{F}}_{{{{\ell }_{2}},{{\ell }_{1}}}}} = a({{\ell }_{1}},{{\ell }_{2}},{{\ell }_{3}})Id.$$
(It is easy to see that a (l 1, ℓ2, ℓ3) does not depend of e1, e2, e3.)


Projective Representation Maslov Index Lagrangian Subspace Plancherel Formula Weil Representation 
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Copyright information

© Springer Science+Business Media New York 1985

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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