Maslov indices on the metaplectic group \(Mp(n)\). (English) Zbl 0705.22013

We use the properties of \(Mp(n)\) to construct functions \(\mu_{\ell}: Mp(n)\to {\mathbb{Z}}_ 8\) associated with the elements \(\ell\) of the lagrangian grassmannian \(\Lambda(n)\) which generalize the Maslov index on Mp(n) defined by J. Leray in his “Lagrangian Analysis” (1981; Zbl 0483.35002). We deduce from these constructions the identity between Mp(n) and a subset of \(Sp(n)\times {\mathbb{Z}}_ 8\), equipped with appropriate algebraic and topological structures.
Reviewer: M.de Gosson


22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53D50 Geometric quantization


Zbl 0483.35002
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[1] [G1] , La définition de l’indice de Maslov sans hypothèse de transversalité, C.R. Acad. Sci. Paris, t. 310, Série I (1990), 279-282. · Zbl 0705.22012
[2] [G2] , La relation entre Sp∞, revêtement universel du groupe symplectique Sp et Sp × ℤ, C.R. Acad. Sci. Paris, t. 310, Série I (1990), 245-248. · Zbl 0732.22001
[3] [G3] , The structure of q-symplectic geometry, to appear in : Journal des Mathématiques Pures et Appliquées, Paris, 1990. · Zbl 0829.58015
[4] [GS] , , Geometric Asymptotics, Math. Surveys 14, A.M.S., Providence, R.I., 1977. · Zbl 0364.53011
[5] [L] , Lagrangian Analysis and Quantum Mechanics, The M.I.T. Press, Cambridge, London, 1981, (Analyse Lagrangienne, R.C.P. 25, Strasbourb, 1978 ; Collège de France 1976-1977).
[6] [LV] , , The Weil representation, Maslov index and Theta series, Birkhäuser (Progress in Mathematics), Boston, Basel, Bruxelles, 1980. · Zbl 0444.22005
[7] [W] , Sur certains groupes d’opérateurs unitaires, Acta Math., 111, 1964. · Zbl 0203.03305
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