Maslov indices on the metaplectic group $Mp(n)$

Gosson, Maurice De

doi:10.5802/aif.1223

Maslov indices on the metaplectic group

M p (n)

Gosson, Maurice De

Annales de l'Institut Fourier, Tome 40 (1990) no. 3, pp. 537-555.

Résumé
Abstract

Nous utilisons des propriétés de $M p (n)$ pour construire des fonctions $μ_{ℓ} : M p (n) \to Z_{8}$ associées aux éléments $ℓ$ de la grassmanienne lagrangienne $Λ$ (n), qui généralisent l’indice sur $M p (n)$ défini par Jean Leray dans son “Analyse Lagrangienne”. À l’aide de ces constructions, nous identifions $M p (n)$ avec un sous-ensemble de $S p (n) \times Z_{8}$ , muni d’une topologie et d’une structure algébrique convenables.

We use the properties of $M p (n)$ to construct functions $μ_{ℓ} : M p (n) \to Z_{8}$ associated with the elements $ℓ$ of the lagrangian grassmannian $Λ$ (n) which generalize the Maslov index on Mp(n) defined by J. Leray in his “Lagrangian Analysis”. We deduce from these constructions the identity between $M p (n)$ and a subset of $S p (n) \times Z_{8}$ , equipped with appropriate algebraic and topological structures.

MR Zbl | 1 citation dans Numdam

DOI : 10.5802/aif.1223

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     title = {Maslov indices on the metaplectic group $Mp(n)$},
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Gosson, Maurice De. Maslov indices on the metaplectic group $Mp(n)$. Annales de l'Institut Fourier, Tome 40 (1990) no. 3, pp. 537-555. doi : 10.5802/aif.1223. https://www.numdam.org/articles/10.5802/aif.1223/

Bibliographie
Cité par

[G1] M. De Gosson, La définition de l'indice de Maslov sans hypothèse de transversalité, C.R. Acad. Sci. Paris, t. 310, Série I (1990), 279-282. | MR | Zbl

[G2] M. De Gosson, 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

[G3] M. De Gosson, The structure of q-symplectic geometry, to appear in : Journal des Mathématiques Pures et Appliquées, Paris, 1990. | Zbl

[GS] V. Guillemin, S. Sternberg, Geometric Asymptotics, Math. Surveys 14, A.M.S., Providence, R.I., 1977. | MR | Zbl

[L] J. Leray, 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).

[LV] G. Lion, M. Vergne, The Weil representation, Maslov index and Theta series, Birkhäuser (Progress in Mathematics), Boston, Basel, Bruxelles, 1980. | Zbl

[W] A. Weil, Sur certains groupes d'opérateurs unitaires, Acta Math., 111, 1964. | MR | Zbl

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