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Restriction de la représentation de Weil à un sous-groupe compact maximal ou à un tore maximal elliptique

Weil's representation is a basic object in representation theory which plays a crucial role in many places: construction of unitary irreducible representations in the frame of the orbit method, Howe correspondence, Theta series,... The decomposition in irreducible of the restriction of Weil's representation to maximal compact subgroups or anisotropic tori of the metaplectic group is thus an important information in representation theory. Except for SL(2), this was not known in the p-adic case. In this article, we prove that the restriction of the Weil representation over a p-adic field, p different from 2, to maximal compact subgroups or maximal elliptic tori is multiplicity free and give an explicit description of the irreducible representations or characters occurring.