Weil Representation and Metaplectic Groups over an Integral Domain

Abstract

Given F a locally compact, nondiscrete, non-archimedean field of characteristic ≠ 2 and R an integral domain such that a non-trivial smooth character χ: F → R ^× exists, we construct the (reduced) metaplectic group attached to χ and R. We show that it is in the expected cases a double cover of the symplectic group over F. Finally, we define a faithful infinite dimensional R-representation of the metaplectic group analogue to the Weil representation in the complex case.

Key Words:

• Howe correspondence
• ℓ-modular representations
• Metaplectic group
• Weil representation

2010 Mathematics Subject Classification:

• 11F27

ACKNOWLEDGMENTS

We would like to heartily thank Vincent Sécherre and Ariane Mézard for useful discussions, proofreading, and suggestions, and the whole Equipe d'Algèbre et Géométrie at the Laboratoire de Mathématiques de Versailles in whose midst the idea of this work arose.

Notes

Communicated by P. Tiep.