Vol. 7, No. 7, 2013

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Weil representation and transfer factor

Teruji Thomas

Vol. 7 (2013), No. 7, 1535–1570
Abstract

This paper concerns the Weil representation of the semidirect product of the metaplectic and Heisenberg groups. First we present a canonical construction of the metaplectic group as a central extension of the symplectic group by a subquotient of the Witt group. This leads to simple formulas for the character, for the inverse Weyl transform, and for the transfer factor appearing in J. Adams’s work on character lifting. Along the way, we give formulas for outer automorphisms of the metaplectic group induced by symplectic similitudes. The approach works uniformly for finite and local fields.

Keywords
metaplectic group, Weil representation, Weyl transform, transfer factor, Cayley transform, Maslov index
Mathematical Subject Classification 2010
Primary: 11F27
Secondary: 20C15
Milestones
Received: 14 August 2011
Revised: 5 June 2012
Accepted: 5 July 2012
Published: 12 October 2013
Authors
Teruji Thomas
School of Mathematics
University of Edinburgh
JCMB, The King’s Buildings
Mayfield Road
Edinburgh
EH9 3JZ
United Kingdom