On factor representations of discrete rational nilpotent groups and the Plancherel formula

Lawrence Jay Corwin and Carolyn Pfeffer Johnston

Vol. 162 (1994), No. 2, 261–275

DOI: 10.2140/pjm.1994.162.261

Abstract

The purpose of this paper is to extend the Kirillov orbit picture of representation theory for nilpotent Lie groups to discrete groups G_Q defined over the rationals Q , following a program begun by Roger Howe. Let Ad^∗ be the coadjoint action of G_Q on the Pontryagin dual g_Q of the Lie algebra of G_Q. It is shown that each coadjoint orbit closure is a coset of the annihilator of an ideal of g_Q, that a certain induced representation canonically associated with an orbit closure is a traceable factor, and that there is an orbital integral formula which gives the trace. Finally, a Plancherel formula is proved.

Mathematical Subject Classification 2000

Primary: 22D12

Secondary: 22E25

Milestones

Received: 18 February 1992

Published: 1 February 1994