Intégrabilité locale des caractères de SL_n(D)

Bertrand Lemaire

Vol. 222 (2005), No. 1, 69–131

DOI: 10.2140/pjm.2005.222.69

Abstract

Let F be a nonarchimedean locally compact field, G be the multiplicative group of a finite dimensional central simple F-algebra, and G′ be the kernel of the reduced norm det′ : G → F^×. We prove in this paper that for all distinguished open subgroup H ⊂ G and all irreducible (smooth complex) representation π of H, the character Θ_π = trace(π) is a locally integrable distribution on H, locally constant on the set of regular elements of H. Then we deduce that for all irreducible representation π′ of G′, the character Θ_π′ = trace(π′) is a locally integrable distribution on G′, locally constant on the set of regular elements of G′.

Keywords

local field, central simple algebra, reduced norm, smooth complex representation, character of SL_n(D), twisted character of GL_n(D), Fourier transform, local integrability

Mathematical Subject Classification 2000

Primary: 22E50

Milestones

Received: 18 November 2003

Revised: 2 November 2004

Accepted: 6 November 2004

Published: 1 November 2005