Comparison of compact induction with parabolic induction

Guy Henniart and Marie-France Vigneras

Vol. 260 (2012), No. 2, 457–495

DOI: 10.2140/pjm.2012.260.457

Abstract

Let F be a nonarchimedean locally compact field of residual characteristic p, let G be a connected reductive F-group, and let K be a special parahoric subgroup of G(F). We choose a parabolic F-subgroup P of G with Levi decomposition P = M@N in good position with respect to K. Let C be an algebraically closed field of characteristic p, and V an irreducible smooth C-representation of K. We investigate the natural intertwiner from the compact induced representation c-Ind_K^G(F)V to the parabolic induced representation Ind_P(F)^G(F)c-Ind_M(F)∩K^M(F)V _N(F)∩K. Under a regularity condition on V , we show that the intertwiner becomes an isomorphism after localization at a specific Hecke operator. When F has characteristic 0, G is F-split and K is hyperspecial, the result was essentially proved by Herzig. We define the notion of K-supersingularity for an irreducible smooth C-representation of G(F) which extends Herzig’s definition for admissible irreducible representations and we give a list of irreducible representations which are neither supercuspidal nor K-supersingular.

Keywords

representations modulo p of reductive p-adic groups, compact induction, parabolic induction, Satake isomorphism

Mathematical Subject Classification 2010

Primary: 22E50

Secondary: 11F70

Milestones

Received: 17 July 2012

Revised: 6 October 2012

Accepted: 6 October 2012

Published: 30 November 2012