Vol. 260, No. 2, 2012

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Comparison of compact induction with parabolic induction

Guy Henniart and Marie-France Vigneras

Vol. 260 (2012), No. 2, 457–495

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-IndKG(F)V to the parabolic induced representation IndP(F)G(F)(c-IndM(F)KM(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.

representations modulo p of reductive p-adic groups, compact induction, parabolic induction, Satake isomorphism
Mathematical Subject Classification 2010
Primary: 22E50
Secondary: 11F70
Received: 17 July 2012
Revised: 6 October 2012
Accepted: 6 October 2012
Published: 30 November 2012
Guy Henniart
Laboratoire de Mathématiques d’Orsay
Université Paris-Sud
CNRS, UMR 8628, Orsay Cedex F-91405
Marie-France Vigneras
Institut de Mathématiques de Jussieu
Université de Paris 7
175 rue du Chevaleret
Paris 75013