A Distributional Treatment of Relative Mirabolic Multiplicity One

2014/06/12 by Maxim Gurevich, Gurevich, Maxim
Mathematics · #20G25 #22E50 #FOS: Mathematics #Representation Theory (math.RT) #math.RT #msc:20G25 #msc:22E50

paper · pdf · doi:10.48550/arxiv.1406.3154

15 pages

arxiv created 2014/07/22 · arxiv updated 2014/07/23

Abstract

We study the role of the mirabolic subgroup P of G=GL_n(F) (F a p-adic field) in smooth irreducible representations of G that possess a non-zero invariant functional relative to a subgroup of the form H_k = GL_k(F)× GL_n-k(F). We show that if a non-zero H₁-invariant functional exists on a representation, then every P∩ H₁-invariant functional must equal to a scalar multiple of it. When k>1, we give a reduction of the same problem to a question about invariant distributions on the nilpotent cone of the tangent space of the symmetric space G/H_k. Some new distributional methods, which are suitable for a setting of non-reductive groups, are developed.

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