A Distributional Treatment of Relative Mirabolic Multiplicity One

摘要

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) x GL(n-k)(F). We show that if a non-zero H-1-invariant functional exists on a representation, then every PnH1-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.