A distributional treatment of relative mirabolic multiplicity one

We study the role of the mirabolic subgroup P of G = GLn(F) (F a p-adic field) for smooth irreducible representations of G that are distin-guished relative to a subgroup of the form Hk = GLk(F) × GLn-k(F) . We show that if a non-zero H1 -invariant linear form exists on a representation, then the a priori larger space of P ∩ H1 -invariant forms is one-dimensional. When k > 1, we give a reduction of the same problem to a question about invariant distributions on the nilpotent cone tangent to the symmetric space G=Hk . Some new distributional methods for non-reductive groups are developed. Mathematics Subject Classification 2010: 20G25, 22E50.