A Distributional Treatment of Relative Mirabolic Multiplicity One

Abstract

We study the role of the mirabolic subgroup $P$ of $G={\mathbf{GL}}_n(F)$ ($F$ a $p$-adic field) in smooth irreducible representations of $G$ that possess a non-zero invariant functional relative to a subgroup of the form $H_k={\mathbf{GL}}_k(F)\times {\mathbf{GL}}_{n-k}(F)$. We show that if a non-zero $H_1$-invariant functional exists on a representation, then every $P\cap H_1$-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.

arXiv comment15 pages