
Journal of Lie Theory 27 (2017), No. 2, 397417 Copyright Heldermann Verlag 2017 A Distributional Treatment of Relative Mirabolic Multiplicity One Maxim Gurevich Dept. of Mathematics, Weizmann Institute of Science, Rehovot 7610001, Israel max.gurevich@weizmann.ac.il [Abstractpdf] We study the role of the mirabolic subgroup $P$ of $G={\bf GL}_n(F)$ ($F$ a $p$adic field) for smooth irreducible representations of $G$ that are distinguished relative to a subgroup of the form $H_{k} ={\bf GL}_k(F)\times {\bf GL}_{nk}(F)$. We show that if a nonzero $H_1$invariant linear form exists on a representation, then the a priori larger space of $P\cap H_1$invariant forms is onedimensional. 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 $G/H_k$. Some new distributional methods for nonreductive groups are developed. Keywords: Distinguished representations, padic symmetric spaces, mirabolic subgroup, invariant distributions. MSC: 20G25, 22E50 [ Fulltextpdf (369 KB)] for subscribers only. 