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Journal of Lie Theory 27 (2017), No. 2, 397--417
Copyright Heldermann Verlag 2017

A Distributional Treatment of Relative Mirabolic Multiplicity One

Maxim Gurevich
Dept. of Mathematics, Weizmann Institute of Science, Rehovot 7610001, Israel


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 $H_{k} ={\bf GL}_k(F)\times {\bf GL}_{n-k}(F)$. We show that if a non-zero $H_1$-invariant linear form exists on a representation, then the a priori larger space of $P\cap H_1$-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 $G/H_k$. Some new distributional methods for non-reductive groups are developed.

Keywords: Distinguished representations, p-adic symmetric spaces, mirabolic subgroup, invariant distributions.

MSC: 20G25, 22E50

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