A Distributional Treatment of Relative Mirabolic Multiplicity One

M. Gurevich

Journal of Lie theory, Tome 27 (2017) no. 2, pp. 397-417

Résumé

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.

Détail

Classification : 20G25, 22E50
Mots-clés : Distinguished representations, p-adic symmetric spaces, mirabolic subgroup, invariant distributions