A Distributional Treatment of Relative Mirabolic Multiplicity One
Journal of Lie Theory, Tome 27 (2017) no. 2, pp. 397-417
Voir la notice de l'article provenant de la source Heldermann Verlag
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.
Classification :
20G25, 22E50
Mots-clés : Distinguished representations, p-adic symmetric spaces, mirabolic subgroup, invariant distributions
Mots-clés : Distinguished representations, p-adic symmetric spaces, mirabolic subgroup, invariant distributions
Affiliations des auteurs :
Maxim Gurevich 1
Maxim Gurevich. A Distributional Treatment of Relative Mirabolic Multiplicity One. Journal of Lie Theory, Tome 27 (2017) no. 2, pp. 397-417. http://geodesic.mathdoc.fr/item/JOLT_2017_27_2_a5/
@article{JOLT_2017_27_2_a5,
author = {Maxim Gurevich},
title = {A {Distributional} {Treatment} of {Relative} {Mirabolic} {Multiplicity} {One}},
journal = {Journal of Lie Theory},
pages = {397--417},
year = {2017},
volume = {27},
number = {2},
url = {http://geodesic.mathdoc.fr/item/JOLT_2017_27_2_a5/}
}