Journal of Lie Theory

no. 2
A Distributional Treatment of Relative Mirabolic Multiplicity One
Journal of Lie Theory, Volume 27 (2017) no. 2, pp. 397-417
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.
DOI: 10.5802/jolt.952
Classification: 20G25, 22E50
Keywords: Distinguished representations, p-adic symmetric spaces, mirabolic subgroup, invariant distributions 
@article{JOLT_2017_27_2_a5,
     author = {M. Gurevich},
     title = {A {Distributional} {Treatment} of {Relative} {Mirabolic} {Multiplicity} {One}},
     journal = {Journal of Lie Theory},
     pages = {397--417},
     year = {2017},
     volume = {27},
     number = {2},
     doi = {10.5802/jolt.952},
     zbl = {1431.22018},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.952/}
}
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M. Gurevich. A Distributional Treatment of Relative Mirabolic Multiplicity One. Journal of Lie Theory, Volume 27 (2017) no. 2, pp. 397-417. doi: 10.5802/jolt.952

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