Distinction for Unipotent p-Adic Groups

Abstract

Let F be a p-adic field and $\varvec{{\mathrm{U}}}$ be a unipotent group defined over F, and set ${\mathrm{U}}=\varvec{{\mathrm{U}}}(F)$. Let $\sigma$ be an involution of $\varvec{{\mathrm{U}}}$ defined over F. Adapting the arguments of Yves Benoist (J Funct Anal 59(2):211–253, 1984; Mem Soc Math France 15:1–37, 1984) in the real case, we prove the following result: an irreducible representation $\pi$ of ${\mathrm{U}}$ is ${\mathrm{U}}^{\sigma }$-distinguished if and only if it is $\sigma$-self-dual and in this case ${\text {Hom}}_{{\mathrm{U}}^\sigma }(\pi ,\mathbb {C})$ has dimension one. When $\sigma$ is a Galois involution, these results imply a bijective correspondence between the set ${\text {Irr}}({\mathrm{U}}^\sigma )$ of isomorphism classes of irreducible representations of ${\mathrm{U}}^\sigma$ and the set ${\text {Irr}}_{{\mathrm{U}}^\sigma -\mathrm {dist}}({\mathrm{U}})$ of isomorphism classes of distinguished irreducible representations of ${\mathrm{U}}$.