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 $${\mathrm{U}}$$ .