Multiplicity one theorems over positive characteristic

Abstract In Aizenbud et al. (2010, Annals of Mathematics 172, 1407–1434), a multiplicity one theorem is proved for general linear groups, orthogonal and unitary groups ( $GL, O,$ U ) over p -adic local fields. That to say that when we have pair such $G_n{\subseteq } G_{n+1}$ , any restriction an irreducible smooth representation $G_{n+1}$ $G_n$ multiplicity-free. This property already known $GL$ field positive characteristic, in this paper, also give proof $O,U$ $SO$ fields odd characteristic. These theorems are shown Gan, Gross, Prasad (2012, Sur les Conjectures de Gross Prasad. I, Société Mathématique France imply the uniqueness Bessel models, Chen Sun (2015, International Research Notice 2015, 5849–5873) Rankin–Selberg models. We prove simultaneously Fourier–Jacobi following outlines American Journal 134, 1655–1678). By Gelfand–Kazhdan criterion, $H\leq G$ follows from statement distribution on G invariant conjugations by H some anti-involution preserving . O$ 1407–1434). An adaptation works characteristic given Mezer (2020, Mathematische Zeitschrift 297, 1383–1396). similar adaptations proofs as well special symplectic groups. Our methods synergy used 0 (Aizenbud [2010, 1407–1434]; [2012, 1655–1678]; Waldspurger Astérisque 346, 313–318]) those