Multiplicity one theorems over positive characteristic

Abstract

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