Multiplicity one theorem for the unitary and orthogonal groups in positive characteristic

Dor Mezer

Published 2020-10-30Version 1

In [AGRS] a multiplicity one theorem is proven 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 $O,U$ over local fields of positive odd characteristic. By the Gelfand-Kazhdan criterion, this theorem reduces to the statement that any $G_n$-invariant distribution on $G_{n+1}$ is also invariant to transposition. This statement for $GL, O$, and $U$ over over $p$-adic local fields is proven in [AGRS]. An adaptation of the proof for $GL$ that works over of local fields of positive odd characteristic is given in [Mez]. In this paper we make this adaptation also for the orthogonal and unitary groups. Our methods are a synergy of the methods of [AGRS] and of [Mez].