Uniqueness of generalized Waldspurger model for \(GL(2n)\)
Let \(F\) be a non-archimedean local field of characteristic zero. Let \(D\) be either the quaternion field over \(F\) or the algebra of two by two matrices over \(F\). Denote by \(G\) the group \(GL(n,D)\) and by \(H\) its subgroup \(GL(n,E)\), where \(E\) is a quadratic extension of \(F\) imbedded in \(D\). It is proved in the paper that for any \(g\in G\) one has \(g^{-1} \in HgH\). As a consequence, there can be at most one \(H\)-invariant linear form (up to a scalar factor) on the space of an irreducible admissible representation of \(G\). For the case where \(D\) is the algebra of two by two matrices and \(E=F\times F\) [see \textit{H. Jacquet} and \textit{S. Rallis}, Uniqueness of linear periods, Compos. Math. 102, 65-123 (1996; Zbl 0855.22018)].
space of an irreducible admissible representation
scientific article; zbMATH DE number 1136629