Multiplicity one for the pair (GL(n,D),GL(n,E))

Abstract

Let F be a local field of characteristic zero. Let D be a quaternion algebra over F. Let E be a quadratic field extension of F. Let {\mu} be a character of GL(1,E). We study the distinction problem for the pair (GL(n,D), GL(n,E)) and we prove that any bi-(GL(n,E), {\mu})-equivariant tempered generalized function on GL(n,D) is invariant with respect to an anti-involution. Then it implies that dimHom({\pi},{\mu}) is at most 1 by the generalized Gelfand-Kazhdan criterion. Thus we give a new proof to the fact that (GL(2n,F),GL(n,E)) is a Gelfand pair when {\mu} is trivial and D splits.