MULTIPLICITY ONE FOR THE PAIR (GLn(D);GLn(E))

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 μ be a character of E × . We study the distinction problem for the pair (GL n ( D );GL n ( E )) and we prove that any bi-(GL n ( E ); μ )-equivariant tempered generalized function on GL n ( D ) is invariant with respect to an anti-involution. Then it implies that dimHom GL n ( E ) (π; μ ) ≤ 1 by the generalized Gelfand-Kazhdan criterion. Thus we give a new proof to the fact that (GL 2 n ( F );GL n ( E )) is a Gelfand pair when μ is trivial and D splits.