Irreducible representations of a product of real reductive groups

Let G1;G2 be real reductive groups and (π, V ) be a smooth admissible representation of G1 - G2 . We prove that (π,V ) is irreducible if and only if it is the completed tensor product of (π,i Vi) , i = 1; 2, where (πI, Vi) is a smooth, irreducible, admissible representation of moderate growth of Gi , i = 1; 2. We deduce this from the analogous theorem for Harish-Chandra modules, for which one direction was proven in A. Aizenbud and D. Gourevitch, Multiplicity one theorem for (GLn+1(ℝ);GLn(ℝ)) , Selecta Mathematica N. S. 15 (2009), 271-294, and the other direction we prove here. As a corollary, we deduce that strong Gelfand property for a pair H ⊂ G of real reductive groups is equivalent to the usual Gelfand property of the pair ΔH ⊂ G × H.