Saito--Tunnell theorem is a local version of Waldspurger's  formula, relating the existence of E^\times invariant linear forms on representation of GL_2 to local root numbers. I present a  generalization of this which relates the existence of GL_n(E) invariant linear forms on GL_2n(F) to local root numbers. The proof  relies on a "relative version" of Harish-Chandra' theory of local  character expansions.