  Let $E/F$ be a cyclic extension of $p$-adic fields and $n$ a positive
integer. Arthur and Clozel constructed a base change process $\pi\mapsto \pi_E$
which associates to a smooth irreducible representation of $GL_n(F)$ a smooth
irreducible representation of $GL_n(E)$, invariant under $Gal(E/F)$. When $\pi$
is tempered, $\pi_E$ is tempered and is characterized by an identity (the
Shintani character relation) relating the character of $\pi$ to the character
of $\pi_E$ twisted by the action of $Gal(E/F)$. In this paper we show that the
Shintani relation also holds when $\pi$ is unitary or elliptic. We prove
similar results for the extension $C/R$. As a corollary we show that for a
cyclic extension $E/F$ of number fields the base change for automorphic
residual representations of the ad\`ele group $GL_n(A_F)$ respects the Shintani
relation at each place of $F$.
