Shintani relation for base change: unitary and elliptic representations

Let $E\mathrm{/}F$ 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 $G{L}_n(F)$ a smooth irreducible representation of $G{L}_n(E)$, invariant under $Gal(E\mathrm{/}F)$. 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 $Gal(E\mathrm{/}F)$. In this paper we show that the Shintani relation also holds when $\pi$ is unitary or elliptic. We prove similar results for the extension $C\mathrm{/}R$. As a corollary we show that for a cyclic extension $E\mathrm{/}F$ of number fields the base change for automorphic residual representations of the ad`ele group $G{L}_n(A_F)$ respects the Shintani relation at each place of $F$.