  Let $q$ be a power of a prime, $l$ a prime not dividing $q$, $d$ a positive
integer coprime to both $l$ and the multiplicative order of $q\mod l$ and $n$ a
positive integer. A. Watanabe proved that there is a perfect isometry between
the principal $l-$blocks of $GL_n(q)$ and $GL_n(q^d)$ where the correspondence
of characters is give by Shintani descent. In the same paper Watanabe also
prove that if $l$ and $q$ are odd and $l$ does not divide $GL_n(q^2)|/|U_n(q)|$
then there is a perfect isometry between the principal $l-$blocks of $U_n(q)$
and $GL_n(q^2)$ with the correspondence of characters also given by Shintani
descent. R. Kessar extended this first result to all unipotent blocks of
$GL_n(q)$ and $GL_n(q^d)$. In this paper we extend this second result to all
unipotent blocks of $U_n(q)$ and $GL_n(q^2)$. In particular this proves that
any two unipotent blocks of $U_n(q)$ at unitary primes (for possibly different
$n$) with the same weight are perfectly isometric. We also prove that this
perfect isometry commutes with Deligne-Lusztig induction at the level of
characters.
