A note on perfect isometries between finite general linear and unitary groups at unitary primes

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.