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 qmodℓ and n a positive integer. A. Watanabe proved that there is a perfect isometry between the principal ℓ  -blocks of GLn(q)GLn(q) and GLn(qd)GLn(qd) where the correspondence of characters is given by Shintani descent. In the same paper Watanabe also proved that if ℓ and q are odd and ℓ   does not divide |GLn(q2)|/|Un(q)||GLn(q2)|/|Un(q)| then there is a perfect isometry between the principal ℓ  -blocks of Un(q)Un(q) and GLn(q2)GLn(q2) with the correspondence of characters also given by Shintani descent. R. Kessar extended this first result to all unipotent blocks of GLn(q)GLn(q) and GLn(qd)GLn(qd). In this paper we extend this second result to all unipotent blocks of Un(q)Un(q) and GLn(q2)GLn(q2). In particular this proves that any two unipotent blocks of Un(q)Un(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.