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 given by Shintani descent. In the same paper Watanabe also proved 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. [ABSTRACT FROM AUTHOR]