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

TL;DR

This paper extends known perfect isometries between blocks of finite general linear and unitary groups at unitary primes, demonstrating that unipotent blocks with the same weight are perfectly isometric and that these isometries commute with Deligne-Lusztig induction.

Contribution

It generalizes the perfect isometry results to all unipotent blocks of unitary groups, showing they are perfectly isometric when sharing the same weight, and establishes compatibility with Deligne-Lusztig induction.

Findings

Unipotent blocks of $U_n(q)$ with the same weight are perfectly isometric.

The perfect isometry commutes with Deligne-Lusztig induction.

Extension of isometry results to all unipotent blocks of $U_n(q)$.

Abstract

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.