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

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• 作者：Michael Livesey ^{michael.livesey@manchester.ac.uk" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Representation theory ; Character theory ; Perfect isometries ; Finite general linear groups ; Finite unitary groups ; Unipotent blocks
• 刊名：Journal of Algebra
• 出版年：2017
• 出版时间：1 January 2017
• 年：2017
• 卷：469
• 期：Complete
• 页码：109-119
• 全文大小：322 K

文摘

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 $q\mathrm{mod}\ell$ and n a positive integer. A. Watanabe proved that there is a perfect isometry between the principal ℓ  -blocks of GL_n(q)${\mathrm{GL}}_n(q)$ and GL_n(q^d)${\mathrm{GL}}_n(q^d)$ 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(q²)|/|U_n(q)|$|{\mathrm{GL}}_n(q^2)|/|{\mathrm{U}}_n(q)|$ then there is a perfect isometry between the principal ℓ  -blocks of dd3a015541455dc" title="Click to view the MathML source">U_n(q)${\mathrm{U}}_n(q)$ and GL_n(q²)${\mathrm{GL}}_n(q^2)$ with the correspondence of characters also given by Shintani descent. R. Kessar extended this first result to all unipotent blocks of GL_n(q)${\mathrm{GL}}_n(q)$ and GL_n(q^d)${\mathrm{GL}}_n(q^d)$. In this paper we extend this second result to all unipotent blocks of dd3a015541455dc" title="Click to view the MathML source">U_n(q)${\mathrm{U}}_n(q)$ and GL_n(q²)${\mathrm{GL}}_n(q^2)$. In particular this proves that any two unipotent blocks of dd3a015541455dc" title="Click to view the MathML source">U_n(q)${\mathrm{U}}_n(q)$ 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.