The character table of GLn(Fq)⋊<σ>

Volume 403,  by Cheng Shu

Abstract

We compute the character table of GLn(Fq)⋊<σ>, σ being an order 2 exterior automorphism. We begin by giving the parametrisation of σ-stable irreducible characters of GLn(q) and that of the conjugacy classes contained in GLn(q)σ. Our main theorem is a formula expressing the extension of a σ-stable character of GLn(q) to GLn(q)σ as a linear combination of induced cuspidal functions. This formula is built upon a key result of J-L. Waldspurger concerning character sheaves. Using the main theorem, the determination of the character table is then reduced to computations in various Weyl groups and the generalised Green functions of classical groups. Finally, we explicitly determine the table for n=2 and 3.

Published: 16 July 2022

在该论文中，有限群GLn(q)⋊<σ> 的特征标表被完全决定。主定理是把GLn(q)的σ-不动的不可约表示在GLn(q)σ上的延拓表达成诱导的cuspidal functions的线性组合的公式。该公式依赖于一个J-L. Waldspurger关于character sheaves的关键结果。通过该公式，特征标表的计算被约化到一些Weyl群和格林函数的计算。这是character sheaf的理论被用在不可约表示的计算的第一个例子。