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

In this paper, the character table of the finite group GLn(q)⋊<σ> is determined. The main theorem is a formula expressing the extension of a σ-stable irreducible 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. This is the first instance where the theory of character sheaf is used to explicitly compute irreducible characters.