Description

The groups GL_n(F_q) of n × n invertible matrices over a finite field are some of the most fundamental objects in algebra. Their characters have been known explicitly for a long time thanks to the work of J. A. Green. Later on their representations were also constructed, starting with the construction of Deligne and Lusztig.

A natural step up is to consider the groups G_n,r := GL_n(ℤ∕p^rℤ) where r is an integer such that r ≥ 1. When r = 1 these coincide with GL_n(F_p), but for r ≥ 2 they behave quite differently. It is a challenging problem to try to describe the characters or representations of G_r. In some sense one can never hope to get a full explicit list of the representations, so the issue is to try to construct as many representations as possible. The reason why these representations are interesting is that they are closely connected with representations of the group GL_n(ℤ_p), where ℤ_p is the ring of p-adic integers. This in turn plays a fundamental role in modern number theory via the local Langlands program. These representations are also at the centre of recent developments in representation zeta functions.

For n = 2 we have the groups G_2,r = GL₂(ℤ∕p^rℤ), and in this case all the representations can be worked out explicitly (see [1]). For any n ≥ 2, the most general construction is Hill’s construction of so-called regular representations (see [2]). After covering the properties of groups of the type G_n,r and some general techniques of representations of finite groups, the project will proceed with a study of the case n = 2. After that, we will cover the main steps of Hill’s construction of regular representations. Finally, the project can branch off in different directions depending on your interests. Examples of topics include:

• Explicit computation of the characters of G_2,r (for p≠2).
• The representations of the groups G_3,2 and G_4,2.
• Representations of the groups GL_n(F_p[x]∕(x^r)) and Onn’s conjecture.
• The Deligne-Lusztig construction for SL₂(F_q).
• The representation zeta functions of SL₂(ℤ∕p^rℤ) and SL₃(ℤ∕p^rℤ).

Prerequisites

Resources

[1] Stasinski, The smooth representations of GL_2(O), link.

[2] Hill, Regular elements and regular characters of GL_2(O), Journal of Algebra 174 (1995), 610-635.

[3] Singla, On Representations of General Linear Groups over Principal Ideal Local Rings of Length Two, link.