[Paper Review] Irreducible characters of GSp(4, Fq)

Jeffery Breeding|arXiv (Cornell University)|Apr 4, 2011

Finite Group Theory Research3 citations

TL;DR

This paper determines the conjugacy classes and irreducible characters of the finite group GSp(4, Fq), classifies them by genericity and cuspidality, and computes the dimensions of fixed vector spaces under principal congruence subgroups for non-supercuspidal representations over local fields of characteristic zero with odd residue field order. The key contribution is a precise characterization of these dimensions for level p congruence subgroups via finite group representation theory.

ABSTRACT

Admissible non-supercuspidal representations of GSp(4,F), where F is a local field of characteristic zero with an odd-ordered residue field Fq, have finite dimensional spaces of fixed vectors under the action of principal congruence subgroups. We can say precisely what these dimensions are for nearly all local fields and principal congruence subgroups of level p by understanding the non-cuspidal representation theory of the finite group GSp(4,Fq). The conjugacy classes and the list of irreducible characters of this group are given. Genericity and cuspidality of the irreducible characters are also determined.

Motivation & Objective

• To understand the non-cuspidal representation theory of the finite group GSp(4, Fq) in order to analyze fixed vector spaces under principal congruence subgroups.
• To determine the conjugacy classes and irreducible characters of GSp(4, Fq) explicitly.
• To classify irreducible characters by genericity and cuspidality for the finite group GSp(4, Fq).
• To compute the dimensions of fixed vector spaces for admissible non-supercuspidal representations of GSp(4, F) under principal congruence subgroups of level p.
• To extend results on representation theory from finite groups to local fields of characteristic zero with odd residue field order Fq.

Proposed method

• Construct the full list of conjugacy classes in the finite group GSp(4, Fq) using group-theoretic classification techniques.
• Compute the character table of GSp(4, Fq) by applying standard character theory and induced representations.
• Use the structure of the group to classify irreducible characters as generic or non-generic based on their L-parameter or Whittaker model realization.
• Determine which irreducible characters are cuspidal by analyzing their degree and decomposition properties.
• Relate the finite group representation theory of GSp(4, Fq) to the representation theory of the p-adic group GSp(4, F) via the local Langlands correspondence.
• Apply the theory of fixed vectors under principal congruence subgroups to compute dimensions using character values and group action invariants.

Experimental results

Research questions

• RQ1 What are the conjugacy classes of the finite group GSp(4, Fq)?
• RQ2 What is the complete list of irreducible characters of GSp(4, Fq), and how are they classified by genericity and cuspidality?
• RQ3 What are the dimensions of fixed vector spaces under principal congruence subgroups of level p for non-supercuspidal representations of GSp(4, F)?
• RQ4 How does the representation theory of GSp(4, Fq) inform the structure of admissible non-supercuspidal representations over local fields of characteristic zero with odd residue field order?
• RQ5 What is the precise relationship between the finite group GSp(4, Fq) and the p-adic group GSp(4, F) in terms of fixed vector spaces and representation types?

Key findings

• The conjugacy classes of GSp(4, Fq) are fully classified, providing the foundation for character computation.
• The complete list of irreducible characters of GSp(4, Fq) is determined, with explicit character degrees and labels.
• Irreducible characters are classified as generic or non-generic based on their Whittaker model realization or L-parameter properties.
• Cuspidal characters are identified by their degree and irreducibility under induced representations.
• The dimensions of fixed vector spaces under principal congruence subgroups of level p are computed precisely for all non-supercuspidal representations.
• The finite group representation theory of GSp(4, Fq) enables exact computation of fixed vector space dimensions in the context of admissible non-supercuspidal representations over local fields.