Irreducible characters of GSp(4, q) and dimensions of spaces of fixed vectors

TL;DR

This paper computes the irreducible characters and conjugacy classes of GSp(4,q), classifies their cuspidality and genericity, and applies these results to determine dimensions of fixed vector spaces in certain representations over local fields.

Contribution

It provides a detailed classification of irreducible characters of GSp(4,q) and links these to the structure of fixed vector spaces in non-supercuspidal representations.

Findings

Classification of conjugacy classes and irreducible characters of GSp(4,q)

Identification of non-cuspidal and generic characters

Explicit dimensions of fixed vector spaces in representations

Abstract

In this paper, we compute the conjugacy classes and the list of irreducible characters of GSp(4,q), where q is odd. We also determine precisely which irreducible characters are non-cuspidal and which are generic. These characters are then used to compute dimensions of certain subspaces of fixed vectors of smooth admissible non-supercuspidal representations of GSp(4,F), where F is a non-archimedean local field of characteristic zero with residue field of order q.