Evaluation of characters of smooth representations of $GL(2,\mathcal {O})$: I. Strongly primitive representations of even level

TL;DR

This paper classifies strongly primitive smooth representations of even level for $GL(2,\mathcal{O})$ over local fields, providing explicit character formulas using sums like Gauss and Kloosterman sums, extending prior work from $\mathbb{Q}_p$.

Contribution

It explicitly describes even level strongly primitive irreducible representations of $GL(2,\mathcal{O})$ and derives character formulas using classical exponential sums, generalizing previous results.

Findings

Explicit classification of even level strongly primitive representations.

Character formulas expressed via Gauss, Kloosterman, and Salié sums.

Extension of prior work from $\mathbb{Q}_p$ to general local fields.

Abstract

Let $F$ be a local field, let ${\mathcal O}$ be its integer ring and $\varpi$ a uniformizer of its maximal ideal. To an irreducible complex finite dimensional smooth representation $\pi$ of $GL(2,{\mathcal O})$ is associated a pair of positive integers $k, k'$ called the level and the sublevel of $\pi.$ The level is the smallest integer $k$ such that $\pi$ factorizes through the finite group $GL(2,{\mathcal O}/\varpi^k {\mathcal O})$, whereas the sublevel is the smallest integer $k'\leq k$ such that there exists $\chi,$ one dimensional representation of $GL(2,{\mathcal O}),$ such that $\pi\otimes \chi$ factorizes through the finite group $GL(2,{\mathcal O}/\varpi^{k'} {\mathcal O}).$ A representation of $GL(2,{\mathcal O})$ is said strongly primitive if the level and sublevel are equal. The classification of smooth finite dimensional representations of $GL(2,{\mathcal O})$ is equivalent to the classification of strongly primitive irreducible representations of $GL(2,{\mathcal O}).$ In this first article we describe explicitely the even level strongly primitive irreducible finite dimensional complex representations of $GL(2,{\mathcal O})$ along the lines of A.Stasinski and R.Barrington-Leigh, G.Cliff and Q.Wen using Clifford theory. In the case where the characteristic $p$ of the residue field is not equal to $2,$ we give exact formulas for the characters of these representations in most cases by reducing them to the evaluation of Gauss sums, Kloosterman sums and Sali\'e sums for the finite ring ${\mathcal O}/\varpi^k{\mathcal O}.$ It generalizes the work of R.Barrington-Leigh, G.Cliff and Q.Wen which was devoted to $F={\mathbb Q}_p.$ A second article will give the evaluation of characters in the odd level case and the exact expressions for certain generalized Zeta function representations of $PGL(2,{\mathcal O})$.