arXiv2018
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 $π$ of $GL(2,{\mathcal O})$ is associated a pair of positive integers $k, k'$ called the level and the sublevel of $π.$ The level is the smallest integer $k$ such that $π$ 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 $χ,$ one dimensional representation of $GL(2,{\mathcal O}),$ such that $π\otimes χ$ 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 Salié 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 $PGL(2,{\mathcal O})$.