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

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 $k,k^{\prime }$ 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 $k^{\prime }\le k$ such that there exists $\chi ,$ one dimensional representation of $GL(2,\mathcal{O}),$ such that $\pi \otimes \chi$ factorizes through the finite group $GL(2,\mathcal{O}/{\varpi }^{k^{\prime }}\mathcal{O}).$ 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 $PGL(2,\mathcal{O})$.

arXiv comment38 pages