A remark on conductor, depth and principal congruence subgroups

Abstract

In this paper we study a relation between conductor, depth, and the level of principal congruence subgroups for irreducible admissible representations of GL n (F ) for a non-archimedean local field F of characteristic zero. Some global and local applications are also discussed.

The density conjecture for principal congruence subgroups

“…The following result classifies representations with C p (p m )-fixed vectors in terms of conductors. This is certainly well-known to experts, but has been explicitly written down in general only recently in the preprint [MY,Theorem 1.2]. Lemma 6.2.…”

“…Proof. The dimension formula is given in [MY,Lemma 3.1] or [NV,Lemma 23.2]. The rest is a direct consequence of Lemma 6.5 and the Berstein-Zelevinsky classification.…”

The sup-norm problem beyond the newform

“…We start with the case n π = 2m π . By [17,Lemma 4.4] we find that a basis for π Kp(mπ ) in the Kirillov model is given by {ξ (mπ) χ : χ ∈ X mπ }.…”