Depth and the Local Langlands Correspondence

Abstract

Abstract. Let G be an inner form of a general linear group over a non-archimedean local field. We prove that the local Langlands correspondence for G preserves depths. We also show that the local Langlands correspondence for inner forms of special linear groups preserves the depths of essentially tame Langlands parameters.

On Satake Parameters for Representations with Parahoric Fixed Vectors

“…13.1.1] or [Sch]). The local Langlands correspondence for the inner form G is also well-understood, and presumably the property LLC+ similarly holds for G. This can likely be extracted from some recent works such as [ABPS,Bad1,HiSa]. We will not verify that G satisfies LLC+ here, and instead we leave this task to another occasion.…”

Comparison of the depths on both sides of the local Langlands correspondence for Weil-restricted groups (with an appendix by Jessica Fintzen)

“…In the special case when M is F -split, the group W F acts trivially on M ∨ , and α φ is a homomorphism, which, by definition, coincides with the restriction of φ to W F . Hence Lemma 2.13 shows that dep(φ) coincides with the definition of the depth of φ, as defined for instance in [ABPS1,§2.3].…”

“…• for GL n (E) and its inner forms, we do have preservation of depth under the LLC for any representation, see [ABPS1]; • for SL n (E) and its inner forms, we have preservation of depth under the LLC for any essentially tame representation, see [ABPS1,Theorem 3.8]; • in large residual characteristic, we have preservation of depth under the LLC for quasi-split classical groups and for arbitrary unitary groups, [Oi1], [Oi2]; • for tamely ramified tori, we have preservation of depth under the LLC for any character, see Yu [Yu1]; • for SL 2 (E) and its inner form, the depth changes under the LLC for any representation π E of G(E) of positive depth that is not essentially tame in the following way: dep(λ G (π E )) < dep(π E ), see [AMPS].…”

“…Proof. Since λ M and λ G are depth preserving (see [ABPS1,Theorem 2.9]), using Lemma 4.3, we get dep(π) = dep(φ) = dep(Ind…”

Some aspects of the geometric structure of the smooth dual of $p$-adic reductive groups

“…A.-M. AUBERT the conference on joint works with Paul, Maarten, Roger and my former student Ahmed Moussaoui, which link the Baum-Connes conjecture to the Langlands program. Paul himself gave a beautiful talk on our work [ABPS10], the most recent of a long list of joint papers [ABPS1], [ABPS2], [ABPS3], [ABPS4], [ABPS5], [ABPS6], [ABPS7], [ABPS8], [ABPS9]. I enjoy always very much attending Paul's talks: I really appreciate his ability in presenting the key ideas of a rather technical subject in a very simple and limpid manner.…”