The inertial Jacquet-Langlands correspondence

Abstract

We give a parametrization of the simple Bernstein components of inner forms of a general linear group over a local field by invariants constructed from type theory, and explicitly describe its behaviour under the Jacquet-Langlands correspondence. Along the way, we prove a conjecture of Broussous, Sécherre and Stevens on preservation of endo-classes.

Breuil-Mézard conjectures for central division algebras

“…In this paragraph we let G = GL m (D) be an inner form of GL n (F ), for D a central division algebra over F of reduced degree d. We write A = M m (D). We summarize the parametrization of simple inertial classes of representations of G from the point of view of [Dot17], building upon the work of Bushnell-Kutzko [BK93] and Broussous, Sécherre and Stevens in a series of papers (see for instance [BSS12] and [SS16]). Recall that these are the inertial classes whose supercuspidal support is inertially equivalent to rπ 0 for some positive divisor r|m and some supercuspidal representation π 0 of GL m/r (D).…”

“…Granting this, one transfers the result from GL n (F ) to D × by composing with JL p . In order to prove theorem 5.2 we need a complete description of the Jacquet-Langlands correspondence in terms of type theory, which was obtained in [Dot17]. We deduce our result from this, a base change procedure to unramified extensions of F originating in [BH96], and explicit computations with a number of character formulas.…”

“…Yao makes similar conjectures in the case of more general inner forms, as this definition of JL K makes sense for GL r (D) when formulated for those inertial types (τ, N ) extending to a Langlands parameter for GL r (D). At least in the case of discrete series parameters, it seems that our methods extend to this situation without too much trouble: the inertial Jacquet-Langlands correspondence is proved in full generality in [Dot17], and there is a natural candidate for the JL p map, namely Lusztig induction for the twisted Levi subgroup GL r (d) of GL n (f ). We have chosen to focus on the simpler case of D × , but we remark that from the viewpoint of a Jacquet-Langlands correspondence for maximal compact subgroups one expects somewhat weaker results for GL r (D).…”

Explicit local Jacquet-Langlands correspondence: The non-dyadic wild case

“…With this result to hand, the way is open to follow the framework of [6] and [8] (but without complications arising from the transfer factors of automorphic induction [13]) to an explicit description of the Jacquet-Langlands correspondence for representations π ∈ A (G) with d(π) = n. The more difficult case is that where d(π) < n while π is cuspidal. With the newly available Endo-class Transfer Theorem of [22] and [11] recalled below, that general case is substantially less mysterious than hitherto. However, it seems unlikely that one will resolve the question finally without the detail of the complementary special case treated here.…”

“…Using these results, for varying ℓ, Sécherre and Stevens show that the Endo-class Transfer Theorem holds in general provided it holds when d(π i ) = n and π 1 is totally ramified [22], that is, χπ 1 ∼ = π 1 when χ is a non-trivial unramified character of F × . Using a neat device combining properties of certain simple characters, relative to unramified base field extension, and trace comparisons of a sort familiar from [6] or [8], Dotto [11] reduces to the split groups and so despatches the outstanding special case. That method also yields the relation between the simple types contained in corresponding representations π i of parametric degree n.…”