Rigid inner forms over local function fields

Abstract

We generalize the concept of rigid inner forms, defined by Kaletha in [Kal16], to the setting of a local function field F in order state the local Langlands conjectures for arbitrary connected reductive groups over F . To do this, we define for a connected reductive group G overfppf (F, u) for a certain canonically-defined profinite commutative group scheme u, building up to an analogue of the classical Tate-Nakayama duality theorem. We define a relative transfer factor for an endoscopic datum serving a connected reductive group G over F , and use rigid inner forms to extend this to an absolute transfer factor, enabling the statement of endoscopic conjectures relating stable virtual characters and ṡ-stable virtual characters for a semisimple ṡ associated to a tempered Langlands parameter.

Local Jacquet-Langlands correspondence for regular supercuspidal representations

“…In [Kal19a, §5], the characteristic ch F of F is also assumed to be zero. This assumption can now be removed, thanks to the work of Dillery [Dil20] extending the formalism of rigid inner twists in [Kal16b] to general characteristic (see also Section 2.1 and the beginning of Section 5.2 in [Kal19a]).…”

“…As discussed in [Kal19a, §5.1], there is a natural bijection between the set of Γ Fstable G * (F sep )-conjugacy classes of embeddings S ֒→ G * over F sep and the set of Γ F -stable " G * -conjugacy classes of embeddings S ֒→ " G * . Thus, given an embedding j as above, its " [Dil20,Lemma 7.6]. Although the torus S above is given abstractly, many conditions imposed are formulated via this embedding j.…”

“…For more on rigid inner twists and their applications to the local Langlands correspondence, see [Kal16a] as well as [Kal16b] and [Dil20].…”

Untitled

“…It is also interesting to consider Theorem 1 for all inner twists of a given quasisplit group simultaneously. That is done best with the rigid inner twists from [Kal1,Dil]. In that setting we replace S φ by a slightly different component group S + φ and we write Φ + ( L G) = (φ, ρ + ) : φ ∈ Φ(G), ρ + ∈ Irr(S + φ ) .…”

Representations of reductive groups over local fields

“…When F has positive characteristic the simplified concept of a Galois gerbe as an extension of the absolute Galois group becomes inadequate, due to the possible non-smoothness of u. Despite this difficulty, Peter Dillery [Dil20] has found a way to construct a suitable analog of E rig . In fact, his construction works uniformly for all non-archimedean local fields and recovers E rig when F has characteristic zero.…”