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 $G$ over $F$ a new cohomology set $H^1(\mathcal{E},Z\to G)\subset H_{\text{fpqc}}^1(\mathcal{E},G)$ for a gerbe $\mathcal{E}$ attached to a class in $H_{\text{fppf}}^2(F,u)$ 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 $\dot{s}$-stable virtual characters for a semisimple $\dot{s}$ associated to a tempered Langlands parameter.

v4: accepted version. Changes to exposition, definition of inverse limit of gerbes, and derived inverse limit calculations, all coming from referee's comments. To appear in Adv. Math. 81 pages

Cited by in corpus (2)

• On formal degrees of unipotent representations
• Functoriality for supercuspidal L-packets