Rigid inner forms over local function fields

Abstract

We generalize the concept of rigid inner forms, defined by Kaletha in [12], 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 H1(E,Z→G)⊂Hfpqc1(E,G) for a gerbe E attached to a class in Hfppf2(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 s˙-stable virtual characters for a semisimple s˙ associated to a tempered Langlands parameter.