Rigid Inner Forms Over Function Fields

Abstract

We generalize the concept of rigid inner forms, defined by Kaletha in [Kal16] and [Kal18], to the setting of a local or global function field F in order to study endoscopy over F and state conjectures regarding representations of an arbitrary connected reductive group G over F . To do this, we define for such G a new cohomology set H1(E,Z → G) ⊂ H1 (E,G), where E is an fpqc A-gerbe over F attached to a class in H2 (F,A) for an explicit profinite commutative fppf group scheme A depending only on F (not on G), and extend the classical Tate-Nakayama duality theorem (locally), Tate’s global duality (cf. [Tat66]) result for tori, and their reductive analogues to these new expanded cohomology sets.

We define a relative transfer factor for an endoscopic datum serving a connected reductive group G over local 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 (local) Langlands parameter. Using global rigid inner forms, a localization map from the local gerbe to its global counterpart allows us to organize sets of local rigid inner forms into coherent families, allowing for a definition of global L-packets and a conjectural formula for the multiplicity of an automorphic representation π in the discrete spectrum of G in terms of these L-packets. We also show that, for a connected reductive group G over a global function field F , the adelic transfer factor ∆A for the ring of adeles A of global F serving an endoscopic datum for G decomposes as the product of the normalized local transfer factors.