On $$\mathsf {G} $$-isoshtukas over function fields

Abstract

In this paper we classify isogeny classes of global $$\mathsf {G} $$ G -shtukas over a smooth projective curve $$C/{\mathbb {F}}_q$$ C / F q (or equivalently $$\sigma $$ σ -conjugacy classes in $$\mathsf {G} (\mathsf {F} \otimes _{{\mathbb {F}}_q} \overline{{\mathbb {F}}_q})$$ G ( F ⊗ F q F q ¯ ) where $$\mathsf {F} $$ F is the field of rational functions of C) by two invariants $${\bar{\kappa }},{\bar{\nu }}$$ κ ¯ , ν ¯ extending previous works of Kottwitz. This result can be applied to study points of moduli spaces of $$\mathsf {G} $$ G -shtukas and thus is helpful to calculate their cohomology.

Representations of the Kottwitz gerbes

“…Note that G-isoshtukas for connected reductive groups G were recently studied by Hamacher and Kim [20] from a somewhat different perspective. Namely, they defined isoshtukas as G-torsors V over F ⊗ Fq F together with an isomorphism Frob * V ≃ V and classified them very explicitly [20,Theorem 1.3]. Since G is connected reductive, then H 1 (F ⊗ Fq F, G) = 0 by Steinberg theorem (see [45]).…”