Rationality of the Gaitsgory-Lafforgue-Raskin canonical Arthur filtration

Under the stated characteristic assumptions, the canonical Arthur filtration on finitely supported unramified automorphic functions is defined over $\mathbb Q$, for split connected semisimple $G$, including the noncuspidal part and every closed invariant nilpotent support. Using the companion decomposition at trivial level, each cuspidal filtration step is identified with the scalar extension of the corresponding rational summands (the common-scope form of GLR Conjectures 3.4.9 and 3.4.11). It does not treat small characteristic or nonsplit groups.

Under the stated characteristic assumptions, the canonical Arthur filtration on finitely supported unramified automorphic functions is defined over $\mathbb Q$, for split connected semisimple $G$, including the noncuspidal part and every closed invariant nilpotent support. Using the companion decomposition at trivial level, each cuspidal filtration step is identified with the scalar extension of the corresponding rational summands (the common-scope form of GLR Conjectures 3.4.9 and 3.4.11). It does not treat small characteristic or nonsplit groups.