The Gaitsgory-Lafforgue-Raskin Ramanujan-Arthur decomposition of cuspidal automorphic functions over function fields

Theorem 1.1: for every $X,G,\ell$ and every effective divisor $D$, the rational and $\overline{\mathbb Q}_\ell$ cuspidal spaces at full level $D$ are the direct sums of their orbit-indexed summands; $D=0$ gives GLR Conjectures 3.4.5-3.4.6. Corollary 1.2: for split adjoint absolutely simple $G$, a cuspidal representation generic and unramified at one place is tempered at every unramified place. The October 5 companion constructs, from this, a commuting algebraic $SL_2$ enhancing each occurring cuspidal excursion parameter at full level; it does not assert ellipticity, Arthur packets or multiplicities, so V. Lafforgue's Arthur conjecture is not settled.

Theorem 1.1: for every $X,G,\ell$ and every effective divisor $D$, the rational and $\overline{\mathbb Q}_\ell$ cuspidal spaces at full level $D$ are the direct sums of their orbit-indexed summands; $D=0$ gives GLR Conjectures 3.4.5-3.4.6. Corollary 1.2: for split adjoint absolutely simple $G$, a cuspidal representation generic and unramified at one place is tempered at every unramified place. The October 5 companion constructs, from this, a commuting algebraic $SL_2$ enhancing each occurring cuspidal excursion parameter at full level; it does not assert ellipticity, Arthur packets or multiplicities, so V. Lafforgue's Arthur conjecture is not settled.