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

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Companion: Global Arthur Enhancements of Cuspidal Excursion Parameters evidence for this event

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Gaitsgory-Lafforgue-Raskin, arXiv:2603.26925 evidence for this event

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Rationality of the Gaitsgory-Lafforgue-Raskin canonical Arthur filtration evidence for this event

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. parent of this event

Companion: Ramanujan-Arthur Decompositions of Cuspidal Functions at Full Finite Level evidence for this event

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