Ramanujan–Arthur cusp decomposition at full finite level
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For split connected semisimple groups over global function fields, do rational and ℓ-adic cuspidal-function spaces decompose into pieces indexed by dual-group nilpotent orbits, compatibly at all unramified Hecke places? The target includes arbitrary finite level.
Why it matters and what counts as progress
Why it matters: Such a decomposition would realize a predicted organization of cusp forms by orbit geometry beyond level zero.
What counts as progress: A direct-sum decomposition in both coefficient settings for every effective level divisor, including multiplicities, with one orbit label governing every unramified place.
How the problem developed
Dennis Gaitsgory, Vincent Lafforgue, and Sam Raskin Formulated the Ramanujan–Arthur cusp-decomposition conjectures for nilpotent-orbit pieces, including their rational form; the later result at trivial level addresses Conjectures 3.4.5 and 3.4.6.
- 24 Sept 2026 · Reported claim [2]
OpenAI The manuscript claims rational and ℓ-adic decompositions for split semisimple groups over global function fields at every full finite level, with arbitrary divisor multiplicities; the trivial-level case proves the cited GLR conjectures.
AI contribution
Ramanujan-Arthur Decompositions of Cuspidal Functions at Full Finite Level
For split connected semisimple groups over global function fields and every effective divisor, including arbitrary multiplicity, rational and ℓ-adic cuspidal-function spaces split into direct sums indexed by dual-group nilpotent orbits. The same orbit controls all unramified Hecke places.
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OpenAI reports the Ramanujan–Arthur cusp decomposition at full finite level