For any reductive group G over a global function field, we use the cohomology G-shtukas with multiple modifications and the geometric Satake equivalence prove the global Langlands correspondence for G in the "automorphic to" direction. Moreover we obtain a canonical decomposition of the spaces cuspidal automorphic forms indexed by global Langlands parameters. The proof not rely at all on the Arthur-Selberg trace formula.
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Vincent Lafforgue (2012) studied this question.