In joint work with George Lusztig, Deligne and Lusztig applied étale cohomology to construct representations of finite groups of Lie type; with Michael Rapoport, Deligne worked on the moduli spaces from the 'fine' arithmetic point of view, with application to modular forms.
In 1993, he—along with Gérard Laumon and Michael Rapoport—proved the local Langlands conjectures for the general linear group GLn(K) for positive characteristic local fields K.
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