On non-abelian Lubin–Tate theory via vanishing cycles

We give a purely local proof, in the depth 0 case, of the result by Harris-Taylor which asserts that the local Langlands correspondence for GL_n over a p-adic field K realizes itself inside the vanishing cycle cohomology of the deformation space of formal O_K-modules of height n. Our proof is given by establishing the direct geometric link with the Deligne-Lusztig theory for GL_n over the residue field.