On non-abelian Lubin–Tate theory via vanishing cycles

Abstract

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 $GL_n$ is realized in the vanishing cycle cohomology of the deformation spaces of one-dimensional formal modules of height $n$. Our proof is given by establishing the direct geometric link with the Deligne–Lusztig theory for $GL_n (\mathbb{F}_q)$.