  In this article, we consider the representations of the general linear group
over a non-archimedean local field obtained from the vanishing cycle cohomology
of the Lubin-Tate tower. We give an easy and direct proof of the fact that no
supercuspidal representation appears as a subquotient of such representations
unless they are obtained from the cohomology of the middle degree. Our proof is
purely local and does not require Shimura varieties.
