Non-cuspidality outside the middle degree of \(\ell \)-adic cohomology of the Lubin-Tate tower
Let \(F\) be a non-Archimedean field. Let \(d\geq 1\) be an integer. Let \(W_F\) be the Weil group of \(F\). We write \(D\) for the \(F\)-central division algebra with invariant \(1/d\). We can consider the Lubin-Tate tower attached to \(F\) and \(d\). Using the \(k\)th vanishing cycle cohomology of the Lubin-Tate tower, we obtain a representation \(H^k_{\text{LT}}\) of \(\text{GL}_d(F)\times D^\times\times W_F\). The representations \(H^k_{\text{LT}}\) are important in the local Langlands correspondence. By non-abelian Lubin-Tate theory (also known as a conjecture of Deligne-Carayol) for every supercuspidal representation \(\pi\) of \(\text{GL}_d(F)\) the corresponding \(\pi\)-isotypic component in \(H^{d-1}_{\text{LT}}\) can be described by the local Jacquet-Langlands correspondence. The main result of the paper gives the proof of the fact that \(H^k_{\text{LT}}\), \(k\neq d-1\), considered as a \(\text{GL}_d(F)\)-representation has no supercuspidal irreducible subquotient. The goal of the paper is to give an easy and direct proof. Earlier, the same fact was proved by Faltings (\(\text{char}(F)=0\)) and Boyer (\(\text{char}(F)\neq 0\)).
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