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On non-abelian Lubin-Tate theory via vanishing cycles. (English) Zbl 1257.11103
Nakamura, Iku (ed.) et al., Algebraic and arithmetic structures of moduli spaces. Proceedings of the conference, Sapporo, Japan, September 2007. Tokyo: Mathematical Society of Japan (MSJ) (ISBN 978-4-931469-59-4/hbk). Advanced Studies in Pure Mathematics 58, 361-402 (2010).
In this paper a purely local proof of the local Langlands correspondence for \(\mathrm{GL}_n\) is given in the case of depth \(0\) supercuspidal representations.
Let \(K\) be a finite extension of \(\mathbb{Q}_p\) with ring of integers \(\mathcal{O}\) and residue field \(k=\mathcal{O}/\mathfrak{p}\) and fix \(n\geq 1\). Let \(K^{\mathrm{ur}}\) be the maximal unramified extension of \(K\) and let \(W\) be the completion of the ring of integers \(\mathcal{O}^{\mathrm{ur}}\) of \(K^{\mathrm{ur}}\). Denote by \(\eta\), resp. \(\overline{\eta}\) the spectrum of \(\mathrm{Frac}W\) and of its algebraic closure, respectively. Further, let \(X\) denote the spectrum of the deformation ring of a formal \(\mathcal{O}\)-module of height \(n\) with level \(\mathfrak{p}\) structure. The \(\ell\)-adic étale cohomology groups \(H^i(X_{\overline{\eta}},\overline{\mathbb{Q}_{\ell}})\) (\(\ell\neq\mathrm{char}(k)=p\)) of the geometric generic fibre \(X_{\overline{\eta}}=X\times_{\mathrm{Spec}W}\overline{\eta}\) are finite dimensional representations of \(\mathrm{GL}_n(k)\times I_K\) where \(I_K\) denotes the inertia in the absolute Galois group of \(K\). On the other hand, let \(DL\) be the Deligne-Lusztig variety for \(\mathrm{GL}_n(k)\), associated to the element of the Weyl group of \(\mathrm{GL}_n\) corresponding to the cyclic permutation \((1,\dots,n)\) in the symmetric group, or equivalently to a non-split torus \(T\) with \(T(k)=k_n^{\times}\) where \(k_n\) is the unique extension of \(k\) of degree \(n\). This \(DL\) is a smooth affine variety over \(\overline{k}\) with actions of \(\mathrm{GL}_n(k)\) and \(T(k)=k_n^{\times}\)—hence the \(\ell\)-adic cohomology groups \(H^i_c(DL,\overline{\mathbb{Q}_\ell})\) are representations of \(\mathrm{GL}_n(k)\times I_K\) via the natural surjection \(I_K\twoheadrightarrow k_n^{\times}\).
The main result of the paper is that the Euler characteristics \[ H^*(X_{\overline{\eta}}):=\sum_i(-1)^i[H^i(X_{\overline{\eta}},\overline{\mathbb{Q}_{\ell}})];\qquad H^*_c(DL):=\sum_i(-1)^i[H^i_c(DL,\overline{\mathbb{Q}_{\ell}})] \] in the Grothendieck group of the category of finite dimensional \(\mathrm{GL}_n(k)\times I_K\)-representations are equal: \(H^*(X_{\overline{\eta}})=H^*_c(DL)\). Moreover, among the \(H^i(X_{\overline{\eta}},\overline{\mathbb{Q}_\ell})\), cuspidal representations \(\pi\) of \(\mathrm{GL}_n(k)\) and generic inertia characters \(\chi\) of \(I_K\) occur only in \(H^{n-1}(X_{\overline{\eta}},\overline{\mathbb{Q}_\ell})\), where they are coupled as \(\bigoplus \pi_\chi\otimes\chi\) by the Deligne-Lusztig correspondence \(\chi\leftrightarrow\pi_\chi\) characterized by \[ \pi_\chi\otimes \mathrm{St} = \mathrm{Ind}^{\mathrm{GL}_n(k)}_{T(k)} \chi\;, \] where \(\mathrm{St}\) is the Steinberg representation of \(\mathrm{GL}_n(k)\).
This above correspondence is a direct consequence of the main theorems of [M. Harris and R. Taylor, The geometry and cohomology of some simple Shimura varieties. Princeton, NJ: Princeton University Press (2001; Zbl 1036.11027)]. As a by-product, the author also obtains results on the geometry of the variety \(X\).
For the entire collection see [Zbl 1193.14002].

11S37 Langlands-Weil conjectures, nonabelian class field theory
11G18 Arithmetic aspects of modular and Shimura varieties
22E50 Representations of Lie and linear algebraic groups over local fields
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