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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].

##### MSC:
 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
##### Keywords:
Lubin-Tate formal group; vanishing cycle; local Langlands
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