## Lefschetz operator and local Langlands modulo $$\ell$$: the limit case.(English)Zbl 1329.11126

Let $$K$$ be a finite extension of $$\mathbb Q_p$$ with residue field $$\mathbb F_q$$. Let $$\ell$$ be a prime such that $$q\equiv 1$$ (mod $$\ell$$). The author investigates the cohomology of the Lubin-Tate towers of $$K$$ with coefficients in $$\overline{\mathbb F}_\ell$$, and it shows how this cohomology encodes Vignéras’ Langlands correspondence for unipotent $$\overline{\mathbb F}_\ell$$-representations. This paper is part of a project, outlined by the author in [Compos. Math. 148, No. 2, 507–530 (2012; Zbl 1247.14025)], that aims at providing a geometric interpretation of the Vignéras correspondence for modulo-$$\ell$$ representations of $$p$$-adic linear groups.
The introduction to this paper, which occupies almost a quarter of the paper, explains in detail the problem: “…M.-F. Vignéras [Invent. Math. 144, No. 1, 177–223 (2001; Zbl 1031.11068)] established a bijection between (classes of) irreducible smooth $$\overline{\mathbb F}_\ell$$-representations of $$\mathrm{GL}(d,K)$$ and (classes of) $$d$$-dimensional Weil-Deligne $$\overline{\mathbb F}_\ell$$-representations for $$K$$. On the one hand we have fairly natural ‘automorphic objects’, but on the other hand we get fairly unnatural ‘Galois objects’. Indeed, the nilpotent part of a Weil-Deligne $$\overline{\mathbb F}_\ell$$-representation has no obvious Galois interpretation, in contrast with $$\overline{\mathbb F}_\ell$$-representations, where it is related to the infinitesimal action of the tame inertia subgroup on some associated continuous Arthur’s second $$\mathrm{SL}(2)$$ factor in the theory of automorphic forms representation of the Weil group. Therefore in the $$\overline{\mathbb F}_\ell$$ case, this nilpotent part appears as an ‘extra datum’, from the arithmetic point of view. In fact, Vignéras’ correspondence was obtained by purely representation-theoretic arguments (a classification theorem à la Zelevinsky), and our aim is to find a geometric interpretation for it.” The paper proceeds to describe the author’s approach, using Carayol’s formulation of “nonabelian Lubin-Tate theory”, and the cohomology of the Lubin-Tate tower, and the role of Arthur’s second $$\mathrm{SL}(2)$$ factor in the theory of automorphic forms. In addition to the results, the author describes previous results, those of the present paper, and sketches the argument.

### MSC:

 11S37 Langlands-Weil conjectures, nonabelian class field theory 11F70 Representation-theoretic methods; automorphic representations over local and global fields 14G35 Modular and Shimura varieties

### Citations:

Zbl 1247.14025; Zbl 1031.11068
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