##
**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.

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.

Reviewer: Yuval Z. Flicker (Columbus)

### 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 |