Cuspidal representations in the cohomology of Deligne-Lusztig varieties for \(\mathrm{GL}(2)\) over finite rings
type: Item
instance of: scholarly article
title: Cuspidal representations in the cohomology of Deligne-Lusztig varieties for \(\mathrm{GL}(2)\) over finite rings
zbMATH Open document ID: 1400.20040
author: Tetsushi Ito
author: Takahiro Tsushima
published in: Israel Journal of Mathematics
publication date: 2018-08-22T00:00:00Z
review text: Let \(K\) be a non-Archimedean local field, \(O\) its ring of integers, \(P\) its unique maximal ideal, \(k=O/P\) the residue field. Assume that \(K\) has the characteristic of \(k\). Let \(G\) be a reductive group over \(k\). \textit{G. Lusztig} generalized the Deligne-Lusztig theory for \(G(O/P)\) by constructing a variety over \(\bar{k}\) whose étale cohomology classes realize some irreducible representations of \(G(O/P^n)\) for general \(n\). These results appear in [Proc. Symp. Pure Math. 33, 171--175 (1979; Zbl 0421.22009); Represent. Theory 8, 1--14 (2004; Zbl 1085.20029)]. The authors refer to the theory in these papers as Deligne-Lusztig theory over finite rings. In the [2004, loc. cit.], the Deligne-Lusztig variety for \(\mathrm{SL}_2(O/P^2)\) has been explicitly studied. In the 1979 paper, Lusztig describes a construction for division algebras as well. As the cohomology of this variety for any division algebra contains many irreducible components with lower conductor, it is difficult to study them in general. In the present work, the authors study the étale cohomology of certain closed subvarieties of Deligne-Lusztig varieties of \(\mathrm{GL}_2\) over such finite rings and prove that cuspidal representations appear in the cohomology. This generalizes some of the results in [Lusztig, 2004, loc. cit.]. In addition, the authors also prove analogous results for the Deligne-Lusztig variety corresponding to the quaternion division algebra case. A product of such varieties may be regarded as an affine bundle over a curve where the base curve appears as an open subscheme of irreducible components of the stable reduction of the Lubin-Tate curve in a special case. The authors state a conjecture on a part of the stable reduction using the above varieties. This work can be thought of as trying to understand bad reduction of Lubin-Tate curves using Deligne-Lusztig varieties. The conjecture is described as follows. Let \(\pi\) be an irreducible, cuspidal representatio…
reviewed by: Balasubramanian Sury
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 14F20
Mathematics Subject Classification ID: 11G20
Mathematics Subject Classification ID: 11F70
zbMATH DE Number: 6923341
zbMATH Keywords: Lubin-Tate Curves
zbMATH Keywords: Deligne-Lusztig varieties
zbMATH Keywords: local Langlands
zbMATH Keywords: etale cohomology
zbMATH Keywords: unramified representations
MaRDI profile type: Publication
full work available at URL: s11856-018-1717-x
OpenAlex ID: W2806726302
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DOI: 10.1007/S11856-018-1717-X
description: scientific article; zbMATH DE number 6923341