Abstract
We define closed subvarieties of some Deligne–Lusztig varieties for GL(2) over finite rings and study their ´etale cohomology. As a result, we show that cuspidal representations appear in it. Such closed varieties are studied in [Lus2] in a special case. We can do the same things for a Deligne–Lusztig variety associated to a quaternion division algebra over a non-archimedean local field. A product of such varieties can be regarded as an affine bundle over a curve. The base curve appears as an open subscheme of a union of irreducible components of the stable reduction of the Lubin–Tate curve in a special case. Finally, we state some conjecture on a part of the stable reduction using the above varieties. This is an attempt to understand bad reduction of Lubin–Tate curves via Deligne–Lusztig varieties.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. M. Aubert, U. Onn, A. Prasad and A. Stasinski, On cuspidal representations of general linear groupsover discrete valuation rings, Israel Journal of Mathematics 175 (2010), 391–420.
C. J. Bushnell and A. Frohlich, Gauss Sums and p-adic Division Algebras, Lecture Notes in Mathematics, Vol. 987, Springer-Verlag, Berlin–New York, 1983.
C. J. Bushnell and G. Henniart, The Local Langlands Conjecture for GL(2), Grundlehren der Mathematischen Wissenschaften, Vol. 335, Springer-Verlag, Berlin, 2006.
M. Boyarchenko and J. Weinstein, Maximal varieties and the local Langlands correspondence for GLn, Journal of the American Mathematical Society 29 (2016), 177–236.
H. Carayol, Non-abelian Lubin–Tate theory, in Automorphic Forms, Shimura varieties, and L-functions, Vol. II (Ann Arbor, MI, 1988), Perspectives in Mathematics, Vol. 11, Academic Press, Boston, MA, 1990, pp. 15–39.
C. Chan, The cohomology of semi-infinite Deligne–Lusztig varieties, preprint, arXiv:1606.01795v1.
Z. Chen and A. Stasinski, The algebraisation of higher Deligne–Lusztig representations, Selecta Mathematica 23 (2017), 2907–2926.
R. Coleman and K. McMurdy, Stable reduction of X0(p3), Algebra & Number Theory 4 (2010), 357–431.
P. Deligne, Cohomologie étale, Lecture Notes in Mathematics, Vol. 569, Springer-Verlag, Berlin, 1977.
P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Annals of Mathematics 103 (1976), 103–161.
N. Imai and T. Tsushima, Stable models of Lubin–Tate curves with level three, Nagoya Mathematical Journal 225 (2017), 100–151.
A. B. Ivanov, Affine Deligne–Lusztig varieties of higher level and the local Langlands correspondence for GL2, Advances in Mathematics 299 (2016), 640–686.
G. Lusztig, Some remarks on the supercuspidal representations of p-adic semisimple groups, in Automorphic Forms, Representations and L-functions. Part 1, Proceedings of Symposia in Pure Mathematics, Vol. 33, American Mathematical Society, Providence, RI, 1979, pp. 171–175.
G. Lusztig, Representations of reductive groups over finite rings, Representation Theory 8 (2004), 1–14.
U. Onn, Representations of automorphism groups of finite o-modules of rank two, Advances in Mathematics 219 (2008), 2058–2085.
M. Rapoport, A guide to the reduction modulo p of Shimura varieties, Astérisque 298 (2005), 271–318.
J. P. Serre, Linear Representations of Finite Groups, Graduate Texts inMathematics, Vol. 42, Springer, New York–Heidelberg, 1977.
A. Stasinski, The smooth representations of GL2(O), Communications in Algebra 37 (2009), 4416–4430.
A. Stasinski, Extended Deligne–Lusztig varieties for general and special linear groups, Advances in Mathematics 226 (2011), 2825–2853.
T. Tsushima, On the stable reduction of the Lubin–Tate curve of level two in the equal characteristic case, Journal of Number Theory 147 (2015), 184–210.
T. Tsushima, On non-abelian Lubin–Tate theory for GL(2) in the odd equal characteristic case, preprint, arXiv:1604.08857.
T. Yoshida, On non-abelian Lubin–Tate theory via vanishing cycles, in Algebraic and Arithmetic Structures of Moduli Spaces (Sapporo 2007), Advanced Studied in Pure Mathematics, Vol. 58, Mathematical Society of Japan, Tokyo, 2010, pp. 361–402.
J. Weinstein, Explicit non-abelian Lubin–Tate theory for GL(2), preprint, arXiv:0910.1132v1.
J. Weinstein, The local Jacquet–Langlands correspondence via Fourier analysis, Journal de Théorie des Nombres de Bordeaux 22 (2010), 483–512.
J. Weinstein, Semistable models for modular curves of arbitrary level, Inventiones Mathematicae 205 (2016), 459–526.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ito, T., Tsushima, T. Cuspidal representations in the cohomology of Deligne–Lusztig varieties for GL(2) over finite rings. Isr. J. Math. 226, 877–926 (2018). https://doi.org/10.1007/s11856-018-1717-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-018-1717-x