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Maximal varieties and the local Langlands correspondence for $GL(n)$

Authors: Mitya Boyarchenko and Jared Weinstein
Journal: J. Amer. Math. Soc. 29 (2016), 177-236
MSC (2010): Primary 11S37, 11G25, 14G22; Secondary 11G18
Published electronically: April 3, 2015
MathSciNet review: 3402698
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Abstract: The cohomology of the Lubin-Tate tower is known to realize the local Langlands correspondence for $GL(n)$ over a nonarchimedean local field. In this article we make progress toward a purely local proof of this fact. To wit, we find a family of formal schemes $\mathcal {V}$ such that the generic fiber of $\mathcal {V}$ is isomorphic to an open subset of Lubin-Tate space at infinite level, and such that the middle cohomology of the special fiber of $\mathcal {V}$ realizes the local Langlands correspondence for a broad class of supercuspidals (those whose Weil parameters are induced from an unramified degree $n$ extension). The special fiber of $\mathcal {V}$ is related to an interesting variety $X$, defined over a finite field, which is “maximal” in the sense that the number of rational points of $X$ is the largest possible among varieties with the same Betti numbers as $X$. The variety $X$ is derived from a certain unipotent algebraic group, in an analogous manner as Deligne-Lusztig varieties are derived from reductive algebraic groups.

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Additional Information

Mitya Boyarchenko
Affiliation: Department of Mathematics, University of Michigan, 2074 East Hall, 530 Church Street, Ann Arbor, Michigan 48109
MR Author ID: 710764

Jared Weinstein
Affiliation: Department of Mathematics and Statistics, Boston University, 111 Cummington Mall, Boston, Massachusetts 02215
MR Author ID: 867611

Received by editor(s): November 8, 2011
Received by editor(s) in revised form: July 15, 2013, and December 17, 2014
Published electronically: April 3, 2015
Additional Notes: The second author is supported by NSF Award DMS-1303312.
Article copyright: © Copyright 2015 American Mathematical Society