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15 September 2010 Arithmetic invariants of discrete Langlands parameters
Benedict H. Gross, Mark Reeder
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Duke Math. J. 154(3): 431-508 (15 September 2010). DOI: 10.1215/00127094-2010-043


The local Langlands correspondence can be used as a tool for making verifiable predictions about irreducible complex representations of p-adic groups and their Langlands parameters, which are homomorphisms from the local Weil-Deligne group to the L-group. In this article, we refine a conjecture of Hiraga, Ichino, and Ikeda which relates the formal degree of a discrete series representation to the value of the local gamma factor of its parameter. We attach a rational function in x with rational coefficients to each discrete parameter, which specializes at x=q, the cardinality of the residue field, to the quotient of this local gamma factor by the gamma factor of the Steinberg parameter. The order of this rational function at x=0 is also an important invariant of the parameter—it leads to a conjectural inequality for the Swan conductor of a discrete parameter acting on the adjoint representation of the L-group. We verify this conjecture in many cases. When we impose equality, we obtain a prediction for the existence of simple wild parameters and simple supercuspidal representations, both of which are found and described in this article.


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Benedict H. Gross. Mark Reeder. "Arithmetic invariants of discrete Langlands parameters." Duke Math. J. 154 (3) 431 - 508, 15 September 2010.


Published: 15 September 2010
First available in Project Euclid: 7 September 2010

MathSciNet: MR2730575
zbMATH: 1207.11111
Digital Object Identifier: 10.1215/00127094-2010-043

Primary: 11S15 , 11S37
Secondary: 22E50

Rights: Copyright © 2010 Duke University Press


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Vol.154 • No. 3 • 15 September 2010
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