Duke Mathematical Journal

Arithmetic invariants of discrete Langlands parameters

Benedict H. Gross and Mark Reeder

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Abstract

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.

Article information

Source
Duke Math. J. Volume 154, Number 3 (2010), 431-508.

Dates
First available: 7 September 2010

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1283865310

Digital Object Identifier
doi:10.1215/00127094-2010-043

Zentralblatt MATH identifier
05816148

Mathematical Reviews number (MathSciNet)
MR2730575

Subjects
Primary: 11S15: Ramification and extension theory 11S37: Langlands-Weil conjectures, nonabelian class field theory [See also 11Fxx, 22E50]
Secondary: 22E50: Representations of Lie and linear algebraic groups over local fields [See also 20G05]

Citation

Gross, Benedict H.; Reeder, Mark. Arithmetic invariants of discrete Langlands parameters. Duke Mathematical Journal 154 (2010), no. 3, 431--508. doi:10.1215/00127094-2010-043. http://projecteuclid.org/euclid.dmj/1283865310.


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