Duke Mathematical Journal
- Duke Math. J.
- Volume 154, Number 3 (2010), 431-508.
Arithmetic invariants of discrete Langlands parameters
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.
Duke Math. J. Volume 154, Number 3 (2010), 431-508.
First available in Project Euclid: 7 September 2010
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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]
Gross, Benedict H.; Reeder, Mark. Arithmetic invariants of discrete Langlands parameters. Duke Math. J. 154 (2010), no. 3, 431--508. doi:10.1215/00127094-2010-043. https://projecteuclid.org/euclid.dmj/1283865310.