The local Langlands correspondence can be used as a tool for making verifiable predictions about irreducible complex representations of -adic groups and their Langlands parameters, which are homomorphisms from the local Weil-Deligne group to the -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 with rational coefficients to each discrete parameter, which specializes at , 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 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 -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.
Benedict H. Gross. Mark Reeder. "Arithmetic invariants of discrete Langlands parameters." Duke Math. J. 154 (3) 431 - 508, 15 September 2010. https://doi.org/10.1215/00127094-2010-043