Arithmetic invariants of discrete Langlands parameters

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.