Orthogonal Root Numbers of Tempered Parameters

TL;DR

This paper proves a formula for orthogonal root numbers of tempered L-parameters, decomposing them into simpler components, thereby resolving a conjecture and advancing understanding of root numbers in representation theory.

Contribution

It establishes a new formula for orthogonal root numbers of tempered parameters, linking them to principal parameters and Langlands's central character, confirming a conjecture of Gross and Reeder.

Findings

Decomposition of orthogonal root numbers into two factors.

Resolution of Gross and Reeder's conjecture.

Application to Weil-Deligne representations in Plancherel measure.

Abstract

We show that an orthogonal root number of a tempered L-parameter decomposes as the product of two other numbers: the orthogonal root number of the principal parameter and the value on a certain involution of Langlands's central character for the parameter. The formula resolves a conjecture of Gross and Reeder and computes root numbers of Weil-Deligne representations arising in the work of Hiraga, Ichino, and Ikeda on the Plancherel measure.