  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.
