The Hiraga-Ichino-Ikeda Conjecture for Principal Series of Split p-adic Groups

Given a p-adic connected split reductive group G, we use the local Langlands correspondence as defined by Reeder and by Aubert, Baum, Plymen and Solleveld, to prove the HII conjecture for irreducible discrete series representations contained in a principal series of G. We verify the predicted formula relating the formal degree of such representations to the adjoint $γ$-factor of their associated Langlands parameter. First, we prove it under the assumption that the center of G is connected, and then we generalize the result.