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

Given a $p$-adic connected split reductive group $\mathcal{G},$ 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 $\mathcal{G}$. We verify the predicted formula relating the formal degree of such representations to the adjoint $\gamma$-factor of their associated Langlands parameter. First, we prove it under the assumption that the center of $\mathcal{G}$ is connected, and then we generalize the result.