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

TL;DR

This paper proves the Hiraga-Ichino-Ikeda conjecture for certain principal series representations of split p-adic groups by relating formal degrees to adjoint gamma factors, confirming predictions from the local Langlands correspondence.

Contribution

It establishes the HII conjecture for irreducible discrete series in principal series of split p-adic groups using the local Langlands correspondence, including cases with disconnected centers.

Findings

Verified the HII conjecture for connected center groups.

Extended the proof to groups with disconnected centers.

Confirmed the predicted relation between formal degrees and gamma factors.

Abstract

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.