The Hiraga-Ichino-Ikeda conjecture on formal degrees for classical groups

TL;DR

This paper proves a conjecture linking formal degrees of certain representations to gamma factors for symplectic and even orthogonal groups, using advanced harmonic analysis and the local Langlands correspondence.

Contribution

It extends the proof of the Hiraga-Ichino-Ikeda conjecture to symplectic and even orthogonal groups over non-Archimedean fields, employing twisted endoscopy and orbital integrals.

Findings

Confirmed the conjecture for symplectic groups

Extended the approach to even orthogonal groups

Provided a method adaptable to other classical groups

Abstract

We prove a conjecture of Hiraga-Ichino-Ikeda relating formal degrees of square-integrable representations to adjoint gamma factors for symplectic and even orthogonal groups over characteristic zero non-Archimedean local fields. The proof is based on the twisted endoscopic characterization of the local Langlands correspondence for such groups and extends an approach already appearing in the original paper of Hiraga-Ichino-Ikeda itself inspired from earlier work of Shahidi. Namely, the argument consists in computing in two different ways a residue of certain standard intertwining operators suitably extended to a large representation realized on an explicit space of functions. This gives rise to the spectral decomposition of certain singular twisted orbital integrals. The main theorem follows from a comparison of this identity with the Plancherel formula of a symplectic or even orthogonal group. The proof can be adapted to deal with other classical groups, namely odd orthogonal and unitary groups, but the formal degree conjecture was already established in these cases.