Multiplicities and Plancherel formula for the space of nondegenerate Hermitian matrices

TL;DR

This paper provides an explicit spectral decomposition and multiplicity formula for the space of nondegenerate Hermitian matrices over local fields, confirming conjectures and extending previous results in the representation theory of p-adic groups.

Contribution

It offers a new explicit Plancherel decomposition and a multiplicity formula for generic representations, advancing understanding of harmonic analysis on Hermitian matrix spaces.

Findings

Confirmed Sakellaridis-Venkatesh conjecture for this case.

Derived a multiplicity formula extending prior work.

Utilized relative trace formulas for proofs.

Abstract

This paper contains two results concerning the spectral decomposition, in a broad sense, of the space of nondegenerate Hermitian matrices over a local field of characteristic zero. The first is an explicit Plancherel decomposition of the associated $L^2$ space thus confirming a conjecture of Sakellaridis-Venkatesh in this particular case. The second is a formula for the multiplicities of generic representations in the $p$-adic case that extends previous work of Feigon-Lapid-Offen. Both results are stated in terms of Arthur-Clozel's quadratic local base-change and the proofs are based on local analogs of two relative trace formulas previously studied by Jacquet and Ye and known as (relative) Kuznetsov trace formulas.