Local root numbers and spectrum of the local descents for orthogonal groups: $p$-adic case

Abstract

We investigate the local descents for special orthogonal groups over $p$-adic local fields of characteristic zero, and obtain explicit spectral decomposition of the local descents at the first occurrence index in terms of the local Langlands data via the explicit local Langlands correspondence and explicit calculations of relevant local root numbers. The main result can be regarded as a refinement of the local Gan–Gross–Prasad conjecture (2012).