# Local Root Numbers and Spectrum of the Local Descents for Orthogonal Groups: p-adic case

## TL;DR

This paper provides an explicit spectral decomposition of local descents for special orthogonal groups over p-adic fields, refining the local Gan-Gross-Prasad conjecture through detailed analysis of local root numbers and Langlands data.

## Contribution

It offers a new explicit description of local descents and their spectral decomposition in terms of local Langlands parameters, advancing understanding of the local Gan-Gross-Prasad conjecture.

## Key findings

- Explicit spectral decomposition of local descents at first occurrence index
- Refinement of the local Gan-Gross-Prasad conjecture
- Calculation of local root numbers and their relation to Langlands data

## Abstract

We investigate the local descents for special orthogonal groups over p-adic local fields of characteristic zero, and obtain an 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.