Non-tempered Gan-Gross-Prasad conjecture for Archimedean general linear groups

TL;DR

This paper proves a key direction of the non-tempered Gan-Gross-Prasad conjecture for Archimedean general linear groups, establishing when certain periods are non-zero based on representation relevance.

Contribution

It confirms the 'period implies relevance' part of the conjecture for Archimedean fields, completing the proof when combined with recent results.

Findings

Proved the 'period implies relevance' direction of the conjecture.

Confirmed the Gan-Gross-Prasad conjecture for Archimedean general linear groups.

Utilized annihilator varieties of distinguished representations.

Abstract

The local non-tempered Gan-Gross-Prasad conjecture suggests that, for a pair of irreducible Arthur type representations of two successive general linear groups, they have a non-zero Rankin-Selberg period if and only if they are "relevant". In this paper, we prove the "period implies relevance" direction of this conjecture for general linear groups over Archimedean local fields. Our proof is based on a previous result of Gourevitch-Sayag-Karshon on the annihilator varieties of distinguished representations. Combining with a recent result of P. Boisseau on the direction "relevance implies period'', this conjecture for general linear groups over Archimedean local fields is settled.