Quotient branching laws and Gan-Gross-Prasad relevance for general linear groups

TL;DR

This paper establishes branching laws for all unitarizable representations of general linear groups over non-Archimedean fields, introduces an explicit algorithm for identifying GGP relevant pairs, and classifies non-zero Hom spaces involving Speh representations.

Contribution

It extends Gan-Gross-Prasad relevance to all unitarizable representations and provides a computable method to determine relevant pairs and Hom space non-vanishing.

Findings

Proved branching laws for all unitarizable representations of GL(n)

Developed an explicit algorithm to identify GGP relevant pairs

Classified cases where Hom spaces are non-zero involving Speh representations

Abstract

This paper proves the branching laws for the full class of unitarizable representations of general linear groups in non-Archimedean local fields, extending the original notion of Gan-Gross-Prasad relevant pair for Arthur-type representations \cite{GGP2, Gur, Cha_crelle}. Further, we provide an explicit computable algorithm to determine the generalized GGP relevant pair, as developed in \cite{Cha_qbl}. In particular, we show that if $\pi$ and $\pi'$ are any irreducible smooth representations of $\mathrm{GL_{n+1}(F)}$ and $\mathrm{GL_{n}(F)}$ respectively, and their Langlands data or Zelevinsky data are given in terms of multisegments, then through an algorithmic process we can determine whether the space $\mathrm{Hom}_{\mathrm{GL_n(F)}}(\pi, \pi')$ is non-zero. Finally, when one of the represntations $\pi$ and $ \pi'$ is a generalized Speh representation, we give a complete classification for the other one for which the Hom space is non-zero.