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

Basudev Pattanayak

Paper

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

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

and

are any irreducible smooth representations of

and

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

is non-zero. Finally, when one of the represntations

and

is a generalized Speh representation, we give a complete classification for the other one for which the Hom space is non-zero.