Multiplicity free Jacquet modules

TL;DR

This paper proves that the Jacquet functor from representations of GL(n+k,F) to GL(n,F) x GL(k,F) is multiplicity free, meaning it produces at most one copy of any irreducible representation, using adapted classical methods.

Contribution

It establishes the multiplicity free property of the Jacquet functor for specific groups, extending classical techniques to functors rather than just representations.

Findings

Jacquet functor is multiplicity free for certain groups

Method adapts Gelfand-Kazhdan classical approach

Discussion on multiplicity free property of other Jacquet functors

Abstract

Let F be a non-Archimedean local field or a finite field. Let n be a natural number and k be 1 or 2. Consider G:=GL(n+k,F) and let M:=GL(n,F) x GL(k,F)<G be a maximal Levi subgroup. Let U< G be the corresponding unipotent subgroup and let P=MU be the corresponding parabolic subgroup. Let J denote the Jacquet functor from representations of G to representations of M (i.e. the functor of coinvariants w.r.t. U). In this paper we prove that J is a multiplicity free functor, i.e. dim Hom(J(\pi),\rho)<= 1, for any irreducible representations \pi of G and \rho of M. To do that we adapt the classical method of Gelfand and Kazhdan that proves "multiplicity free" property of certain representations to prove "multiplicity free" property of certain functors. At the end we discuss whether other Jacquet functors are multiplicity free.