Multiplicity one for the pair (GL(n,D),GL(n,E))

TL;DR

This paper proves a multiplicity one result for the pair (GL(n,D),GL(n,E)) over local fields, showing that certain equivariant functions are invariant under an anti-involution, leading to a Gelfand pair property.

Contribution

It introduces a new proof of the multiplicity one property for (GL(2n,F),GL(n,E)) when D splits, using invariance under an anti-involution and generalized Gelfand-Kazhdan criteria.

Findings

Any bi-(GL(n,E),μ)-equivariant tempered generalized function on GL(n,D) is anti-involution invariant.

The dimension of Hom(π,μ) is at most 1 for relevant representations.

(GL(2n,F),GL(n,E)) forms a Gelfand pair when μ is trivial and D splits.

Abstract

Let F be a local field of characteristic zero. Let D be a quaternion algebra over F. Let E be a quadratic field extension of F. Let {\mu} be a character of GL(1,E). We study the distinction problem for the pair (GL(n,D), GL(n,E)) and we prove that any bi-(GL(n,E), {\mu})-equivariant tempered generalized function on GL(n,D) is invariant with respect to an anti-involution. Then it implies that dimHom({\pi},{\mu}) is at most 1 by the generalized Gelfand-Kazhdan criterion. Thus we give a new proof to the fact that (GL(2n,F),GL(n,E)) is a Gelfand pair when {\mu} is trivial and D splits.