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

Research paper by **Hengfei Lu**

Indexed on: **26 May '21**Published on: **23 May '21**Published in: **arXiv - Mathematics - Representation Theory**

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#### 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.