TY - JOUR
T1 - (GLn+1(F), GLn(F)) is a Gelfand pair for any local field F
AU - Aizenbud, Avraham
AU - Gourevitch, Dmitry
AU - Sayag, Eitan
N1 - Funding Information:
During the preparation of this work, Eitan Sayag was partially supported by ISF grant # 147/05.
PY - 2008/11/1
Y1 - 2008/11/1
N2 - Let F be an arbitrary local field. Consider the standard embedding GL n(F) → GLn+1(F) and the two-sided action of GL n(F)× GLn(F) on GLn+1(F). In this paper we show that any GLn(F) × GLn(F)-invariant distribution on GLn+1(F) is invariant with respect to transposition. We show that this implies that the pair (GLn+1(F), GLn(F)) is a Gelfand pair. Namely, for any irreducible admissible representation (π,E) of GLn+1(F), HomGLn(F)(E,ℂ) ≤ 1. For the proof in the archimedean case, we develop several tools to study invariant distributions on smooth manifolds.
AB - Let F be an arbitrary local field. Consider the standard embedding GL n(F) → GLn+1(F) and the two-sided action of GL n(F)× GLn(F) on GLn+1(F). In this paper we show that any GLn(F) × GLn(F)-invariant distribution on GLn+1(F) is invariant with respect to transposition. We show that this implies that the pair (GLn+1(F), GLn(F)) is a Gelfand pair. Namely, for any irreducible admissible representation (π,E) of GLn+1(F), HomGLn(F)(E,ℂ) ≤ 1. For the proof in the archimedean case, we develop several tools to study invariant distributions on smooth manifolds.
KW - Mathematics - Representation Theory
KW - 22E,22E45,20G05,20G25,46F99
UR - http://www.scopus.com/inward/record.url?scp=56749177561&partnerID=8YFLogxK
U2 - 10.1112/S0010437X08003746
DO - 10.1112/S0010437X08003746
M3 - Article
AN - SCOPUS:56749177561
VL - 144
SP - 1504
EP - 1524
JO - Compositio Mathematica
JF - Compositio Mathematica
SN - 0010-437X
IS - 6
ER -