A strong multiplicity one theorem for $\operatorname{SL_2}$

Jingsong Chai and Qing Zhang

Vol. 285 (2016), No. 2, 345–374

DOI: 10.2140/pjm.2016.285.345

Abstract

It is known that the multiplicity one property holds for ${SL}_2$ while the strong multiplicity one property fails. However, in this paper we show that if we require further that a pair of cuspidal representations $\pi$ and ${\pi }^{\prime }$ of SL${}_2$ have the same local components at the archimedean places and the places above 2, and they are generic with respect to the same additive character, then they also satisfy the strong multiplicity one property. The proof is based on a local converse theorem for SL${}_2$.

Keywords

strong multiplicity one theorem, local converse theorem, Howe vectors.

Mathematical Subject Classification 2010

Primary: 11F70, 22E50

Milestones

Received: 19 November 2015

Revised: 9 May 2016

Accepted: 27 May 2016

Published: 21 November 2016

Correction: 17 October 2017