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Multiplicity formula for restriction of representations of $ \widetilde{GL_{2}}(F)$ to $ \widetilde{SL_{2}}(F)$

Authors: Shiv Prakash Patel and Dipendra Prasad
Journal: Proc. Amer. Math. Soc. 144 (2016), 903-908
MSC (2010): Primary 22E35; Secondary 22E50
Published electronically: May 28, 2015
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Abstract: In this note we prove a certain multiplicity formula regarding the restriction of an irreducible admissible genuine representation of a 2-fold cover $ \widetilde {\textup {GL}}_{2}(F)$ of $ \textup {GL}_{2}(F)$ to the 2-fold cover $ \widetilde {\textup {SL}}_{2}(F)$ of $ \textup {SL}_{2}(F)$, and find in particular that this multiplicity may not be one, a result that was recently observed for certain principal series representations in the work of Szpruch (2013). The proofs follow the standard path via Waldspurger's analysis of theta correspondence between $ \widetilde {\textup {SL}}_{2}(F)$ and $ {\rm PGL}_{2}(F)$.

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Shiv Prakash Patel
Affiliation: Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India

Dipendra Prasad
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400005, India

Keywords: Covering groups, multiplicity formula, restriction of representations
Received by editor(s): September 10, 2014
Received by editor(s) in revised form: December 31, 2014
Published electronically: May 28, 2015
Communicated by: Kathrin Bringmann
Article copyright: © Copyright 2015 American Mathematical Society