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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
DOI: https://doi.org/10.1090/proc12721
Published electronically: May 28, 2015
MathSciNet review: 3430864
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Abstract | References | Similar Articles | Additional Information

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)$.


References [Enhancements On Off] (What's this?)

  • [1] J.-P. Labesse and R. P. Langlands, $ L$-indistinguishability for $ {\rm SL}(2)$, Canad. J. Math. 31 (1979), no. 4, 726-785. MR 540902 (81b:22017), https://doi.org/10.4153/CJM-1979-070-3
  • [2] Dani Szpruch, Some results in the theory of genuine representations of the metaplectic double cover of $ GSp_{2n}(F)$ over p-adic fields, J. Algebra 388 (2013), 160-193. MR 3061683, https://doi.org/10.1016/j.jalgebra.2013.05.001
  • [3] Jean-Loup Waldspurger, Correspondances de Shimura et quaternions, Forum Math. 3 (1991), no. 3, 219-307 (French). MR 1103429 (92g:11054), https://doi.org/10.1515/form.1991.3.219

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Additional Information

Shiv Prakash Patel
Affiliation: Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India
Email: shiv@math.iitb.ac.in

Dipendra Prasad
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400005, India
Email: dprasad@math.tifr.res.in

DOI: https://doi.org/10.1090/proc12721
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

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