Multiplicity formula for restriction of representations of ̃𝐺𝐿₂(𝐹) to ̃𝑆𝐿₂(𝐹)

Patel, Shiv Prakash; Prasad, Dipendra

doi:10.1090/proc12721

Multiplicity formula for restriction of representations of $\widetilde {\mathrm {GL}_{2}}(F)$ to $\widetilde {\mathrm {SL}_{2}}(F)$
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by Shiv Prakash Patel and Dipendra Prasad

Proc. Amer. Math. Soc. 144 (2016), 903-908

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

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 $\widetilde {\mathrm {GL}}_{2}(F)$ of $\mathrm {GL}_{2}(F)$ to the 2-fold cover $\widetilde {\mathrm {SL}}_{2}(F)$ of $\mathrm {SL}_{2}(F)$, 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 $\widetilde {\mathrm {SL}}_{2}(F)$ and $\textrm {PGL}_{2}(F)$.

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