## Multiplicity formula for restriction of representations of $\widetilde {\mathrm {GL}_{2}}(F)$ to $\widetilde {\mathrm {SL}_{2}}(F)$

HTML articles powered by AMS MathViewer

- 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
- PDF | Request permission

## 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 Waldspurgerβs analysis of theta correspondence between $\widetilde {\mathrm {SL}}_{2}(F)$ and $\textrm {PGL}_{2}(F)$.## References

- J.-P. Labesse and R. P. Langlands,
*$L$-indistinguishability for $\textrm {SL}(2)$*, Canadian J. Math.**31**(1979), no.Β 4, 726β785. MR**540902**, DOI 10.4153/CJM-1979-070-3 - 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), 160β193. MR**3061683**, DOI 10.1016/j.jalgebra.2013.05.001 - Jean-Loup Waldspurger,
*Correspondances de Shimura et quaternions*, Forum Math.**3**(1991), no.Β 3, 219β307 (French). MR**1103429**, DOI 10.1515/form.1991.3.219

## Bibliographic 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
- MR Author ID: 291342
- Email: dprasad@math.tifr.res.in
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**144**(2016), 903-908 - MSC (2010): Primary 22E35; Secondary 22E50
- DOI: https://doi.org/10.1090/proc12721
- MathSciNet review: 3430864