Multiplicity formula for restriction of representations of $\widetilde{{\rm GL}}_{2}(E)$ to $\widetilde{{\rm SL}}_{2}(E)$

Shiv Prakash Patel; Dipendra Prasad

Multiplicity formula for restriction of representations of $\widetilde{{\rm GL}}_{2}(E)$ to $\widetilde{{\rm SL}}_{2}(E)$

Representation Theory 2014-05-26 v1

Authors: Shiv Prakash Patel , Dipendra Prasad

Abstract

In this note we prove a certain multiplicity formula regarding the restriction of an irreducible admissible genuine representation of a 2-fold cover $\widetilde{{\rm GL}}_{2}(E)$ of ${\rm GL}_{2}(E)$ to the 2-fold cover $\widetilde{{\rm SL}}_{2}(E)$ of ${\rm SL}_{2}(E)$ , and find in particular that this multiplicity may not be one, a result that seems to have been noticed before. The proofs follow the standard path via Waldspurger's analysis of theta correspondence between $\widetilde{{\rm SL}}_{2}(E)$ and ${\rm PGL}_{2}(E)$ .

Keywords

multiplicity formula restriction of representations covering groups theta correspondence irreducible admissible representations Waldspurger analysis

Cite

@article{arxiv.1405.6023,
  title  = {Multiplicity formula for restriction of representations of $\widetilde{{\rm GL}}_{2}(E)$ to $\widetilde{{\rm SL}}_{2}(E)$},
  author = {Shiv Prakash Patel and Dipendra Prasad},
  journal= {arXiv preprint arXiv:1405.6023},
  year   = {2014}
}

Multiplicity formula for restriction of representations of $\widetilde{{\rm GL}}_{2}(E)$ to $\widetilde{{\rm SL}}_{2}(E)$

Abstract

Keywords

Cite

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