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Shimura lifts are correspondence between integer weight and half-integral weight automorphic forms. Half integral weight things are associated to representation of a double cover of $G =SL_2(\mathbb{R})$, where as integer things live already in a rep of $G$.

Q: Is there a conceptual (representation-theoretic) explanation for this phenomenon? Is there a local (in terms of places) analogue of this phenomenon?

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Yes, this comes from the pair $SO(2,1)\times Mp(2)$ inside Mp(6)$, both globally/automorphically and locally. –  paul garrett May 2 '13 at 12:49
    
What phenomenon exactly? Are you talking about the Shimura correspondence? –  Eren Mehmet Kiral Jul 25 '13 at 4:10
    
    
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