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As a place-holder answer: while, unfortunately I do not know a good reference offhand, I believe (based on a memory of having done the computation at least twice myself) that direct computation shows that the restriction to meta-U(n) does descend to U(n).

Conceivably various expository papers of Steve Kudla include either-or-both discussion of the various pairs inside metaplectic groups (as your meta-version of U(1)xU(n)) and the sort of splitting property of interest.

Edit: Oops! Indeed, as commented and amplified in another answer, the descent is just to SU(n).

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As a place-holder answer: while, unfortunately I do not know a good reference offhand, I believe (based on a memory of having done the computation at least twice myself) that direct computation shows that the restriction to meta-U(n) does descend to U(n).

Conceivably various expository papers of Steve Kudla include either-or-both discussion of the various pairs inside metaplectic groups (as your meta-version of U(1)xU(n)) and the sort of splitting property of interest.