The answer of Theo seems to be right. Nevertheless, it is not explained that "the Mp(2,R)-representation that gave rise to V factors through SL(2,C) and on the other hand Mp(2,R) -> SL(2,C) factors through SL(2,R)". This cannot be true in general. E.g. in the case of SO(n) and Spin(n), i.e., when SL(2,R) is replaced by SO(n) and Mp(2,R) by Spin(n). Note that Spin(n,C) is (as SL(2,C)) simply connected as well (so topologically, it works). I think a bit incomplete part of the answer - for me - can be in the complexifications. Note that any representation of SL(2,R) comes from a representation of SL(2,C) but not every representation of SO(n) comes from a representation of SO(n,C). The distingushing property is that sl(2,R) is the split real form of sl(2,C), but so(n) is the compact real form of so(n,C). Could somebody make a comment on it?

The answer of Theo seems to be right. Nevertheless, it is not explained that "the Mp(2,R)-representation that gave rise to V factors through SL(2,C) and on the other hand Mp(2,R) -> SL(2,C) factors through SL(2,R)". This cannot be true in general. E.g. in the case of SO(n) and Spin(n), i.e., when SL(2,R) is replaced by SO(n) and Mp(2,R) by Spin(n). Note that Spin(n,C) is (as SL(2,C)) simply connected as well (so topologically, it works). I think a bit incomplete part of the answer - for me - can be in factorizations related to the complexifications. (SO(n) is a compact real of SO(n,C), but SL(n,R) is split real of SL(n,C) - this should explain the difference bteween the SO and Sp-cases).