Again the same paper:

https://projecteuclid.org/euclid.cmp/1103858969

On page 104, they argue that Eq. (6.4) follows immediately from (6.3) on page 103 and (E1) on page 102, with $V$ replaced by $\bar{U}$, because $\Lambda(U,\bar{U})$ leaves the zero component invariant.

Here are the equations:

$$(E1)\sigma_{\nu k}(x_1,\ldots , x_n) = \sum_\mu S_k(U^{-1},V^{-1})^\mu_\nu \sigma_{\mu k}(Rx_1+a,\ldots, Rx_n+a)$$

$$(6.3)S_{\nu k}(\xi_1,\ldots , \xi_n)=\int e^{-\sum_{k=1}^n (\xi_k^0 q_k^0 - i \xi_k q_k)}\tilde{W}(q_1,\ldots, q_n)d^{4n}q$$

I don't see how this is immediate, can someone clear this matter to me?