I'm studying QFT on Weinberg's book. And I have a question about its notation for Lorentz transformation property of the free fields (Chap.5).

In Sec. 5.1 of Vol. 1, annihilation fields $\psi_{\ell}^{+}(x)$ and creation fields $\psi_{\ell}^{-}(x)$ are given (as (5.1.4), (5.1.5)) so that they satisfy the transformation properties below: \begin{align} &U_{0}(\Lambda, a) \psi_{\ell}^{+}(x) U_{0}^{-1}(\Lambda, a)=\sum_{\bar{\ell}} D_{\ell \bar{\ell}}\left(\Lambda^{-1}\right) \psi_{\ell}^{+}(\Lambda x+a) \quad (5.1.6)\\ &U_{0}(\Lambda, a) \psi_{\ell}^{-}(x) U_{0}^{-1}(\Lambda, a)=\sum_{\bar{\ell}} D_{\ell \bar{\ell}}\left(\Lambda^{-1}\right) \psi_{\ell}^{-}(\Lambda x+a) \quad (5.1.7) \end{align} $U_0$ is the transformation operators, and $D$ is the representation matrices for the homogeneous Lorentz transformation.

My question is why we choose $\Lambda^{-1}$ rather than $\Lambda$, as the argument of $D$. Is it consistent with the transformation rule for one-particle states given in the scattering theory part (such as (3.1.1))?