In Weinberg's QFT book

In chapter 4, the change of creation operator under Lorentz transformation is describe by (4.2.12),

$$\begin{aligned} U_{0}(\Lambda, \alpha) a^{\dagger}(\mathbf{p} \sigma n) U_{0}^{-1}(\Lambda, \alpha)=& \mathrm{e}^{-i\left(\Lambda_{P}\right) \cdot \alpha} \sqrt{(\Lambda p)^{0} / p^{0}} \\ & \times \sum_{\overline{\sigma}} D_{\overline{\sigma} \sigma}^{(j)}(W(\Lambda, p)) a^{\dagger}\left(\mathbf{p}_{\Lambda} \overline{\sigma} n\right) \end{aligned}\tag{4.2.12}$$

However in chapter 5, equation (5.1.12) gives

$$U_{0}(\Lambda, b) a^{\dagger}(\mathbf{p}, \sigma, n) U_{0}^{-1}(\Lambda, b)=\exp (-\mathrm{i}(\Lambda p) \cdot b) \sqrt{(\Lambda p)^{0} / p^{0}}$$ $$\times \sum_{\overline{\sigma}} D_{\sigma \overline{\sigma}}^{\left(j_{n}\right) *}\left(W^{-1}(\Lambda, p)\right) a^{\dagger}\left(\mathbf{p}_{\Lambda}, \overline{\sigma}, n\right)\tag{5.1.12}$$

we could notice that the little group W has changed, but why?


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