Local translations in curved spacetime A global Poincare transformation on a scalar field induces
$$\delta(a, \lambda)\phi(x) = [a^{\mu}+\lambda^{\mu\nu}x_{\nu}]\partial_{\mu}\phi(x). \tag{11.46}$$
In curved spacetime we replace $a^{\mu} \rightarrow \xi^{\mu}(x)$, but I read that in fact this new spacetime dependent parameter $\xi^{\mu}$ eats up the effect of the "orbital part of global Lorentz", so that in fact $\xi^{\mu}(x)=a^{\mu}(x)+\lambda^{\mu\nu}(x)x_{\nu}$, and we effectively have to treat $\xi^{\mu}$ and $\lambda^{ab}$ (the spin part of Lorentz) as the basis for gauge transformations.
I don't understand why this should be, any comments that might help? Why does the new local translation parameter include the effect of what globally used to be rotations/boosts?
References:


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*D.Z. Freedman & A. Van Proeyen, SUGRA, 2012; p. 225.

 A: *

*Ref. 1 defines a local translation on spacetime $M$ as a diffeomorphism. 

*Note that the words local and global in this physics context mean point-dependent (=$x$-dependent) and point-independent (=$x$-independent), respectively. [Be aware that mathematicians in other contexts typically use the words local and global to refer to objects defined in a local neighborhood $U\subseteq M$, or globally on the whole manifold $M$, respectively. Ref. 1 does not dwell on this latter distinction, and implicitly assumes that the definitions of objects can be extended/restricted appropriately, if needed.]

*An infinitesimal diffeomorphisms $\xi^{\mu}(x)$ can be identified with a vector field. 

*Note that in flat Minkowski spacetime the affine transformations are global$^1$ Poincare transformations.  However, Ref. 1 concerns GR and curved spacetime. Global$^1$ Lorentz transformations, global$^1$ Poincare transformations, and affine transformations become obsolete/meaningless notions in GR. Such transformations are all just special cases of local translations. 

*On the other hand, in the terminology of Ref. 1, the local Lorentz transformations do not denote (and should not be confused with) a subset of diffeomorphisms. Instead a local Lorentz transformation $\Lambda^a{}_b(x)$ is a transformations between choices of frames in the tangent bundle. In other words, it transforms the flat indices $a,b,\ldots$, (but not the curved indices $\mu,\nu,\ldots$), of various tensors, e.g. the vielbein $e^a_{\mu}(x)$.

*Together the local translations and local Lorentz transformations form the local Poincare transformations.

*See also this & this related Phys.SE posts.
References:


*

*D.Z. Freedman & A. Van Proeyen, SUGRA, 2012; p. 225.


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$^1$ We repeat that the word global means here $x$-independent, cf. pt. 2.
