Question on gauge fields "acting on different representations" 
-First of all, in the end of this question, unfortunatelly, I'll kind of request the solution of an exercise. But, this isn't for any kind of classroom evaluation. It is just that I kind of grasp the heuristic picture, but I don't know how to perform the technical calculations, due to my poor knowledge on lie algebra and lie group theory.

*

*I'm following the text book $[1]$.


*The question is written in section VII$)$, on equation $(8)$.

Now, standard model of particle physics is a (chiral) gauge theory. Therefore, it is a theory which is enconded in the language of Principal fiber bundles and associated fibre bundles. I will write a series of topics that explain better my problem here.
I) Connection $1$-$\mathrm{forms}$
Now, in Salam-Weinberg model, the gauge group of the theory is $G =  SU(2)_{L} \times U(1)_{Y}$. The connection constructed in the principal bundle, the $(SU(2)_{L} \times U(1)_{Y})$-$\mathrm{bundle}$, is the connection $1$-$\mathrm{form}$:
$$A = W + B.\tag{1}$$
$A$ is not the electromagnetic gauge field, rather, the connection $1$-$\mathrm{form}$ of the principal bundle. $W$ is the weak gauge field and $B$ the hypercharge gauge field.
II) Local Connection $1$-$\mathrm{forms}$
A local version of $(1)$ (the local gauge field) that "puts the gauge field on spacetime" is given by:
$$A_{s} = s^{*}A.\tag{2}$$
Where, $s$ is a section on the principal bundle (the local gauge choice), and the $s^{*}$ is the pullback of the section (when we apply this map on $A$, we bring the information of the gauge field for a region located on the base manifold $\mathcal{M}$). Furthermore, since our algebraic landscape deals with groups and lie groups, the action of $A_{s}$ on a vector field $X \in T_{p}\mathcal{M}$ lies on the lie algebra: $A_{s}(X) \in \mathfrak{g}$.
III) Representations, Spinors and Multiplets
The map, $\rho: G \to GL(V)$ is the representation of the Lie group (gauge group)  Therefore, using this map we produce matrices that acts on vectors of a vector space $V$.
Furthermore, given a representation $\rho$, it is possible to define its differential representation:
$$\rho_{*}: \mathfrak{g} \to \mathrm{End}(V),$$
where the operation $\cdot_{*}$ is the pushfoward.
The necessity of dealing with spinors as multiplets introduces a algebraic structure called: "Twisted Spinor Bundle" $[1]$:
$$TS = S \otimes E = S \otimes (P\times_{\rho}V). \tag{3}$$
Where, S is the spinor bundle, and E the associated bundle (the $P$ is the principal bundle and $\rho: G \to GL(V)$ the representation).

The tensor structure $(3)$ tells us: "we have spinor fields in $S$ and the fact that we construct a tensor product with $E$ we construct the well-known multiplets $\psi$".

Actually, Twisted Spinor Bundles are also called Gauge Multiplet Spinor Bundles.
IV) Covariant Derivatives 1
In $(3)$ we can construct the covariant derivative of the theory acting on multiplets (spinors) as:
$$D^{A}_{\mu}\psi = \partial_{\mu}\psi + \rho_{*}(A_{s}(X))\psi = \partial_{\mu}\psi - \frac{ig}{2}W^{a}_{\mu}\sigma_{a}\psi - \frac{ig'}{2}B_{\mu}\psi. \tag{4}$$
V) Chirality
An important feature of the standard model is its chirality. Following $[1]$, this means that the whole twisted spinor bundles "slipts" in right part $R$ and left part $L$ as:
$$ S \otimes E = (S_{L} \otimes E_{L}) \oplus (S_{R} \otimes E_{R}). \tag{5} $$
The spinor field notion is then defined with the one observes that first $S$ splits as $S_R\oplus S_L$ and that we are hence free to define a field $\psi_R$ as a section of some $S_R \otimes E_R$ and a field $\psi_L$ as a section of some $S_L \otimes E_L$ where $E_R$ and $E_L$ have no a priori relation. Then we define $\psi = \psi_R \oplus \psi_L$
The structure $(5)$ is called "Twisted Chiral Spinor Bundle".
Due to this mathematical structure, and knowing that $E_{L}$ and $E_{R}$ depends on representations. There are two (possibly distinct) representations of $G$ on complex vector spaces $V_{R}$ and $V_{L}$, i.e., the whole formal bundle that I'm talking about is (with all structures explicitly showed):
$$TS_{\mathrm{chiral}} = S \otimes E = (S_{L} \otimes E_{L}) \oplus (S_{R} \otimes E_{R})  = (S_{L} \otimes (P\times_{\rho_{L}}V_{L})) \oplus (S_{R} \otimes (P\times_{\rho_{R}}V_{R})). \tag{6}$$
The map, $\rho_{L}: G \to GL(V_{L})$ is the representation of the Lie group (gauge group), that produce matrices that acts on vectors of a vector space $V_{L}$. Similarly, the map, $\rho_{R}: G \to GL(V_{R})$ is the representation of the Lie group (gauge group), that produce matrices that acts on vectors of a vector space $V_{R}$. Therefore, we have two distinct induced representations as well: $\rho_{L*}: \mathfrak{g} \to \mathrm{End}(V_{L})$ and $\rho_{R*}: \mathfrak{g} \to \mathrm{End}(V_{R})$.
VI) Covariant Derivatives 2
In same fashion, we can construct the covariant derivative of the theory acting on chiral multiplets (spinors) as:
$$D^{A}_{\mu}\psi= \partial_{\mu}\psi + \rho_{L*}(A_{s}(X))\psi_{L} + \rho_{R*}(A_{s}(X))\psi_{R} \implies$$
$$D^{A}_{\mu}\psi \equiv D^{A}_{\mu}(\psi_{L} + \psi_{R}) =\partial_{\mu}\psi_{L} + \rho_{L*}(A_{s}(X))\psi_{L} + \partial_{\mu}\psi_{R} + \rho_{R*}(A_{s}(X))\psi_{R}\tag{7}$$
VII) My Question
Explicitly, I have the following data:

*

*$G = SU(2)_{L} \times U(1)_{Y}$

*$\mathfrak{g} = \mathfrak{su}(2)_{L} \oplus \mathfrak{u}(1)_{Y}$

*$V_{L} = \mathbb{C}_{L}^2 \otimes \mathbb{C}_{Y}$

*$V_{R} = \mathbb{C}_{L} \otimes \mathbb{C}_{Y}$
The subscripts are nothing but labels, the $\mathbb{C}$'s are your standard linear algebra complex vector spaces. Also, for your convenience:

*

*$\rho_{L*}:\mathfrak{su}(2)_{L} \oplus \mathfrak{u}(1)_{Y} \to \mathrm{End}(\mathbb{C}_{L}^2 \otimes \mathbb{C}_{Y})$

*$\rho_{R*}:\mathfrak{su}(2)_{L} \oplus \mathfrak{u}(1)_{Y} \to \mathrm{End}(\mathbb{C}_{L} \otimes \mathbb{C}_{Y})$
So my question is: how can I show the explicit calculation (i.e. could you please write the calculation) that:
$$\rho_{R*}(A_{s}(X)) = \rho_{R*}(W_{s}(X)+B_{s}(X)) = \rho_{R*}(W_{s}(X))+\rho_{R*}(B_{s}(X)) = \rho_{R*}(B_{s}(X))? \tag{8}$$
There are three different, but equivalent, ways to ask question:

$1)$ Exercise $8.11.8$ of $[1]$, page $525$.
$2)$ What happens, explicitly, with $\rho_{R*}(W_{s}(X))$? I mean, this term seems to simply "vanish".
$3)$ In physics (local chart) language: concerning the gauge group of Salam-Weinberg theory, how can I show that:
$$\rho_{L*}(A_{s}(X)) = - \frac{ig}{2}W^{a}_{\mu}\sigma_{a} - \frac{ig'}{2}B_{\mu} \tag{9}$$
and
$$\rho_{R*}(A_{s}(X))= - \frac{ig'}{2}B_{\mu}?\tag{10}$$
Where $\sigma_{a}$ are the Pauli Matrices.


$[1]$ Mark J.D. Hamilton Mathematical Gauge Theory, Springer, 2017.
 A: To answer via the second suggestion, $\rho_{R^*}$ is $0$ on $\mathfrak{su}(2)$ because it is effectively just the trivial action on $\mathbb{C}$: $$\rho_{R^*}(W+B)(X\otimes Y) = \rho_1(W)X \otimes Y + X\otimes \rho_2(B)Y,$$ where $\rho_1, \rho_2$ are the actions of $\mathfrak{su}(2)$ on $\mathbb{C}_L$ and $\mathfrak{u}(1)$ on $\mathbb{C}_Y$ respectively. (This is simply the way that any direct sum of Lie algebras acts on a tensor product of their representations)
There are many possibilities for the action of $\rho_2$ (I assume one is picked implictly earlier on) but $\rho_1$ must be trivial as $\mathfrak{su}(2)$ has no non-trivial representations on a 1-dimensional space (the same is true of any semisimple Lie algebra).
Note: this is entirely based on the representation theory of Lie algebras (especially semisimple ones) and doesn't engage with the larger context you have mentioned.
Edit: I will admit, it is unclear to me why the constructions are not symmetric and for example why $V_L$ and $V_R$ have different dimensions.
The same fact will not hold for $\rho_{L^*}$ as defined since we cannot then assume the corresponding $\rho_1$ is trivial. Perhaps there is some motivating physics reason for making these choices but, the way you have described it, it feels like $S \otimes E $ should split into $S_R\otimes E \oplus S_L\otimes E$ and I don't see where $E_R, E_L$ come from
