You don't need any character theory to do this. Let $V$ be an irreducible representation of any group $G$ (the group is not necessarily finite and $V$ is not necessarily finite-dimensional) and let $L$ be a $1$-dimensional representation. I claim that $V \otimes L$ is still irreducible. The reason is that tensoring with $L$ is invertible: the natural map $L^{\ast} \otimes L \to 1$ (where $1$ is the trivial representation) is an isomorphism, so
$$(V \otimes L) \otimes L^{\ast} \cong V.$$
Consequently, if $W$ is a proper nonzero submodule of $V \otimes L$, then $W \otimes L^{\ast}$ is a proper nonzero submodule of $V$. More abstractly, tensoring with $L$ is an automorphism of the category of representations of $G$, and automorphisms of categories preserve categorical properties of their objects like irreducibility.
To sum up the discussion in the comment into an answer: It is indeed true that for each $A \in \mathfrak g$, $\rho_W(A)$ is a homomorphism of $S_N$-modules. However, Schur's Lemma talks about homomorphisms of simple (a.k.a irreducible) $S_N$-modules, and as OP said, generally $W$ is not irreducible, but decomposes as a direct sum of irreducible $X_i$.
It is still possible to infer something from Schur's Lemma, namely:
For each such irreducible $X_i$, the restriction of $\rho_W(A)$ to $X_i$ is either
- $\rho_W(A)_{\vert X_i} = 0$
or
- $\rho_W(A)_{\vert X_i}$ induces an isomorphism (of $S_N$-modules) to another $X_j$ (where in general $j\neq i$).
The part of Schur's lemma whose sloppy interpretation is that "our endomorphism is a scalar" actually says that the endomorphism ring of a simple $S_N$-module is a skew field, finite dimensional over the ground field $k$ we've tacitly been working with all the time. If that field was $\mathbb C$, then that endomorphims ring is necessarily $\mathbb C$, but if our field was $\mathbb R$, it could in principle be $\mathbb R, \mathbb C$ or $\mathbb H$. However, I think that for the symmetric group $S_N$ it's actually known (cf. MO/10635) that all real (or even rational) representations have Schur index $1$ ($\Leftrightarrow$ Frobenius-Schur indicator $1$), meaning that indeed $End_{\mathbb R[S_N]}(X_i) \simeq \mathbb R$ and hence
- if in the second case above $i=j$, then $\rho_W(A)_{\vert X_i}$ is given by multiplication with some $\lambda_i \in \mathbb R^*$.
To see this in an example, let $k=\mathbb R, \mathfrak g = \mathfrak{sl}_2(\mathbb R), V=$ the standard representation of $\mathfrak g$ on $\mathbb R^2$, and $N=2$. Then $W= V^{\otimes 2}$ as $\mathbb R[S_2]$-module decomposes into four $1$-dimensional components; namely, set $x_i = e_i \otimes e_i $ for $i=1,2$, $x_3 = e_1 \otimes e_2 +e_2\otimes e_1$, and $x_4 = e_1 \otimes e_2 -e_2\otimes e_1 $, and let $X_i := \mathbb Rx_i$. Then $X_4$ is the alternating (sign) representation, the three other $X_i$ are isomorphic to the trivial representation.
Now e.g. for $A=\pmatrix{0 &1\\0&0}$ we have $\rho_W(A)_{\vert X_i} = \begin{cases} 0 \text{ if } i=1,4 \\ x_2 \mapsto x_3 \text{ if } i=2, \text{ giving an iso } X_2 \simeq X_3 \\ x_3 \mapsto 2 x_1 \text{ if } i=3, \text{ giving an iso } X_3 \simeq X_1 \end {cases}$
whereas for $H=\pmatrix{1 &0\\0&-1}$ we have $\rho_W(H)_{\vert X_i} = \begin{cases} x_1 \mapsto 2x_1 \text{ if } i=1, \text{ i.e. } \lambda_1=2 \\x_2 \mapsto -2x_2 \text{ if } i=2, \text{ i.e. } \lambda_2=-2 \\ 0 \text{ if } i=3,4\end {cases}$
etc.
Best Answer
You can decompose $W$ into irreducibles, $W = \oplus_{i = 1}^n V_i$, and use $Hom_G(V, \oplus V_i) = \oplus Hom_G(V,V_i)$ - a function into a product is determined by its coordinate projections.
At this point the desired result follows from Schur's lemma, as Qiaochu says in the comments.
A pedantic linguistic quibble -- does it make sense to say 'the number of copies of V in W'? If $W = V \oplus V$, there are infinitely many copies of $V$ inside of $W$ (can you describe all of them?). The concept you want instead is 'the dimension of the V-isotypic subspace.' Alternatively, it would be accurate to say "the number of copies of V in a decomposition of W into irreducibles" or "the multiplicity of $V$ in $W$."