Action of $\mathfrak{sl}_2(\Bbb{C})$ on $\textrm{Sym}^2 V$

I am reading Fulton and Harris and on page 150, there is the following passage (in the second paragraph) that I don't understand:

"Similarly, a basis for the symmetric square $W = \textrm{Sym}^2V = \textrm{Sym}^2 \Bbb{C}^2$ is given by $\{x^2,xy,y^2\}$ and we have $$\begin{eqnarray*} H(x \cdot x) &=& x \cdot H(x) + H(x) \cdot x = 2 x \cdot x\\ H(x\cdot y) &=& x\cdot H(y) + H(x) \cdot y = 0\\ H(y \cdot y) &=& y \cdot H(y) + H(y) \cdot y = 2y \cdot y\end{eqnarray*}$$ so the representation $W = \Bbb{C}\cdot x^2 \oplus \Bbb{C}\cdot xy \oplus \Bbb{C}\cdot y^2 = W_{-2} \oplus W_0 \otimes W_2$ is the representation $V^{(2)}$ above."

I should say $x,y$ are the standard basis vectors of $\Bbb{C}^2$ and $H$ is the matrix $diag(1,-1)$.

My question: What are $W_0$, $W_{-2}$ and $W_2$ in the last line of that paragraph? Also, is it a typo where they write the tensor product of $W_0$ and $W_2$ instead of direct sum? Furthermore, what is this $V^{(2)}$ mentioned? I have looked through the text and there is mentioned things like $V_\alpha$, but none with superscripts.

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There is no typo. The Lie algebra acts on a tensor product as a derivation (i.e., using the product rule from calculus). So, we have: $H ( v \otimes w ) = H(v) \otimes w + v \otimes H(w)$ – Bombyx mori Sep 19 '12 at 0:48
$V^{(n)}$ is shorthand for $\mathrm{Sym}^nV$, the $n$th symmetric power. Are the $W_{\#}$'s weight spaces? – anon Sep 19 '12 at 1:02

(1) For a representation $V$ and a weight $\alpha$, $V_{\alpha}$ is the weight space of weight $\alpha$ (see (11.3) on p. 147). So $W_{-2}, W_0, W_2$ are the weight spaces of $W$. The subscripts denote the weights, while of course $W$ refers to the representation at hand.
(3) The representation $V^{(n)}$ is defined on the very bottom of p. 149 as the (unique!) irreducible representation of $\mathfrak{sl}_2(\mathbb{C})$ with highest weight $n$, i.e. the $(n+1)$-dimensional representation whose weights are $-n, -n + 2, \dots, n-2, n$. In fact, at this point they haven't really constructed these irreps (other than "use the formulas provided and check that they work"), so the point of the discussion on p. 150 is that $V^{(n)} \cong \text{Sym}^n(V)$ (as representations of $\mathfrak{sl}_2(\mathbb{C})$) where $V = V^{(1)}$ is the standard representation of $\mathfrak{sl}_2(\mathbb{C})$ on $V = \mathbb{C}^2$. This is the "standard" construction of the irreps of $\mathfrak{sl}_2(\mathbb{C})$.