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.


 A: (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.
(2) Yes, the tensor product should be a direct sum.  Interestingly, in my version ('Springer study edition', 2004 copyright) this typo is corrected.
(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 $\mathfrak{sl}_2(\mathbb{C})$.
