Lifting representation of $\mathfrak{so}(3,\mathbb{R})$ to representation of $SO(3,\mathbb{R})$

Question

I have a couple questions regarding representations of $\mathfrak{so}(3,\mathbb{R})$ , $\mathfrak{sl}(2,\mathbb{C})$, $SO(3,\mathbb{R})$ and the symmetric power representation $S^{k}(V) = V^{\otimes k} / \{v_{1} \otimes \dots \otimes v_{k} - v_{\sigma(1)} \otimes \dots \otimes v_{\sigma(k)} : \sigma \in S_{k} \}$ for $V = \mathbb{C}^{2}$

$1.)$ How do I know that every representation of $\mathfrak{sl}(2,\mathbb{C})$ can be considered as a representation of $\mathfrak{so}(3,\mathbb{R})$?

$2.)$ How can I determine when a representation of $\mathfrak{sl}(2,\mathbb{C})$ lifts to representation of $SO(3,\mathbb{R})$, or when a representation of $\mathfrak{so}(3,\mathbb{R})$ lifts to representation of $SO(3,\mathbb{R})$?

$3.)$ Is there a general way to have a representation of $\mathfrak{g} = Lie(G)$ lift to a representation of $G$?

$4.)$ In general $S^{k}(V)$ is a representation of $\mathfrak{sl}(2,\mathbb{C})$, I need to show that $S^{k}(V)$ lifts to a representation of $SO(3,\mathbb{R})$ if and only if $k$ is even.

(Not a question) If $E = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$, $F = \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}$ is the basis of $\mathfrak{sl}(2,\mathbb{C})$, and $H = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$ and $e_{1} ,e_{2}$ the standard basis of $\mathbb{C}^{2}$ and $e_{1}^{i} e_{2}^{k-i} : i \in [k] \cup \{0\}$ is the basis of $S^{k}(V)$. The actions are $E \cdot (e_{1}^{i} e_{2}^{k-i}) = (k-i) e_{1}^{i+1} e_{2}^{k-i-1}$, $F \cdot (e_{1}^{i} e_{2}^{k-i}) = ie_{1}^{i-1} e_{2}^{k-i+1}$, and $H \cdot (e_{1}^{i} e_{2}^{k-i}) = (2i-k) e_{1}^{i} e_{2}^{k-i}$.

Somewhere I saw that for $S^{k}(V)$ to lift to a representation of $SO(3,\mathbb{R})$ we require that $e^{\pi i \rho(h)} = Id$ (matrix exponential). How do we know that this condition is sufficient for the representation to be lifted? Assuming this, then $e^{\pi i \rho(h)} = Id$ is satisfied precisely when $k$ is even. The link is here

Answer 1

For question (1): You can start from the $2:1$ spin map $\pi: \text{SU(2)}\rightarrow \text{SO(3)}$ to derive their Lie algebra $\mathfrak{su}(2)\cong\mathfrak{so}_3\mathbb{R}$ are isomorphic. Also note $\mathfrak{su}(2)\otimes_{\mathbb{R}}\mathbb{C}=\mathfrak{su}(2) \oplus i\cdot\mathfrak{su}(2)=\mathfrak{sl}(2,\mathbb{C})$. Here, you might need to use a general fact: there is one-to-one correspondence between representation of a $\textbf{real}$ Lie algebra and representation of its complexification.( You can find this proposition from textbook written by Hall, GTM222. If you like, I can elaborate later.) Thus, every representation of $\mathfrak{sl}(2,\mathbb{C})$ can be considered as representation of $\mathfrak{su}(2)\cong\mathfrak{so}_3\mathbb{R}$.

For question(2): Let me cite $\textbf{Lie correspondence}$ here: If $G$ is a simply connected Lie group, then every representation of its Lie algebra $\mathfrak{g}$ can be lifted to $G$ by exponential map. In particular, when $G$ is a matrix group, then exponential map is given by $\exp{X} = e^{X}=\sum_{k=0}^{\infty} \frac{X^{k}}{k !}, \text{where}\quad X \in \mathfrak{g}$. Since $\pi_1(\text{SO(3)}) = \mathbb{Z}_2, \pi_1(\text{SL}(2,\mathbb{C}) )= \{1\}$, when you exponent all irreducible representation $S^k(V)$ of $\mathfrak{sl}(2,\mathbb{C})$, only half of them are representations of $\mathfrak{so}(3,\mathbb{R})$. This is because spinor rotate two times faster than vector. When $k$ is odd, you can check $\exp(\rho(2\pi X)) = -I$ and $\hat{\rho}(\exp(2\pi X))) = I$, where $X = \left(\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{array}\right), \rho$ is representation of $\mathfrak{so}(3)$ and $\hat{\rho}$ is representation of SO$(3)$. This is a contradiction since the diagram should commutates if $\hat{\rho}$ is a representaion of SO$(3)$.