The finite-dimensional irreducible rational representations of $SL_2(\mathbb C)$ are
$$
V_n=\operatorname{Sym}^n(\mathbb C^2),\qquad n\geq0,
$$
of dimension $n+1$. The central element $-I$ acts on $V_n$ by $(-1)^n$. Therefore precisely the even-indexed representations
$$
V_{2m},\qquad m\geq0,
$$
descend to irreducible representations of $PGL_2(\mathbb C)$. This is the <Descent of an irreducible SL2 representation to PGL2>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 102 / 1 / iii / Solution

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The irreducible rational representations of $SL_2(\mathbb C)$ are $\operatorname{Sym}^n(\mathbb C^2)$. Such a representation descends through $SL_2\to PGL_2$ exactly when the central element $-I$ acts trivially, which holds exactly when $n$ is even.


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