What is Schur's Lemma ? and why is it valid only for the algebraically closed field ?
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A different version of Schur's lemma (which involves algebraically closed fields) is the following:
The reason you want an algebraically closed field is that, as Benjamin Lim noted in his comment, the map $f$ has an eigenvalue $\lambda$. Then the map $f-\lambda I$ is a $G$-equivariant map, which is not invertible, and so (as noted in Julian's answer) it is the zero map. Therefore, we conclude that $f=\lambda I$.
Schur's Lemma says:
The difference for algebraically closed fields is that skew fields have to be the ground field (at least for finite dimensional modules).