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If I have a group $G$ and a complex irreducible representation $g:G\rightarrow GL_n(\mathbb{C})$.

I am trying to use schur's lemma to show that for $x\in Z(G)$ we have that $g(z)=\lambda_z I_n$.

Now if we define a map $f_z:\mathbb{C}\rightarrow \mathbb{C}$ to be a homomorphism of represenations from $(g,\mathbb{C})$ to $(g,\mathbb{C})$ given by $f(c)=g(z)c$

Then from Schur's Lemma we have that this is an isomorphism.

Am I on the right track? Where do I go from here?


marked as duplicate by Arnaud D., Lord Shark the Unknown, Scientifica, Shailesh, José Carlos Santos Nov 1 '18 at 10:54

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Yes you are on the right track. Now you use a corollary of Schur's lemma that over an algebraically closed field and for an irreducible module $V$ the skew field $End(V)$ is not just any skew field but the field of scalars.


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