What is the central idempotent of a representation? The article I am reading says 

Let $P_\lambda \in Z(G)$ be the central idempotent corresponding to the representation $\lambda$.

Could someone explain what this sentence means?
 A: Let $R$ be an arbitrary ring. The elements $r \in Z(R)$ of the center of $R$, essentially by definition, act as $R$-module homomorphisms on all $R$-modules simultaneously (and in fact all such homomorphisms come from elements of the center). So central idempotents act as $R$-module projections on all $R$-modules simultaneously.
Now, if $R = \mathbb{C}[G]$ is a group algebra of a finite group (say), then there's a distinguished class of $R$-module projections that you care about, namely the ones that project each representation
$$V = \bigoplus n_{\lambda} V_{\lambda}$$
(where $V_{\lambda}$ are the irreducible representations of $G$ and $n_{\lambda}$ are the multiplicities with which they appear in $V$) onto their $\lambda$-isotypic components $n_{\lambda} V_{\lambda}$. These projections are given by multiplication by suitable central idempotents in $\mathbb{C}[G]$.
Example. Let $G = C_2$ be the cyclic group of order $2$, and call the elements of the group $1$ and $g$. There are two irreducible representations, the trivial representation and the sign representation, and the corresponding central idempotents are $\frac{1 + g}{2}$ and $\frac{1 - g}{2}$.
More generally, if $V_{\lambda}$ has character $\chi_{\lambda}$, then the corresponding central idempotent is
$$\frac{\chi_{\lambda}(1)}{|G|} \sum_{g \in G} \overline{\chi_{\lambda}}(g) g \in Z(\mathbb{C}[G]).$$
This is a corollary of the orthogonality relations.
