Proposition 9

Let M and $M'$ be semisimple A-modules, and consider descriptions $((V_{\lambda})_{\lambda\in\mathscr{S}}, \alpha )$ and $((V'_{\lambda})_{\lambda\in\mathscr{S}}, \alpha ')$ of M and $M'$, respectively. For every family $\boldsymbol{f}= (f_{\lambda})_{\lambda\in\mathscr{S}}$ in $\prod_{\lambda\in\mathscr{S}}$ Hom$_{D_{\lambda}}(V_{\lambda},V_{\lambda}')$, there exists a unique A-linear mapping $\varphi (\boldsymbol{f})\in$ Hom$_A(M,M')$ for which the following diagram commutes:

$\bigoplus(S_{\lambda}\otimes_DV_{\lambda})^{\alpha}$ // M

$\lambda \in \mathscr{S}$

$\bigoplus(S_{\lambda}\otimes_DV'_{\lambda})^{\alpha'}$ // $M'$.

$\lambda \in \mathscr{S}\lambda$

The mapping $\varphi :\prod_{\lambda\in\mathscr{S}}$ Hom$_{D_{\lambda}}(V_{\lambda},V'_{\lambda})\rightarrow$ Hom$_A(M,M')$ defined this way is a group isomorphism. When we have $M = M',V_{\lambda}= V'_{\lambda}$ for every $\lambda \in \mathscr{S}$, and $\alpha =\alpha '$, the mapping $\varphi$ is a ring isomorphism from $\prod_{\lambda\in\mathscr{S}}$ End$_{D_{\lambda}}(V_{\lambda})$ to End$_A(M)$.

In view of the description of the isotypical components of M and $M'$ given in Proposition 8, a), this follows from Theorem 3 of VIII, p. 64 and Proposition 5, b) of VIII, p. 66.