Proposition 8

Let M be a semisimple A-module and $((V_{\lambda})_{\lambda\in\mathscr{S}}, \alpha )$ a description of M.

a) For every $\lambda \in \mathscr{S}$, the mapping $\alpha$ induces an isomorphism from the A-module $S_{\lambda}\otimes_{D_{\lambda}}V_{\lambda}$ to the isotypical component of M of type $\lambda$.

b) For every $\lambda \in \mathscr{S}$, the mapping $\beta_{\lambda}: V_{\lambda}\rightarrow$ Hom$_A(S_{\lambda},M)$ defined by $\beta_{\lambda}(v)(s) =\alpha (s\otimes v)$ is $D_{\lambda}$-linear and bijective.

c) Let N be a submodule of M. There exists a unique family $(W_{\lambda})_{\lambda\in\mathscr{S}}$ with the following properties: $W_{\lambda}$ is a $D_{\lambda}$-linear subspace of $V_{\lambda}$ for every $\lambda \in \mathscr{S}$, and N is the image under $\alpha$ of the module $\bigoplus_{\lambda\in\mathscr{S}}(S_{\lambda}\otimes_{D_{\lambda}}W_{\lambda})$ identified with its canonical image in the module $\bigoplus_{\lambda\in\mathscr{S}}(S_{\lambda}\otimes_{D_{\lambda}}V_{\lambda})$. For every $\lambda \in \mathscr{S},W_{\lambda}$ is the set of elements $v\in V_{\lambda}$ such that $\alpha (s\otimes v)$ belongs to N for every $s\in S_{\lambda}$.

The A-module $S_{\lambda}\otimes_{D_{\lambda}}V_{\lambda}$ is isotypical of type $\lambda$ for every $\lambda \in \mathscr{S}$. Assertion a) then follows from the facts that $\alpha$ is an isomorphism and that M is the direct sum of the family $(M_{\lambda})_{\lambda\in\mathscr{S}}$ (VIII, p. 65, Proposition 4, a)).

Let $\lambda \in \mathscr{S}$. Denote the restriction of $\alpha$ to $S_{\lambda}\otimes_{D_{\lambda}}V_{\lambda}$ by $\alpha_{\lambda}: S_{\lambda}\otimes_{D_{\lambda}}V_{\lambda}\rightarrow$ M. With the notation of No. 3 applied to the $(A,D_{\lambda}$)-bimodule $S_{\lambda}$, we have

where the last equality follows from VIII, p. 60. Hence, $\beta_{\lambda}$ is the composition of the $D_{\lambda}$-linear homomorphism $\mathscr{H}(\alpha_{\lambda})$ and $\beta_{V_{\lambda}}$. Assertion b) then follows from Proposition 3, b) of VIII, p. 62.

Let N be a submodule of M. We have $N =\bigoplus_{\lambda\in\mathscr{S}}(N\cap M_{\lambda})$ (VIII, p. 66, Corollary), so c) follows from Theorem 2 of VIII, p. 62.