Proposition 11

Let $\lambda \in \mathscr{S}$. The isotypical component $\mathfrak{a}_{\lambda}$ of A is both the isotypical component of $A_s$ of type $S_{\lambda}$ and the isotypical component of $A_d$ of type $S^*_{\lambda}$. Moreover, we have

(1) $[A_s: S_{\lambda}] = [A_d: S^*_{\lambda}] =$ dim$_{D_{\lambda}}S_{\lambda}$

and

(2) long(A) = long(A$^o$) $=\sum_{\lambda\in\mathscr{S}}$ dim$_{D_{\lambda}}S_{\lambda}$.

The first assertion is the specific case $M = A_s$ of Proposition 6, c) of VIII, p. 139. The equality $[A_s: S_{\lambda}] = [A_d: S^*_{\lambda}]$ follows from Proposition 7 of VIII, p. 140 because the dual of the left A-module $A_s$ is isomorphic to the right A-module $A_d$. By Propositions 8, a) and 9, c), the mapping $a\mapsto a_{S_{\lambda}}$ defines an isomorphism of left A-modules from $\mathfrak{a}_{\lambda}$ to End$_{D_{\lambda}}(S_{\lambda})$. Since $[A_s: S_{\lambda}]$ is, by definition, the length of the left A-module $\mathfrak{a}_{\lambda}$, the relation $[A_s: S_{\lambda}] =$ dim$_{D_{\lambda}}S_{\lambda}$ follows from Lemma 2 of VIII, p. 121. Finally, relation (2) is obtained from (1) by taking the sum over $\lambda$.