Proposition 11

a) The multiplicity $[f_*(T) : S]$ of the simple A-module S in the semisimple A-module $f_*(T)$ is equal to the dimension of H viewed as a right vector space over D.

b) The multiplicity $[f^*(S) : T]$ is finite and equal to the dimension of H viewed as a left vector space over E.

Assertion a) follows from formula (11) of VIII, p. 72.

The B-module $f^*(S)$ is semisimple and finitely generated, hence has finite length. By formula (12) of loc. cit., we have

(20) $[f^*(S) : T] =$ dim$_E$ Hom$_B(f^*(S),T)$.

Now, in II, §5, No. 1, p. 277–278, formula (2) and Remark 2, we defined an E-linear bijection from Hom$_A(S, f_*(T))$ to Hom$_B(f^*(S),T)$. Assertion b) then follows from formula (20).