Suppose $G$ is a complex algebraic group, $L$ a proper Levi subgroup, and $\lambda$ an irreducible character of the subgroup $L^F$ of $F$-stable points in $L$, which is contained in $F$-stable parabolic subgroups $P$ and $Q$.
It's a theorem that the parabolic induction and restriction funtors $R^G_{L\subset P},\ ^\ast R^G_{L\subset P}$, etc. are independent of $P$ and $Q$. A line in a proof by Digne and Michel says,
By the Mackey Formula and the adjunction, $$ \langle R^G_{L\subset P}\lambda;R^G_{L\subset Q}\lambda\rangle_{G^F}=\sum\langle\ ^\ast R^L_{L\cap\ ^xL\subset L\cap\ ^xQ}\lambda;\ ^\ast R^{\ ^xL}_{L\cap\ ^xL\subset P\cap\ ^xL}\ ^x\lambda\rangle_{L^F\cap\ ^xL^F} $$
Can anyone explain how exactly they're using Mackey and the adjunction to get this equality? The Mackey formula gives a decomposition of the composition $^\ast R^G_{L\subset P}\circ R^G_{L\subset Q}$, so I'm not sure how to use it to look at the inner product of two induction functors.
This is a pretty standard argument. Look at the hom space between the two induced representations, use Frobenius reciprocity to turn this into a hom from $\lambda$ to the restriction-induction of $\lambda$, use Mackey on the latter of these, and then Frobenius reciprocity again.