So far our approach to induction has been strictly based on characters. We proved using Frobenius reciprocity that if $\chi$ is the character of a representation $(\pi, V)$ of the subgroup $H$ of the finite group $G$, then $\chi^G$ is the character of a representation $(\pi^G, V^G)$ of the group $G$, where $\chi^G$ is given by the explicit formula
\[ \chi^G (g) = \frac{1}{|H|} \sum_{x \in G} \dot{\chi} (x g x^{- 1}) . \] (4.04.0.1)

And we saw evidence that this is a powerful tool. It is remarkable that one can do so much without actually constructing $\pi^G$. Still, this approach is a little unsatisfying. It obscures the fact that the representation $(\pi^G, V^G)$ is a concrete thing, which can be given a concrete definition that will work not only for finite groups, but also for Lie groups.

Since we have only a little time left, we will be a little sketchy about details. Some of these will be put into Exercises. These have a different meaning from the Exercises in earlier chapters. Those were problems that I hoped you would write up and turn in. They were usually not integral parts of the text. In this Chapter, Exercises include facts that I don't fully have time to cover or write up. By doing these you will fill gaps in the exposition.