Example 3

Given a linear representation $(M, \pi )$ of G, the contragredient or dual representation $\pi^{\vee}$ of $\pi$ is the representation of G in the K-module dual to M defined by the relation $\pi^{\vee}(g) =^t\pi (g^{-1})$ for every $g\in G$ (cf. II, §2, No. 5, p. 234). If M is a finite-dimensional free K-module, then so is its dual, and we have $\chi_{\pi^{\vee}}(g) =\chi_{\pi}(g^{-1})$ for every $g\in G$.