= Solution
{c}
Let $g$ be a <p-regular element>, with <eigenvalues> $\lambda_i$ on $V$. On both $g^{-1}$ acting on $V$ and $g$ acting on the <dual representation> $V^*$, the <eigenvalues> are $\lambda_i^{-1}$. Multiplicative lifts of <roots of unity> satisfy $\widehat{\lambda_i^{-1}}=\widehat\lambda_i^{-1}=\overline{\widehat\lambda_i}$. Hence
$$
\boxed{\chi_V(g^{-1})=\overline{\chi_V(g)}=\chi_{V^*}(g)}.
$$
All these identities concern the domain of a <Brauer character>, namely the <p-regular elements>. If necessary, extend the <p-modular system> to contain the relevant <roots of unity> and use one consistent choice of lifts.
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