Fourier transform on finite groups in characteristic $p>0$ Is there a Fourier theory for finite groups in characteristic $p>0$? Assume that $p$ divides the order $|G|$ of finite groups (or just work with $p$-groups), i.e., in a modular representation-theoretic setting.
If no such full theory, maybe just for abelian case? Thank you for any relevant references.
 A: Here is a brief outline of the modular theory developed by R. Brauer : First of all, is necessary to take complex valued class functions. Then (after certain choices of prime ideal containing the rational prime $p$, etc. ), there are $m$ irreducible Brauer characters $\{\phi_{i}: 1 \leq i \leq m \}$, where $G$ has $m$ conjugacy classes of $p$-regular elements. There is a "dual basis" of Brauer characters of projective indecomposable modules $\{ \Phi_i : 1 \leq i \leq m \}$. Any complex valued class-function $\psi$ defined only on $p$-regular elements is uniquely expressible in the form $\sum_{i = 1}^{m} \langle \psi, \Phi_{i} \rangle \phi_{i}$, where for two complex valued class functions $\alpha, \beta$ defined only on $p$-regular elements, we define
$\langle \alpha, \beta \rangle $ to be the complex number $|G|^{-1}\sum_{g \in G_{p^{\prime}}} \alpha(g) \beta(g^{-1})$, where $G_{p^{\prime}}$ is the set of $p$-regular $g \in G$.
Each $\phi_{i}$ is obtained from an irreducible representation $\sigma_{i}$ of $G$ over an algebraically closed field $F$ of characteristic $p$. For $g \in G$ of order prime to $p$, $\phi_{i}(g)$ is obtained by lifting the eigenvalues to $g$ in that representation to $p^{\prime}$-roots of unity in $\mathbb{C}$ in a consistent fashion and adding them.
Each $\Phi_{i}$ is defined in an analogous manner, except that we must use the projective cover (as $FG$-module) of the irreducible module associated to $\sigma_{i}$. It is the case that we always have $\langle \phi_{i}, \Phi_{j} \rangle = \delta_{i,j}$ for $1 \leq i,j \leq m$.
It is worth remarking that if we try to do this for a prime $p$ which does not divide $|G|$, then we find that $\phi_{i} = \Phi_{i}$ for each $i$, and that the $\phi_{i}$ obtained are just the complex irreducible characters of $G$ (and all elements of $G$ are $p$-regular in this case). This is essentially because all $FG$-modules are projective when $p$ does not divide $|G|$.
