For a finite group $G$ there is the Fourier transform $\displaystyle \hat{f}(\rho)=\sum_{g \in G} f(g)\rho(g)$ with inverse $$\displaystyle f(g)=\frac{1}{|G|}\sum_{\rho}d_{\rho}\operatorname{Tr}\left(\rho(g^{-1})\hat{f}(\rho)\right).$$
This follows from the decomposition of the group algebra $\mathbb{C}G \cong \bigoplus_{i}V_i$ where $V_i$ are the irreducible representations of $G$ over $\mathbb{C}$, the map given by $\sum_g a_g g \mapsto \sum_g a_g\rho_i(g): V_i \mapsto V_i$.
In modular representation theory, there is the decomposition $R=kG \cong \bigoplus_i B_i$ where each $B_i$ is an indecomposable ideal and the composition factors of each $B_i$ are an equivalence class of simple modules. One can also decompose the identity as $1=e_1+\dotsb+e_n$ into primitive central orthogonal idempotents and write $R=\bigoplus_iRe_i$.
Is the Fourier transform for $kG$ just the obvious map? That is, $B_i=Re_i$ is an $R$-module and so a $k$-vector space, so we can write $\rho_i: Re_i \mapsto Re_i$ and $\sum_g a_g g \mapsto \sum_g a_g\rho_i(g): B_i \rightarrow B_i$?
