Brauer reciprocity

Write $P_{\phi }$ as the image of an idempotent matrix over $kG$. Because $\mathcal{O}$ is complete by [F3], idempotents lift through the ideal $\mathfrak{m}{M}_n(\mathcal{O}G)$. A lifted idempotent has image a projective $\mathcal{O}G$-lattice ${\widehat{P}}_{\phi }$ whose reduction is $P_{\phi }$.

Let $L_{\chi }$ be a stable lattice affording $\chi$. Since ${\widehat{P}}_{\phi }$ is projective, formation of $\mathrm{Hom}$ from it commutes with extension to $K$ and reduction to $k$. Hence ${\dim }_K{\mathrm{Hom}}_{KG}(K\otimes {\widehat{P}}_{\phi },V_{\chi })={\dim }_k{\mathrm{Hom}}_{kG}(P_{\phi },L_{\chi }\mathrm{/}\mathfrak{m}{L}_{\chi })\mathrm{.}$ The left side is the multiplicity of $V_{\chi }$ in the split semisimple module $K\otimes {\widehat{P}}_{\phi }$.

The functor ${\mathrm{Hom}}_{kG}(P_{\phi },-)$ is exact, and on a simple module $S_{\psi }$ it has dimension $1$ for $\psi =\phi$ and $0$ otherwise, because every map from $P_{\phi }$ to a simple module factors through its head $S_{\phi }$. The right side of step 2.1 is therefore the composition multiplicity of $S_{\phi }$ in $L_{\chi }\mathrm{/}\mathfrak{m}{L}_{\chi }$, namely $d_{\chi \phi }$ by [F1].

Thus the multiplicity of every $\chi$ in the ordinary character of $K\otimes {\widehat{P}}_{\phi }$ is $d_{\chi \phi }$. These coefficients depend only on $P_{\phi }$, so the character is independent of the chosen lift and equals ${\mathrm{\Phi }}_{\phi }={\sum }_{\chi }{d}_{\chi \phi }\chi$.