Local block projection controls p-section character support

Choose a $K$-basis $w_1,\dots ,w_n$ of $V_{\chi }$ and set $M={\sum }_{g\in G,1\le i\le n}\mathcal{O}g{w}_i\mathrm{.}$ This is a finitely generated, $G$-stable, torsion-free $\mathcal{O}$-module spanning $V_{\chi }$ over $K$, hence is finite free because $\mathcal{O}$ is a DVR. Thus $M$ is an $\mathcal{O}G$-lattice affording $\chi$. Since $\chi$ belongs to $B$, F1 says that $\widehat{B}$ acts as the identity on $V_{\chi }$, and therefore $\widehat{B}M=M$.

Apply Nagao with $D=⟨u⟩$ and $H=C_G(u)$. Its hypotheses hold because $D$ is central in $H$ and $D{C}_G(D)=H\le N_G(D)$. Suppose $c^G\mathrm{\ne }B$. Each indecomposable summand of $M_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}}$ belongs by F2 to a block $d$ with $d^G=B$. Thus $d\mathrm{\ne }c$, so orthogonality of the lifted block idempotents from F1 gives $\widehat{c}{M}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{r}}=0$. Consequently [F1, F2] $M_c=\widehat{c}M=\widehat{c}{M}_{\mathrm{e}\mathrm{r}\mathrm{r}},$ which is a direct summand of $M_{\mathrm{e}\mathrm{r}\mathrm{r}}$. Decompose $M_c$ into finitely many indecomposable $\mathcal{O}H$-lattices. Each is therefore an indecomposable summand of $M_{\mathrm{e}\mathrm{r}\mathrm{r}}$, so F3 makes its character zero at $uv$.

Scalar extension commutes with the idempotent projection: $K{\otimes }_{\mathcal{O}}{M}_c\cong \widehat{c}(K{\otimes }_{\mathcal{O}}M)=V_c\mathrm{.}$ Adding the finitely many zero traces from step 1.2 proves ${\chi }_c(uv)=0$. This also shows that the character component is independent of the chosen stable lattice. If $M_c=0$ the character is zero identically; if $u=1$, Nagao has no error part, so the antecedent $c^G\mathrm{\ne }B$ forces this zero case. Algebraic closedness and AC are used exactly through F3 and the AC-stated block contracts.