Regular defect-group bimodule inside a block ($kD\mid {\mathrm{Res}}_{D\times D}^{G\times G}(kGb)$)

For a block idempotent $b$ with defect $D$, the Brauer quotient of a module of its modular block bimodule at $\Delta D$ is nonzero: on the fixed basis $C_G(D)$, the block projection acts by ${\mathrm{Br}}_D(b)$. Restriction to $D\times D$ splits into transitive permutation modules; their stabilizers have order at most $|D|$. A nonzero quotient at $\Delta D$ therefore forces a summand with stabilizer exactly $\Delta D$, which is the regular $kD$ bimodule. Tensoring on the right with any $kD$-module $U$ gives $U\mid {\mathrm{Res}}_D^G(b{\mathrm{Ind}}_D^{G}U)$. Consequently finite representation type of the modular block forces finite representation type of $kD$.