Let $\mathcal{ D}$ be the Drinfeld double of the bosonization ${ \mathfrak B}(V)\#\Bbbk G$ of a finite-dimensional Nichols algebra ${ \mathfrak B}(V)$ over a finite group $G$. It is known that the simple $\mathcal{ D}$-modules are parametrized by the simple modules over $\mathcal{ D}(G)$, the Drinfeld double of $G$. This parametrization can be obtained by considering the head $\mathsf{ L}(\lambda)$ of the Verma module $\mathsf{ M}(\lambda)$ for every simple $\mathcal{ D}(G)$-module $\lambda$. In the present work, we show that the projective $\mathcal{ D}$-modules are filtered by Verma modules and the BGG Reciprocity $[\mathsf{ P}(\mu):\mathsf{ M}(\lambda)]=[\mathsf{ M}(\lambda):\mathsf{ L}(\mu)]$ holds for the projective cover $\mathsf{ P}(\mu)$ of $\mathsf{ L}(\mu)$. We use graded characters to proof the BGG Reciprocity and obtain a graded version of it. Also, we show that a Verma module is simple if and only if it is projective. We also describe the tensor product between projective modules.
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