Character formulae in category $\mathcal O$ for exceptional Lie superalgebras $D(2|1;\zeta)$

We establish character formulae for representations of the one-parameter family of simple Lie superalgebras $D(2|1;\zeta)$. We provide a complete description of the Verma flag multiplicities of the tilting modules and the projective modules in the BGG category $\mathcal O$ of $D(2|1;\zeta)$-modules of integral weights, for any complex parameter $\zeta$. The composition factors of all Verma modules in $\mathcal O$ are then obtained.