Character formulae in category $\mathcal{O}$ for exceptional Lie superalgebras $D(2\mathrm{\mid }1;\zeta )$

We establish character formulae for representations of the one-parameter family of simple Lie superalgebras $D(2\mathrm{\mid }1;\zeta )$. 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 $D(2\mathrm{\mid }1;\zeta )$-modules of integral weights, for any complex parameter $\zeta$. The composition factors of all Verma modules in $\mathcal{O}$ are then obtained.