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In this paper, we study the representations of the periplectic Liesuperalgebra using the Duflo-Serganova functor. Given a simple$\mathfrak{p}(n)$-module $L$ and a certain element $x\in \mathfrak{p}(n)$ ofrank $1$, we give an explicit description of the composition factors of the$\mathfrak{p}(n-1)$-module $DS_x(L)$, which is defined as the homology of thecomplex $$\Pi M \xrightarrow{x} M \xrightarrow{x} \Pi M.$$ In particular, weshow that this $\mathfrak{p}(n-1)$-module is multiplicity-free. We then usethis result to give a simple explicit combinatorial formula for thesuperdimension of a simple integrable finite-dimensional$\mathfrak{p}(n)$-module, based on its highest weight. In particular, thisreproves the Kac-Wakimoto conjecture for $\mathfrak{p}(n)$, which was provedearlier by the authors.

Duflo-Serganova functor and superdimension formula for the periplectic Lie superalgebra