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

In this paper, we study the representations of the periplectic Lie superalgebra using the Duflo–Serganova functor. Given a simple p(n)-module L and a certain odd element x ∈ p(n) of rank 1, we give an explicit description of the composition factors of the p(n−1)-module DS_x (L), which is defined as the homology of the complex ∏M^x−→ M^x−→ ∏M, where ∏ denotes the parity-change functor (−) ⊗ ℂ^0|1. In particular, we show that this p(n−1)-module is multiplicity-free. We then use this result to give a simple explicit combinatorial formula for the superdimension of a simple integrable finite-dimensional p(n)-module, based on its highest weight. In particular, this reproves the Kac–Wakimoto conjecture for p(n), which was proved earlier by the authors.