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

Abstract

In this paper, we study the representations of the periplectic Lie superalgebra using the Duflo–Serganova functor. Given a simple $\mathfrak{𝔭}(n)$-module $L$ and a certain odd element $x\in \mathfrak{𝔭}(n)$ of rank $1$, we give an explicit description of the composition factors of the $\mathfrak{𝔭}(n-1)$-module ${\text{DS}}_x(L)$, which is defined as the homology of the complex

$$\mathrm{\Pi }M\underset{}{\overset{x}{\to }}M\underset{}{\overset{x}{\to }}\mathrm{\Pi }M,$$

where $\mathrm{\Pi }$ denotes the parity-change functor $(-)\otimes {ℂ}^{0|1}$.

In particular, we show that this $\mathfrak{𝔭}(n-1)$-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 $\mathfrak{𝔭}(n)$-module, based on its highest weight. In particular, this reproves the Kac–Wakimoto conjecture for $\mathfrak{𝔭}(n)$, which was proved earlier by the authors.