#### Vol. 16, No. 3, 2022

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

### Inna Entova-Aizenbud and Vera Serganova

Vol. 16 (2022), No. 3, 697–729
##### Abstract

In this paper, we study the representations of the periplectic Lie superalgebra using the Duflo–Serganova functor. Given a simple $\mathfrak{𝔭}\left(n\right)$-module $L$ and a certain odd element $x\in \mathfrak{𝔭}\left(n\right)$ of rank $1$, we give an explicit description of the composition factors of the $\mathfrak{𝔭}\left(n-1\right)$-module ${\mathrm{DS}}_{x}\left(L\right)$, 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 $\left(-\right)\otimes {ℂ}^{0|1}$.

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

 To Pavel Etingof for his 50th birthday.
##### Keywords
Lie superalgebra, periplectic Lie superalgebra, superdimension, Duflo–Serganova functor
##### Mathematical Subject Classification
Primary: 17B10, 17B55