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 ${\mathrm{DS}<!--FUNCTION\; APPLICATION-->}_x(L)$, which is defined as the homology of the complex

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.

在本文中，我们使用 Duflo-Serganova 函子研究了周波李超代数的表示。给定一个简单的$\mathfrak{𝔭}(n)$-模块$\mathrm{大号}$和某个奇怪的元素$X\in \mathfrak{𝔭}(n)$等级$1$，我们给出了组成因素的明确描述$\mathfrak{𝔭}(n-1)$-模块${\mathrm{DS}<!--FUNCTION\; APPLICATION-->}_X(\mathrm{大号})$, 定义为复合体的同源性

在哪里$\mathrm{\Pi }$表示奇偶变化函子$(-)\otimes {ℂ}^{0|1}$.

特别是，我们证明了这$\mathfrak{𝔭}(n-1)$-module 是无多重性的。

然后我们使用这个结果给出一个简单的可积有限维的超维的简单显式组合公式$\mathfrak{𝔭}(n)$-模块，基于其最高权重。特别是，这证明了 Kac-Wakimoto 猜想$\mathfrak{𝔭}(n)$，作者之前已经证明了这一点。