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 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 $D{S}_x(L)$, which is defined as the homology of the complex $$\Pi M\stackrel{x}{\to }M\stackrel{x}{\to }\Pi M.$$ 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.

ver 2: proof of prop. 3.2.2 significantly shortened

References in corpus (3)

• On associated variety for Lie superalgebras
• On representations of the Lie superalgebra p(n)
• On DS functor for affine Lie superalgebras

Cited by in corpus (5)

• Depths and cores in the light of DS-functors
• On classical tensor categories attached to the irreducible representations of the General Linear Supergroups $GL(n|n)$
• Bipartite extension graphs and the Duflo--Serganova functor
• Parabolic category ${\mathcal{O}}^{\mathfrak{p}}$ for periplectic Lie superalgebras $\mathfrak{pe}(n)$
• On semisimplicity of Jantzen middles for the periplectic Lie superalgebra