Balanced and neat elements in quasi-reductive Lie superalgebras

Abstract

Let $G$ be a quasi-reductive supergroup (so its underlying algebraic group $G_{\bar 0}$ is reductive). We consider two trivially intersecting classes of odd elements: neat elements and balanced elements. Neat elements are always $ad$-nilpotent and may be embedded into subalgebras that are isomorphic to $\mathfrak{osp}(1|2)$, a simple Lie superalgebra whose underlying Lie algebra is $\mathfrak{sl}_2$. Balanced odd elements, on the other hand, are a natural generalization of the notion of a self-commuting element (an element $x\in Lie(G)_{\bar 1}$ for which $[x,x]=0$). Balanced elements are used to define homology-type functors on the category of representations of $G$. We show that any element $x\in Lie(G)_{\bar 1}$ may be written as a sum of a neat and a balanced odd element which commute with each other. This theorem has a categorical application. Let $\mathfrak{g}^{(1|1)}$ be the $(1|1)$-dimensional Lie superalgebra generated by $x \in Lie(G)_{\bar 1}$. The semisimplification of the category of finite-dimensional super-representations of $\mathfrak{g}^{(1|1)}$ is a functor $S: Rep(\mathfrak{g}^{(1|1)}) \to Rep(SOSp(1|2))$. Any $x\in Lie(G)_{\bar 1}$ induces a homomorphism $ i_x:\mathfrak{g}^{(1|1)}\to Lie(G)$. Let $$Φ_x=S\circ (-)\downarrow_{i_x}:Rep(G)\to Rep(SOSp(1|2))$$ be the composition of the restriction functor $(-)\downarrow_{i_x}$ and the functor $S $. We show that the functor $Φ_x$ may be described explicitly using the homology-type functor $Φ_{x_{bal}}$ corresponding to the balanced part of $x$ in the above decomposition. These homology-type functors are known as Duflo-Serganova functors. Finally, we provide a full classification of distinguished odd elements in simple quasi-reductive Lie superalgebras and show that in all cases except $\mathfrak{spe}(n)$, such elements are either balanced or neat.