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 $${\Phi }_x=S\circ (-){\downarrow }_{i_x}:Rep(G)\to Rep(SOSp(1|2))$$ be the composition of the restriction functor $(-){\downarrow }_{i_x}$ and the functor $S$. We show that the functor ${\Phi }_x$ may be described explicitly using the homology-type functor ${\Phi }_{x_{bal}}$ 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.

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