Heisenberg order of differential operators on the superspaces $\mathbb{R}^{2l+1|n}$

TL;DR

This paper investigates three types of filtrations of differential operators on superspaces with contact structures, generalizing classical models and analyzing the induced module structures on associated symbol spaces.

Contribution

It introduces and studies Heisenberg and bifiltrations of differential operators on superspaces, extending classical results to the super case and analyzing module structures.

Findings

Existence of three filtrations on differential operators on superspaces.

The Heisenberg filtration induces a module structure on associated symbol spaces.

Generalization of classical models to the super setting.

Abstract

We study in this paper, the existence of tree types of filtrations of the space $\mathcal{D}_{\lambda\mu}(\R^{2l+1|n})$ of differential operators on the superspaces $\R^{2l+1|n}$ endowed with the standard contact structure $\alpha$. On this space $\mathcal{D}_{\lambda\mu}(\R^{2l+1|n})$, we have the first filtration called canonical and because of the existence of the contact structure on superspaces $\R^{2l+1|n}$ we obtain the second filtration on the space $\mathcal{D}_{\lambda\mu}(\R^{2l+1|n})$ called filtration of Heisenberg and thus the space $\mathcal{D}_{\lambda\mu}(\R^{2l+1|n})$ is therefore denoted by $\mathcal{H}_{\lambda\mu}(\R^{2l+1|n})$. We have also a new filtration induced on $\mathcal{D}_{\lambda\mu}(\R^{2l+1|n})$ by the two filtrations and it calls bifiltration. Explicitly, the space $\mathcal{D}_{\lambda\mu}(\R^{2l+1|n})$ of differential operators is filtered canonically by the order of its differential operators and the order is $k\in \N$. When it is filtered by order of Heisenberg, the order of any differential operator is equal to $d\in\frac{1}{2}\N$. This study is the generalization, in super case, of the model studied by C.H.Conley and V.Ovsienko in \cite{CoOv12}. Finally, we show that the $\spo(2l+2|n)$-module structure on the space $\mathcal{D}_{\lambda\mu}(\R^{2l+1|n})$ of differential operators is induced on the space $\mathcal{H}_{\lambda\mu}(\R^{2l+1|n})$ and therefore on the associated space $\mathcal{S}_\delta(\R^{2l+1|n})$ of normal symbols, on the space $\mathcal{P}_\delta(\R^{2l+1|n})$ of symbols of Heisenberg and on the space of fine symbol $\Sigma_\delta(\R^{2l+1|n})$.