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

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 $\Sigma_\delta(\R^{2l+1|n})$.