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 D λµ (R 2l+1|n ) of differential operators on the superspaces R 2l+1|n endowed with the standard contact structure α. On this space D λµ (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 D λµ (R 2l+1|n ) called filtration of Heisenberg and thus the space D λµ (R 2l+1|n ) is therefore denoted by H λµ (R 2l+1|n ). We have also a new filtration induced on D λµ (R 2l+1|n ) by the two filtrations and it calls bifiltration. Explicitly, the space D λµ (R 2l+1|n ) of differential operators is filtered canonically by the order of its differential operators and the order is k ∈ N. When it is filtered by order of Heisenberg, the order of any differential operator is equal to d ∈ 1 2 N. This study is the generalization, in super case, of the model studied by C.H.Conley and V.Ovsienko in [3]. Finally, we show that the spo(2l + 2|n)-module structure on the space D λµ (R 2l+1|n ) of differential operators is induced on the space H λµ (R 2l+1|n ) and therefore on the associated space S δ (R 2l+1|n ) of normal symbols, on the space P δ (R 2l+1|n ) of symbols of Heisenberg and on the space of fine symbol Σ δ (R 2l+1|n ).