A Lie conformal algebra is an algebraic structure that encodes the singular part of the operator product expansion of chiral fields in conformal field theory. A Lie pseudoalgebra is a generalization of this structure, for which the algebra of polynomials $\mathbf{k}[\partial ]$ in the indeterminate ∂ is replaced by the universal enveloping algebra $U(\mathfrak{d})$ of a finite-dimensional Lie algebra $\mathfrak{d}$ over the base field k. The finite (i.e., finitely generated over $U(\mathfrak{d})$) simple Lie pseudoalgebras were classified in our 2001 paper [1]. The complete list consists of primitive Lie pseudoalgebras of type $W,S,H$, and K, and of current Lie pseudoalgebras over them or over simple finite-dimensional Lie algebras. The present paper is the third in our series on representation theory of simple Lie pseudoalgebras. In the first paper, we showed that any finite irreducible module over a primitive Lie pseudoalgebra of type W or S is either an irreducible tensor module or the image of the differential in a member of the pseudo de Rham complex. In the second paper, we established a similar result for primitive Lie pseudoalgebras of type K, with the pseudo de Rham complex replaced by a certain reduction, called the contact pseudo de Rham complex. This reduction in the context of contact geometry was discovered by M. Rumin [11]. In the present paper, we show that for primitive Lie pseudoalgebras of type H, a similar to type K result holds with the contact pseudo de Rham complex replaced by a suitable complex. However, the type H case in more involved, since the annihilation algebra is not the corresponding Lie–Cartan algebra, as in other cases, but an irreducible central extension. When the action of the center of the annihilation algebra is trivial, this complex is related to work by M. Eastwood [6] on conformally symplectic geometry, and we call it conformally symplectic pseudo de Rham complex.

李保形代数是一种代数结构，它对保形场论中手征场的算子乘积展开的奇异部分进行编码。甲烈pseudoalgebra是这种结构的概括，为此，多项式的代数$\mathbf{克}[\partial ]$ 在不确定 ∂ 中被通用包络代数取代 $你(\mathfrak{d})$ 有限维李代数的 $\mathfrak{d}$在基场k 上。有限的（即，有限地生成在$你(\mathfrak{d})$) 简单的李赝代数在我们 2001 年的论文中被分类 [1]。完整列表由类型的原始 Lie 伪代数组成$宽,秒,H$, 和K , 和当前李赝代数或在简单的有限维李代数上。本文是我们关于简单李赝代数表示论系列的第三篇论文。在第一篇论文中，我们证明了W或S类型的原始 Lie伪代数上的任何有限不可约模要么是不可约张量模，要么是伪 de Rham 复数成员中微分的像。在第二篇论文中，我们对K类型的原始 Lie伪代数建立了类似的结果，将伪 de Rham 复形替换为一定的约简，称为接触伪 de Rham 复形. M. Rumin [11] 发现了接触几何环境中的这种减少。在本文中，我们展示了对于H型的原始 Lie伪代数，与K型结果相似，接触伪 de Rham 复形被合适的复形代替。然而，H型情况更复杂，因为湮灭代数不是对应的 Lie-Cartan 代数，就像在其他情况下一样，而是一个不可约的中心扩展。当湮灭代数中心的作用微不足道时，这个复形与 M. Eastwood [6] 在共形辛几何上的工作有关，我们称它为共形辛伪 de Rham 复形。