与共形李代数抛物相关的非紧李代数的不变微分算子
Invariant Differential Operators for Non-Compact Lie Algebras Parabolically Related to Conformal Lie Algebras
摘要：
本文继续系统地构造非紧半单李群的不变微分算子。我们的出发点是一类代数，我们称之为“共形李代数”（CLA），它与闵可夫斯基时空的共形代数具有非常相似的性质，尽管我们的目标是以一种自然的方式超越这一类。为此，我们在两个具有相同复化的非紧半单李代数g和g'之间引入了{ \it抛物关系}的新概念，这两个半单李代数g和g'具有相同复化的极大抛物子代数.因此，我们考虑了与CLA E_{7(-25)}抛物相关的例外代数E_{7(7)}，抛物子代数包括E_{6(6)}和E_{6(-6)}。其他有趣的例子是正交代数so（p，q），所有这些都是抛物相关的共形代数so（n，2）与p+q=n+2，抛物子代数包括洛伦兹子代数so（n-1，1）和它的类似物so（p-1，q-1）。我们也考虑E_{6(6)}和E_{6(2)}，它们是与厄米对称情形E_{6(-14)}抛物相关的，即包含sl（6）的实形式的抛物子代数。我们还给出了一个公式表示的主要多重有效的CLA和所有的代数是抛物线相关的。在所有考虑的情况下，我们给出了不可分解的基本表示，包括所有相关的不变微分算子的必要数据的主要多重。在so（p，q）的情况下，我们也给出了约化的多重态。我们应该强调的是，在{ \it影子场}对中以最经济的方式给出多重态。此外，我们应该强调，所有不变微分算子的分类包括作为特殊情况的所有可能的{ \it守恒律}和{ \it守恒流}，酉或非酉。
英文摘要：
In the present paper we continue the project of systematic construction of invariant differential operators for non-compact semisimple Lie groups. Our starting points is the class of algebras, which we call 'conformal Lie algebras' (CLA), which have very similar properties to the conformal algebras of Minkowski space-time, though our aim is to go beyond this class in a natural way. For this we introduce the new notion of {\it parabolic relation} between two non-compact semisimple Lie algebras g and g' that have the same complexification and possess maximal parabolic subalgebras with the same complexification. Thus, we consider the exceptional algebra E_{7(7)} which is parabolically related to the CLA E_{7(-25)}, the parabolic subalgebras including E_{6(6)} and E_{6(-6)} . Other interesting examples are the orthogonal algebras so(p,q) all of which are parabolically related to the conformal algebra so(n,2) with p+q=n+2, the parabolic subalgebras including the Lorentz subalgebra so(n-1,1) and its analogs so(p-1,q-1). We consider also E_{6(6)} and E_{6(2)} which are parabolically related to the hermitian symmetric case E_{6(-14)}, the parabolic subalgebras including real forms of sl(6). We also give a formula for the number of representations in the main multiplets valid for CLAs and all algebras that are parabolically related to them. In all considered cases we give the main multiplets of indecomposable elementary representations including the necessary data for all relevant invariant differential operators. In the case of so(p,q) we give also the reduced multiplets. We should stress that the multiplets are given in the most economic way in pairs of {\it shadow fields}. Furthermore we should stress that the classification of all invariant differential operators includes as special cases all possible {\it conservation laws} and {\it conserved currents}, unitary or not.
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