Invariant Differential Operators for Non-Compact Lie Algebras Parabolically Related to Conformal Lie Algebras

摘要

In the present paper we continue the project of systematic construction ofinvariant differential operators for non-compact semisimple Lie groups. Ourstarting points is the class of algebras, which we call 'conformal Liealgebras' (CLA), which have very similar properties to the conformal algebrasof Minkowski space-time, though our aim is to go beyond this class in a naturalway. For this we introduce the new notion of {\it parabolic relation} betweentwo non-compact semisimple Lie algebras g and g' that have the samecomplexification and possess maximal parabolic subalgebras with the samecomplexification. Thus, we consider the exceptional algebra E_{7(7)} which isparabolically related to the CLA E_{7(-25)}, the parabolic subalgebrasincluding E_{6(6)} and E_{6(-6)} . Other interesting examples are theorthogonal algebras so(p,q) all of which are parabolically related to theconformal algebra so(n,2) with p+q=n+2, the parabolic subalgebras including theLorentz subalgebra so(n-1,1) and its analogs so(p-1,q-1). We consider alsoE_{6(6)} and E_{6(2)} which are parabolically related to the hermitiansymmetric case E_{6(-14)}, the parabolic subalgebras including real forms ofsl(6). We also give a formula for the number of representations in the mainmultiplets valid for CLAs and all algebras that are parabolically related tothem. In all considered cases we give the main multiplets of indecomposableelementary representations including the necessary data for all relevantinvariant differential operators. In the case of so(p,q) we give also thereduced multiplets. We should stress that the multiplets are given in the mosteconomic way in pairs of {\it shadow fields}. Furthermore we should stress thatthe classification of all invariant differential operators includes as specialcases all possible {\it conservation laws} and {\it conserved currents},unitary or not.