Abstract
AbstractIn 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 ofparabolic relationbetween two non-compact semisimple Lie algebrasGandG′that have the same complexification and possess maximal parabolic subalgebras with the same complexification. Thus, we consider the exceptional algebraE7(7)which is parabolically related to the CLAE7(−25), the parabolic subalgebras includingE6(6)andE6(−26). Other interesting examples are the orthogonal algebras so(p, q) all of which are parabolically related to the conformal algebraso(n, 2) withp+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 E6(6)and E6(2)which are parabolically related to the hermitian symmetric case E6(−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 ofso(p,q) we give also the reduced multiplets. We should stress that the multiplets are given in the most economic way in pairs ofshadow fields. Furthermore we should stress that the classification of all invariant differential operators includes as special cases all possibleconservation lawsandconserved currents, unitary or not.
Highlights
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 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
Summary
Let G be a semisimple non-compact Lie group, and K a maximal compact subgroup of G. Since our ERs may be induced from finite-dimensional representations of M the Verma modules are always reducible. It is more convenient to use generalized Verma modules V Λ such that the role of the highest/lowest weight vector v0 is taken by the (finite-dimensional) space Vμ v0. The data for each intertwining differential operator consists of the pair (β, m), where β is a (non-compact) positive root of GC, m ∈ N, such that the BGG Verma module reducibility condition (for highest weight modules) is fulfilled:. This embedding is realized by a singular vector vs expressed by a polynomial Pm,β(G−) in the universal enveloping algebra (U (G−)) v0 , G− is the subalgebra of GC generated by the negative root generators [169]. These parameters are redundant, since they are expressed in terms of the Dynkin labels, some statements are best formulated in their terms.
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