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

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.


Introduction
Invariant differential operators play very important role in the description of physical symmetries -starting from the early occurrences in the Maxwell, d'Allembert, Dirac, equations, (for more examples cf., e.g., [1]), to the latest applications of (super-)differential operators in conformal field theory, supergravity and string theory (for reviews, cf. e.g., [2], [3].
Finally, among our motivations are the mathematical developments -see the relevant mathematical references: , and others throughout the text.
Thus, it is important for the applications in physics to study systematically such operators. In a recent paper [72] we started the systematic explicit construction of invariant differential operators. We gave an explicit description of the building blocks, namely, the parabolic subgroups and subalgebras from which the necessary representations are induced. Thus we have set the stage for study of different non-compact groups.
Since the study and description of detailed classification should be done group by group we had to decide which groups to study first. A natural choice would be non-compact groups that have discrete series of representations. By the Harish-Chandra criterion [73] these are groups where holds: where K is the maximal compact subgroup of the non-compact group G. Another formulation is to say that the Lie algebra G of G has a compact Cartan subalgebra.
In the present paper we are mainly interested in non-compact Lie algebras (and groups) that are 'parabolically' related to the conformally Lie algebras.
Another interesting example are the algebras so * (2m) and so(p, q) which have a series of maximal parabolics with M-factors [72],: , whose complexifications coincide for p + q = 2m (1.9) As we know only for m = 2n, i.e., for so * (4n) is fulfilled relation (1.5), with M = M n = = su * (2n) from (1.8), (recalling that K ′ ∼ = su(2n)). Obviously, so(p, q) is parabolically related to so * (4n) with this M-factor only when p = q = 2n, i.e., G ′ = so(2n, 2n) with M ′ n = sl(2n, R) (which is outside the range of (1.9)). We leave the classification of the parabolic relations between the non-compact semisimple Lie algebras for a subsequent publication. In the present paper we consider mainly algebras parabolically related to conformal Lie algebras with maximal parabolics fulfilling (1.5). We summarize the relevant cases in the following table: Table of conformal Lie algebras (CLA) G with M-factor fulfilling (1.5) and the corresponding parabolically related algebras G ′ below not CLA ! E 6(−14) so(10) su(5, 1) E 6(6) sl(6, R) where we have included also the algebra E 6(−14) ; we display only the semisimple part K ′ of K; sl(n, C) R denotes sl(n, C) as a real Lie algebra, (thus, (sl(n, C) R ) C = sl(n, C) ⊕ sl(n, C)); e 6 denotes the compact real form of E 6 ; and we have imposed restrictions to avoid coincidences or degeneracies due to well known isomorphisms: -5 -After this extended introduction we give the outline of the paper. In Section 2 we give the preliminaries, actually recalling and adapting facts from [72]. We add a remark on conservation laws and conserved currents which are an integral part of our approach. In Section 3 we consider the case of the pseudo-orthogonal algebras so(p, q) which are parabolically related to the conformal algebra so(n, 2) for p + q = n + 2. We add historical remarks and a remark on shadow representations. In Section 4 we consider the algebras su * (4k) and sl(4k, R) as parabolically related to the CLA su(2k, 2k). In Section 5 we consider the algebra sp(r, r) as parabolically related to the CLA sp(2r) (of rank 2r). In Section 6 we consider the algebra E 7(7) as parabolically related to the CLA E 7(−25) . In Section 7 we consider the algebras E 6(6) and E 6(2) as parabolically related to the hermitian symmetric case E 6(−14) . In Section 8 we give Summary and Outlook.

Preliminaries
Let G be a semisimple non-compact Lie group, and K a maximal compact subgroup of G. Then we have an Iwasawa decomposition G = KA 0 N 0 , where A 0 is Abelian simply connected vector subgroup of G, N 0 is a nilpotent simply connected subgroup of G preserved by the action of A 0 . Further, let M 0 be the centralizer of A 0 in K. Then the subgroup Further, let G, K, P, M, A, N denote the Lie algebras of G, K, P, M, A, N , resp. For our purposes we need to restrict to maximal parabolic subgroups P = M AN , i.e. rank A = 1, resp. to maximal parabolic subalgebras P = M ⊕ A ⊕ N with dim A = 1.
Let ν be a (non-unitary) character of A, ν ∈ A * , parameterized by a real number d, called the conformal weight or energy.
Further, let µ fix a discrete series representation D µ of M on the Hilbert space V µ , or the finite-dimensional (non-unitary) representation of M with the same Casimirs.
We call the induced representation χ = Ind G P (µ ⊗ ν ⊗ 1) an elementary representation of G [16]. (These are called generalized principal series representations (or limits thereof) in [96].) Their spaces of functions are: The representation action is the left regular action: • An important ingredient in our considerations are the highest/lowest weight representations of G C . These can be realized as (factor-modules of) Verma modules V Λ over G C , where Λ ∈ (H C ) * , H C is a Cartan subalgebra of G C , weight Λ = Λ(χ) is determined uniquely from χ [97].
Actually, since our ERs may be induced from finite-dimensional representations of M the Verma modules are always reducible. Thus, it is more convenient to use generalized Verma modulesṼ Λ such that the role of the highest/lowest weight vector v 0 is taken by the (finitedimensional) space V µ v 0 . For the generalized Verma modules (GVMs) the reducibility is controlled only by the value of the conformal weight d. Relatedly, for the intertwining differential operators only the reducibility w.r.t. non-compact roots is essential.
• One main ingredient of our approach is as follows. We group the (reducible) ERs with the same Casimirs in sets called multiplets. The multiplet corresponding to fixed values of the Casimirs may be depicted as a connected graph, the vertices of which correspond to the reducible ERs and the lines (arrows) between the vertices correspond to intertwining operators. The explicit parametrization of the multiplets and of their ERs is important for understanding of the situation. The notion of multiplets was introduced in [98], [99] and applied to representations of SO o (p, q) and SU (2, 2), resp., induced from their minimal parabolic subalgebras. Then it was applied to the conformal superalgebra [100], to infinite-dimensional (super-)algebras [101], to quantum groups [102]. 2 Remark: Note that the multiplets of Verma modules include in general more members, since there enter Verma modules which are induced from infinite-dimensional representations of M but nevertheless have the same Casimirs. The main multiplets in this case contain as many members as the Weyl group W (G C ) of G C . For example, for su(2, 2) the maximal multiplets contain 24 members (|W (sl(ℓ, C))| = ℓ!), which were considered in [99] and the su(2, 2) sextets of ERs induced from the maximal parabolic with M = sl(2, C) are submerged in the 24-member multiplets.♦ In fact, the multiplets contain explicitly all the data necessary to construct the intertwining differential operators. Actually, the data for each intertwining differential operator consists of the pair (β, m), where β is a (non-compact) positive root of G C , m ∈ N, such that the BGG Verma module reducibility condition (for highest weight modules) is fulfilled: where ρ is half the sum of the positive roots of G C . When the above holds then the Verma module with shifted weight V Λ−mβ (orṼ Λ−mβ for GVM and β non-compact) is embedded in the Verma module V Λ (orṼ Λ ). This embedding is realized by a singular vector v s expressed by a polynomial P m,β (G − ) in the universal enveloping algebra (U (G − )) v 0 , G − is the subalgebra of G C generated by the negative root generators [104]. More explicitly, [97], v s m,β = P m,β v 0 (or v s m,β = P m,β V µ v 0 for GVMs). 3 Then there exists [97] an intertwining differential operator of order m = m β : given explicitly by: 2 For other applications we refer to [103]. 3 For explicit expressions for singular vectors we refer to [105].
where G − denotes the right action on the functions F .
Thus, in each such situation we have an invariant differential equation of order m = m β : In most of these situations the invariant operator D m,β has a non-trivial invariant kernel in which a subrepresentation of G is realized. Thus, studying the equations with trivial RHS: is also very important. For example, in many physical applications in the case of first order differential operators, i.e., for m = m β = 1, equations (2.7) are called conservation laws, and the elements f ∈ ker D m,β are called conserved currents.
The above construction works also for the subsingular vectors v ssv of Verma modules. Such a vector is also expressed by a polynomial P ssv (G − ) in the universal enveloping algebra: v s ssv = P ssv (G − ) v 0 , cf. [106]. Thus, there exists a conditionally invariant differential operator given explicitly by: D ssv = P ssv ( G − ), and a conditionally invariant differential equation, for many more details, see [106]. (Note that these operators/equations are not of first order.) Below in our exposition we shall use the so-called Dynkin labels: where Λ = Λ(χ), ρ is half the sum of the positive roots of G C .
We shall use also the so-called Harish-Chandra parameters: where β is any positive root of G C . These parameters are redundant, since they are expressed in terms of the Dynkin labels, however, some statements are best formulated in their terms. 4 3 The pseudo-orthogonal algebras so(p, q)
(3.1) (There are more choices when p = q.) We would like to consider a case, which would relate parabolically all G = so(p, q) for p + q -fixed. Thus, in order in order to include the case q = 1 (where there is only one parabolic which is both minimal and maximal), we choose the case j = 1 : Then we have: With this choice we get for the conformal algebra exactly the Bruhat decomposition in (1.3).
We label the signature of the ERs of G as follows: where the last entry of χ labels the characters of A , and the first h entries are labels of the finite-dimensional nonunitary irreps of M ∼ = so(p − 1, q − 1).
The reason to use the parameter c instead of d will become clear below.

Main multiplets
Following results of [107][108][109], [91] we present the main multiplets (which contain the maximal number of ERs with this parabolic) with the explicit parametrization of the ERs in the multiplets in a simple way (helped by the use of the signature entry c): . Clearly, the multiplets correspond 1-to-1 to the finitedimensional irreps of so(p + q, C) with signature {n 1 , . . . , n h , n h+1 } and we are able to use previous results due to the parabolic relation between the so(p, q) algebras for p + q -fixed.
Note that the two representations in each pair χ ± were called shadow fields in the 1970s, see more on this towards the end of this Section.
Further, the number of ERs in the corresponding multiplets is equal to 2[ p+q 2 ] = 2(h+1). This maximal number is equal to the following ratio of numbers of elements of Weyl groups: where H C , H C m are Cartan subalgebras of G C , M C , resp. The above formula actually holds for all conformal Lie algebras and those parabolically related to them. More precisely, we have: • The number of elements of the main multiplets of a conformal Lie algebra G with M-factor fulfilling (1.5) is given by (3.6). The same number holds for any algebra G ′ parabolically related to G w.r.t. M.♦ Further, we denote by C ± i the representation space with signature χ ± i . We first give the multiplets pictorially in Figures 1 and 2 for p + q even and odd, resp., and then explain notations and results: The ERs in the multiplet are related by intertwining integral and differential operators.
The integral operators were introduced by Knapp and Stein [110]. They correspond to elements of the restricted Weyl group of G. In fact, these operators are defined for any ER, not only for the reducible ones, the general action being in the context of (3.4), (3.5) : These operators intertwine the pairs C ± i (cf. (3.5)): In the conformal setting (both Euclidean q = 1 and Minkowskian q = 2) the integral kernel of the Knapp-Stein operator is given by the conformal two-point function [16].
The intertwining differential operators correspond to non-compact positive roots of the root system of so(p + q, C), cf. [97]. In the current context, compact roots of so(p + q, C) are those that are roots also of the subalgebra so(p + q − 2, C), the rest of the roots are non-compact. In more detail, we briefly recall the root systems: For p + q = 2h + 2 even, the positive root system of so(2h + 2, C) may be given by vectors The non-compact roots may be taken as For p + q = 2h + 3 odd, the positive root system of so(2h + 3, C) may be given by vectors ε i ± ε j , 1 ≤ i < j ≤ h + 1, ε k , 1 ≤ k ≤ h + 1. The non-compact roots may be taken as ε 1 ± ε i , ε 1 . The root ε 1 − ε i corresponds to the operator d i−1 , the root ε 1 + ε i corresponds to the operator d ′ i−1 . The root ε 1 has a special position since it intertwines the same ERs that are intertwined by the Knapp-Stein integral operator G + h+1 . The latter means that G + h+1 degenerates to the differential operator d h+1 , and this degenerations determines that d h+1 ∼ n 1 , (for n 1 ∈ N), where is the d'Alembert operator, as explained explicitly for the case so (3,2) in [111]. (The latter phenomenon happens for the Knapp-Stein integral operators at critical points, but usually there is no non-compact root involved, cf., e.g., [16].) The degrees of these intertwining differential operators are given just by the differences of the c entries [109]: where d ′ h is omitted from the first line for (p + q) even. By our construction they are equal to the Harish-Chandra parameters for the non-compact roots: Matters are arranged so that in every multiplet only the ER with signature χ − 1 contains a finite-dimensional nonunitary subrepresentation in a subspace E. The latter corresponds to the finite-dimensional unitary irrep of so(p + q) with signature {n 1 , . . . , n h , n h+1 }.
The subspace E is annihilated by the operator G + 1 , and is the image of the operator Although the diagrams are valid for arbitrary so(p, q) (p + q ≥ 5) the contents is very different. We comment only on the ER with signature χ + 1 . In all cases it contains an UIR of so(p, q) realized on an invariant subspace D of the ER χ + 1 . That subspace is annihilated by the operator G − 1 , and is the image of the operator G + 1 . (Other ERs contain more UIRs.) If pq ∈ 2N the mentioned UIR is a discrete series representation. Other ERs contain more discrete series UIRs. The number of discrete series is given by the formula [96]: where H C is a Cartan subalgebra of both G C and K C .
And if q = 2 the invariant subspace D is the direct sum of two subspaces D = D + ⊕D − , in which are realized a holomorphic discrete series representation and its conjugate antiholomorphic discrete series representation, resp. These are contained only in χ + 1 and count for two series in the formula (3.12). Furthermore, any holomorphic discrete series representation is infinitesimally equivalent to a lowest weight GVM of the conformal algebra so(p, 2), while an anti-holomorphic discrete series representation is infinitesimally equivalent to a highest weight GVM. Highest/lowest weight GVMs are related to other pairs besides χ + 1 . A detailed analysis of these occurrences is done for the conformal algebra so (3,2) in [91] and for so (4,2) in [108], [91].

Reduced multiplets
Besides the main multiplets which are 1-to-1 with the finite-dimensional irreps of so(p + q, C), there are other multiplets which we describe here.
• We start with the case p + q even. In this case there are h + 1(= (p + q)/2) multiplets Clearly, the signatureχ ± i may be obtained from χ ± i by setting the corresponding Harish-Chandra parameter equal to zero: Although written compactly as (3.5) no pair is related to any other pair. This may be seen easily as follows. Consider (3.5) and set formally n h+1 = n h . The signatures χ ± 1 and χ ± 2 coincide are become equal toχ ± 1 , but the rest of the signatures χ ± i , i ≥ 3 are not allowed in our class, e.g., is not allowed since it violates (3.4) due to equality of two M-signature entries (n h ). Thus, from the whole multiplet only the pairχ ± 1 remains in our class. Similarly for the rest of the pairs. Inside a fixed pairχ ± i , i = 1, . . . , h + 1, act two operators: a Knapp-Stein integral operator fromχ + i toχ − i , and a differential operator fromχ − i toχ + i . In more detail: • Let first i = 1, . . . , h − 1. Inside a fixed pairχ ± i , acts the Knapp-Stein integral operator G − i (3.8) (coinciding with G − i+1 for this signature), and a differential operatord i of degree 2n h+1−i which is a degeneration of the Knapp-Stein integral operator G + i (coinciding with G + i+1 for this signature). For this differential operator for n 1 = 0 we have:d i ∼ n h+1−i , (n h+1−i ∈ N). 6 • Inside the fixed pairχ ± h acts the Knapp-Stein integral operator G − h (3.8) (coinciding with G − h+1 for this signature), and the differential operator d ′ h of degree 2n 1 (cf. the previous subsection) which in addition is a degeneration of the Knapp-Stein integral operator G + h (coinciding with G + h+1 for this signature). • Inside the fixed pairχ ± h+1 acts the Knapp-Stein integral operator G − h+1 (3.8) (coinciding with G + h for this signature), and the differential operator d h of degree 2n 2 which in addition is a degeneration of the Knapp-Stein integral operator G + h+1 (coinciding with G − h for this signature).
• We continue with the case p + q odd. In this case there are h doublets 7 with signatureŝ χ ± given similarly to the even case as follows: The signatureχ ± i may be obtained from χ ± i by setting the corresponding Harish-Chandra parameter equal to zero: Inside a fixed pairχ ± i , i = 1, . . . , h, acts the Knapp-Stein integral operator G − i (3.8) (coinciding with G − i+1 for this signature), and a differential operatord i of degree 2n h+1−i which is a degeneration of the Knapp-Stein integral operator G + i (coinciding with G + i+1 for this signature). For the differential operators we haved i ∼ n h+1−i , (when n h+1−i ∈ N). The difference with the even situation is only for i = h, where the degeneration of G + h+1 was present already in the main multiplet. If pq ∈ 2N the representationsχ + 1 ,χ + 1 , contain an UIR called limits of the discrete series representations. And if q = 2 that UIR is the direct sum of two subspaces in which are realized limits of holomorphic discrete series representation and its conjugate limits of anti-holomorphic discrete series representation, resp. The latter do not happen in any other doublet, while limits of discrete series representations happen in other doublets. (For more on this see [91] for so (3,2) and [108], [91] for so(4, 2).) . This was used already in [16] since the multiplets were given more economically in terms of pairs of ERs in which the parameter c just changes sign. (Also mathematicians use the parameter c due to the fact that in its terms the representation parameter space looks simple: the principal unitary series representation induced from a maximal parabolic is given by c = iρ, ρ ∈ R; the supplementary series of unitary representations is given by −s < c < s, s ∈ R, etc.)

Remarks on shadow fields and history
Otherwise in the current context we should use for each Knapp-Stein operators conjugated doublet of shadow fields : The reason the representations χ ± in the 1970s were called "shadow fields" in the context of the conformal algebra so(n, 2) is that the sum of their conformal weights equals the dimension n of Minkowski space-time -isomorphic to N orÑ , cf. (3.3). This continues to be true for general so(p, q) : and also for all conformal Lie algebras considered in the next Sections.
Shadow fields appear all the time in conformal field theory. For example, in [112] we showed that in the generic case each field on the AdS bulk has two boundary fields which are shadow fields being related by a integral Knapp-Stein operator. Later Klebanov-Witten [113] showed that these two boundary fields are related by a Legendre transform.
For a current discussion on shadow fields we refer to [114].
• The diagram for p + q even appeared first for the Euclidean conformal group in fourdimensional space-time SU * (4) ∼ = Spin(5, 1) in [107]. Later it was generalised to the Minkowskian conformal group in four-dimensional space-time SO(4, 2) in [108]. In both cases, the three (= (p + q)/2) doublets (from the previous subsection) were also given together the corresponding degeneration of the Knapp-Stein integral operators.
The exposition above including Figures 1 & 2 follows the exposition for Euclidean case so(n + 1, 1) in [109]. Later the results were generalised to the Minkowskian case so(n, 2) [91].
• Actually, the case of Euclidean conformal group in arbitrary dimensions SO(p, 1) was studied in [16] for representations of M = so(p − 1) which are symmetric traceless tensors. This means in (3.4) we should set n 1 = n 2 = · · · = n h−1 = 0, and then only the first two pairs χ ± 1 , χ ± 2 in (3.5) are possible. Thus from the two figures only the upper quadrants are relevant, and were given in [16], cf. Fig. 1 there.
In the case so(3, 1) ∼ = sl(2, C) the multiplet in general contains only four ERs, and is in fact representable by the diagram in the case of symmetric traceless tensors of so(p, 1), p > 3, cf. [16], Appendix B.
The case so(2, 1) ∼ = sl(2, R) is special and must be treated separately. But in fact, it is contained in what we presented already. In that case the multiplets contain only two ERs -17 -which may be depicted by the top pair χ ± 1 in both Figures. (Formally, set h = 0 in both Figures.) They have the properties that we described, including the (anti)holomorphic discrete series which are present in this case. That case was the first given already in 1946-7 independently by Gel'fand et al [115] and Bargmann [116]. 4 The Lie algebras su * (2n) and sl(n, R)
The algebra G = su * (2n) has n − 1 maximal parabolic subalgebras with M-factors (cf. (5.8) from [72]): with complexification: We would like to relate parabolically this algebra with the appropriate conformal Lie algebra, namely, with su(n, n). It was considered in [92] with M-factor: M ′ = sl(n, C) R which has complexification: M ′C = sl(n, C) ⊕ sl(n, C) . Clearly, the latter expression can match (4.2) only if n = 2k, i.e., n must be even.
Thus, we set n = 2k and consider:

Case sl(n, R)
Let sl(n, R). Its maximal compact subalgebra is K = so(n), and thus it does not have discrete series representations. It has [ n 2 ] maximal parabolic subalgebras obtained by deleting a node from its standard Dynkin diagram and taking into account the symmetry (cf. [72]):
The signature of the ERs of both G may be denoted as: same as for su(2k, 2k).
In our diagrams we need also the Harish-Chandra parameters for the non-compact roots using the following notation: The number of ERs in the corresponding multiplets is according to (3.6): (4.11) (which was given for su(n, n) in [92]).
Below we give the diagrams for the cases k = 1, 2. Of course, the case k = 1 is known long time ago, first as su * (4) ∼ = so(5, 1), cf. [107], then as su(2, 2) ∼ = so(4, 2), cf. [108], and also as sl(4, R) ∼ = so(3, 3), as we recalled already in the previous section on so(p, q) algebras. We present it here using a new diagram look which can handle the more complicated cases that follow further. In this new look only the invariant differential operators are presented explicitly. The integral Knapp-Stein operators, more precisely the restricted Weyl reflection action is understood by a symmetry of the picture, either w.r.t. a central point, or w.r.t. middle line.
Thus, in Figure 3 we give the case k = 1, where the Knapp-Stein symmetry is w.r.t. to the bullet in the middle of the figure. Then in Figure 4 we give the diagram Figure 1 for the special case h = 2, as given originally for so (5,1) in [107], and so(4, 2) in [108], stressing that both Figures 3 and 4 have the same content.
Next we give the case k = 2, in Figure 5, which applies to su * (8), sl(8, R) and su (4,4). (For reduced multiplets we refer to [92].) The diagram is very complicated and just to be able to depict all the relevant information we must use the following condensing conventions. Each intertwining differential operator is represented by an arrow accompanied by a symbol i j...ℓ encoding the root β j...ℓ and the number m β j...ℓ which is involved in the BGG criterion. This notation is used to save space, but it can be used due to the fact that only intertwining differential operators which are non-composite are displayed, and that the data β, m β , which is involved in the embedding V Λ −→ V Λ−m β ,β turns out to involve only the m i corresponding to simple roots, i.e., for each β, m β there exists i = i(β, m β , Λ) ∈ {1, . . . , r}, (r = rank G), such that m β = m i . Hence the data β j...ℓ ,m β j...ℓ is represented by i j...ℓ on the arrows. Fig. 3. Main multiplets for su * (4) ∼ = so(5, 1), su(2, 2) ∼ = so(4, 2) and sl(4, R) ∼ = so (3,3) with parabolic factor M C = sl(2, C) ⊕ sl(2, C).

The Lie algebras sp(p, r)
Let G = sp(p, r), p ≥ r. It has maximal compact subalgebra K = sp(p) ⊕ sp(r) and has discrete series representations (as rank K = p + r = rank G). It has r maximal parabolic subalgebras with M-factors (cf. (9.8) from [72]): with complexification: We would like to match this algebra with the appropriate conformal Lie algebra, namely, with sp(n, R). It was considered in [93] with M-factor: M ′ = sl(n, R) with complexification M ′C = sl(n, C). Obviously, the latter can match (5.2) only if n is even and p = r = j = n/2. Thus, we shall consider The signature of the ERs of G is: The Knapp-Stein restricted Weyl reflection acts as follows: In terms of an orthonormal basis ε i , i = 1, . . . , n, the positive roots of sp(2r, C) are: the simple roots are: the positive non-compact roots are: the Harish-Chandra parameters: m β ≡ (Λ + ρ, β) for the noncompact roots are: The correspondence between the signatures χ and the highest weight Λ is: n i = m i , c = − 1 2 (mα + m 2r ) = − 1 2 (m 1 + · · · + m 2r−1 + 2m 2r ) (5.10) whereα = β 11 is the highest root.
The number of ERs in the corresponding multiplets is according to (3.6): (which was given for sp(n, R) in [93]).
Below we give pictorially the multiplets for sp(r, r) for r = 1, 2, valid also for sp(2r, R). (The case r = 3, together with the reduced multiplets and sp(5, R are given in [93].) In fact, the case r = 1 is known long time as sp(1, 1) ∼ = so(4, 1), cf. [16], then later as sp(2, R) ∼ = so(3, 2), cf. [111], as we recalled already in the previous section on so(p, q) algebras. We present it here using the new diagram look which we already used in the previous Section. Thus, in Figure 6 we give the case r = 1, where the Knapp-Stein symmetry is w.r.t. to the bullet in the middle of the figure. Thus, it is seen that the action of the differential operator indexed by 1 12 is the same as the Knapp-Stein operator from Λ ′− to Λ ′+ , so that the latter operator degenerates as discussed in Section 1. Then in Figure 7 we give the diagram Figure 2 for the special case h = 1, stressing that both Figures 6 and 7 have the same content.
It has the following Dynkin-Satake diagram (same as for E 7 ) [118]: The real algebra E 7(7) has seven maximal parabolics which are obtained by deleting one node as explained in [72]. We choose the one which is most suitable w.r.t. the maximal compact subgroup K = su(8), as will become clear below. This parabolic is obtained by deleting the root α 7 from the Dynkin-Satake diagram (6.1), i.e., we shall use as M-factor E 6(6) (the split real form of E 6 ).
We label the signature of the ERs of G as follows: where the last entry of χ labels the characters of A , and the first 6 entries are labels of the finite-dimensional nonunitary irreps of M , (or of the finite-dimensional unitary irreps of the compact e 6 ).
Further, we need the root system of the complex algebra E 7 . With Dynkin diagram enumerating the simple roots α i as in (6.1), the positive roots are: first there are 21 roots forming the positive root system of sl(7) (with simple roots α 1 , α 3 , α 4 , α 5 , α 6 , α 7 ), then 21 positive roots which are positive roots of the E 6 subalgebra including the non-sl(7) root α 2 , and finally the following 21 roots including the -28 -non-E 6 root α 7 : α 2 + α 4 + α 5 + α 6 + α 7 , α 2 + α 3 + α 4 + α 5 + α 6 + α 7 , (6.4) The differential intertwining operators that give the multiplets correspond to the noncompact roots, and since we shall use the latter extensively, we introduce more compact notation for them. Namely, the non-simple roots will be denoted in a self-explanatory way as follows: i.e., the non-compact roots will be written as: where the first six roots in (6.6a) are from the sl(7) subalgebra, and the 21 in (6.6b) are those from (6.4).
Since the algebras E 7(7) and E 7(−25) are parabolically related they have the same signatures, and thus the same main multiplets.
The number of ERs in the corresponding main multiplets is according to (3.6): (which was given for E 7(−25) in [94]).
Below we give the main multiplets valid for both algebras in Figure 9. For reduced multiplets cf. [94]. 46 1 1,37 6 27,45 3 17           Fig. 9. Main type of multiplets for E 7(7) and E 7(−25) with parabolic factor M C = E 6 7 Two real forms of E 6 7.1 The Lie algebra E 6 (6) Let G = E 6(6) . This is the split real form of E 6 denoted also as E ′ 6 or E I . The maximal compact subgroup is K ∼ = sp(4). This real form does not have discrete series representations (as rank G = rank K).
We use the following Dynkin-Satake diagram (same as for E 6 ): The real algebra E 6(6) has four maximal parabolics which are obtained by deleting one node as explained in [72]. (Note that deleting node 1 or node 6 produces the same parabolic, same for deleting node 3 or node 5.) We choose the parabolic obtained by deleting node 2.

The Lie algebra E 6(2)
Let G = E 6(2) . This is another real form of E 6 sometimes denoted as E ′′ 6 , or E II . The maximal compact subalgebra is K ∼ = su(6) ⊕ su (2). This real form has discrete series representations.
The Satake diagram is: The real algebra E 6(2) has four maximal parabolics which are obtained by deleting one node as explained in [72] (taking into account E 6 symmetry as in the previous case). We choose the parabolic obtained by deleting node 2.

Representations and multiplets
We note that the M-factors of the two real forms of E 6 discussed in the previous subsections have the same complexification: sl(6, R) C = su(3, 3) C = sl(6, C) -32 -i.e., they are parabolically related and we can discuss them together.
Relative to our parabolic subalgebra, the roots in (7.5) are non-compact, while the rest are compact. As before we introduce more condensed notation for the noncompact roots: Now we should connect our considerations with the case of another real form of E 6 , namely, the Lie algebra E 6(−14) , cf. [95]. In that paper we chose as maximal parabolic P ′ = M ′ ⊕ A ′ ⊕ N ′ , where M ′ ∼ = su(5, 1), dim R N = 21, cf. (11.21) of [72].
Since both the algebras and the maximal parabolics have the same complexification, this means that they are parabolically related, thus, we have the same non-compact roots, the same signatures, and the same multiplets. We show only the main multiplet in Figure  10, referring to [95] for the diagrams of reduced multiplets. The main multiplet has 70 -33 -members and the figure has the standard E 6 symmetry, namely, conjugation exchanging indices 1 ←→ 6, 3 ←→ 5. The Knapp-Stein operators act pictorially as reflection w.r.t. the dotted line separating the H − ... members from the H + ... members. Note that there are five cases when the embeddings correspond to the highest rootα : V Λ − −→ V Λ + , Λ + = Λ − − mαα . In these five cases the weights are denoted as: Λ ± k ′′ , Λ ± k ′ , Λ ± k , Λ ± k , Λ ± k o , then: mα = m 1 , m 3 , m 4 , m 5 , m 6 , resp. We recall that Knapp-Stein operators G + intertwine the corresponding ERs T − χ and T + χ . In the above five cases the Knapp-Stein operators G + degenerate to differential operators as we discussed earlier.
- 34 -In the present paper we continued the project of systematic construction of invariant differential operators for non-compact semisimple Lie groups. Our aim in this paper was to extend our considerations beyond the class of algebras, which we call 'conformal Lie algebras' (CLA). 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. Thus, we considered the algebras so(p, q) all of which are parabolically related to the conformal algebra so(n, 2) with p + q = n + 2, then the algebras su * (4k) and sl(4k, R) parabolically related to the CLA su(2k, 2k), then sp(r, r) as parabolically related to the CLA sp(2r) (of rank 2r), then the exceptional Lie algebra E 7(7) which is parabolically related to the CLA E 7(−25) , finally the exceptional Lie algebras E 6(6) and E 6(2) parabolically related to the hermitian symmetric case E 6(−14) .
We have given 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 have given 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 have given also the reduced multiplets. We note that the multiplets are given in the most economic way in pairs of shadow fields related by the Knapp-Stein restricted Weyl symmetry (and the corresponding integral operators).
Finally, we should stress that the classification of all invariant differential operators includes as special cases all possible conservation laws and conserved currents, unitary or not.
We plan also to extend these considerations to the supersymmetric cases and also to the quantum group setting. Such considerations are expected to be very useful for applications to string theory and integrable models. It is interesting to note that almost all of the algebras that appear in Table 1 of [39] are treated in the present paper, though our motivations and approach are different (see also [119]).