Available at: http://www.ictp.it/~pub− off IC/2007/015
United Nations Educational, Scientific and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS
INVARIANT DIFFERENTIAL OPERATORS FOR NON-COMPACT LIE GROUPS: PARABOLIC SUBALGEBRAS
V.K. Dobrev Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 72 Tsarigradsko Chaussee, 1784 Sofia, Bulgaria and The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.
Abstract
In the present paper we start the systematic explicit construction of invariant differen- tial operators by giving explicit description of one of the main ingredients - the cuspidal parabolic subalgebras. We explicate also the maximal parabolic subalgebras, since these are also important even when they are not cuspidal. Our approach is easily generalised to the supersymmetric and quantum group settings and is necessary for applications to string theory and integrable models.
MIRAMARE – TRIESTE March 2007
1 1. Introduction Invariant differential operators play very important role in the description of physical sym- metries - 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 a recent review, cf. e.g., [2]). Thus, it is important for the applications in physics to study systematically such operators. In the present paper we start with the classical situation, with the representation theory of semisimple Lie groups, where there are lots of results by both mathematicians and physi- cists, cf., e.g. [3-38]. We shall follow a procedure in representation theory in which such operators appear canonically [22] and which has been generalized to the supersymmetry setting [39] and to quantum groups [40]. We should also mention that this setting is most appropriate for the classification of unitary representations of superconformal symmetry in various dimensions, [41],[42],[43],[44], for generalization to the infinite-dimensional set- ting [45], and is also an ingredient in the AdS/CFT correspondence, cf. [46]. (For a recent paper with more references cf. [47].) Although the scheme was developed some time ago there is still missing explicit descrip- tion of the building blocks, namely, the parabolic subgroups and subalgebras from which the representations are induced. Just in passing, we shall mention that parabolic subalgebras found applications in quan- tum groups, (in particular, for the quantum deformations of noncompact Lie algebras), cf. e.g., [40,48,49,50,51,52], and in integrable systems, cf. e.g., [53,54,55,56]. In the present paper the focus will be on the role of parabolic subgroups and subalgebras in representation theory. In the next section we recall the procedure of [22] and the prelim- inaries on parabolic subalgebras. Then, in Sections 3-11 we give the explicit classification of the cuspidal parabolic subalgebras which are the relevant ones for our purposes. The cuspidal parabolic subalgebras are also summarized in table form in an Appendix.
2. Preliminaries
2.1. General setting Let G be a noncompact semisimple Lie group. Let K denote the maximal compact sub- group of G. Then we have the Iwasawa decomposition G = KAN, where A is abelian simply connected, the so-called vector subgroup of G, N is a nilpotent simply connected subgroup of G preserved by the action of A. Further, let M be the centralizer of A in
K. Then the subgroup P0 = MAN is called the minimal parabolic subgroup of G. A ′ ′ ′ parabolic subgroup P = M A N is any subgroup of G which contains P0 . The number of
2 r non-conjugate parabolic subgroups (counting also P0 and G)is 2 , where r = rank A, cf., ′ ′ e.g., [6]. Note that in general M is a reductive Lie group with structure: M = MdMsMa, where Md is a finite group, Ms is a semisimple Lie group, Ma is an abelian Lie group central in M ′. The importance of the parabolic subgroups stems from the fact that the representa- tions induced from them generate all (admissible) irreducible representations of G [7]. In fact, it is enough to use only the so-called cuspidal parabolic subgroups P = M ′A′N ′, singled out by the condition that rank M ′ = rank M ′ K; thus M ′ has discrete series ∩ representations [8],[12].1 However, often induction from a non-cuspidal parabolic is also convenient. Let P = M ′A′N ′ be a parabolic subgroup. Let ν be a (non-unitary) character of A′, ν ′∗. If P is cuspidal, let µ fix a discrete series representation Dµ of M ′ on the ∈A Hilbert space Vµ , or the so-called limit of a discrete series representation (cf. [21]). If P is non-cuspidal µ denotes a finite-dimensional irreducible representation Dµ of M ′ on 2 the linear space Vµ . We call the induced representation χ = IndG(µ ν 1) an elementary representation P ⊗ ⊗ of G [13]. (These are called generalized principal series representations (or limits thereof) in [21].) Their spaces of functions are:
= C∞(G, V ) (gman) = eν(H) Dµ(m−1) (g) (2.1) Cχ {F ∈ µ | F · F } where a = exp(H), H . The special property of the functions of is called right ∈ A Cχ covariance [13],[22] (or equivariance).3 Because of this covariance the functions actually F do not depend on the elements of the parabolic subgroup P = M ′A′N ′. The elementary representation (ER) χ acts in as the left regular representation T Cχ (LRR) by: ( χ(g) )(g′) = (g−1g′) , g,g′ G . (2.2) T F F ∈
1 ′ The simplest example of cuspidal parabolic subgroup is P0 when M = M is compact. In all other cases M ′ is non-compact. 2 In general, µ is a actually a triple (ǫ, σ, δ), where ǫ is the signature of the character of Md , σ
gives the unitary character of Ma , δ fixes a discrete or finite-dimensional irrep of Ms on Vµ (the latter depends only on δ). 3 It is well known that when Vµ is finite-dimensional Cχ can be thought of as the space of smooth sections of the homogeneous vector bundle (called also vector G-bundle) with base space
G/P and fibre Vµ , (which is an associated bundle to the principal P -bundle with total space G). We shall not need this description for our purposes.
3 One can introduce in a Fr´echet space topology or complete it to a Hilbert space Cχ (cf. [6]). We shall need also the infinitesimal version of LRR:
. d (X )(g) = (exp( tX)g) , (2.3) LF dtF − |t=0
where, X CI , , g G. ∈ G F ∈ Cχ ∈ The ERs differ from the LRR (which is highly reducible) by the specific representation spaces . In contrast, the ERs are generically irreducible. The reducible ERs form a Cχ measure zero set in the space of the representation parameters µ, ν. (Reducibility here is topological in the sense that there exist nontrivial (closed) invariant subspace.) The irreducible components of the ERs (including the irreducible ERs) are called subrepresen- tations. The other feature of the ERs which makes them important for our considerations is a highest (or lowest) weight module structure associated with them [22]. For this we shall use the right action of CI (the complexification of ) by the standard formula: G G . d (Xˆ )(g) = (g exp(tX)) , (2.4) F dtF |t=0
where, X CI , , g G, which is defined first for X and then is extended to ∈ G F ∈ Cχ ∈ ∈ G CI by linearity. Note that this action takes out of for some X but that is exactly G F Cχ why it is used for the construction of the intertwining differential operators. We illustrate this special property in the case of the minimal parabolic subalgebra and in all cases when Vµ is finite dimensional. Then it has a highest weight vector v0. Using this we introduce CI-valued realization ˜ χ of the space by the formula: T Cχ
ϕ(g) v , (g) , (2.5) ≡ h 0 F i
where , is the M-invariant scalar product in V . On these functions the right action of h i µ CI is defined by: G (Xϕˆ )(g) v , (Xˆ )(g) . (2.6) ≡ h 0 F i Part of the main result of our paper [22] is: Proposition. The functions of the CI-valued realization ˜ χ of the ER satisfy : T Cχ
Xϕˆ = Λ(X) ϕ, X CI , Λ ( CI )∗ (2.7a) · ∈ H ∈ H Xϕˆ = 0 ,X CI , (2.7b) ∈ G+ 4 where Λ = Λ(χ) is built canonically from χ ,4 CI , CI , are the positive, negative root G+ G− spaces of CI , i.e., we use the standard triangular decomposition CI = CI CI CI . 5 G G G+ ⊕ H ⊕ G− Note that conditions (2.7) are the defining conditions for the highest weight vector of a highest weight module (HWM) over CI with highest weight Λ. Of course, it is enough to G + impose (2.7b) for the simple root vectors Xj . Furthermore, special properties of a class of highest weight modules, namely, Verma modules, are immediately related with the construction of invariant differential operators. To be more specific let us recall that a Verma module is a highest weight module V Λ with highest weight Λ, induced from one-dimensional representations of the Borel subalgebra = CI +. Thus, V Λ = U( CI )v , where v is the highest weight vector, U( CI ) is the B H ⊕ G ∼ G− 0 0 G− universal enveloping algebra of CI . 6 Verma modules are universal in the following sense: G− every irreducible HWM is isomorphic to a factor-module of the Verma module with the same highest weight. Generically, Verma modules are irreducible, however, we shall be mostly interested in the reducible ones since these are relevant for the construction of differential equations. We recall the Bernstein-Gel’fand-Gel’fand [5] criterion (for semisimple Lie algebras) according to which the Verma module V Λ is reducible iff
2 Λ + ρ, β m β, β = 0 , (2.8) h i − h i holds for some β ∆+, m IN, where ∆+ denotes the positive roots of the root system ∈ ∈ ( CI , CI ), ρ is half the sum of the positive roots ∆+. G H Λ Whenever (2.8) is fulfilled there exists [11] in V a unique vector vs, called singular vector, such that v = v and it has the properties (2.7) of a highest weight vector with s 6 0 shifted weight Λ mβ : − Xvˆ = (Λ mβ)(X) v ,X CI , (2.9a) s − · s ∈ H Xvˆ = 0 ,X CI , (2.9b) s ∈ G+ The general structure of a singular vector is [22] :
− − vs = Pmβ(X1 ,...,Xℓ )v0 , (2.10)
4 It contains all the information from χ, except about the character ǫ of the finite group Md. In the case of G being a complex Lie group we need two weights to encode χ, as explained in Section 3. 5 Note that we are working here with highest weight modules instead of the lowest weight modules used in [22]; also the weights are shifted by ρ with respect to the notation of [22]. 6 For more mathematically precise definition, cf. [11].
5 where P is a homogeneous polynomial in its variables of degrees mk , where k ZZ mβ i i ∈ + come from the decomposition of β into simple roots: β = k α , α ∆ , the system i i i ∈ S of simple roots, X− are the root vectors corresponding to the negative roots ( α ), α j P − j j being the simple roots, ℓ = rank CI = dim CI is the (complex) rank of CI . 7 CI G CI H G It is obvious that (2.10) satisfies (2.9a), while conditions (2.9b) fix the coefficients of
Pmβ up to an overall multiplicative nonzero constant. Now we are in a position to define the differential intertwining operators for semisimple Lie groups, corresponding to the singular vectors. Let the signature χ of an ER be such that the corresponding Λ = Λ(χ) satisfies (2.8) for some β ∆+ and some m IN.8 Then there exists an intertwining differential ∈ ∈ operator [22] ′ : ˜ χ ˜ χ , (2.11) Dmβ T −→ T where χ′ is such that Λ′ = Λ′(χ′) = Λ mβ. − The important fact is that (2.11) is explicitly given by [22] :
ϕ(g) = P (Xˆ −,..., Xˆ −) ϕ(g) , (2.12) Dmβ mβ 1 ℓ ˆ − where Pmβ is the same polynomial as in (2.10) and Xj denotes the action (2.4). One important simplification is that in order to check the intertwining properties of the operator in (2.12) it is enough to work with the infinitesimal versions of (2.1) and (2.2), i.e., work with representations of the Lie algebra. This is important for using the same approach to superalgebras and quantum groups, and to any other (infinite-dimensional) (super-)algebra with triangular decomposition.
2.2. Generalities on parabolic subalgebras Let G be a real linear connected semisimple Lie group.9 Let be the Lie algebra of G, G θ be the Cartan involution in , and = be the Cartan decomposition of , so G G K ⊕ P G that θX = X,X , θX = X,X ; is the maximal compact subalgebra of . ∈ K − ∈ P K G Let be the maximal subspace of which is an abelian subalgebra of ; r = dim is A P G A the split (or real) rank of , 1 r ℓ = rank . The subalgebra is called a Cartan G ≤ ≤ G A subspace of . P 7 CI A singular vector may also be written in terms of the full Cartan-Weyl basis of G−. 8 If β is a real root, (i.e., β| IC = 0, where Hm is the Cartan subalgebra of M), then some Hm conditions are imposed on the character ǫ representing the finite group Md [15]. 9 The results are easily extended to real linear reductive Lie groups with a finite number of components.
6 Let ∆A be the root system of the pair ( , ): G A . ∗ λ λ . ∆A = λ λ =0, =0 , = X [Y,X] = λ(Y )X , Y . (2.13) { ∈A | 6 GA 6 } GA { ∈G| ∀ ∈ A}
10 λ The elements of ∆A are called -restricted roots. For λ ∆A , are called - A ∈ GA A restricted root spaces, dim λ 1. Next we introduce some ordering (e.g., the lexico- R GA ≥ graphic one) in ∆A . Accordingly the latter is split into positive and negative restricted + − roots: ∆A = ∆ ∆ . Now we can introduce the corresponding nilpotent subalgebras: A ∪ A ± . λ = A . (2.14) N ⊕ ± G λ∈∆A
With this data we can introduce the Iwasawa decomposition of : G = , = ± . (2.15) G K⊕A⊕N N N . Next let be the centralizer of in , i.e., = X [X,Y ]=0, Y . In M A K M { ∈K| ∀ ∈ A} general is a compact reductive Lie algebra, and we can write = s a , where . M M M ⊕M =[ , ] is the semisimple part of , and is the abelian subalgebra central in Ms M M M Ma . M We mention also that a Cartan subalgebra of is given by: = , Hm M Hm Hs ⊕Ma where is a Cartan subalgebra of . Then a Cartan subalgebra of is given by: Hs Ms H G = .11 H Hm ⊕A Next we recall the Bruhat decomposition [57]:
= + − , (2.16) G N ⊕M⊕A⊕N . and the subalgebra = − called a minimal parabolic subalgebra of . (Note P0 M⊕A⊕N G that we may take equivalently + instead of − .) N N ± Naturally, the -subalgebras , , , , s, a, 0 are the Lie algebras of the G K A N M M M P ± G-subgroups introduced in the previous subsection K,A,N ,M,Ms,Ma,P0 , resp. We mention an important class of real Lie algebras, the split real forms. For these we can use the same basis as for the corresponding complex simple Lie algebra CI , but G over IR. Restricting CI IR one obtains the Bruhat decomposition of (with = 0) −→ G M 10 The terminology comes from the fact that things may be arranged so that these roots are obtained as restriction to A of the noncompact roots of the root system ∆ of the pair (GCI , HCI ). Noncompact (compact) roots of ∆ are those that are non-zero (zero) when restricted to A. 11 It is the most noncompact among the non-conjugate Cartan subalgebras of G.
7 from the triangular decomposition of CI = + CI −, and obtains the minimal G G ⊕ H ⊕ G parabolic subalgebras from the Borel subalgebra = CI +, (or − instead of +). P0 B H ⊕ G G G Furthermore, in this case dim = dim ±. IR K IR N A standard parabolic subalgebra is any subalgebra ′ of containing . The number P G P0 of standard parabolic subalgebras, including and , is 2r. P0 G Remark: In the complex case a standard parabolic subalgebra is any subalgebra ′ of P CI containing . The number of standard parabolic subalgebras, including and CI , is G B B G 2ℓ, ℓ = rank . CI G ♦ Thus, if r = 1 the only nontrivial parabolic subalgebra is . P0 Thus, further in this section r > 1. Any standard parabolic subalgebra is of the form:
′ = ′ ′ ′− , (2.17) P M ⊕A ⊕N so that ′ , ′ , ′− − ; ′ is the centralizer of ′ in (mod ′) ; ′− M ⊇ M A ⊆ A N ⊆ N M A G A is comprised from the negative root spaces of the restricted root system ∆A′ of N ( , ′). The decomposition (2.17) is called the Langlands decomposition of ′ . One also G A P has the analogue of the Bruhat decomposition (2.16):
= ′+ ′ ′ ′− , (2.18) G N ⊕A ⊕M ⊕N where ′+ = θ ′−. N N The standard parabolic subalgebras may be described explicitly using the restricted simple root system ∆S = ∆+ ∆− , such that if λ ∆+, (resp. λ ∆−), one has: A A ∪ A ∈ A ∈ A r λ = n λ , λ ∆S , all n 0, (resp. all n 0) . (2.19) i i i ∈ A i ≥ i ≤ i=1 X We shall follow Warner [6], where one may find all references to the original mathematical work on parabolic subalgebras. For a short formulation one may say that the parabolic S r subalgebras correspond to the various subsets of ∆A - hence their number 2 . To formalize this let us denote: S = 1, 2,...,r , and let Θ denote any subset of S . Let ∆± r { } r Θ ∈ ∆A denote all positive/negative restricted roots which are linear combinations of the simple restricted roots λ , i Θ. Then a standard parabolic subalgebra corresponding i ∀ ∈ to Θ will be denoted by and is given explicitly as: PΘ
+ + . λ Θ = 0 (Θ) , (Θ) = A . (2.20) P P ⊕ N N ⊕ + G λ∈∆Θ
8 Clearly, = , = , since +( ) = 0, +(S ) = + . Further, P∅ P0 PSr G N ∅ N r N we need to bring (2.20) in the form (2.17). First, define (Θ) as the algebra generated . G. by +(Θ) and −(Θ) = θ +(Θ) . Next, define (Θ) = (Θ) , and as the N N N A G ∩A AΘ orthogonal complement (relative to the Euclidean structure of ) of (Θ) in . Then A A A = (Θ) . Note that dim (Θ) = Θ , dim = r Θ . Next, define: A A ⊕AΘ A | | AΘ −| | + . λ − . + Θ = A , Θ = θ Θ . (2.21) N +⊕ + G N N λ∈∆A−∆Θ . Then ± = ±(Θ) ± . Next, define = (Θ) +(Θ) −(Θ). Then N N ⊕NΘ MΘ M⊕A ⊕N ⊕N is the centralizer of in (mod ). Finally, we can derive: MΘ AΘ G AΘ = +(Θ) = − +(Θ) = PΘ P0 ⊕N M⊕A⊕N ⊕N − − + = (Θ) Θ (Θ) Θ (Θ) = M⊕A ⊕A ⊕N ⊕N ⊕N (2.22) = (Θ) −(Θ) +(Θ) − = M⊕A ⊕N ⊕N ⊕AΘ ⊕NΘ = − . MΘ ⊕AΘ ⊕NΘ Thus, we have rewritten explicitly the standard parabolic in the desired form (2.17). PΘ The associated (generalized) Bruhat decomposition (2.18) is given now explicitly as:
+ + + + = 0 = (Θ) 0 = Θ = G N ⊕ P NΘ ⊕N ⊕ P NΘ ⊕ P (2.23) = + − . NΘ ⊕MΘ ⊕AΘ ⊕NΘ
Another important class are the maximal parabolic subalgebras which correspond to Θ of the form: Θmax = S j , 1 j r . (2.24) j r\{ } ≤ ≤ max dim (Θ ) = r 1, dim Θmax = 1. A j − A j Reminder 1: We recall for further use the fundamental result of Harish-Chandra [3] that has discrete series representations iff rank = rank . G G K ♦ Reminder 2: We recall for further use the well known fact that ( , ) is a Hermitian G K symmetric pair when the maximal compact subalgebra contains a u(1) factor. Then K has highest and lowest weight representations. All these algebras have discrete series G representations. ♦
9 3. The complex simple Lie algebras considered as real Lie algebras Let be a complex simple Lie algebra of dimension d and (complex) rank ℓ. We need Gc the triangular decomposition:
= + − . (3.1) Gc N ⊕H⊕N We have dim = d, rank = dim = ℓ, dim ± = (d ℓ)/2. Considered as CI Gc CI Gc CI H CI N − real Lie algebras we have: dim = 2d, rank = dim = 2ℓ, dim = d, IR Gc IR Gc IR H IR K rank = ℓ, dim ± = d ℓ. Note that the maximal compact subalgebra of is IR K IR N − K Gc isomorphic to the compact real form of . Gk Gc Thus, the complex simple Lie algebras do not have discrete series representations (and highest/lowest weight representations).
Let Hj , j = 1,...,ℓ, be the basis of , i.e., = c.l.s. Hj , j = 1,...,ℓ , where H . H { } c.l.s. stands for complex linear span. Let = = r.l.s. H , j = 1,...,ℓ , where A HIR { j } r.l.s. stands for real linear span. Then the Iwasawa decomposition of is: Gc = , = ± . (3.2) Gc K⊕A⊕N N N The commutant of in is given by: M A K = u(1) u(1) , ℓ factors . (3.3) M ⊕···⊕ In fact, the basis of consists of the vectors iH , j =1,...,ℓ . The Bruhat decompo- M { j } sition of is: Gc = + − . (3.4) Gc N ⊕M⊕A⊕N Comparing (3.1) and (3.4) we see that
= . (3.5) H M⊕A
The restricted root system ( , ) looks the same as the complex root system ( , ), Gc A Gc H but the restricted roots have multiplicity 2, since dim ± = 2 dim ±. IR N CI N Let Θ be a string subset of S of length s. The -factor of the corresponding ℓ MΘ parabolic subalgebra is:
= u(1) u(1) , ℓ s factors , (3.6) MΘ Gs ⊕ ⊕···⊕ − where is a complex simple Lie algebra of rank s isomorphic to a subalgebra of . Gs Gc Thus, the complex simple Lie algebras, considered as real noncompact Lie algebras, do not have non-minimal cuspidal parabolic subalgebras.
10 Thus, it is enough to consider elementary representations induced from the minimal parabolic subgroup P = MAN, where M = U(1) ... U(1), (ℓ factors), A = 0 ∼ × × ∼ SO(1, 1) ... SO(1, 1), (ℓ factors), N = exp ±.12 Thus, the signature χ = [µ, ν], × × ∼ N consists of ℓ integer numbers µ ZZ giving the unitary character µ =(µ ,...,µ ) of M, i ∈ 1 ℓ and of ℓ complex numbers ν CI giving the character ν =(ν ,...,ν ) of A, ν = ν(H ). i ∈ 1 ℓ j j Thus, if H = σ H , σ IR, is a generic element of , then for the corresponding j j j j ∈ A factor in (2.1) we have eν(H) = exp σ ν . Analogously, if m = exp i φ H M, P j j j j j j ∈ φ IR, then we have Dµ(m−1) = exp i φ µ . Thus, the right covariance property j ∈ P j j j P (2.1) becomes: P (gman) = exp (σ ν + iφ µ ) (g) (3.7) F j j j j ·F j X To relate with the general setting of the previous subsection we must introduce two
weight functionals: Λ, Λ,˜ such that Λ(Hj) = λj , Λ(˜ Hj) = λ˜j . Let us use (3.5) and H = (σ + iφ )H . Then (3.7) can be represented as: j j j j ∈ H P (gman) = exp Λ(H) + Λ(˜ H¯ ) (g) = F ·F (3.8) = exp (σ + iφ )λ +(σ iφ )λ˜ (g) j j j j − j j ·F j X provided: ν = λ + λ˜ , µ = λ λ˜ (3.9) j j j j j − j The ERs for which Λ˜ = 0 are called holomorphic, and those for which Λ = 0 are called antiholomorphic. Thus, we see that the complex case is richer than the real one. Indeed, there are two Verma modules associated with an ER, one ’holomorphic’ V Λ and one ’antiholomorphic’ V Λ˜ . The ER is reducible when either V Λ or V Λ˜ are reducible, or equivalently, if for at least one j 1,...,ℓ holds: λ ZZ , or λ˜ ZZ . ∈{ } j ∈ + j ∈ + More information can be found in [8] from where we mention some important statements: All irreducible representations of a complex semisimple Lie group are obtained as subrepre- sentations of the elementary representations induced from the minimal parabolic subgroup. All finite-dimensional irreps are obtained as subrepresentations when all λ , λ˜ ZZ . j j ∈ + To summarize, the elementary representations (in particular, the right covariance con- ditions) for a complex semisimple Lie group Gc are given by:
∞ Λ(X)+Λ(˜ X¯) ˜ = C (G ) (gxn) = e (g) , Λ(X) Λ(˜ X) ZZ, (3.10) CΛ,Λ {F ∈ c | F ·F } − ∈ 12 We should note that the minimal parabolic subgroup P0 is isomorphic to a Borel subgroup of
Gc, due to the obvious isomorphism between the abelian subgroup MA and the Cartan subgroup
H of Gc.
11 where x = exp(X), X , n G+ = exp( +), and the last condition in (3.10) is ∈ H ∈ c N necessary to ensure uniqueness on the Cartan subgroup H = exp( ) of G . c H c The maximal parabolic subalgebras have -factors as follows MΘ = u(1), i =1,...,ℓ , (3.11) MΘ Gi ⊕ where is a complex simple Lie algebra of rank ℓ 1 which may be obtained from Gi − Gc by deleting the i-th node of the Dynkin diagram of . Gc
4. AI : SL(n, IR) In this section G = SL(n, IR), the group of invertible n n matrices with real elements × and determinant 1. Then = sl(n, IR) and the Cartan involution is given explicitly by: G θX = tX, where tX is the transpose of X . Thus, = so(n), and is spanned by − ∈ G K ∼ matrices (r.l.s. stands for real linear span):
= r.l.s. X e e , 1 i < j n , (4.1) K { ij ≡ ij − ji ≤ ≤ }
where eij are the standard matrices with only nonzero entry (=1) on the i-th row and j-th column, (e ) = δ δ . Note that does not have discrete series representations ij kℓ ik jℓ G if n > 2. Indeed, the rank of sl(n, IR) is n 1, and the rank of its maximal compact − subalgebra so(n) is [n/2] and the latter is smaller than n 1 unless n = 2. − Further, the complementary space is given by: P = r.l.s. Y e +e , 1 i < j n, H e e , 1 j n 1 . (4.2) P { ij ≡ ij ji ≤ ≤ j ≡ jj − j+1,j+1 ≤ ≤ − } The split rank is r = n 1, and from (4.2) it is obvious that in this setting one has: − = r.l.s. H , 1 j n 1 = r . (4.3) A { j ≤ ≤ − } Since is a maximally split real form of CI = sl(n, CI), then = 0, and the minimal G G M parabolic subalgebra and the Bruhat decomposition, resp., are given as a Borel subalgebra and triangular decomposition of CI , but over IR: G = + − , = − , (4.4) G N ⊕A⊕N P0 A⊕N where +, −, resp., are upper, lower, triangular, resp.: N N + = r.l.s. e , 1 i < j n , − = r.l.s. e , 1 j < i n . (4.5) N { ij ≤ ≤ } N { ij ≤ ≤ } 12 The simple root vectors are given explicitly by: . . X+ = e ,X− = e , 1 j n 1 = r . (4.6) j j,j+1 j j+1,j ≤ ≤ − Note that matters are arranged so that
[X+,X−] = H , [H ,X±] = 2X± , (4.7) j j j j j ± j
and further we shall denote by sl(2, IR)j the sl(2, IR) subalgebra of spanned by ± G Xj ,Hj . The parabolic subalgebras may be described by the unordered partitions of n.13 Ex- . s plicitly, letν ¯ = ν ,...,ν , s n, be a partition of n: ν = n. Then the set { 1 s} ≤ j=1 j Θ corresponding to the partitionν ¯ and denoted by Θ(¯ν) consists of the numbers of the P entries νj that are bigger than 1:
Θ(¯ν) = j : ν > 1 . (4.8) { j } Note that in the case s = n all ν are equal to 1 - this is the partitionν ¯ = 1,..., 1 j 0 { } corresponding to the empty set: Θ(¯ν ) = (corresponding to the minimal parabolic). 0 ∅ Then the factor in (2.22) and (2.23) is: MΘ(¯ν)
Θ(¯ν) = sl(νj, IR) = sl(νj, IR) , sl(1, IR) 0 (4.9) M 1≤⊕j≤s 1≤⊕j≤s ≡ νj >1
Certainly, some partitions give isomorphic (though non-conjugate!) subalgebras. MΘ(¯ν) The parabolic subalgebras in these cases are called associated, and this is an equivalence relation. The parabolic subalgebras up to this equivalence relation correspond to the ordered partitions of n. The most important for us cuspidal parabolic subalgebras correspond to those partitions ν¯ = ν ,...,ν for which ν 2, j. Indeed, if some ν > 2 then will not have { 1 s} j ≤ ∀ j MΘ(¯ν) discrete series representations since it contains the factor sl(νj, IR). A more explicit description of the cuspidal cases is given as follows. It is clear that the cuspidal parabolic subalgebras are in 1-to-1 correspondence with the sequences of r numbers: . n¯ = n ,...,n , (4.10) { 1 r } such that nj = 0, 1, and if for fixed j we have nj = 1, then nj+1 = 0, (clearly from the latter follows also nj−1 = 0, but we shall use this notation also in other contexts). In the
13 The parabolic subalgebras may also be described by the various flags of IRn, F., e.g., [6], but we shall not use this description.
13 language above to each nj = 1 there is an entry νj =2inν ¯ bringing an sl(2, IR) factor to , i.e., MΘ Θ(¯n) = j : n =1, n =0 . (4.11) { j j+1 } More explicitly, the cuspidal parabolic subalgebras are given as follows:
Θ(¯n) = sl(2, IR)jt , njt =1, 1 j1 < j2 < < jk r , jt < jt+1 1 . M 1≤⊕t≤k ≤ ··· ≤ − (4.12) The corresponding and ± have dimensions: AΘ(¯n) NΘ(¯n) dim = n 1 k , dim ± = 1 n(n 1) k , (4.13) AΘ(¯n) − − NΘ(¯n) 2 − − where k = Θ(¯n) was introduced in (4.12). | | Note that the minimal parabolic subalgebra is obtained when all nj =0,n ¯0 = 0,..., 0 , then Θ(¯n ) = , = 0, k = 0. { } 0 ∅ MΘ(¯n0) Interlude: The number of cuspidal parabolic subalgebras of sl(n, IR), n 2, including ≥ also the case = ′ = sl(n, IR) when n = 2, is equal to F (n+1), where F (n), n ZZ , P M ∈ + are the Fibonacci numbers. Proof: First we recall that the Fibonacci numbers are determined through the relations F (m) = F (m 1) + F (m 2), m 2 + ZZ , together with the boundary values: F (0) = − − ∈ + 0, F (1) = 1. We shall count the number of sequences of r numbers ni, introduced above (r = n 1). Let us denote by N(r) the number of the above-described sequences. − Let us divide these sequences in two groups: the first with n1 = 1 and the others with n = 0. Obviously the number of sequences with n = 1 is equal to the N(r 2) since 1 1 − n = 0, and then we are left with the above-described sequences but of r 2 numbers. 2 − Analogously, the number of sequences with n = 0 is equal to the N(r 1) since we 1 − are left with all above-described sequences of r 1 numbers. Thus, we have proved that − N(r) = N(r 1) + N(r 2). This is the Fibonacci recursion relation and we have only − − to adjust the boundary conditions. We have N(1) = 2, N(2) = 3, i.e., N(r) = F (r + 2), or in terms of n = r + 1: N(n 1) = F (n + 1). − ♦ For further use we recall that there is explicit formula for the Fibonacci numbers: [(n−1)/2] xn (1 x)n n F (n) = − − = 21−n 5s , (4.14) 2s +1 (5) s=0 X where x is the golden ratio : px2 = x +1, i.e., x = (1 √5)/2. ± Finally, we mention that the maximal parabolic subalgebras corresponding to Θ from (2.24) have the following factors: = sl(j, IR) sl(n j, IR) , 1 j n 1 , MΘj ⊕ − ≤ ≤ − ± (4.15) dim Θ = 1, = j(n j) A j NΘj − 14 (Note that the cases j and n j are isomorphic, or coinciding when n is even and − j = 1 n.) Only one of the maximal ones is cuspidal, namely, for = sl(4, IR), n = 4 and 2 G j = 2 we have = sl(2, IR) sl(2, IR) . (4.16) MΘ2 ⊕
5. AII : SU ∗(2n) The group G = SU ∗(2n), n 2, consists of all matrices in SL(2n, CI) which commute with ≥ a real skew-symmetric matrix times the complex conjugation operator C :
∗ . 0 1n SU (2n) = g GL(2n, CI) JnCg = gJnC , Jn , det g =1 . (5.1) { ∈ | ≡ 1n 0 } − The Lie algebra = su∗(2n) is given by: G . su∗(2n) = X gl(2n, CI) J CX = XJ C , tr X =0 = { ∈ | n n } a b (5.2) = X = , a,b gl(n, CI) , tr(a +¯a)=0 . { ¯b a¯ ∈ } − dim =4n2 1. R G − ∗ We consider n 2 since su (2) ∼= su(2), and we note that the case n = 2 (of split rank ≥ ∗ 1) will appear also below: su (4) ∼= so(5, 1), cf. the corresponding Section. The Cartan involution is given by: θX = X†. Thus, = sp(n): − K ∼ a b = X = , a,b gl(n, CI) , a† = a , tb = b . (5.3) K { b† ta ∈ − } − − Note that su∗(2n) does not have discrete series representations (rank = n < K rank su∗(2n)=2n 1). The complimentary space is given by: − P a b = X = , a,b gl(n, CI) , a† = a , tb = b, tr a =0 . (5.4) P { b† ta ∈ − } The split rank is n 1 and the abelian subalgebra is given explicitly by: − A a 0 = X = a = diag(a ,...,a ), a IR , tr a =0 . (5.5) A { 0 a | 1 n j ∈ } The subalgebras ± which form the root spaces of the root system ( , ) are of real N G A dimension 2n(n 1). The subalgebra is given by: − M a b = X = a = i diag (φ ,...,φ ), φ IR , M { ¯b a | 1 n j ∈ − − (5.6) b = diag(b ,...,b ), b CI = 1 n j ∈ } ∼ = su(2) su(2) , n factors . ∼ ⊕···⊕ 15 Claim: All non-minimal parabolic subalgebras of su∗(2n) are not cuspidal. Proof: Necessarily n > 2. Let Θ enumerate a connected string of restricted simple roots: Θ = S = i,...,j , where 1 i j 6. AIII,AIV : SU(p, r) In this section G = SU(p, r), p r, which standardly is defined as follows: ≥ . † 1p 0 SU(p, r) = g GL(p + r, CI) g β0g = β0 , β0 , det g =1 , (6.1) { ∈ | ≡ 0 1r } − where g† is the Hermitian conjugate of g. We shall use also another realization of G differing from (6.1) by unitary transformation: 1 0 0 1 0 0 p−r 1 p−r β β 0 01 = Uβ U −1 , U 0 1 1 . (6.2) 0 2 r 0 √ r r 7→ ≡ 0 1 0 ≡ 2 0 1 1 r − r r The Lie algebra = su(p, r) is given by (β = β , β ): G 0 2 . su(p, r) = X gl(p + r, CI) X†β + βX =0 , tr X =0 . (6.3) { ∈ | } The Cartan involution is given explicitly by: θX = βXβ. Thus, = u(1) su(p) su(r), K ∼ ⊕ ⊕ and more explicitly is given as (β = β0): u1 0 † = X = uj = u, j =1, 2; tr u1 + tr u2 =0 . (6.4) K { 0 u2 | − } Note that su(p, r) has discrete series representations since rank = 1+rank su(p) + K rank su(r) = p + r 1 = rank su(p, r), and highest/lowest weight representations. − The split rank is equal to r and the abelian subalgebra may be given explicitly by A (β = β2): u = r.l.s. H e − − e , 1 j r . (6.5) A { j ≡ p r+j,p r+j − p+j,p+j ≤ ≤ } At this moment we need to consider the cases p = r and p>r separately, since the minimal parabolic subalgebras are different. 16 6.1. The case SU(n,n), n > 1 In this subsection G = SU(n,n). We consider n > 1 since SU(1, 1) ∼= SL(2, IR), which was already treated. The subalgebra = u(1) u(1), (n 1 factors), and is explicitly given as (β = β ): M ∼ ⊕···⊕ − 2 u 0 = X = u = i diag (φ ,...,φ ), φ IR ; tr u =0 . (6.6) M { 0 u | 1 n j ∈ } The subalgebras ± which form the root spaces of the root system ( , ) are of real N G A dimension n(2n 1). The simple root system ( , ) looks as that of the symplectic algebra − G A Cn, however, the root spaces of the short roots have multiplicity 2. Further, we choose the long root of the Cn simple root system to be αn . Claim: The nontrivial cuspidal parabolic subalgebras are given by Θ of the form: Θ = j +1,...,n , 1 j Proof: First note that we exclude j = 0 since Θ0 = Sn . Consider now any Θ which contains a subset S = i,...,j , where 1 i j = su(n j, n j) u(1) u(1), j factors (6.8) MΘj ∼ − − ⊕ ⊕···⊕ cf. the Remark above. ♦ max The maximal parabolic subalgebras correspond to the sets Θj , j = 1,...,n, cf. (2.24). The corresponding subalgebras are of the form: MΘ max = sl(j, CI) su(n j, n j) u(1) u(1), j factors , (6.9) Mj ⊕ − − ⊕ ⊕···⊕ max where we use the convention: sl(1, CI) = 0. Note that Θ1 = Θ1 and that the only cuspidal maximal parabolic subalgebra is . PΘ1 6.2. The case SU(p, r), p>r 1 ≥ In this subsection G = SU(p, r). We include also the case r = 1 although we noted that the case of split rank 1 is clear in general. 17 The subalgebra = su(p r) u(1) u(1), (r factors), and is explicitly given as M ∼ − ⊕ ⊕···⊕ (β = β2): up−r 0 0 † = X = 0 u 0 u = u − , u = i diag (φ ,...,φ ), φ IR , p−r p r 1 n j M { 0 0 u | − ∈ tr u − + 2tr u =0 p r } (6.10) The subalgebras ± which form the root spaces of the root system ( , ) are of real N G A dimension r(2p 1). The restricted simple root system ( , ) looks as that of the orthog- − G A onal algebra Br, however, the root spaces of the long roots have multiplicity 2, the short simple root, say α , has multiplicity 2(p r), and there is also a root 2α with multiplicity r − r 1. Similarly to the su(n,n) case one can prove that the nontrivial cuspidal parabolic sub- algebras are given by Θ of the form: Θ = j +1,...,r , 1 j = su(p j, r j) u(1) u(1), j factors . (6.12) MΘj ∼ − − ⊕ ⊕···⊕ The maximal parabolic subalgebras, (cf. (2.24)), contain the subalgebras are of the MΘ form: max = sl(j, CI) su(p j, r j) u(1) u(1), j factors . (6.13) Mj ⊕ − − ⊕ ⊕···⊕ Thus, the only cuspidal maximal parabolic subalgebra is . PΘ1 7. BDI,BDII : SO(p, r) In this section G = SO(p, r), p r, which standardly is defined as follows: ≥ . † 1p 0 SO(p, r) = g SO(p + r, CI) g β0g = β0 , β0 , (7.1) { ∈ | ≡ 0 1r } − where g† is the Hermitian conjugate of g. We shall use also another realization of G differing from (7.1) by unitary transformation: 1 0 0 1 0 0 p−r 1 p−r β β 0 01 = Uβ U −1 , U 0 1 1 . (7.2) 0 2 r 0 √ r r 7→ ≡ 0 1 0 ≡ 2 0 1 1 r − r r 18 The Lie algebra = so(p, r) is given by (β = β , β ): G 0 2 . so(p, r) = X so(p + r, CI) X†β + βX =0 . (7.3) { ∈ | } The Cartan involution is given explicitly by: θX = βXβ. Thus, = so(p) so(r), and K ∼ ⊕ more explicitly is given as (β = β0): u1 0 = X = u1 so(p), u2 so(r) . (7.4) K { 0 u2 | ∈ ∈ } ∗ Note that so(5, 1) ∼= su (4), so(4, 2) ∼= su(2, 2), so(3, 3) ∼= sl(4, IR), so(4, 1) ∼= sp(1, 1), so(3, 2) = sp(2, IR), so(3, 1) = sl(2, CI), so(2, 2) = sl(2, IR) sl(2, IR), so(2, 1) = sl(2, IR), ∼ ∼ ∼ ⊕ ∼ (so(1, 1) is abelian). Thus, below we can restrict to p + r > 4, since the cases p + r = 5 are not treated yet. Note that so(p, r) has discrete series representations except when both p, r are odd numbers, since then rank = rank so(p) + rank so(r) = 1 (p + r 2) < rank so(p, r) = K 2 − 1 (p + r). It has highest/lowest weight representations when p r = 2 and p =2,r = 1. 2 ≥ The split rank is equal to r and the abelian subalgebra may be given explicitly by A (β = β2): u = r.l.s. H e − − e , 1 j r . (7.5) A { j ≡ p r+j,p r+j − p+j,p+j ≤ ≤ } The subalgebra = so(p r) and is explicitly given as (β = β ): M ∼ − 2 u 0 = X = u so(p r) (7.6) M { 0 0 | ∈ − } The subalgebras ± which form the root spaces of the root system ( , ) are of real N G A dimension r(p 1). Except in the case p = r the restricted simple root system ( , ) − G A looks as that of the orthogonal algebra Br, however, the short simple root, say αr , has multiplicity p r. − Thus, we consider first the case p>r> 1. First we note that the parabolic subalgebras given by Θ = j ,j 19 subalgebra so(p j, r j). Clearly, if both p, r are even (odd), then also j must be even − − (odd), while if one of p, r is even and the other odd, i.e., p + r is odd, then j takes all values from (7.7). To be more explicit we first introduce the notation: . n¯ = n ,...,n , 1 s r , (7.8) s { 1 s } ≤ ≤ (note thatn ¯r =n ¯ from (4.10)). Then we shall use the notation introduced for the sl(n, IR) case, namely, Θ(¯ns) from (4.11). Then the cuspidal parabolic subalgebras are given by the noncompact factors from (4.12): MΘ s =1, 2,...,r 1 p + r odd − s = Θ(¯ns) so(p s,r s) , s =2, 4,...,r 2 p, r even (7.9) M M ⊕ − − ( s =1, 3,...,r − 2 p, r odd − Next we note that we can include the case when the second factor in is compact by MΘ just extending the range of s to r. Thus, all cuspidal parabolic subalgebras of so(p, r) in the case p>r will be determined by the following subalgebras: MΘ s =1, 2,...,r p + r odd s = Θ(¯ns) so(p s,r s) , s =2, 4,...,r p,r even (7.10) M M ⊕ − − ( s =1, 3,...,r p,r odd Finally, we note that the maximal parabolic subalgebras corresponding to (2.24) have -factors given by: MΘ max = sl(j, IR) so(p j, r j) , j =1, 2,...,r . (7.11) Mj ⊕ − − Thus, the maximal parabolic subalgebras are cuspidal (and can be found in (7.10)) when j =1, 2 and the number (p j)(r j) is even. − − Now we consider the split cases p = r 4. (Note that the other split-real cases, i.e., ≥ when p = r +1, were considered above without any peculiarities. The split cases p = r < 4 are not representative of the situation and were treated already: so(3, 3) ∼= sl(4, IR), so(2, 2) = so(2, 1) so(2, 1), so(1, 1) is not semisimple.) We accept the convention that ∼ ⊕ the simple roots αr−1 and αr form the fork of the so(2r, CI) simple root system, while αr−2 is the simple root connected to the simple roots αr−3 , αr−1 and αr . Special care is needed only when Θ includes these four special roots, i.e., . Θ Θˆ = s,...,r , 1 s r 4 . (7.12) ⊃ s { } ≤ ≤ − 20 In these cases, we have factor of the form so(r s,r s), i.e., there will be no cuspidal M − − parabolic if r s is odd. − For all other Θ the parabolic subalgebras would be like those of sl(r, IR), when r / Θ ∈ or r 1 / Θ, or like those of sl(r 2, IR) with possible addition of one or two sl(2, IR) − ∈ − factors, (when r 2 / Θ). To describe the latter cases we need a modification of the − ∈ notation (7.8): Θo(¯n) = j : n =1, n =0 if j = r 1 . (7.13) { j j+1 6 − } Thus, the cuspidal parabolic subalgebras are determined by the following factors: MΘ ˆ s =2, 4,...,r 4 r even Θ(¯ns) so(r s,r s) , Θ Θs , − = M ⊕ − − ⊃ s =1, 3,...,r 4 r odd MΘ − ˆ Θo(¯n) , Θ / Θs M ⊃ (7.14) The maximal parabolic subalgebras corresponding to (2.24) have -factors given by: MΘ sl(r, IR) j = r 1,r sl(r 2, IR) sl(2, IR) sl(2, IR) j = r − 2 max = − ⊕ ⊕ − (7.15) Mj sl(r 3, IR) sl(4, IR) j = r 3 sl(j,− IR) so⊕(r j, r j) j r − 4 ⊕ − − ≤ − Thus the maximal parabolic subalgebras which are cuspidal occur for j = 1 and odd r 5, ≥ (4th case), or j = 2 and either r = 4, (2nd case), or even r 6, (4th case): ≥ max = so(r 1,r 1) , r =5, 7,... M1 − − sl(2, IR) sl(2, IR) sl(2, IR) r =4 (7.16) max = ⊕ ⊕ M2 sl(2, IR) so(r 2,r 2) r =6, 8,... ⊕ − − 8. CI : Sp(n, IR), n > 1 In this section G = Sp(n, IR) - the split real form of Sp(n, CI). Both are standardly defined by: . Sp(n, F ) = g GL(2n, F ) tgJ g = J , det g =1 , F = IR, CI. (8.1) { ∈ | n n } Correspondingly, the Lie algebras are given by: sp(n, F ) = X gl(2n, F ) : tXJ + J X =0 . (8.2) { ∈ n n } Note that dim sp(n, F ) = n(2n + 1). The general expression for X sp(n, F ) is F ∈ AB X = , A,B,C gl(n, F ), tB = B, tC = C . (8.3) C tA ∈ − 21 A basis of the Cartan subalgebra of sp(n, CI) is: H Ai 0 Hi = , i =1,...,n 1, Ai = diag(0,...0, 1, 1, 0,..., 0), 0 Ai − − − (8.4) A′ 0 H = n , A′ = (0,..., 0, 2). n 0 A′ n − n The same basis over IR spans the subalgebra of = sp(n, IR), since A G rankF sp(n, F ) = n. Note that sp(2, IR) ∼= so(3, 2), sp(1, IR) ∼= sl(2, IR). The maximal compact subalgebra of = sp(n, IR) is = u(n), thus sp(n, IR) has G K ∼ discrete series representations (and highest/lowest weight representations). Explicitly, A 0 = X = , A gl(n, CI), . (8.5) K { 0 A† ∈ } − The subalgebras ± which form the root spaces of the root system ( , ) are of real N G A dimension n2. Further, we choose the long root of the Cn simple root system to be αn . The parabolic subalgebras corresponding to Θ such that n / Θ are the same as the ∈ parabolic subalgebras of sl(n, IR). The parabolic subalgebras corresponding to Θ such that n Θ contain a string Θ′ = s +1,...,n . This string brings in a factor ∈ s { } MΘ sp(n s, IR), which has discrete series representations. Thus cuspidality depends on the − rest of the possible choices and are the same as the parabolic subalgebras of sl(j, IR). Thus, we have: ′ Θ = Θ(¯n − ) Θ , s =1,...,n, (8.6) s s 1 ∪ s with the convention that Θ(¯n ) = , Θ′ = . Then the -factors of the cuspidal 0 ∅ n ∅ MΘ parabolic subalgebras of sp(n, IR) are given as follows: = sp(n s, IR) , s =1,...,n . (8.7) MΘs MΘ(¯ns−1) ⊕ − The minimal parabolic subalgebra for which = 0 is obtained for s = n since MΘ then enumerates all cuspidal parabolic subalgebras of sl(n, IR), including the MΘ(¯nn−1) minimal case = 0. MΘ The maximal parabolic subalgebras, cf. (2.24), 1 j n, contain subalgebras of ≤ ≤ MΘ the form: max = sl(j, IR) sp(n j, IR) , (8.8) Mj ⊕ − i.e., the only maximal cuspidal are those for j =1, 2. 22 9. CII : Sp(p, r) In this section G = Sp(p, r), p r, which standardly is defined as follows: ≥ . † β0 0 Sp(p, r) = g Sp(p + r, CI) g γ0g = γ0 , γ0 = , (9.1) { ∈ | } 0 β0 and correspondingly the Lie algebra = sp(p, r) is given by G . sp(p, r) = X sp(p + r, CI) X†γ + γ X =0 . (9.2) { ∈ | 0 0 } The Cartan involution is given explicitly by: θX = γ Xγ . Thus, = sp(p) sp(r), 0 0 K ∼ ⊕ and has discrete series representations (but not highest/lowest weight representations). G More explicitly: u1 0 0 0 0 u2 0 0 = X = † u1 sp(p), u2 sp(r) . (9.3) K { 0 0 u 0 | ∈ ∈ } − 1 0 0 0 u† − 2 The split rank is equal to r and the abelian subalgebra may be given explicitly by: A s = r.l.s. H e − − e e + e , 1 j r . A { j ≡ p r+j,p r+j − p+j,p+j − 2p+j,2p+j 2p+r+j,2p+r+j ≤ ≤ } (9.4) The subalgebras ± which form the root spaces of the root system ( , ) are of real N G A dimension r(4p 1). − The subalgebra = sp(p r) sp(1) sp(1), r factors. M ∼ − ⊕ ⊕···⊕ Here, just for a moment we distinguish the cases p>r and p r, since the restricted − root systems are different. When p>r the restricted simple root system ( , ) looks as that of B , however, the G A r short simple root say, α , has multiplicity 4(p r), the long roots have multiplicity 4. r − When p = r > 1 the restricted simple root system ( , ) looks as that of C , however, G A r the long root say, αr , has multiplicity 3, the short roots have multiplicity 4. (We consider r > 1 since sp(1, 1) ∼= so(4, 1).) In spite of these differences from now on we can consider the two subcases together, i.e., we take p r. ≥ There are two types of parabolic subalgebras depending on whether r / Θ or r Θ. ∈ ∈ Let r / Θ. Then the parabolic subalgebras are like those of su∗(2n). Let Θ enumerate ∈ a connected string of restricted simple roots: Θ = S = i,...,j , where 1 i j 10. DIII : SO∗(2n) The group G = SO∗(2n) consists of all matrices in SO(2n, CI) which commute with a real skew-symmetric matrix times the complex conjugation operator C : . SO∗(2n) = g SO(2n, CI) J Cg = gJ C . (10.1) { ∈ | n n } The Lie algebra = so∗(2n) is given by: G . so∗(2n) = X so(2n, CI) J CX = XJ C = { ∈ | n n } a b (10.2) = X = , a,b gl(n, CI), ta = a, b† = b . { ¯b a¯ ∈ − } − dim = n(2n 1), rank = n. R G − G 24 Note that so∗(8) = so(6, 2), so∗(6) = su(3, 1), so∗(4) = so(3) so(2, 1), so∗(2) = so(2). ∼ ∼ ∼ ⊕ ∼ Further, we can restrict to n 4 since the other cases are not representative. ≥ The Cartan involution is given by: θX = X†. Thus, = u(n): − K ∼ a b = X = , a,b gl(n, CI), ta = a = a,¯ b† = b = ¯b , (10.3) K { b a ∈ − − } − and = so∗(2n) has discrete series representations (and highest/lowest weight represen- G tations). The complimentary space is given by: P a b = X = , a,b gl(n, CI) , ta = a =¯a, b† = b = ¯b . (10.4) P { b a ∈ − − } − dim = n(n 1). The split rank is r [n/2]. R P − ≡ The subalgebras ± which form the root spaces of the root system ( , ) are of real N G A dimension n(n 1) [n/2]. − − Here, just for a moment we distinguish the cases n even and n odd since the subalgebras and the restricted root systems are different. M For n =2r the split rank is equal to r 2, and the restricted root system is as that of ≥ Cr , but the short roots have multiplicity 4, while the long simple root αr has multiplicity 1. The subalgebra = so(3) so(3), r factors. M ∼ ⊕···⊕ For n = 2r +1 the split rank is equal to r 2, and the restricted root system is as ≥ that of Br , but all simple roots have multiplicity 4, and there is a restricted root 2αr of multiplicity 1, where αr is the short simple root. The subalgebra = so(2) so(3) so(3), r factors. M ∼ ⊕ ⊕···⊕ In spite of these differences from now on we can consider the two subcases together. There are two types of parabolic subalgebras depending on whether r / Θ or r Θ. ∈ ∈ If r / Θ then the parabolic subalgebras are like those of su∗(2r), and they are not ∈ cuspidal. Let r Θ and consider the various strings containing r : ∈ Θ = j +1,...,r , 1 j = so∗(2n 4j) so(3) so(3), j factors , j =1,...,r 1 . (10.6) Mj − ⊕ ⊕···⊕ − 25 The other factors in the cuspidal parabolic subalgebras have dimensions: dim = j, Aj dim ± = j(4n 4j 3). Extending the range of j we include the minimal parabolic Nj − − case for j = r =[n/2], and the case ′ = = for j = 0 which is also cuspidal. M P G The maximal parabolic subalgebras enumerated by Θmax from (2.24) have -factors j MΘ as follows: max = so∗(2n 4j) su∗(2j) , j =1,...,r . (10.7) Mj − ⊕ ± ± max The factors in the maximal parabolic subalgebras have dimensions: dim ( j ) = N max N j(4n 6j 1). Only the case j = 1 is cuspidal, noting that 1 coincides with 1 from − −∗ M M (10.6), (su (2) ∼= su(2) ∼= so(3)). 11. Real forms of the exceptional simple Lie algebras We start with the real forms of the exceptional simple Lie algebras. Here we can not be so explicit with the matrix realizations. To compensate this we use the Satake dia- grams [58],[6], which we omitted until now.14 A Satake diagram has a starting point the Dynkin diagram of the corresponding complex form. For a split real form it remains the same. In the other cases some dots are painted in black - these considered by themselves are Dynkin diagrams of the compact semisimple factors of the minimal parabolic sub- M algebras. Further, there are arrows connecting some nodes which use the ZZ2 symmetry of some Dynkin diagrams. Then the reduced root systems are described by Dynkin-Satake diagrams which are obtained from the Satake diagrams by dropping the black nodes, identifying the arrow-related nodes, and adjoining all nodes in a connected Dynkin-like diagram, but in addition noting the multiplicity of the reduced roots (which is in general different from 1). More details can be seen in [6], and we have tried to make the exposition transparent (by repeating things). ′ 11.1. EI : E6 ′ The split real form of E6 is denoted as E6 , sometimes as E6(+6) . The maximal compact subgroup is = sp(4), dim = 42, dim ± = 36. This real form does not have K ∼ IR P IR N discrete series representations. For a split real form the Satake diagram coincides with the Dynkin diagram of the corresponding complex Lie algebra [6]. 14 We could do this, since by being explicit we could consider simultaneously cases with different Satake diagrams. 26 In the present case this Dynkin-Satake diagram is taken as follows: α6 ◦ | (11.1) α◦1 −−−α ◦2 −−−α ◦3 −−−α ◦4 −−−α ◦5 Taking into account the above enumeration of simple roots the cuspidal parabolic sub- algebras have -factors as follows: MΘ 0 Θ = , minimal sl(2, IR) , Θ = ∅j , j =1,..., 6 j { } sl(2, IR)j sl(2, IR)k , Θ = j, k : j +1 < k, j, k = 3, 6 ; ⊕ { } { } 6 { } Θ = (j, k)=(5, 6), M sl(2, IR)j sl(2, IR)k sl(2, IR)ℓ , Θ = j,k,ℓ , (j,k,ℓ)=(1, 3, 5), (1, 4, 6), ⊕ ⊕ (1{, 5, 6),}(2, 4, 6), (2, 5, 6) so(4, 4) , Θ = 2, 3, 4, 6 , { } (11.2) where sl(2, IR) denotes the sl(2, IR) subalgebra of spanned by X± ,H , (using the j G j j same notation as in the Section on sl(n, IR)). The dimensions of the other factors are, respectively: 6 36 5 35 ± dim Θ = 4 , dim Θ = 34 (11.3) A N 3 33 2 24 Taking into account (2.24) the maximal parabolic subalgebras are determined by: max = max = so(5, 5) , dim( ±)max = 16 M1 ∼ M5 ∼ NΘ max max ± max 2 = 5 = sl(5, IR) sl(2, IR) , dim( ) = 25 M ∼ M ∼ ⊕ NΘ (11.4) max = sl(3, IR) sl(3, IR) sl(2, IR) , dim( ±)max = 29 M3 ∼ ⊕ ⊕ NΘ max = sl(6, IR) , dim( ±)max = 21 M6 ∼ NΘ Clearly, no maximal parabolic subalgebra is cuspidal. ′′ 11.2. EII : E6 ′′ Another real form of E6 is denoted as E6 , sometimes as E6(+2) . The maximal compact subgroup is = su(6) su(2), dim = 40, dim ± = 36. This real form has K ∼ ⊕ IR P IR N discrete series representations. The split rank is equal to 4, while = u(1) u(1). M ∼ ⊕ 27 The Satake diagram is: α6 ◦ | (11.5) α◦1 −−−α ◦2 −−−α ◦3 −−−α ◦4 −−−α ◦5 Thus, the reduced root system is presented| {z by a Dynkin-Satak} e diagram looking like the | {z } F4 Dynkin diagram: = (11.6) λ◦1 −−−λ ◦2 ⇒λ ◦3 −−−λ ◦4 but the short roots have multiplicity 2 (the long - multiplicity 1). It is obtained from (11.5) by identifying α1 and α5 and mapping them to λ4, identifying α2 and α4 and mapping them to λ3, while the roots α3,α6 are mapped to the F4-like long simple roots λ2, λ1 , resp. Using the above enumeration of F simple roots we give the -factors of all parabolic 4 MΘ subalgebras: u(1) u(1) , Θ = , minimal ⊕ ∅ sl(2, IR)j u(1) u(1) , Θ = j , j =1, 2 sl(2, CI) ⊕u(1) ,⊕ Θ = {j}, j =3, 4 j ⊕ { } sl(3, IR) u(1) u(1) , Θ = 1, 2 ⊕ ⊕ { } sl(2, IR) sl(2, CI) u(1) , Θ = 1, j , j =3, 4 1 ⊕ j ⊕ { } sl(4, IR) u(1) , Θ = 2, 3 Θ = ⊕ { } (11.7) M sl(2, IR)2 sl(2, CI)4 u(1) , Θ = 2, 4 sl(3, CI) , ⊕ ⊕ Θ = {3, 4} { } so(4, 4) u(1) , Θ = 1, 2, 3 , ⊕ { } sl(3, IR) u(1) sl(2, CI)4 , Θ = 1, 2, 4 ⊕ ⊕ { } sl(2, IR)1 sl(3, CI) , Θ = 1, 3, 4 ⊕ { } sl(6, IR) , Θ = 2, 3, 4 { } The dimensions of the other factors are, respectively: 4 36 3 35 3 34 2 33 2 33 2 ± 30 dim = , dim = (11.8) AΘ 2 NΘ 33 2 30 1 24 1 31 1 29 1 21 28 The maximal parabolic subalgebras are given the last four lines in the above lists corre- max sponding to Θj , j =4, 3, 2, 1, cf. (2.24). The cuspidal parabolic subalgebras are those containing: u(1) u(1) , Θ = , minimal = sl(2,⊕ IR) u(1) u(1) , Θ = ∅j , j =1, 2 (11.9) Θ j M so(4, 4) ⊕u(1) , ⊕ Θ = {1}, 2, 3 ⊕ { } ′′′ 11.3. EIII : E6 ′′′ Another real form of E6 is denoted as E6 , sometimes as E6(−14) . The maximal compact subgroup is = so(10) so(2), dim = 32, dim ± = 30. This real form has K ∼ ⊕ IR P IR N discrete series representations (and highest/lowest weight representations). The split rank is equal to 2, while = so(6) so(2). M ∼ ⊕ The Satake diagram is: α6 ◦ | (11.10) α◦1 −−−α •2 −−−α •3 −−−α •4 −−−α ◦5 Thus, the reduced root system is presented by a Dynkin-Satake diagram looking like the | {z } B2 Dynkin diagram but the long roots (incl. λ1) have multiplicity 6, while the short roots (incl. λ2) have multiplicity 8, and there are also the roots 2λ of multiplicity 1, where λ is any short root. It is obtained from (11.10) by dropping the black nodes, (they give rise to ), identifying α and α and mapping them to λ , while the root α is mapped to M 1 5 2 6 the long simple root λ1 . The non-minimal parabolic subalgebras are given by: so(7, 1) so(2) , Θ = 1 ± 24 Θ = ⊕ { } , dim = (11.11) M sl(6, IR) , Θ = 2 NΘ 21 { } n Both are maximal (dim = 1) and not cuspidal. AΘ iv 11.4. EIV : E6 iv Another real form of E6 is denoted as E6 , sometimes as E6(−26) . The maximal compact subgroup is = f , dim = 26, dim ± = 24. This real form does not K ∼ 4 IR P IR N have discrete series representations. The split rank is equal to 2, while = so(8). M ∼ The Satake diagram is: α6 • | (11.12) α◦1 −−−α •2 −−−α •3 −−−α •4 −−−α ◦5 29 Thus, the reduced root system is presented by a Dynkin-Satake diagram looking like the A2 Dynkin diagram but all roots have multiplicity 8. It is obtained from (11.12) by dropping the black nodes, while α1 ,α5 , resp., are mapped to the A2-like simple roots λ1 , λ2 . The two non-minimal parabolic subalgebras are isomorphic and given by: = so(9, 1) , Θ = j , j =1, 2 , dim ± = 16 . (11.13) MΘ { } NΘ Both are maximal (dim = 1) and not cuspidal. AΘ ′ 11.5. EV : E7 ′ The split real form of E7 is denoted as E7 , sometimes as E7(+7) . The maximal compact subgroup = su(8), dim = 70, dim ± = 63. This real form has discrete series K ∼ IR P IR N representations. We take the Dynkin-Satake diagram as follows: α7 ◦ | (11.14) α◦1 −−−α ◦2 −−−α ◦3 −−−α ◦4 −−−α ◦5 −−−α ◦6 Taking into account the above enumeration of simple roots the cuspidal parabolic sub- algebras have -factors as follows: MΘ 0 Θ = , minimal sl(2, IR) , Θ = ∅j , j =1,..., 7 j { } sl(2, IR)j sl(2, IR)k , Θ = j, k : j +1 < k, ⊕ { } j, k = 3, 7 ; j, k = 6, 7 , { } 6 { } { } { } sl(2, IR)j sl(2, IR)k sl(2, IR)ℓ , Θ = j,k,ℓ = 1, 3, 5 , 1, 3, 6 , ⊕ ⊕ { } { } { } Θ = 1, 4, 6 , 1, 4, 7 , 1, 5, 7 , 1, 6, 7 , M { } { } { } { } 2, 4, 6 , 2, 4, 7 , 2, 5, 7 , 2, 6, 7 sl(2, IR) sl(2, IR) sl(2, IR) sl(2, IR) { } { } { } { } 1 ⊕ 4 ⊕ 6 ⊕ 7 sl(2, IR) sl(2, IR) sl(2, IR) sl(2, IR) 2 ⊕ 4 ⊕ 6 ⊕ 7 so(4, 4) , Θ = 2, 3, 4, 7 , { } so(6, 6) , Θ = 2, 3, 4, 5, 6, 7 , { } (11.15) The dimensions of the other factors are, respectively: 7 63 6 62 5 61 4 60 dim = , dim ± = (11.16) AΘ 3 NΘ 59 3 59 3 51 1 33 30 Taking into account (2.24) the maximal parabolic subalgebras are determined by: max = so(6, 6) , dim( ±)max = 33 M1 ∼ NΘ max = sl(6, IR) sl(2, IR) , dim( ±)max = 47 M2 ∼ ⊕ NΘ max = sl(4, IR) sl(3, IR) sl(2, IR) , dim( ±)max = 53 M3 ∼ ⊕ ⊕ NΘ max = sl(5, IR) sl(3, IR) , dim( ±)max = 60 (11.17) M4 ∼ ⊕ NΘ max = so(5, 5) sl(2, IR) , dim( ±)max = 42 M5 ∼ ⊕ NΘ max = E′ , dim( ±)max = 27 M6 ∼ 6 NΘ max = sl(7, IR) , dim( ±)max = 42 M7 ∼ NΘ Clearly, the only maximal cuspidal parabolic subalgebra is the one containing max . M1 ′′ 11.6. EVI : E7 ′′ Another real form of E7 is denoted as E7 , sometimes as E7(−5) . The maximal compact subgroup is = so(12) su(2), dim = 64, dim ± = 60. This real form has K ∼ ⊕ IR P IR N discrete series representations. The split rank is equal to 4, while = su(2) su(2) su(2). M ∼ ⊕ ⊕ The Satake diagram is: α7 • | (11.18) α◦1 −−−α ◦2 −−−α ◦3 −−−α •4 −−−α ◦5 −−−α •6 Thus, the reduced root system is presented by a Dynkin-Satake diagram looking like the F4 Dynkin diagram, cf. (11.6), but the short roots have multiplicity 4 (the long - multi- plicity 1). Going to this Dynkin-Satake diagram we drop the black nodes, (they give rise to ), while α ,α ,α ,α , are mapped to λ , λ , λ , λ , resp., of (11.6). M 1 2 3 5 1 2 3 4 Using the above enumeration of F simple roots we shall give the -factors of all 4 MΘ parabolic subalgebras: = su(2) su(2) su(2) , Θ = , minimal M ⊕ ⊕ ∅ sl(2, IR)j , Θ = j , j =1, 2 ∗ ⊕M { } su (4) su(2)j+3 , Θ = j , j =3, 4 ⊕ { } sl(3, IR) , Θ = 1, 2 ⊕M ∗ { } sl(2, IR)1 su (4) su(2)j+3 , Θ = 1, j , j =3, 4 ⊕ ⊕ { } so(6, 2) su(2) , Θ = 2, 3 Θ = ⊕ ∗ { } (11.19) M sl(2, IR)2 su (4) su(2)7 , Θ = 2, 4 su∗(6) , ⊕ ⊕ Θ = {3, 4} { } so(7, 3) su(2)6 , Θ = 1, 2, 3 , ⊕ ∗ { } sl(3, IR) su (4) su(2)7 , Θ = 1, 2, 4 ⊕ ∗ ⊕ { } sl(2, IR)1 su (6) , Θ = 1, 3, 4 ∗ ⊕ { } so (12) , Θ = 2, 3, 4 { } 31 The dimensions of the other factors are, respectively: 4 60 3 59 3 56 2 57 2 55 2 50 dim = , dim ± = (11.20) AΘ 2 NΘ 55 2 48 1 42 1 53 1 47 1 33 The maximal parabolic subalgebras are the last four in the above list corresponding to max Θj , j =4, 3, 2, 1, cf. (2.24). The cuspidal parabolic subalgebras are those containing = su(2) su(2) su(2) , Θ = , minimal M ⊕ ⊕ ∅ sl(2, IR)j , Θ = j , j =1, 2 Θ = ⊕M { } (11.21) M so(6, 2) su(2) , Θ = 2, 3 so∗(12) ⊕, Θ = {2, 3}, 4 { } ′′′ 11.7. EVII : E7 ′′′ Another real form of E7 is denoted as E7 , sometimes as E7(−25) . The maximal compact subgroup is = e so(2), dim =54, dim ± = 51. This real form has discrete K ∼ 6 ⊕ IR P IR N series representations (and highest/lowest weight representations). The split rank is equal to 3, while = so(8). M ∼ The Satake diagram is: α7 • | (11.22) α◦1 −−−α •2 −−−α •3 −−−α •4 −−−α ◦5 −−−α ◦6 Thus, the reduced root system is presented by a Dynkin-Satake diagram looking like the C3 Dynkin diagram: = (11.23) λ◦1 ⇒λ ◦2 −−−λ ◦3 but the short roots have multiplicity 8 (the long - multiplicity 1). Going to the C3 diagram we drop the black nodes, (they give rise to ), while α ,α ,α , are mapped to λ , λ , λ , M 1 5 6 3 2 1 resp., of (11.23). 32 Using the above enumeration of C simple roots we shall give the -factors of all 3 MΘ parabolic subalgebras: so(8) , Θ = , minimal so(9, 1) , Θ = ∅j , j =1, 2 { } sl(2, IR)3 so(8) , Θ = 3 Θ = ⊕ { } (11.24) M eiv , Θ = 1, 2 6 { } sl(2, IR)3 so(9, 1) , Θ = 1, 3 so(10, 2) ,⊕ Θ = {2, 3} { } The dimensions of the other factors are, respectively: 3 51 2 43 2 50 dim = , dim ± = (11.25) AΘ 1 NΘ 27 1 42 1 33 The last three give rise to the maximal parabolic subalgebras, while the cuspidal ones arise from: so(8) , Θ = , minimal = sl(2, IR) so(8) , Θ = ∅3 (11.26) Θ 3 M so(10, 2) ,⊕ Θ = {2,}3 { } ′ 11.8. EVIII : E8 ′ The split real form of E8 is denoted as E8 , sometimes as E8(+8) . The maximal compact subgroup = so(16), dim = 128, dim ± = 120. This real form has discrete K ∼ IR P IR N series representations. We take the Dynkin-Satake diagram as follows: α8 ◦ | (11.27) α◦1 −−−α ◦2 −−−α ◦3 −−−α ◦4 −−−α ◦5 −−−α ◦6 −−−α ◦7 Taking into account the above enumeration of simple roots the cuspidal parabolic sub- 33 algebras have -factors as follows: MΘ 0 Θ = , minimal ∅ sl(2, IR)j , Θ = j , j =1,..., 8 sl(2, IR) sl(2, IR) , Θ = {j,} k : j +1 < k, j ⊕ k { } j, k = 3, 8 ; j, k = 7, 8 , { } 6 { } { } { } sl(2, IR) sl(2, IR) sl(2, IR) , Θ = j,k,ℓ = 1, 3, 5 , 1, 3, 6 , j ⊕ k ⊕ ℓ { } { } { } 1, 3, 7 , 1, 4, 6 , 1, 4, 7 , 1, 4, 8 , { } { } { } { } 1, 5, 7 , 1, 5, 8 , 1, 6, 8 , 2, 4, 6 , { } { } { } { } 2, 4, 7 , 2, 4, 8 , 2, 5, 7 , 2, 5, 8 , { } { } { } { } 2, 6, 8 , 2, 7, 8 , 3, 5, 7 , 4, 6, 8 , { } { } { } { } 4, 7, 8 , 5, 7, 8 Θ = { } { } M sl(2, IR)1 sl(2, IR)3 sl(2, IR)5 sl(2, IR)7 ⊕ ⊕ ⊕ sl(2, IR)1 sl(2, IR)4 sl(2, IR)6 sl(2, IR)8 sl(2, IR) ⊕ sl(2, IR) ⊕ sl(2, IR) ⊕ sl(2, IR) 1 ⊕ 4 ⊕ 7 ⊕ 8 sl(2, IR) sl(2, IR) sl(2, IR) sl(2, IR) 1 ⊕ 5 ⊕ 7 ⊕ 8 sl(2, IR) sl(2, IR) sl(2, IR) sl(2, IR) 2 ⊕ 4 ⊕ 6 ⊕ 8 sl(2, IR) sl(2, IR) sl(2, IR) sl(2, IR) 2 ⊕ 4 ⊕ 7 ⊕ 8 sl(2, IR) sl(2, IR) sl(2, IR) sl(2, IR) 2 ⊕ 5 ⊕ 7 ⊕ 8 so(4, 4) , Θ = 2, 3, 4, 8 { } so(6, 6) , Θ = 2, 3, 4, 5, 6, 8 { } E′ , Θ = 1, 2, 3, 4, 5, 6, 8 7 { } (11.28) The dimensions of the other factors are, respectively: 8 120 7 119 6 118 5 117 4 116 4 116 4 116 dim = , dim ± = (11.29) AΘ 4 NΘ 116 4 116 4 116 4 116 4 108 2 90 1 57 34 Taking into account (2.24) the maximal parabolic subalgebras are determined by: max = so(7, 7) , dim( ±)max = 78 M1 ∼ NΘ max = sl(7, IR) sl(2, IR) , dim( ±)max = 98 M2 ∼ ⊕ NΘ max = sl(5, IR) sl(3, IR) sl(2, IR) , dim( ±)max = 106 M3 ∼ ⊕ ⊕ NΘ max ± max 4 = sl(5, IR) sl(4, IR) , dim( ) = 104 M ∼ ⊕ NΘ (11.30) max = so(5, 5) sl(3, IR) , dim( ±)max = 97 M5 ∼ ⊕ NΘ max = E′ sl(2, IR) , dim( ±)max = 83 M6 ∼ 6 ⊕ NΘ max = E′ , dim( ±)max = 57 M7 ∼ 7 NΘ max = sl(8, IR) , dim( ±)max = 92 M8 ∼ NΘ Clearly, the only maximal cuspidal parabolic subalgebra is the one containing max . M7 ′′ 11.9. EIX : E8 ′′ Another real form of E8 is denoted as E8 , sometimes as E8(−25) . The maximal compact subgroup is = e su(2), dim = 54, dim ± = 51. This real form has discrete K ∼ 7 ⊕ IR P IR N series representations. The split rank is equal to 4, while = so(8). M ∼ The Satake diagram α8 • | (11.31) α◦1 −−−α •2 −−−α •3 −−−α •4 −−−α ◦5 −−−α ◦6 −−−α ◦7 Thus, the reduced root system is presented by a Dynkin-Satake diagram looking like the F4 Dynkin diagram, cf. (11.6), but the short roots have multiplicity 8 (the long - multi- plicity 1). Going to the F diagram we drop the black nodes, (they give rise to ), while 4 M α1 ,α5 ,α6 ,α7 are mapped to λ4 , λ3 , λ2 , λ1 , resp., of (11.6). Using the above enumeration of F simple roots we shall give the -factors of all 4 MΘ 35 parabolic subalgebras: = so(8) , Θ = , minimal M ∅ sl(2, IR)j , Θ = j , j =1, 2 so(9, 1) , ⊕M Θ = {j}, j =3, 4 { } sl(3, IR) , Θ = 1, 2 ⊕M { } sl(2, IR)1 so(9, 1) , Θ = 1, j , j =3, 4 ⊕ { } so(10, 2) , Θ = 2, 3 Θ = { } (11.32) M sl(2, IR)2 so(9, 1) , Θ = 2, 4 iv ⊕ { } e6 , Θ = 3, 4 so(11, 3) , Θ = {1, 2,}3 , { } sl(3, IR) so(9, 1) , Θ = 1, 2, 4 ⊕ { } sl(2, IR) eiv , Θ = 1, 3, 4 1 ⊕ 6 { } e′′′ , Θ = 2, 3, 4 7 { } The dimensions of the other factors are, respectively: 4 108 3 107 3 100 2 105 2 99 2 90 dim = , dim ± = (11.33) AΘ 2 NΘ 99 2 84 1 78 1 97 1 83 1 57 The maximal parabolic subalgebras are the last four in the above lists corresponding to max Θj , j =4, 3, 2, 1, cf. (2.24). The cuspidal ones arise from: = so(8) , Θ = , minimal M ∅ sl(2, IR)j , Θ = j , j =1, 2 Θ = ⊕M { } (11.34) M so(10, 2) , Θ = 2, 3 e′′′ , Θ = {2, 3}, 4 7 { } ′ 11.10. FI : F4 ′ The split real form of F4 is denoted as F4 , sometimes as F4(+4) . The maximal compact subgroup = sp(3) su(2), dim =28, dim ± = 24. This real form has discrete K ∼ ⊕ IR P IR N series representations. 36 Taking into account the enumeration of simple roots as in (11.6) the parabolic subalge- bras have -factors as follows: MΘ 0 Θ = , minimal ∅ sl(2, IR)j , Θ = j , j =1, 2, 3, 4 sl(3, IR) , Θ = {j,} k = 1, 2 , 3, 4 jk { } { } { } sl(2, IR)j sl(2, IR)k , Θ = j, k = 1, 3 , 1, 4 , 2, 4 Θ = ⊕ { } { } { } { } (11.35) M sp(2, IR) , Θ = 2, 3 so(4, 3) , Θ = {1, 2}, 3 , sl(3, IR) sl(2, IR) , Θ = {1, 2, 4}, 1, 3, 4 , ⊕ { } { } sp(3, IR) , Θ = 2, 3, 4 { } The dimensions of the other factors are, respectively: 4 24 3 23 2 21 2 22 dim = , dim ± = (11.36) AΘ 2 NΘ 20 1 15 1 20 1 15 The maximal parabolic subalgebras are in the last three line s in the above lists corre- max sponding to Θj , j =4, 3, 2, 1, cf. (2.24). The cuspidal parabolic subalgebras arise from: 0 Θ = , minimal sl(2, IR) , Θ = ∅j , j =1, 2, 3, 4 j { } Θ = sl(2, IR)j sl(2, IR)k , Θ = j, k = 1, 3 , 1, 4 , 2, 4 (11.37) M ⊕ { } { } { } { } sp(2, IR) , Θ = 2, 3 sp(3, IR) , Θ = {2, 3}, 4 { } ′′ 11.11. FII : F4 ′′ Another real form of F4 is denoted as F4 , sometimes as F4(−20) . The maximal compact subgroup = so(9), dim = 16, dim ± = 15. This real form has discrete series K ∼ IR P IR N representations. The split rank is equal to 1, while = so(7). M ∼ The Satake diagram is: = (11.38) α•1 −−−α •2 ⇒α •3 −−−α ◦4 Thus, the reduced root system is presented by a Dynkin-Satake diagram looking like the A1 Dynkin diagram but the roots have multiplicity 8. Going to the A1 Dynkin diagram we drop the black nodes, (they give rise to ), while α becomes the A diagram. M 4 1 37 ′ 11.12. G : G2 ′ The split real form of G2 is denoted as G2 , sometimes as G2(+2) . The maximal compact subgroup = su(2) su(2), dim = 8, dim ± = 6. This real form has discrete K ∼ ⊕ IR P IR N series representations. The non-minimal parabolic subalgebras have -factors as follows: MΘ = sl(2, IR) , Θ = j , j =1, 2 (11.39) MΘ j { } They are cuspidal and maximal. 12. Summary and Outlook In the present paper we have started the systematic explicit construction of the invariant differential operators by giving explicit description of one of the main ingredients in our setting - the cuspidal parabolic subalgebras. We explicated also the maximal parabolic subalgebras, since these are important even when they are not cuspidal. In sequels of this paper [59] we shall present the construction of the invariant differential operators and expand the scheme to the supersymmetric case, in view of applications to conformal field theory and string theory. Acknowledgments. The author would like to thank for hospitality the Abdus Salam International Centre for Theoretical Physics, where part of the work was done. This work was supported in part by the Bulgarian National Council for Scientific Research, grant F-1205/02, the Alexander von Humboldt Foundation in the framework of the Clausthal-Leipzig-Sofia Cooperation, and the European RTN ’Forces-Universe’, contract MRTN-CT-2004-005104. 38 Appendix Table of Cuspidal Parabolic Subalgebras dim dim ± G MΘ IR AΘ IR NΘ u(1) u(1) ℓ d ℓ GCI ⊕···⊕ − dim = d ℓ factors CI GCI rank = ℓ CI GCI 1 sl(n, IR) Θ(¯n) = sl(2, IR)jt n 1 k 2 n(n 1) k M 1≤⊕t≤k − − − − 0 k [n/2] ≤ ≤ j < j 1 t t+1 − 1 j , j n 1 ≤ 1 k ≤ − minimal: k = 0, =0 n 1 1 n(n 1) MΘ − 2 − su∗(2n) su(2) su(2) n 1 2n(n 1) ⊕···⊕ − − n factors su(p, r) p r su(p j, r j) u(1) u(1) j j(2(p + r j) 1) ≥ − − ⊕ ⊕···⊕ − − j factors, 1 j 39 dim dim ± G MΘ IR AΘ IR NΘ sp(n, IR) sp(n s, IR) s MΘ(¯ns−1) ⊕ − ≤ s =1,...,n minimal: =0, (s = n) n n2 MΘ sp(p, r) sp(p j, r j) sp(1) sp(1) j j(4p +4r 4j 1) − − ⊕ ⊕···⊕ − − p r j factors, j =1,...,r ≥ minimal: j = r so∗(2n) so∗(2n 4j) so(3) so(3) j j(4n 4j 3) − ⊕ ⊕···⊕ − − j factors, j =1,...,r [n/2] ≡ minimal: j = r [n/2] ≡ ′ EI ∼= E6 0, minimal 6 36 sl(2, IR)j 5 35 j =1,..., 6 sl(2, IR) sl(2, IR) 4 34 j ⊕ k j +1 EII = E′′ u(1) u(1), minimal 4 36 ∼ 6 ⊕ sl(2, IR) u(1) u(1), j =1, 2 3 35 j ⊕ ⊕ so(4, 4) u(1) 1 24 ⊕ EIII = E′′′ so(6) so(2) 2 30 ∼ 6 ⊕ iv EIV ∼= E6 so(8) 2 24 40 dim dim ± G MΘ IR AΘ IR NΘ ′ EV ∼= E7 0, minimal 7 63 sl(2, IR)j , j =1,..., 7 6 62 sl(2, IR) sl(2, IR) 5 61 j ⊕ k j +1 EVI = E′′ su(2) su(2) su(2), minimal 4 60 ∼ 7 ⊕ ⊕ sl(2, IR) su(2) su(2) su(2), j =1, 2 3 59 j ⊕ ⊕ ⊕ so(6, 2) su(2) 2 50 ⊕ so∗(12) 1 33 ′′′ EVII ∼= E7 s0(8), minimal 3 51 sl(2, IR) so(8) 2 50 3 ⊕ so(10, 2) 1 33 ′ FI ∼= F4 0, minimal 4 24 sl(2, IR)j , j =1, 2, 3, 4 3 23 sl(2, IR) sl(2, IR) , (j, k) = (13), (14), (24) 2 22 j ⊕ k sp(2, IR) 2 20 sp(3, IR) 1 15 ′′ FII ∼= F4 so(7) 1 15 ′ G ∼= G2 0, minimal 2 6 sl(2, IR)j , j =1, 2 1 5 41 dim dim ± G MΘ IR AΘ IR NΘ ′ EVIII ∼= E8 0, minimal 8 120 sl(2, IR)j , j =1,..., 8 7 119 sl(2, IR) sl(2, IR) 6 118 j ⊕ k j +1 ′′ EIX ∼= E8 s0(8), minimal 4 108 sl(2, IR) so(8), j =1, 2 3 107 j ⊕ so(10, 2) 2 90 ′ E7 1 57 42 References [1] A.O. 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