arXiv:hep-th/0702152v13 7 Oct 2020 hs ti motn o h plctosi hsc ostud to physics in applications the for ( theory, important is string applicatio it and latest Thus, supergravity the theory, to field [1]), conformal e.g., in Maxwell cf., the examples in more occurrences early (for the from i starting role - important metries very play operators differential Invariant Introduction 1. h uesmercadqatmgopstig n sncsayfrapp easily for is necessary approach is Our models. and integrable settings cuspidal. and subalgebras, group not theory quantum ingredie are parabolic and main they maximal supersymmetric when the the the even of also important one explicate also of We are description explicit subalgebras. giving parabolic by operators tial ntepeetpprw tr h ytmtcepii constructi explicit systematic the start we paper present the In h bu aa nentoa etrfrTertclPhysi Theoretical for Center International Salam Abdus The nain ieeta Operators Differential Invariant nttt o ula eerhadNcerEnergy Nuclear and Research Nuclear for Institute o o-opc i Groups: Lie Non-Compact for 2Tairdk huse 74Sfi,Bulgaria Sofia, 1784 Chaussee, Tsarigradsko 72 aaoi Subalgebras Parabolic ..Bx56 taaCsir 11 Costiera Strada 586, Box P.O. ugra cdm fSciences of Academy Bulgarian praetaddress) (permanent 41 ret,Italy Trieste, 34014 ..Dobrev V.K. Abstract and 1 so spr)ieeta operators (super-)differential of ns h ecito fpyia sym- physical of description the n o eetrve,c.eg,[2]). e.g., cf. review, recent a for ytmtclysc operators. such systematically y ’lebr,Drc equations, Dirac, d’Allembert, , no nain differen- invariant of on t h cuspidal the - nts iain ostring to lications cs eeaie to generalised ic these since 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. [3-41]. We shall follow a procedure in representation theory in which such operators appear canonically [24] and which has been generalized to the supersymmetry setting [42] and to quantum groups [43]. We should also mention that this setting is most appropriate for the classification of unitary representations of superconformal symmetry in various dimensions, [44],[45],[46],[47], for generalization to the infinite-dimensional setting [48], and is also an ingredient in the AdS/CFT correspondence, cf. [49]. (For a recent paper with more references cf. [50].) 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., [43,51,52,53,54,55], and in integrable systems, cf. e.g., [56,57,58,59]. 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 [24] 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 a maximal compact subgroup of G. Then we have an Iwasawa decomposition G = KAN, where A is abelian simply connected, a 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 a minimal parabolic subgroup of G. A parabolic subgroup P = M ′A′N ′ is any subgroup of G (including G itself) which contains a minimal parabolic subgroup. The number of non-conjugate parabolic subgroups is 2r, 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 representations induced from them generate all (admissible) irreducible representations of G [7]. (For the role of parabolic subgroups in the construction of unitary representations we refer to [10],[13].) 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 [8],[14], so that M ′ has discrete series representations [3].1 However, often induction∩ from a non-cuspidal parabolic is also convenient.
1 ′ The simplest example of cuspidal parabolic subgroup is P0 when M = M is compact. In all
2 Let P = M ′A′N ′ be a parabolic subgroup. Let ν be a (non-unitary) character of A′, ν ′∗, where ′ is the Lie algebra of A′. If P is cuspidal, let µ fix a discrete series ∈ A Aµ ′ representation D of M on the Hilbert space Vµ , or the so-called limit of a discrete series representation (cf. [23]).2 Although not strictly necessary, sometimes it is convenient to induce from non-cuspidal P (especially if P is a maximal parabolic). In that case, we use any non-unitary finite- µ ′ dimensional irreducible representation D of M on the linear space Vµ . More than this, except in the case of induction from limits of discrete series, we can always work with finite-dimensional representations Vµ by the so-called translation. Namely, when P is non-minimal and cuspidal, then instead of the inducing discrete series representation of M ′ we can consider the finite-dimensional irrep of M ′ which lies on the same orbit of the Weyl group (in other words, has the same Casimirs). G We call the induced representation χ = IndP (µ ν 1) an elementary representation of G [15]. (These are called generalized principal series⊗ ⊗ representations (or limits thereof) in [23].) Their spaces of functions are:
= C∞(G, V ) (gman) = e−ν(H) Dµ(m−1) (g) (2.1) Cχ {F ∈ µ | F · F } where a = exp(H) A′, H ′, m M ′, n N ′. The special property of ∈ ∈ A ∈ ∈ 3 the functions of χ is called right covariance [15],[24] (or equivariance). Because of this covariance theC functions actually do not depend on the elements of the parabolic subgroup P = M ′A′N ′. F χ The elementary representation (ER) acts in χ as the left regular representation (LRR) by: T C ( χ(g) )(g′) = (g−1g′) , g,g′ G. (2.2) T F F ∈ One can introduce in χ a Fr´echet space topology or complete it to a Hilbert space (cf. [6]). We shall need alsoC the infinitesimal version of LRR:
. d (X )(g) = (exp( tX)g) , (2.3) LF dtF − |t=0
where, χ, g G, X ; then we use complex linear extension to extend the definitionF to ∈ a C representation∈ of∈ GCI . G 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 The ERs differ from the LRR (which is highly reducible) by the specific representation spaces χ. In contrast, the ERs are generically irreducible. The reducible ERs form a measureC 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 [24]. For this we shall use the right action of CI (the complexification of ) by the standard formula: G G . d (X )(g) = (g exp(tX)) , (2.4) R F dtF |t=0 where χ, g G, X ; then we use complex linear extension to extend the F ∈ C ∈ ∈ G CI definition to a representation of . Note that this action takes out of χ for some X but that is exactly why it is usedG for the construction of the iFntertwiningC differential operators.
We can show this property in all cases when Vµ is a highest weight module, e.g., the case of the minimal parabolic subalgebra and when (M ′,M ′ K) is a Hermitian symmetric pair. In fact, we agreed that, except when inducing from limi∩ ts of discrete series, the space Vµ will be finite-dimensional.
Then Vµ 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 . [If M ′ = M is abelian or discrete h i µ 0 then Vµ is one–dimensional and ˜χ coincides with χ; so we set ϕ = .] On these functions the right action of CI Cis defined by:4 C F G (X ϕ)(g) v , (X )(g) . (2.6) R ≡ h 0 R F i
Part of the main result of our paper [24] is: Proposition. The functions of the CI-valued realization ˜ χ of the ER satisfy : T Cχ X ϕ = Λ(X) ϕ, X CI , Λ ( CI )∗ (2.7a) R · ∈ H ∈ H X ϕ = 0 ,X CI , (2.7b) R ∈ G+
4 In the geometric language we have replaced the homogeneous vector bundle with base space
G/P and fibre Vµ with a line bundle again with base space G/P (also associated to the principal P -bundle with total space G). The functions ϕ can be thought of as smooth sections of this line bundle.
4 5 CI where Λ = Λ(χ) is built canonically from χ , ± are from the standard triangular decomposition CI = CI CI CI . 6 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 CI Λ CI CI = +. Thus, V = U( −)v0, where v0 is the highest weight vector, U( −) is the B H ⊕ G ∼ G CI 7 G universal enveloping algebra of − . Verma modules are universal in the following sense: every irreducible HWM is isomorphicG 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 [12] in V a unique vector vs, called singular vector, which has the properties (2.7) of a highest weight vector with shifted weight Λ mβ : −
X v = (Λ mβ)(X) v ,X CI , (2.9a) s − · s ∈ H X v = 0 ,X CI , (2.9b) s ∈ G+
The general structure of a singular vector is [24] :
− − vs = Pmβ(X1 ,...,Xℓ )v0 , (2.10)
where Pmβ is a homogeneous polynomial in its variables of degrees mki, where ki ZZ+ come from the decomposition of β into simple roots: β = k α , α ∆ , the system∈ i i i ∈ S
5 P 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, cf. Section 3. 6 Note that we are working here with highest weight modules instead of the lowest weight modules used in [24]; also the weights are shifted by ρ with respect to the notation of [24]. 7 For more mathematically precise definition, cf. [12].
5 of simple roots, X− are the root vectors corresponding to the negative roots ( α ), α j − j j being the simple roots, ℓ = rank CI = dim CI is the (complex) rank of CI . 8 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.9 Then there exists an intertwining differential operator [24] ∈ ∈ ′ : ˜ χ ˜ χ , (2.11) Dmβ T −→ T where χ′ is such that Λ′ = Λ′(χ′) = Λ mβ. − The most important fact is that (2.11) is explicitly given by [24] :
ϕ(g) = P ((X−) ,..., (X−) ) ϕ(g) , (2.12) Dmβ mβ 1 R ℓ R − where Pmβ is the same polynomial as in (2.10) and (Xj )R 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.10 Let be the Lie algebra of G, θ be a Cartan involution in , and = be a Cartan decompositionG of , so that θX = X,X , θX = X,XG G; isK a ⊕ maximal P compact subalgebra of .G Let be a maximal subspace∈K of −which is∈ anP abelianK subalgebra of ; r = dim is theG splitA(or real) rank of , 1 r Pℓ = rank . The subalgebra is calledG a CartanA subspace of . G ≤ ≤ G A P 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} λ The elements of ∆A are called -restricted roots. For λ ∆A , A are called - restricted root spaces, dim λ 1.A Next we introduce some∈ orderingG (e.g., the lexicographicA R GA ≥
8 CI A singular vector may also be written in terms of the full Cartan-Weyl basis of G−. 9 If β is a real root, (i.e., β| CI = 0, where Hm is the Cartan subalgebra of M), then some Hm conditions are imposed on the character ǫ representing the finite group Md [17]. 10 The results are easily extended to real linear reductive Lie groups with a finite number of components.
6 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 general Mis a compact reductive LieA algebra,K andM we{ can∈K| write = ∀ ∈, A} where . M M Ms ⊕Ma s =[ , ] is the semisimple part of , and a is the abelian subalgebra central in M. M M M M M We mention also that a Cartan subalgebra of is given by: = , Hm M Hm Hs ⊕Ma where s is a Cartan subalgebra of s . Then a Cartan subalgebra of is given by: = H .11 M H G H Hm ⊕A Next we recall the Bruhat decomposition [60]:
= + − , (2.16) G N ⊕M⊕A⊕N . − and the subalgebra 0 = called a minimal parabolic subalgebra of . (Note that we may take equivalentlyP M⊕A⊕N+ instead of − .) G 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 over IR. Restricting CI IR one obtains the Bruhat decomposition of (with G = 0) from the triangular decomposition−→ of CI = + CI −, and obtainsG theM minimal G G ⊕ H ⊕ GCI + − + parabolic subalgebras 0 from the Borel subalgebra = , (or instead of ). Furthermore, in this caseP dim = dim ±. B H ⊕ G G G IR K IR N ′ A standard parabolic subalgebra is any subalgebra of containing 0 . The number of standard parabolic subalgebras, including and P, is 2Gr. P P0 G Remark: In the complex case a standard parabolic subalgebra is any subalgebra ′ of CI containing . The number of standard parabolic subalgebras, including and PCI , is 2Gℓ, ℓ = rank B. B G CI G ♦ Thus, if r = 1 the only nontrivial parabolic subalgebra is . P0 Thus, further in this section r > 1.
11 Note that H is a θ-stable Cartan subalgebra of G such that H ∩ P = A. It is the most noncompact among the non-conjugate Cartan subalgebras of G.
7 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 hasG A the analogue of the Bruhat decomposition (2.16): P
= ′+ ′ ′ ′− , (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 λi , i Θ. Then a standard parabolic subalgebra corresponding 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
+ + + Clearly, ∅ = 0 , Sr = , since ( ) = 0, (Sr) = . Further, we need toP bringP (2.20)P in the formG (2.17). First,N ∅ define (Θ)N as the algebraN generated + − . + G. by (Θ) and (Θ) = θ (Θ) . Next, define (Θ) = (Θ) , and Θ as the orthogonalN complementN (relativeN to the Euclidean structurA eG of )∩A of (Θ) inA . Then = (Θ) . Note that dim (Θ) = Θ , dim = r ΘA. Next,A define:A A A ⊕AΘ A | | AΘ −| | + . λ − . + Θ = A , Θ = θ Θ . (2.21) N +⊕ + G N N λ∈∆A−∆Θ
± ± ± . + − Then = (Θ) Θ . Next, define Θ = (Θ) (Θ) (Θ). Then isN the centralizerN ⊕N of in (mod ).M Finally,M⊕A we can derive:⊕N ⊕N 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 8 Thus, we have rewritten explicitly the standard parabolic Θ in the desired form (2.17). The associated (generalized) Bruhat decomposition (2.18) isP given now explicitly as:
+ + + + = 0 = Θ (Θ) 0 = Θ Θ = G N ⊕ P N ⊕N ⊕ P N ⊕ 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 symmetric pair when the maximal compact subalgebra contains aG uK(1) factor. Then has highest and lowest weight representations. All theseK algebras have discrete series representations.G ♦
3. The complex simple Lie algebras considered as real Lie algebras
Let c be a complex simple Lie algebra of dimension d and (complex) rank ℓ. We need the triangularG 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: dimIR c = 2d, rankIR c = dimIR = 2ℓ, dimIR = d, ± G G H K rankIR = ℓ, dimIR = d ℓ. Note that the maximal compact subalgebra of c is isomorphicK to the compactN real− form of . K G Gk Gc Thus, the complex simple Lie algebras do not have discrete series representations (and highest/lowest weight representations over IR).
Let Hj , j = 1,...,ℓ, be a basis of , i.e., = c.l.s. Hj , j = 1,...,ℓ , (where c.l.s. stands for complex linear span), suchH that eachH ad(H ) has{ only real eigenvalues.} Let . j = IR = r.l.s. Hj , j = 1,...,ℓ , where r.l.s. stands for real linear span. Then the IwasawaA H 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 ⊕···⊕ 9 In fact, the basis of consists of the vectors iH , j =1,...,ℓ . The Bruhat decompo- M { j } sition of c is: G = + − . (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 ( c, ) looks the same as the complex root system ( c, ), but the restricted roots have multiplicityG A 2, since dim ± = 2dim ±. G H IR N CI N Let Θ be a string subset of Sℓ of length s. The Θ-factor of the corresponding parabolic subalgebra is: M = u(1) u(1) , ℓ s factors , (3.6) MΘ Gs ⊕ ⊕···⊕ − where s is a complex simple Lie algebra of rank s isomorphic to a subalgebra of c. Thus, theG complex simple Lie algebras, considered as real noncompact Lie algebras, do notG have non-minimal cuspidal parabolic subalgebras. Thus, it is enough to consider elementary representations induced from the minimal parabolic subgroup P0 = MAN, where M = U(1) ... U(1), (ℓ factors), A = ∼ ± 12 × × ∼ SO(1, 1) ... SO(1, 1), (ℓ factors), N ∼= exp . Thus, the signature χ = [µ, ν], consists of×ℓ integer× numbers µ ZZ giving the unitaryN character µ =(µ ,...,µ ) of M, i ∈ 1 ℓ and of ℓ complex numbers νi CI giving the character ν =(ν1 ,...,νℓ) of A, νj = ν(Hj ). 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 . Thus the elementary representations (in particular, the right j j j j ∈ H covariance conditions) for a complex semisimple Lie group Gc are given by: P ∞ ˜ = C (G ) CΛ,Λ {F ∈ c | (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 ν = λ + λ˜ , µ = λ λ˜ ZZ j j j j j − j ∈ 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.
10 and the last condition in (3.8) stresses that we have uniqueness on the compact subgroup M of the Cartan subgroup Hc = MA of Gc. 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, i.e., when (2.8) holds for either Λ or Λ.˜ 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 ∈ + The maximal parabolic subalgebras have -factors as follows MΘ = u(1), i =1,...,ℓ , (3.9) MΘ Gi ⊕ where i is a complex simple Lie algebra of rank ℓ 1 which may be obtained from c by deletingG the i-th node of the Dynkin diagram of −. G 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: t t G θX = X, where X is the transpose of X . Thus, ∼= so(n), and is spanned by matrices− (r.l.s. stands for real linear span): ∈ G K
=r.l.s. X e e , 1 i [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 ¯ = ν1,...,νs , s n, be a partition of n: j=1 νj = n. Then the set Θ corresponding to{ the partition} ≤ν ¯ 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(¯ν0) = (corresponding to the minimal parabolic). 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!) Θ(¯ν) subalgebras. The parabolic subalgebras in these cases are called associated, and thisM 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. 12 language above to each nj = 1 there is an entry νj =2inν ¯ bringing an sl(2, IR) factor to Θ , i.e., M 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 Θj = sl(j, IR) sl(n j, IR) , 1 j n 1 , M ⊕ − ≤ ≤ − (4.15) ± dim Θ = 1, dim = j(n j) A j NΘj − (Note that the cases j and n j are isomorphic, or coinciding when n is even and 1 − j = 2 n.) Only one of the maximal ones is cuspidal, namely, for = sl(4, IR), n = 4 and j = 2 we have G = sl(2, IR) sl(2, IR) . (4.16) MΘ2 ⊕ 13 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 SL(2n, CI) JnCg = gJnC , Jn . (5.1) { ∈ | ≡ 1n 0 } − The Lie algebra = su∗(2n) is given by: G . su∗(2n) = X sl(2n, CI) J CX = XJ C = { ∈ | 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 < rank su∗(2n)=2n 1). The complimentary space is given by: K − 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 dimension 2n(n N1). The subalgebra is given by: G A − 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 . ∼ ⊕···⊕ 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 14 roots: Θ = Sij = 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) + rank su(r) = p + r 1 = rank su(p, r), and highest/lowest weight representations.K − 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. 15 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 dimension n(2n 1).N The simple root system ( , ) looks as that of the symplecticG A 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 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) 16 The subalgebras ± which form the root spaces of the root system ( , ) are of real dimension r(2p 1).N The restricted simple root system ( , ) looks as thatG A of the orthog- − G A onal algebra Br, however, the root spaces of the long roots have multiplicity 2, the short simple root, say αr , has multiplicity 2(p r), and there is also a root 2αr with multiplicity 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 Θ subalgebras are of the form: M 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 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 | ∈ ∈ } 17 ∗ 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 1 numbers, since then rank = rank so(p) + rank so(r) = 2 (p + r 2) < rank so(p, r) = 1 (p + r). It has highest/lowestK 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 dimension r(p 1).N Except in the case p = r the restricted simple rootG systemA ( , ) − 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 Θj = j ,j (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 − 18 Next we note that we can include the case when the second factor in Θ is compact by just extending the range of s to r. M 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 The algebras Θ have highest/lowest weight representations only when s = r 2 or s = r, since thenM the second factor is so(p r +2, 2), so(p r), resp. − − − 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 max j = 1, 2 and the number (p j)(r j) is even. In addition, j have highest/lowest weight representations only− when −r j =0, 2, (or p j = 2).M − − 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 { } ≤ ≤ − In these cases, we have factor of the form so(r s,r s), i.e., there will be no cuspidal parabolic if r s is odd.M − − − 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) Only the second subcase, namely, o , has highest/lowest weight representations. MΘ (¯n) 19 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,... ⊕ − − max Of these, only 2 for r = 4 has highest/lowest weight representations (it belongs to the second subcaseM of (7.14)). 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 ∈ − 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). 20 The maximal compact subalgebra of = sp(n, IR) is ∼= u(n), thus sp(n, IR) has discrete series representations (and highest/lowestG weight representations).K Explicitly, A 0 = X = A u(n) . (8.5) K { 0 tA | ∈ } − The subalgebras ± which form the root spaces of the root system ( , ) are of real dimension n2. N G A 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 Θs = s +1,...,n . This string brings in Θ a factor sp(n ∈s, IR), which has discrete series{ representations.} Thus cuspidality dependsM 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(¯n0) = , Θn = . Then the Θ-factors of the cuspidal parabolic subalgebras of sp(n, IR) are∅ given as∅ follows: M = 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 Θ(¯nn−1) enumerates all cuspidal parabolic subalgebras of sl(n, IR), including the minimalM case = 0. MΘ The maximal parabolic subalgebras, cf. (2.24), 1 j n, contain Θ subalgebras of the form: ≤ ≤ M max = sl(j, IR) sp(n j, IR) , (8.8) Mj ⊕ − i.e., the only maximal cuspidal are those for j =1, 2. 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 } 21 The Cartan involution is given explicitly by: θX = γ0Xγ0 . Thus, ∼= sp(p) sp(r), and has discrete series representations (but not highest/lowest weightK representations).⊕ MoreG 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. Hj ep−r+j,p−r+j ep+j,p+j e2p+j,2p+j + e2p+r+j,2p+r+j , 1 j r . A { ≡ − − ≤ ≤ (9}.4) The subalgebras ± which form the root spaces of the root system ( , ) are of real dimension r(4p N1). G A − 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 Br, however, the short simple root say, α , has multiplicity 4(p r),G theA 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 = Sij = i,...,j , where 1 i j The corresponding factor in Θ is given by algebra sp(p j,r j) which has discrete series representations. If Θ containsM in addition some other string− then− it would bring some su∗(2(s+1)) factor and the corresponding will not have discrete series representations. MΘ 22 Thus the nontrivial cuspidal parabolic subalgebras are given by Θj from (9.6) and the corresponding is: MΘ = sp(p j,r j) sp(1) sp(1), j factors . (9.7) Mj ∼ − − ⊕ ⊕···⊕ All these j do not have highest/lowest weight representations. The other factors in M ± the cuspidal parabolic subalgebras have dimensions: dim j = j, dim j = j(4p + 4r 4j 1). Extending the range of j we include the minimalA parabolicN subalgebra for j =−r and− the case ′ = = for j = 0. M P G The maximal parabolic subalgebras corresponding to (2.24) have -factors given by: MΘ max = su∗(2j) sp(p j,r j) , 1 j r . (9.8) Mj ⊕ − − ≤ ≤ The ± factors in the maximal parabolic subalgebras have dimensions: dim ( ±)max = N Nj j(4p + 4r 6j + 1). The only cuspidal maximal parabolic subalgebra is Θ1 , using ∗ − P su (2) ∼= su(2) ∼= sp(1) and noting that (9.7) and (9.8) coincide for j = 1. 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 ∗ ∗ ∗ ∗ 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- tations).G 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 | ∈ − − } − 23 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 dimension n(n 1)N [n/2]. G A − − 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 − ⊕ ⊕···⊕ − All these j have highest/lowest weight representations. The other factors in the M ± cuspidal parabolic subalgebras have dimensions: dim j = j, dim j = j(4n 4j 3). Extending the range of j we include the minimal parabolicA case forN j = r =[n/−2],− and the case ′ = = for j = 0 which is also cuspidal. M P G max The maximal parabolic subalgebras enumerated by Θj from (2.24) have Θ-factors as follows: M 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 diagrams 24 [61],[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 subal- M gebras. 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), dimIR = 42, dimIR = 36. This real form does not have discrete seriesK representations. P N For a split real form the Satake diagram coincides with the Dynkin diagram of the corresponding complex Lie algebra [6]. 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)). All these Θ , except the last case (so(4, 4)), have highest/lowest weight representations. M 14 We could do this, since by being explicit we could consider simultaneously cases with different Satake diagrams. 25 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( Θ ) = 25 M ∼ M ∼ ⊕ N (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), dimIR = 40, dimIR = 36. This real form has discrete seriesK representations.⊕ P N The split rank is equal to 4, while = u(1) u(1). M ∼ ⊕ The Satake diagram is: α6 ◦ | (11.5) α◦1 −−−α ◦2 −−−α ◦3 −−−α ◦4 −−−α ◦5 | {z } Thus, the reduced root system| is presented{z by a Dynkin-Satak} e diagram looking like the 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 26 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(5, 3) 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 ⊕ { } su(3, 3) , 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 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 su(3, 3) , ⊕ ⊕ Θ = {2}, 3, 4 { } All these have highest/lowest weight representations. MΘ ′′′ 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), dimIR = 32, dimIR = 30. This real form has discrete seriesK representations⊕ (and highest/lowestP weight representations).N The split rank is equal to 2, while = so(6) so(2). M ∼ ⊕ 27 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 su(5, 1) , Θ = 2 NΘ 21 { } n Both are maximal (dim = 1), the second is 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 ∼= f4, dimIR = 26, dimIR = 24. This real form does not have discrete series representations.K P N 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 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), dimIR = 70, dimIR = 63. This real form has discrete series representations.K P N 28 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) All these Θ , except the last two cases (so(4, 4), so(6, 6)), have highest/lowest weight representations.M 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 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 = 50 (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 29 ′′ 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), dimIR = 64, dimIR = 60. This real form has discrete seriesK representations.⊕ P N 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 F4 simple roots we shall give the Θ-factors of all parabolic subalgebras: M = 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 { } 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 30 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 { } the last one being also maximal. All these Θ have highest/lowest weight representa- tions. M ′′′ 11.7. EVII : E7 ′′′ Another real form of E7 is denoted as E7 , sometimes as E7(−25) . The maximal compact ± subgroup is ∼= e6 so(2), dimIR =54, dimIR = 51. This real form has discrete series representationsK ⊕ (and highest/lowestP weight represN entations). 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 α1 ,α5 ,α6 , are mapped to λ3 , λ2 , λ1 , resp., of (11.23). M Using the above enumeration of C3 simple roots we shall give the Θ-factors of all parabolic subalgebras: M 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 31 The last three give rise to the maximal parabolic subalgebras. The cuspidal parabolic subalgebras are those containing so(8) , Θ = , minimal = sl(2, IR) so(8) , Θ = ∅3 (11.26) Θ 3 M so(10, 2) ,⊕ Θ = {2}, 3 { } the last one being also maximal. All these Θ have highest/lowest weight representa- tions. M ′ 11.8. EVIII : E8 ′ The split real form of E8 is denoted as E8 , sometimes as E8(+8) . The maximal compact ± subgroup ∼= so(16), dimIR = 128, dimIR = 120. This real form has discrete series representations.K P N 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- 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) 32 ′ All these Θ , except the last three cases (so(4, 4), so(6, 6),E7), have highest/lowest weight representations.M 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 Taking into account (2.24) the maximal parabolic subalgebr as 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( Θ ) = 104 M ∼ ⊕ N (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 ∼= e7 su(2), dimIR = 54, dimIR = 51. This real form has discrete series representations.K ⊕ P N 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 33 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 F4 simple roots we shall give the Θ-factors of all parabolic subalgebras: M = 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 ∅ sl(2, IR)j so(8) , Θ = j , j =1, 2 Θ = ⊕ { } (11.34) M so(10, 2) , Θ = 2, 3 e′′′ , Θ = {2, 3}, 4 7 { } the last one being also maximal. All these Θ have highest/lowest weight representa- tions. M 34 ′ 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), dimIR =28, dimIR = 24. This real form has discrete series representations.K ⊕ P N 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 { } sl(2, IR)j sl(2, IR)k , Θ = j,k = 1, 3 , 1, 4 , 2, 4 Θ = ⊕ { } { } { } { } (11.37) M sp(2, IR) , Θ = 2, 3 so(4, 3) , Θ = {1, 2}, 3 { } sp(3, IR) , Θ = 2, 3, 4 { } the last two being also maximal. All these Θ, except the last but one, have high- est/lowest weight representations. M ′′ 11.11. FII : F4 ′′ Another real form of F4 is denoted as F4 , sometimes as F4(−20) . The maximal compact ± subgroup ∼= so(9), dimIR = 16, dimIR = 15. This real form has discrete series representations.K P N 35 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 ′ 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), dimIR =8, dimIR = 6. This real form has discrete series representations.K ⊕ P N 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. All have highest/lowest weight representations. MΘ 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 [62] 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. Acknowledgements. The author would like to thank for hospitality the Abdus Salam International Center 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. The author would like to thank the anonymous referee for numerous remarks which contributed to improving the exposition. 36 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 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 37 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 ⊕ ⊕ su(3, 3) 1 24 EIII = E′′′ so(6) so(2) 2 30 ∼ 6 ⊕ su(5, 1) 1 21 iv EIV ∼= E6 so(8) 2 24 38 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 so(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 so(4, 3) 1 15 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 39 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 so(8), minimal 4 108 sl(2, IR) so(8), j =1, 2 3 107 j ⊕ so(10, 2) 2 90 ′ E7 1 57 40 References [1] A.O. Barut and R. R¸aczka, Theory of Group Representations and Applications, II edition, (Polish Sci. Publ., Warsaw, 1980). [2] J. Terning, Modern Supersymmetry: Dynamics and Duality, International Series of Monographs on Physics # 132, (Oxford University Press, 2005). [3] Harish-Chandra, Ann. Math. 116 (1966) 1-111. [4] A.W. Knapp and E.M. Stein, Ann. Math. 93 (1971) 489-578; II : Inv. Math. 60 (1980) 9-84. [5] I.N. Bernstein, I.M. Gel’fand and S.I. Gel’fand, Funkts. Anal. Prilozh. 5 (1) (1971) 1-9; English translation: Funkts. Anal. Appl. 5 (1971) 1-8. [6] G. Warner, Harmonic Analysis on Semi-Simple Lie Groups I, (Springer, Berlin, 1972). [7] R.P. Langlands, On the classification of irreducible representations of real algebraic groups, Math. Surveys and Monographs, Vol. 31 (AMS, 1988), first as IAS Princeton preprint (1973). [8] D.P. Zhelobenko, Harmonic Analysis on Semisimple Complex Lie Groups, (Moscow, Nauka, 1974, in Russian). [9] B. Kostant, in: Lecture Notes in Math., Vol. 466 (Springer-Verlag, Berlin, 1975) pp. 101-128. [10] J. Wolf, Unitary Representations of Maximal Parabolic Subgroups of the Classical Groups, Memoirs Amer. Math. Soc. 180, (AMS, 1976). [11] D.P. Zhelobenko, Math. USSR Izv. 40 (1976) 1055-1083. [12] J. Dixmier, Enveloping Algebras, (North Holland, New York, 1977). [13] J. Wolf, Classification and Fourier inversion for parabolic subgroups with square inte- grable nilradical, Memoirs Amer. Math. Soc. 225, (AMS, 1979). [14] A.W. Knapp and G.J. Zuckerman, in: Lecture Notes in Math., Vol. 587 (Springer, Berlin, 1977) pp. 138-159; Ann. Math. 116 (1982) 389-501. [15] V.K. Dobrev, G. Mack, V.B. Petkova, S.G. Petrova and I.T. Todorov, Harmonic Anal- ysis on the n-Dimensional Lorentz Group and Its Applications to Conformal Quantum Field Theory, Lecture Notes in Physics, Vol. 63 (Springer-Verlag, Berlin-Heidelberg- New York, 1977). [16] V.K. Dobrev and V.B. Petkova, Reports Math. Phys. 13 (1978) 233-277. [17] B. Speh and D.A. Vogan, Jr., Acta Math. 145 (1980) 227-299. [18] D. Vogan, Representations of Real Reductive Lie Groups, Progr. Math., Vol. 15 (Boston-Basel-Stuttgart, Birkh¨auser, 1981). [19] H. Schlichtkrull, Inv. Math. 68 (1982) 497-516. [20] B. Speh, Inv. Math. 71 (1983) 443-465. [21] R.L. Lipsman, Trans. AMS 285 (1984) 845-854. 41 [22] V.K. Dobrev, J. Math. Phys. 26 (1985) 235-251; V.K. Dobrev, Lett. Math. Phys. 9 (1985) 205-211. [23] A.W. Knapp, Representation Theory of Semisimple Groups (An Overview Based on Examples), (Princeton Univ. Press, 1986). [24] V.K. Dobrev, Rep. Math. Phys. 25 (1988) 159-181; first as ICTP Trieste preprint IC/86/393 (1986). [25] H.P. Jakobsen, in: Lect. Notes in Phys., Vol. 261 (1986) 253. [26] R.J. Baston and M.G. Eastwood, The Penrose Transform, Its Interaction with Rep- resentation Theory, (Oxford Math. Monographs, 1989). [27] C.R. Graham et al, J. London Math. Soc. (2), 46 (1992) 557-565. [28] V.K. Dobrev and P. Moylan, Fort. d. Phys. 42 (1994) 339-392. [29] T. Kobayashi, Inv. Math. 117 (1994) 181-205. [30] T.P. Branson, G. Olafsson and B. Orsted, J. Funct. Anal. 135 (1996) 163-205. [31] M. Eastwood, Suppl. Rend. Circ. Mat. Palermo, Serie II, Numero 43 (1996) 57-76. [32] I. Dimitrov and I. Penkov, J. Transf. Groups, 3 (1998) 241-253. [33] L. Dolan, C.R. Nappi and E. Witten, JHEP 0110 (2001) 016, hep-th/0109096. [34] X. Gomez and V. Mazorchuk, Comm. Algebra, 29 (2001) 5329-5334. [35] A.W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Progr. Math., vol. 140 (Boston-Basel-Stuttgart, Birkh¨auser, 2002). [36] A. Francis, Alg. Colloquium, 10 (2003) 95-100. [37] B. Kostant, Inv. Math. 158 (2004) 181-226. [38] I. Dimitrov, V. Futorny and I. Penkov, Comm. Math. Phys. 250 (2004) 47-63. [39] K. Erdmann, Archiv d. Math. 82 (2004) 385-390. [40] K. Baur and N. Wallach, Represent. Theory, 9 (2005) 1-29. [41] H. Sabourin and R.W.T. Yu, Journal of Lie Theory 16 (2006) 57-65; math.RT/0407354. [42] V.K. Dobrev and V.B. Petkova, Fortschr. d. Phys. 35 (1987) 537-572. [43] V.K. Dobrev, Lett. Math. Phys. 22 (1991) 251-266; V.K. Dobrev, J. Phys. A: Math. Gen. 26 (1993) 1317-1334; first as G¨ottingen University preprint, (July 1991); V.K. Dobrev and P.J. Moylan, Phys. Lett. 315B (1993) 292-298; V.K. Dobrev, J. Phys. A: Math. Gen. 27 (1994) 4841-4857 & 6633-6634, hep-th/9405150. V.K. Dobrev, Phys. Lett. 341B (1994) 133-138 & 346B (1995) 427. [44] V.K. Dobrev and V.B. Petkova, Phys. Lett. 162B (1985) 127-132; V.K. Dobrev and V.B. Petkova, Lett. Math. Phys. 9 (1985) 287-298; V.K. Dobrev and V.B. Petkova, Proceedings, eds. A.O. Barut and H.D. Doebner, Lecture Notes in Physics, Vol. 261 (Springer-Verlag, Berlin, 1986) p. 291-299 and p. 300-308. [45] S. Minwalla, Adv. Theor. Math. Phys. 2 (1998) 781-846, hep-th/9712074. [46] V.K. Dobrev, J. Phys. A35 (2002) 7079-7100, hep-th/0201076; V.K. Dobrev and R.B. Zhang, Phys. Atom. Nuclei, 68 (2005) 1660-1669, hep-th/0402039; V.K. Dobrev, 42 A.M. Miteva, R.B. Zhang and B.S. Zlatev, Czech. J. Phys. 54 (2004) 1249-1256; hep- th/0402056. [47] C. Carmeli, G. Cassinelli, A. Toigo and V.S. Varadarajan, Comm. Math. Phys. 263 (2006) 217-258, hep-th/0501061. [48] V.K. Dobrev, Phys. Lett. 186B (1987) 43-51; V.K. Dobrev and A.Ch. Ganchev, Mod. Phys. Lett. A3, 127 (1988). [49] V.K. Dobrev, Nucl. Phys. B553 (1999) 559-582; hep-th/9812194. [50] V.K. Dobrev, J. Phys. A39 (2006) 5995-6020; hep-th/0512354. [51] V.M. Futorny and D.J. Melville, Alg. Repres. Theory, 1 (1998) 135-153. [52] F. Fauquant-Millet and A. Joseph, J. Transf. Groups, 6 (2001) 125-142. [53] A. Joseph and D. Todoric, Alg. Repr. Theory, 5 (2002) 57-99. [54] V.D. Lyakhovsky, Parabolic twists for algebras sl(n), math.QA/0510295. [55] J. Brundan and A. Kleshchev, Comm. Math. Phys. 254 (2005) 191-220. [56] J. de Boer and L. Feher, Comm. Math. Phys. 189 (1997) 759-793. [57] M. Gerstenhaber and A. Giaquinto, Lett. Math. Phys. 40 (1997) 337-353. [58] J. Dorfmeister, H. Gradl and J. Szmigielski, Acta Appl. Math. 53 (1998) 1-58. [59] E. Karolinsky, A. Stolin and V. Tarasov, Lett. Math. Phys. 71 (2005) 173-178. [60] F. Bruhat, Bull. Soc. Math. France, 84 (1956) 97-205. [61] I. Satake, Ann. Math. 71 (1960) 77-110. [62] V.K. Dobrev, in preparation. 43