Talk at Workshop Quantum Spacetime 16 Zakopane, Poland,

Transcription

1 Talk at Workshop Quantum Spacetime 16 Zakopane, Poland, Invariant Differential Operators: Overview (Including Noncommutative Quantum Conformal Invariant Equations) V.K. Dobrev

2 Invariant differential operators and equations play very important role in the description of physical symmetries. Thus, it is important for the applications in physics to study systematically such operators. In a recent paper we started the systematic explicit construction of invariant differential operators. We gave an explicit description of the building blocks, namely, the parabolic subgroups and subalgebras from which the necessary representations are induced. Thus we

3 have set the stage for study of different non-compact groups and quantum groups obtained by parabolic deformation.

4 We introduce the important notions on the example of the conformal algebras so(n,2) in n-dimensional Minkowski space-time. In that case, there is a maximal Bruhat decomposition that has direct physical meaning: so(n, 2) = M A N Ñ, M = so(n 1, 1), dim A = 1, dim N = dim Ñ = n where so(n 1, 1) is the Lorentz algebra of n-dimensional Minkowski spacetime, the subalgebra A = so(1, 1) represents the dilatations, the conjugated

5 subalgebras N, Ñ are the algebras of translations, and special conformal transformations, both being isomorphic to n-dimensional Minkowski space-time. The subalgebra P = M A N ( = M A Ñ ) is a maximal parabolic subalgebra. There are other special features of the conformal algebra which are important. In particular, the complexification of the maximal compact subalgebra is isomorphic to the complexification of the

6 first two factors of the Bruhat decomposition: K C = so(n, C) so(2, C) = ( ) = so(n 1, 1) C so(1, 1) C = M C A C There are some hermitian-symmetric algebras that share the above-mentioned special properties of so(n, 2) : so(n, 2), sp(n, R), su(n, n), so (4n), E 7( 25)

7 In view of applications to physics, we proposed to call these algebras conformal Lie algebras, (or groups). We have started the study of all algebras in the above class in the framework of the present approach, and we have considered also the algebra E 6( 14).

8 Lately, we discovered an efficient way to extend our considerations beyond this class introducing the notion of parabolically related non-compact semisimple Lie algebras. A semisimple non-compact Lie algebra may have several different parabolic subalgebras P = M A N, where M is reductive Lie algebra, A is abelian, N is nilpotent preserved by the action of A.

9 Definition: Let G, G be two noncompact semisimple Lie algebras with the same complexification G C = G C. We call them parabolically related if they have parabolic subalgebras P = M A N, P = M A N, such that: P C = P C. Certainly, there may be several such parabolic relationships for any given algebra G. Furthermore, two algebras G, G may be parabolically related with different parabolic subalgebras.

10 We summarize the algebras parabolically related to conformal Lie algebras with maximal parabolics fulfilling ( ) in the following table:

11 G K M G M dimv so(n,2) so(n) so(n 1,1) so(p,q), so(p 1,q 1) n 3 n p + q = n + 2; sl(4, R), n = 4 sl(2, R) sl(2, R) su(n,n) su(n) su(n) sl(n, C) R sl(2n, R) sl(n, R) sl(n, R) n 3 n 2 su (2n), n = 2k su (2k) su (2k) sp(2r, R) su(2r) sl(2r, R) sp(r,r) su (2r) rank = 2r 4 r(2r + 1) so (4n) su(2n) su (2n) so(2n,2n) sl(2n, R) n 3 n(2n 1) E 7( 25) e 6 E 6( 26) E 7(7) E 6(6) 27

12 For our purposes we need to restrict to maximal parabolic subalgebras P = M A N with dim A = 1. Let ν be a (non-unitary) character of A, ν A, parameterized by a real number d, called the conformal weight or energy. Further, let µ fix a discrete series representation D µ of M on the Hilbert space V µ, or on a finite-dimensional (non-unitary) representation of M.

13 We call the induced representation χ = Ind G P (µ ν 1) an elementary representation of G [DMPPT], where G, P = M AN are semisimple Lie groups with Lie algebras G, P. Their spaces of functions are: C χ = {F C (G, V µ ) F(gman) = = e ν(h) D µ (m 1 ) F(g)} where a = exp(h) A, H A, m M, n N. The representation action is the left regular action: (T χ (g)f)(g ) = F(g 1 g ), g, g G.

14 An important ingredient in our considerations are the highest/lowest weight representations of G C. These can be realized as (factor-modules of) Verma modules V Λ over G C, where H C is a Cartan subalgebra of G C, the weight ΛΛ(χ) (H C ) is determined uniquely from χ. Another important ingredient of our approach is as follows. We group the (reducible) ERs with the same Casimirs in sets called multiplets. The multiplet corresponding to fixed values of

15 the Casimirs may be depicted as a connected graph, the vertices of which correspond to the reducible ERs and the lines (arrows) between the vertices correspond to intertwining operators. The explicit parametrization of the multiplets and of their ERs is important for understanding of the situation.

16 Actually, the data for each intertwining differential operator consists of the pair (β, m), where β is a (non-compact) positive root of G C, m N, such that the BGG Verma module reducibility condition is fulfilled: (Λ + ρ, β ) = m, β 2β/(β, β) ρ is half the sum of the positive roots of G C.When the above holds then the Verma module with shifted weight V Λ mβ is embedded in the Verma module V Λ. This embedding is realized by a singular vector v s determined by a polynomial P m,β (G ) in the universal enveloping algebra U(G ), G is the

17 subalgebra of G C generated by the negative root generators. More explicitly,, vm,β s = P m,β v 0. Then there exists an intertwining differential operator D m,β : C χ(λ) C χ(λ mβ) given explicitly by: D m,β = P m,β (Ĝ ) where Ĝ denotes the right action on the functions F.

18 Conformal algebras so(n, 2) and parabolically related Let G = so(n, 2), n > 2. We label the signature of the ERs of G as follows: χ = { n 1,..., n h ; c }, n j Z/2, c = d n 2, h [ n 2 ], n 1 < n 2 < < n h, n even, 0 < n 1 < n 2 < < n h, n odd, where the last entry of χ labels the characters of A, and the first h entries are labels of the finite-dimensional nonunitary irreps of M = so(n 1, 1).

19 The reason to use the parameter c instead of d is that the parametrization of the ERs in the multiplets is given in a simple intuitive way: χ ± 1 = {ɛn 1,..., n h ; ±n h+1 }, n h < n h+1, χ ± 2 = {ɛn 1,..., n h 1, n h+1 ; ±n h } χ ± 3 = {ɛn 1,...,n h 2,n h,n h+1 ; ±n h 1 }... χ ± h χ ± h+1 = {ɛn 1, n 3,..., n h, n h+1 ; ±n 2} = {ɛn 2,..., n h, n h+1 ; ±n 1} ɛ = ±, n even 1, n odd

20 Further, we denote by C ± i the representation space with signature χ ± i. The number of ERs in the corresponding multiplets is equal to: W (G C, H C ) / W (M C, Hm C ) = 2(1+ h) where H C, H C m of G C, M C, resp. are Cartan subalgebras At this moment we show the simplest example for the conformal group in 4- dimensional Minkowski space-time so(4, 2).

21 φ µ A µ [λ, ] F [λ,µ] λ J µ µ Φ Simplest example of diagram with conformal invariant operators A µ electromagnetic potential, µ φ = A 0 µ F electromagnetic field, [λ A µ] = λ A µ µ A λ = Fλµ 0 J µ electromagnetic current, λ F λµ = Jµ, 0 µ J µ = Φ 0 superscript 0 indicates that the mapping is not onto

22 φ [0, 0; 0] µ A µ [1, 1; 1] I + I [0, 2; 2] F F + [2, 0; 2] I I + J µ [1, 1; 3] µ Φ [0, 0; 4] Same example accounting the reducibility: F = F + F F ± k = E k ± ih k, E k F k0, H k 1 2 ε klmf lm This diagram shows also the Weyl symmetry realized by integral operators intertwining representations symmetric w.r.t. the bullet where the physically relevant representations T χ of the 4-dimensional conformal algebra su(2, 2) = so(4, 2) are labelled by χ = [n 1, n 2 ; d], where n 1, n 2 are non-negative integers fixing finitedimensional irreducible representations

23 of the Lorentz subalgebra, (the dimension being (n 1 + 1)(n 2 + 1)), and d is the conformal weight. Indeed, (n 1, n 2 ) = (1, 1) is the fourdimensional Lorentz representation, (carried by J µ above), and (n 1, n 2 ) = (2, 0), (0, 2) are the two conjugate threedimensional Lorentz representations, (carried by F ± k above), while the conformal weights are the canonical weights of a current (d = 3), and of the Maxwell field (d = 2).

24 To give explicitly the operators above we pass to line bundles converting the indices to polynomials of two auxiliary spin variables z, z so that a Lorentz representations with signature (n 1, n 2 ) is replaced by a spin-tensor G(z, z) which is a polynomial in z, z of order n 1, n 2, resp. I + = z + + v ) ( zz + + z v + z v + z 1 2 I = z + + v ) ( zz + + z v + z v + z 1 2

25 x ± x 0 ± x 3 v x 1 ix 2, v x 1 + ix 2 ± / x ±, v / v, v / v, F + (z) z 2 (F if 2 + ) 2zF 3 + (F 1 + if 2 + ) F ( z) z 2 (F1 if 2 ) 2 zf 3 (F 1 + if 2 ) J(z, z) zz(j 0 + J 3 ) + z(j 1 ij 2 ) + +z(j 1 + ij 2 ) + (J 0 J 3 )

26 To proceed further we rewrite the I ± operators in the following form: I + = 2 (2I 1 ) 1 I 2 3I 2 I 1 I = 1 ) 2 (2I 3 I 2 3I 2 I 3 (1) where I 1 z, I 2 zz + + z v + z v +, I 3 z The operators I a represent the right action on our functions of the three simple roots of the root system of sl(4, C). Formula (1) means that the operators

27 I ± correspond to singular vectors related to the two non-simple non-highest roots.

28 We pass now to quantum space-time. We use the following version of q-minkowski: x ± v = q ±1 vx ±, x ± v = q ±1 vx ± x + x x x + = λv v, vv = v v where λ q q 1. As expected, the above relations coincide with the commutation relations between the translation generators P µ of the q-conformal algebra.

29 The commutation rules involving the spin variables z, z are: zz = z z x + z = q 1 zx +, x z = qzx λv vz = q 1 zv, vz = qz v λx + zx + = qx + z, zx = q 1 x z + λ v zv = q 1 v z + λx +, z v = q v z The coordinates x ±, v, v, z, z realize a flag manifold Y q of SU q (2, 2). Our functions on Y q are formal power series in x ±, v, v and polynomials in z, z: ˆϕ = i,j,k,l,m,n Z + i n 1, n n 2 z i v j x k xl + vm z n

30 To rewrite the Maxwell equations in these variables we need q-shift operators, q-multiplication operators and q-derivatives acting on the functions ˆϕ(x ±, v, v, z, z). For example: M v ˆϕ = T v ˆϕ = i,j,k,l,m,n Z + i n 1, n n 2 i,j,k,l,m,n Z + i n 1, n n 2 z i v j+1 x k xl + vm z n, z i q j v j x k xl + vm z n and analogously for the other variables. Then the q-difference operators are: ˆD v ˆϕ = 1 λ M 1 v ( Tv Tv 1 ) ˆϕ

31 Then we have for the q-right action of the simple roots: I1 q = ˆD z T z T v T + (T T v ) 1 I2 q ( = q M z ˆD v T 2 + ˆD T + I q 3 = ˆD z T z + M z M z ˆD + T T v T 1 v + + q 1 M z ˆD v λ M v M z ˆD ˆD + T v ) T v T 1 z Now the q-singular vector leads to the q-maxwell operators: ( I q + = 2 1 [2] q I1 q Iq 2 [3] qi2 q Iq 1 Iq = 1 ( 2 [2] q I3 q Iq 2 [3] qi2 q Iq 3 ) )

32 and explicitly: I + q = q2 2 [2] q ( q ˆD v T + + ˆM z ˆD + (T v T + ) 1 T v ) T 2 z (T vt z ) q 2 ( q ˆM z ˆD v T + ˆD + + ˆM z ˆM z ˆD + (T v T + ) 1 T v + + q 1 ˆM z ˆD v (T T + ) 1 λ ˆM v ˆM z ˆD ˆD + (T T + ) 1 T v ) I q ˆD z T z (T v T z ) 1 ; = q ( 2 [2] q q 1 ˆD v (T T + ) ˆM z ˆD + (T v T + ) 1 T v λ ˆM v ˆD ˆD + (T T + ) 1 T v ) T T + T v T z q3 ( q ˆM z ˆD v T + ˆD + + ˆM z ˆM z ˆD + (T v T + ) 1 T v + + q 1 ˆM z ˆD v (T T + ) 1 λ ˆM v ˆM z ˆD ˆD + (T T + ) 1 T v ) ˆD z T T + T v

33 The parametrization of ALL so(4, 2) sextets is: χ ± pνn = { (p, n)± ; 2 ± 1 2 (p + 2ν + n) } χ ± pνn = { (p + ν, n + ν)± ; 2 ± 2 1 (p + n) } χ ± pνn = { (ν, p + ν + n)± ; 2 ± 1 2 (n p) } where (r, s) + = (s, r), (r, s) = (r, s). They correspond to the finite-dimensional irreps of su(2, 2) of dimension: pνn(p + ν)(n + ν)(p + ν + n)/12

34 χ pνn d ν 1 χ pνn d n 2 d p 3 χ pνn χ + pνn d p 3 d n 2 χ + pνn d ν 1 χ + pνn For p = n = ν = 1 we get the simplest example considered above with:

35 φ χ 111, Φ χ+ 111, A µ χ 111, J µ χ + 111, F ± χ ± 111. We return to the general case so(n, 2) :

36 χ 1. d 1 χ 2 χ h 1 d h 1 χ h d h d h+1 χ h+1 χ + h+1 d h+1 χ + h d h 1. χ + h 1 d h χ + 2 d 1 χ + 1 General even case so(p,q), p + q = n + 2, n = 2h even

37 χ 1. d 1 χ 2 χ h d h χ h+1 d h+1 χ + h+1 d h. χ + h χ + 2 d 1 χ + 1 General odd case so(p,q), p + q = n + 2, n = 2h + 1 odd

38 Matters are arranged so that in every multiplet only the ER with signature χ 1 contains a finite-dimensional nonunitary subrepresentation in a subspace E. The latter corresponds to the finite-dimensional unitary irrep of so(n + 2 = p + q) with signature {n 1,..., n h, n h+1 }. Although the diagrams are valid for arbitrary so(p, q) (p+q 5) the content is different. We comment only on the ER with signature χ + 1. In all cases it contains an UIR of so(p, q) realized on

39 an invariant subspace D of the ER χ + 1. (Other ERs contain more UIRs.) If pq 2N the mentioned UIR is a discrete series representation. (Other ERs contain more discrete series UIRs.) And if q = 2 the invariant subspace D is the direct sum of two subspaces D = D + D, in which are realized a holomorphic discrete series representation and its conjugate anti-holomorphic discrete series representation, resp. Note that the corresponding lowest weight

40 VM is infinitesimally equivalent only to the holomorphic discrete series, while the conjugate highest weight VM is infinitesimally equivalent to the antiholomorphic discrete series.

41 Above we restricted to n > 2. The case n = 2 is reduced to n = 1 since so(2, 2) = so(1, 2) so(1, 2). The case so(1, 2) is special and must be treated separately. But in fact, it is contained in what we presented already. In that case the multiplets contain only two ERs which may be depicted by the pair χ ± 1 in all pictures that we presented. And they have the properties that we described. That case was the first given already in 1947 independently by Bargmann and Gel fand et al.

42 The Lie algebra su(n, n) and parabolically related Let G = su(n, n), n 2. The maximal compact subgroup is K = u(1) su(n) su(n), while M = sl(n, C) R. The number of ERs in the corresponding multiplets is equal to ( W (G C, H C ) / W (M C, Hm C 2n ) = n The signature of the ERs of G is: χ = { n 1,..., n n 1, n n+1..., n 2n 1 ; c } ) n j N, c = d n

43 Below we give the diagrams for su(n, n) for n = 3, 4. These are diagrams also for the parabolically related sl(2n, R) and for n = 2k these are diagrams also for the parabolically related su (4k). (We omit su(2, 2) = so(4, 2), su (4) = so(5, 1), sl(4, R) = so(3, 3).) We only have to take into account that the latter two algebras do not have discrete series representations. We use the following conventions. Each intertwining differential operator is represented by an arrow accompanied by a

44 symbol i j...k encoding the root β j...k and the number m βj...k which encode the reducibility of the corresponding Verma module.

45 Λ 0 Λ c Λ + c 3 3 Λ a Λ b Λ b Λ c Λ c Λ d 3 Λ 24 d 5 Λ 35 1 e Λ + e Λ d 24 Λ + d Λ + Λ c c 25 Λ Λ + b Λ + b a 3 15 Λ + 0 Pseudo-unitary symmetry su(3, 3) Pseudo-unitary symmetry su(n,n) is similar to conformal symmetry in n 2 dimensional space, for n = 2 coincides with conformal 4-dimensional case. By parabolic relation the su(3,3) diagram above is valid also for sl(6,r).

46 Λ 0 Λ Λ 10 Λ Λ Λ Λ Λ Λ 21 4 Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ Λ + Λ Λ Λ Λ + 21 Λ Λ Λ + 11 Λ Λ Λ + 01 Λ Λ 4 4 Λ Λ + 0 Pseudo-unitary symmetry su(4, 4) (By parabolic relation the diagram above is valid also for sl(8,r) and su (8).)

47 The Lie algebras sp(n, R) and sp( n 2, n 2 ) (n even) Let n 2. Let G = sp(n, R), the split real form of sp(n, C) = G C. The maximal compact subgroup is K = u(1) su(n), while M = sl(n, R). The number of ERs in the corresponding multiplets is: W (G C, H C ) / W (M C, Hm C ) = 2n The signature of the ERs of G is: χ = { n 1,..., n n 1 ; c }, n j N,

48 Below we give pictorially the multiplets for sp(n, R) for n = 3, 4, 5, 6. For n = 2r these are also multiplets for sp(r, r). (We omit sp(2, R) = so(3, 2) and sp(1, 1) = so(4, 1).) We only have to take into account that sp(r, r) has discrete series representations but no highest/lowest weight representations.

49 Λ Λ a Λ c Λ 3 22 b Λ + 13 b 2 12 Λ + c Λ + a 3 11 Λ + 0 Main multiplets for Sp(3,IR)

50 Λ Λ a Λ 4 33 Λ b c Λ c 1 Λ d Λ 24 d Λ 3 23 e Λ + e Λ d Λ + 13 Λ c + c 4 22 Λ b 3 12 Λ + d Λ + a 4 11 Λ + 0 Main multiplets for Sp(4,IR) and Sp(2,2)

51 Λ Λ a Λ b Λ Λ c c Λ Λ d d Λ e Λ e Λ e Λ f Λ f Λ f Λ g Λ 33 Λ + h 1 15 g Λ 5 33 Λ g h Λ + g Λ + f Λ+ f Λ + f Λ + e 4 23 Λ e Λ d Λ + d Λ + Λ 2 14 c + c 5 22 Λ b Λ + e Λ + a 5 11 Λ + 0 Main multiplets for Sp(5,IR)

52 Λ 0 Λ b Λ c Λ c Λ Λ d d Λ Λ e e Λ e Λ 16 2 f Λ f Λ f Λ f Λ g Λ g 4 35 Λ g Λ Λ j 2 26 h Λ Λ j 1 16 h Λ k Λ Λ 4 35 h j 6 44 Λ 1 Λ 4 16 l k Λ Λ + Λ l Λ m 5 34 m k Λ 25 Λ + l Λ m 6 33 m Λ l Λ Λ + m 3 25 m Λ Λ + k 33 l Λ + Λ + k l 34 Λ Λ + j 33 h Λ k Λ + h Λ + 6 j Λ + 24 h Λ + Λ + j g 3 Λ + Λ + g 14 6 g Λ f 6 Λ+ Λ + 33 f f Λ+ f 22 Λ e 15 5 Λ+ 23 e 6 Λ + e Λ d 6 Λ+ 22 d Λ + 4 c 13 6 Λ+ 22 c Λ + b 5 12 Λ + a 6 11 Λ Λ a 5 56 Main multiplets for Sp(6, R) and Sp(3,3)

53 The Lie algebra so (12) The Lie algebra G = so (2n) is given by: so. (2n) = {X so(2n, C) : J n CX = XJ n C} = { X = a b a, b gl(n, C), b ā t a = a, b = b }. dim R G = n(2n 1), rank G = n. The maximal compact subalgebra is K = u(n). Thus, G = so (2n) has discrete series representations and highest/lowest weight representations. The split rank is r [n/2].

54 For even n = 2r we choose a maximal parabolic P = MAN such that A = so(1, 1), M = M max r = su (n). Further we restrict to our case of study G = so (12). We label the signature of the ERs of G as follows: χ = { n 1, n 2, n 3, n 4, n 5 ; c }, n j Z +, c = d 15 2 where the last entry labels the characters of A, and the first five entries are

55 labels of the finite-dimensional (nonunitary) irreps of M = su (6). The main multiplets are in 1-to-1 correspondence with the finite-dimensional irreps of so (12), i.e., they are labelled by the six positive Dynkin labels m i N. The number of ERs/VMs in the corresponding multiplets is : W (G C, H C ) / W (K C, H C ) = = W (so(12, C)) / W (sl(6, C)) = 32

56 where H is a Cartan subalgebra of both G and K. They are given explicitly in the Figure below. The statements made for the ER with signature χ 0 in the previous cases remain valid also here. Also the conjugate ER χ + 0 contains a unitary discrete series subrepresentation. All the above is valid also for the algebra so(6, 6), cf., however, the latter

57 algebra does not have highest/lowest weight representations.

58 Λ Λ a Λ b Λ Λ c c Λ d Λd Λ e Λ Λ e e Λ Λ f f 6 34 Λ f 4 35 Λ Λ g g Λ 3 25 g Λ g 2 15 Λ + Λ + g 1 16 g 634 Λ + f Λ + f 6 Λ + 2 f Λ + e 2 Λ + 15 e Λ + Λ + d 3 14 d Λ + 3 Λ + 14 c c 5 23 Λ + e Λ + b 4 13 Λ+ a 6 12 Λ+ 0 Fig. 1. SO (12) main multiplets

59 The Lie algebras E 7( 25) and E 7(7) Let G = E 7( 25). The maximal compact subgroup is K = e 6 so(2), while M = E 6( 26). The signatures of the ERs of G are: χ = { n 1,..., n 6 ; c }, n j N. The same can be used for the parabolically related exceptional Lie algebra E 7(7). We only have to take into account that the latter algebra has discrete series representations but not highest/lowest weight representations.

60 Λ Λ a 6 67 Λ b 5 57 Λ c ,47 Λ d 3 37 Λ e Λ e ,47 1 1,37 Λ Λ f f 4 1 1,37 2 2,47 Λ 27 g Λ g 5 1 1,37 Λ 4 27,4 27 h Λ h 1 1, ,4 Λ 6 27,45 j 3 17 Λ Λ j j 7 1 1, , ,4 Λ 27,46 k 3 Λ 17 k Λ k 1 1, , , Λ 4 17,4 Λ Λ l l l 6 27, , , ,4 Λ Λ m m Λ m 7 27, , ,4 5 17, ,34 Λ Λ n n 7 Λ 27,46 n 2 17, , , , ,45 Λ + Λ + n n Λ + 7 n 2 27,46 17, ,46 517, ,35 Λ + Λ + m m 2 Λ + m 17, , , , ,35,4 Λ , ,46 l Λ+ 3 17,35,4 7 27,46 l Λ+ 1 17,25,4 l 5 Λ+ Λ + Λ + 17, ,35,4 6 17,46 1 k k k 17,25,4 7 27,46 Λ + j 3 17,35,4 5 17,36 Λ+ j 1 17,25,4 6 17,46 Λ+ j 4 Λ + Λ + 17,36,4 1 17,25,4 5 17,36 h h 2 Λ + g 17,36, ,25,4 Λ+ g 4 17,36,4 Λ ,25,4 f Λ+ 3 17,26,4 f 2 17,36,45 Λ + e 3 17,26,4 Λ+ e 2 17,36,45 Λ +d 4 17,26,45 Λ + c 5 17,26,45,4 Λ + b 6 17,26,35,4 Λ + a Λ ,16,35,4 Fig. 14. Main multiplets for E 7( 25) and E 7(7).

61 The Lie algebras E 6( 14), E 6(6) and E 6(2) Let G = E 6( 14). The maximal compact subalgebra is K = so(10) so(2), while M = su(5, 1). The signature of the ERs of G is: χ = { n 1, n 3, n 4, n 5, n 6 ; c}, c = d The above can be used for the parabolically related exceptional Lie algebras E 6(6) and E 6(2). We only have to

62 take into account that only the algebra E 6(2) has discrete series representations but no highest/lowest weight representations.

63 Λ Λ a 4 2,4 3 Λ 24 b 52,45 Λ c Λ c , ,46 Λ d Λ d 1 Λ d ,46 2, Λ e Λ 4 e , ,46 1 Λ f 14 Λ f Λ f ,4 Λ 5 26 ẽ Λ 6 2,46 f o 4 Λ 25 f , ,4 2 25,4 Λ g 1 6 2, , ,4 Λ Λ g g h ,4 2, Λ Λ Λ g h 1 o h Λ g 2 25,4 ˆk ,4 Λ Λ j o 4 15,4 6 2, ĥ ,4 Λ j 2 25,4 Λ j Λ j 5 15,34 6 2, , , ,45 Λ k o Λ k Λ k Λ k Λ k 6 α 5 α 4 α 3 α 1 α Λ + k o 5 2,46 6 Λ+ k Λ + k 15, Λ + 15,4 3 26,4 4 j 2 Λ + 16 Λ , j Λ , ,4 ĥ ,4 1 26, j o 16 2 Λ+ 5 15,4 3 26,4 Λ + 16 ˆk + j g o Λ + 1 h 3 6 Λ ,45 Λ + 15,34 16 h g 26, Λ + 15,4 15,4 3 h + g 26,4 Λ g ,45 26, , Λ + g 15, , ,4 Λ ,34 f o Λ + f , ,4 Λ + Λ + ẽ 6 15,34 f Λ + f 1 26,45 Λ + f 4 16, , ,4 26, ,4 Λ + e Λ + e Λ , , , ,45 d Λ + Λ + d d 1 26, , , ,34 Λ + c Λ + Λ + c 3 16,45 b 5 16, ,35 Λ + a 2 16,35,4 Λ + 0 Main Type for E 6( 14), E 6(6) and E 6(2) Λ + k Λ+ k