- Email: [email protected]
On irreducible A2-finite G2-modules
JOURNAL
OF ALGEBRA
117, 72-80 (1988)
On Irreducible
A,-Finite
PEKKA
G,-Modules
KEK~L~NEN
University of Jyviiskylii, Department of Mathematics, Seminaarinkatu 15, SF-40100, Jyviiskylii, Finland Communicated by Jacques Tits Received June 7, 1986
We classify irreducible A,-finite G,-modules using step algebra methods introduced by J. Mickelsson [S]. With step algebra methods, D. P. Zhelobenko has proved a general result for the classification of irreducible Harish-Chandra modules over symmetric pairs [9, lo]. The irreducible linite dimensional representations of a Lie algebra chain G2 1 A, have been studied by M. Perroud [7].
1. STEP ALGEBRAS
Let g be a complex Lie algebra, f a semisimple subalgebra in g, and h a Cartan subalgebra of f. Let b* be the complex dual of b, A c lj* a set of simple roots, and A + the set of dominant integral elements A+ = {iEt)*l
(A,a)E.z+VaEA},
where (2, a) = 2(& tl)/(cc, tl) and (., .): h* x $* + @ is the dual of Killing form of f restricted to 6. We choose a basis {h,, .... h,l for h and define an ordering “<” on by setting 1
where f- corresponds to negative roots and f , to positive roots. Let 72 0021-8693/88 $3.00 Copyright 6 1988 by Academic Press, Inc. All rights of reproductmn in any form reserved.
p
the A+ the the
be
A,-FINITE G,-MODULES
73
an ad I-invariant complement of f in g. We choose a basis {t,, ,.., t,,} for p consisting of weight vectors
Ck t,l = Pi(h) t, such that pi>pj when i>j. We denote by U(a) the enveloping algebra of a Lie algebra a. Let t: U(g) + U(g)/U(g) I, be the quotient mapping. We define
and the step algebra of a pair (g, I)
S(g,f) = s’khf)/s’(g, f) n U(g)f+. An element SE S(g, f) has weight p, if [h, s] =p(h) s for all huh; identify h with t(b) and h* with t(b)*. We split
we
where U, is spanned by terms ty ... t$ u, UE U(b). Let P’: U(g) -+ U, be the projection on the first summand. We define P: S(g, f) + U,, P(s) = P’(s’), where s = t(d), s’ E S’(g, f). It is shown in [4] that P is an injection. Furthermore, from Proposition I, 1.8, in [3] we get PROPOSITION1. For each t,E p, i = 1, .... n, there exists si E S(g, f) such that si has weight pi and
where U;E U(b) is such that u,(A) # 0 if i + pi E A +. If u E U(b) we denote by u(A) the value of a polynomial on b* obtained via the replacement h H A(h). Let S,(g, f ) be the subalgebra of S( g, f) generated by s, , .... S, of Proposition 1 and by U(b). We define D = {s E S,(g, f) 1[h, s] = 0 Vh E 6)
74
PEKKA KEKALkNEN 2. DESCRIPTION OF IRREDUCIBLE g-MoDuLEs BY THE ACTION OF D
A g-module V is f-finite if it is a sum of irreducible finite dimensiona f-modules. If 2 E n +, then Vi, denotes the sum of all f-submodules in V with highest weight 2. We define v; = V,n v+.
Vf = {XE V/f+ x=0},
V+ is in a natural way a S(g, f)-module and VT a D-module. Let Z, c U(f) be the left ideal which annihilates the vector of maximal weight in a finite dimensional irreducible f-module with maximal weight 1. We denote by S the left ideal of S,(g, f) generated by elements si of weight ,U~< 0. We set D:=D/Dn(t(U(g)Z,)+S_).
If V, is a minimal component of V, i.e., V, # 0 and VP= 0 for p
3. STEP ALGEBRA S,(G,,A,)
In this section we study the case g = Gz and f = A,. Let h be a Cartan subalgebra of g and (., .): h* x h* + @ the dual of the Killing form K: g x g + C of g restricted to h. For the root system
we have the normalization
(4 a) = 2,
(4 PI= -1,
(878)=4.
For each p E Z there exist elements eP and e-, in g such that for all h E 6 we have
Ck e,,l = Ah) ep,
EC,,e-J = h,,
75
A*-FINITE G2-MODULES
where h, E h is such that rc(hlc,h) = p(h), and for all p and p EC
The structure constants N,,, satisfy N,., = -NIL P and also N,,, p = -N-p, -p’
N,,, = -N,, -P-P’
With respect to our normalization we can set
N rr+2p.r+/3==
12
N,+xp,.=
-1.
From these we get all the other structure constants using the conditions above. The subalgebra f = A2 is determined by the root subsystem C={fa,
&(~+3fl),
k(2a+38)}
ii=(Lxa+3fi). The Cartan subalgebra of f generated by 2 is h. We choose a basis {h, , h,} for h such that cx(h,)= 1, or(h2)=0, fl(h,)=O, and j?(h2)= 1. If AELI+ we set Ai=A(hj) and I
As in Proposition 1, we associate with each e, E p a step s, (we use the same notation for an element of S(g, f) and for an element of S’(g, f))
76
PEKKA KEKKLAINEN s pcr-p=epz-B
W, -h,Nh,
+cl,e-,(h,
+ 1)-e,+2Se-2,-3fl(2hl
-h2)
+ 1)-e,+21,e-.-3pe-.
s~.~2p=e~.~2ct(h2-h,)(h,+1)+e,+~e..2,~3~(h2-h~+1) +ege-.-3fl(hl
+ 1)-erx+Be-I-38e-a.
Each step s,, has weight ~1.The projection P acting on a step sP gives the first term in sP. The steps s, have been computed using extremal projections explained by Zhelobenko [S] (see also [l, 23). Let Z(g) be the center of U(g). We denote by ZO the subalgebra of Z(g) generated by 21and zO, where
Let Jj, be the left ideal in (h-l(h)4 IhEl)}. LEMMA
U(g)/U(g) f,
generated by the set
1. Let LEA+.
L(i,-A,+l)modJ, &
s-.-28sg3sgs-.~28+s-.-p
S@SP= SPSIC,
when p+p$
(0, &B, k(a+8),
1k(a+2b)).
77
If R, > I,, , then s-
(4+mb-4+1) ?-2ps.+.zp=--s,+2ps~.-2p
(2,
+s
+
1)(~2-~,)
&-A, r+pSpzpp (2E”, -I,+
+sfls-p
+ 1 1)(/I, + 1)
A,+2 (21, -A,+ l)(A,-A,) . II mod J,
s
2A, - il, + 2 l)(A,-2,)
1~/~sI+B--s1+2/I.~~.~2P(~,+
+s
(21, - 4 + 2)(1,, + 2) “+Bs-“-B(2~,-~2+1)(~1+1)
A, +2 +sgs-B (2A, --I,+ 1)(%2-A.,) .Q mod J,
~-P~P--s.+w-~-u --s
(2, + l)(~,-jJ
&-A,+1 x+~S-~~~J (21, -A,+ l)(A, + 1)
(2J”, - I”,)(& - I, + 1) +sDs-B (21,-i,+ 1)(&-A,) -(2i,-n,)(i,-r,+l)(fl,-L,).Q
~j’ II, = A,, then
modJ,.
78
PEKKA KEKjiLAINEN
s pI-ps,+p-z(A,
A, +2 A.,+ 1
+2)+s,+&qs-.Pp-
-;L1(A1+2)(L,+3).Q
modJ,
--- 1 +1 I
s~Bsb-z+s,+p~-,-BA
1 3n,(lz,-2)(n,+3).QmodJ,.
We denote here z = t(z,). Proof: By a direct computation. Because the projection P is an injection, it is sufficient to consider the projections P(s,s,). This simplifies the computations greatly.
4. A*-FINITE G2-Mo~u~~s For g = G, and f = A, let again D=(s~S,(g,~))[h,s]=OVh~b) D:=D/Dn(t(U(g)Z,)+S-).
S is the left ideal of S,(g, I) generated by {sPXPzB,SC,-~, sPs}. THEOREM 1. If A E A + such that A, > 1,) then 0: z C. Proof A general element in D is a linear combination of monomials S=Sp, . ..sPnu, u E U(h), where pi + ... + p,, = 0, because s commutes with h. If pn < 0, then s E 0 mod S . If p,, > 0, let k be the last index, for which ,uk< 0. Using Lemma 1 we can write s=a.s,;..s,,,,s,,
. . . s,” + terms of lower degree mod J,,
where a E R. Using induction on n and n - k we see that elements sP,,pi < 0 can be shifted to the left giving zero modulo S . It follows that any element of D is in @. f mod(t(U(g( 1,) + S). On the other hand Q$Dn(t(U(dZJ+S-). THEOREM 2. If 1 E A + such that A, = A,, then 0: z Z,. Proof: As in the proof of Theorem 1, we can show that any element of D is a polynomial of z modulo (t( U(g) IA) + S). Thus
0s; E GM(&)
n (t(U(d 1,) + S- 1.
Next we shall show that t(Z,) n S = 0. A general element s in S
is a
A ,-FINITE G,-MODULES
79
linear combination of monomials So,,... svm, where p, = --c( - j?. If s~t(Z,)nS~ we get pl+ . . + ~1,= 0, because s must commute with b. Using Lemma 1 we can show that s is of the form
Using induction it is easy to seethat all the highest order terms of s include the factor ez+B. On the other hand the same does not hold for z”. It follows that SE t(Z,) n S only when s =O. It is easy to see that t(Z,) n t( U(g) I;,) = 0. Thus 0: 1 t(Z,) z Z,. From Theorems 1 and 2 and Proposition 2 we now get COROLLARY
1. For each 1,E A + such that A, > I, there exists a unique
equivalence class in V.
qf irreducible
f-finite g-modules V such that V, is minimal
COROLLARY 2. For each A E A + such that AZ= A, and each cOE @ there exists a unique equivalence class of irreducible f-finite g-modules V such that V, is minimal in V and zO is represented by the scalar cO.
ACKNOWLEDGMENT I thank Professor Jouko Mickelsson
for helpful discussions.
REFERENCES 1. R. M. ASHEROVA, Yu. F. SMIRNOV, AND V. N. TOLSTOI, Projection operators for simple Lie groups, Teorer. Mar. Fiz. 8 (1971), 255-271; English transl. in Theoref. and Math. Phys. 8 (1971), 813-825. 2. R. M. ASHEROVA, Yu. F. SMIRNOV, AND V. N. TOLSTOI, Description of a class of projection operators for semisimple complex Lie algebras, Mar. Zametki 26 (1979), 15-25; English transl. in Math. Notes 26 (1979), 499-504. 3. A. VAN DEN HOMBERGH, “Harish-Chandra Modules and Representations of Step Algebras,” Ph.D. thesis, Department of Mathematics, Katholik University of Nijmegen, 1976. 4. J. MICKELSSON,Step algebras of semi-simple subalgebras of Lie algebras, Rep. Math. Phys. 4 (1973), 307-318. 5. J. MICKELSSON,On certain irreducible modules of the Lie algebra gl(4, C), Ann. Acad. Sci. Fenn. Ser. A I Math. 1 (1975), 285-296. 6. J. MICKELSSON,Discrete series of Lie superalgebras, Rep. Math. Phys. 18 (1980), 197-210. 7. M. F'ERROUD, On the irreducible representations of the Lie algebra chain G, 3 A,, J. Math. Phys. 17 (1976), 1998%2006.
80
PEKKA KEKiLiINEN
8. D. P. ZHELOBENKO, S-algebras and Verma modules over reductive Lie algebras, Dokl. Akad. Nauk SSSR 273 (1983), 785-788; English trans.]. in Souier Math. Dokl. 28 (1983) 696700. 9. D. P. ZHELOBENKO, Minimal K-types and classification of irreducible representations of reductive Lie groups, Funktsional. Anal. i Prilozhen. 18 (1984) 79-80; English transl. in Functional Anal. Appl. 18 (1984), 333-335. 10. D. P. ZHELOBENKO, S-algebras and Harish-Chandra modules over reductive Lie algebras, Dokl. Akad. Nuuk SSSR 283 (1985) 13061308; English transl. in Sooief Mufh. Dokl. 32 (1985). 328-331.
OF ALGEBRA
117, 72-80 (1988)
On Irreducible
A,-Finite
PEKKA
G,-Modules
KEK~L~NEN
University of Jyviiskylii, Department of Mathematics, Seminaarinkatu 15, SF-40100, Jyviiskylii, Finland Communicated by Jacques Tits Received June 7, 1986
We classify irreducible A,-finite G,-modules using step algebra methods introduced by J. Mickelsson [S]. With step algebra methods, D. P. Zhelobenko has proved a general result for the classification of irreducible Harish-Chandra modules over symmetric pairs [9, lo]. The irreducible linite dimensional representations of a Lie algebra chain G2 1 A, have been studied by M. Perroud [7].
1. STEP ALGEBRAS
Let g be a complex Lie algebra, f a semisimple subalgebra in g, and h a Cartan subalgebra of f. Let b* be the complex dual of b, A c lj* a set of simple roots, and A + the set of dominant integral elements A+ = {iEt)*l
(A,a)E.z+VaEA},
where (2, a) = 2(& tl)/(cc, tl) and (., .): h* x $* + @ is the dual of Killing form of f restricted to 6. We choose a basis {h,, .... h,l for h and define an ordering “<” on by setting 1
where f- corresponds to negative roots and f , to positive roots. Let 72 0021-8693/88 $3.00 Copyright 6 1988 by Academic Press, Inc. All rights of reproductmn in any form reserved.
p
the A+ the the
be
A,-FINITE G,-MODULES
73
an ad I-invariant complement of f in g. We choose a basis {t,, ,.., t,,} for p consisting of weight vectors
Ck t,l = Pi(h) t, such that pi>pj when i>j. We denote by U(a) the enveloping algebra of a Lie algebra a. Let t: U(g) + U(g)/U(g) I, be the quotient mapping. We define
and the step algebra of a pair (g, I)
S(g,f) = s’khf)/s’(g, f) n U(g)f+. An element SE S(g, f) has weight p, if [h, s] =p(h) s for all huh; identify h with t(b) and h* with t(b)*. We split
we
where U, is spanned by terms ty ... t$ u, UE U(b). Let P’: U(g) -+ U, be the projection on the first summand. We define P: S(g, f) + U,, P(s) = P’(s’), where s = t(d), s’ E S’(g, f). It is shown in [4] that P is an injection. Furthermore, from Proposition I, 1.8, in [3] we get PROPOSITION1. For each t,E p, i = 1, .... n, there exists si E S(g, f) such that si has weight pi and
where U;E U(b) is such that u,(A) # 0 if i + pi E A +. If u E U(b) we denote by u(A) the value of a polynomial on b* obtained via the replacement h H A(h). Let S,(g, f ) be the subalgebra of S( g, f) generated by s, , .... S, of Proposition 1 and by U(b). We define D = {s E S,(g, f) 1[h, s] = 0 Vh E 6)
74
PEKKA KEKALkNEN 2. DESCRIPTION OF IRREDUCIBLE g-MoDuLEs BY THE ACTION OF D
A g-module V is f-finite if it is a sum of irreducible finite dimensiona f-modules. If 2 E n +, then Vi, denotes the sum of all f-submodules in V with highest weight 2. We define v; = V,n v+.
Vf = {XE V/f+ x=0},
V+ is in a natural way a S(g, f)-module and VT a D-module. Let Z, c U(f) be the left ideal which annihilates the vector of maximal weight in a finite dimensional irreducible f-module with maximal weight 1. We denote by S the left ideal of S,(g, f) generated by elements si of weight ,U~< 0. We set D:=D/Dn(t(U(g)Z,)+S_).
If V, is a minimal component of V, i.e., V, # 0 and VP= 0 for p
3. STEP ALGEBRA S,(G,,A,)
In this section we study the case g = Gz and f = A,. Let h be a Cartan subalgebra of g and (., .): h* x h* + @ the dual of the Killing form K: g x g + C of g restricted to h. For the root system
we have the normalization
(4 a) = 2,
(4 PI= -1,
(878)=4.
For each p E Z there exist elements eP and e-, in g such that for all h E 6 we have
Ck e,,l = Ah) ep,
EC,,e-J = h,,
75
A*-FINITE G2-MODULES
where h, E h is such that rc(hlc,h) = p(h), and for all p and p EC
The structure constants N,,, satisfy N,., = -NIL P and also N,,, p = -N-p, -p’
N,,, = -N,, -P-P’
With respect to our normalization we can set
N rr+2p.r+/3==
12
N,+xp,.=
-1.
From these we get all the other structure constants using the conditions above. The subalgebra f = A2 is determined by the root subsystem C={fa,
&(~+3fl),
k(2a+38)}
ii=(Lxa+3fi). The Cartan subalgebra of f generated by 2 is h. We choose a basis {h, , h,} for h such that cx(h,)= 1, or(h2)=0, fl(h,)=O, and j?(h2)= 1. If AELI+ we set Ai=A(hj) and I
As in Proposition 1, we associate with each e, E p a step s, (we use the same notation for an element of S(g, f) and for an element of S’(g, f))
76
PEKKA KEKKLAINEN s pcr-p=epz-B
W, -h,Nh,
+cl,e-,(h,
+ 1)-e,+2Se-2,-3fl(2hl
-h2)
+ 1)-e,+21,e-.-3pe-.
s~.~2p=e~.~2ct(h2-h,)(h,+1)+e,+~e..2,~3~(h2-h~+1) +ege-.-3fl(hl
+ 1)-erx+Be-I-38e-a.
Each step s,, has weight ~1.The projection P acting on a step sP gives the first term in sP. The steps s, have been computed using extremal projections explained by Zhelobenko [S] (see also [l, 23). Let Z(g) be the center of U(g). We denote by ZO the subalgebra of Z(g) generated by 21and zO, where
Let Jj, be the left ideal in (h-l(h)4 IhEl)}. LEMMA
U(g)/U(g) f,
generated by the set
1. Let LEA+.
L(i,-A,+l)modJ, &
s-.-28sg3sgs-.~28+s-.-p
S@SP= SPSIC,
when p+p$
(0, &B, k(a+8),
1k(a+2b)).
77
If R, > I,, , then s-
(4+mb-4+1) ?-2ps.+.zp=--s,+2ps~.-2p
(2,
+s
+
1)(~2-~,)
&-A, r+pSpzpp (2E”, -I,+
+sfls-p
+ 1 1)(/I, + 1)
A,+2 (21, -A,+ l)(A,-A,) . II mod J,
s
2A, - il, + 2 l)(A,-2,)
1~/~sI+B--s1+2/I.~~.~2P(~,+
+s
(21, - 4 + 2)(1,, + 2) “+Bs-“-B(2~,-~2+1)(~1+1)
A, +2 +sgs-B (2A, --I,+ 1)(%2-A.,) .Q mod J,
~-P~P--s.+w-~-u --s
(2, + l)(~,-jJ
&-A,+1 x+~S-~~~J (21, -A,+ l)(A, + 1)
(2J”, - I”,)(& - I, + 1) +sDs-B (21,-i,+ 1)(&-A,) -(2i,-n,)(i,-r,+l)(fl,-L,).Q
~j’ II, = A,, then
modJ,.
78
PEKKA KEKjiLAINEN
s pI-ps,+p-z(A,
A, +2 A.,+ 1
+2)+s,+&qs-.Pp-
-;L1(A1+2)(L,+3).Q
modJ,
--- 1 +1 I
s~Bsb-z+s,+p~-,-BA
1 3n,(lz,-2)(n,+3).QmodJ,.
We denote here z = t(z,). Proof: By a direct computation. Because the projection P is an injection, it is sufficient to consider the projections P(s,s,). This simplifies the computations greatly.
4. A*-FINITE G2-Mo~u~~s For g = G, and f = A, let again D=(s~S,(g,~))[h,s]=OVh~b) D:=D/Dn(t(U(g)Z,)+S-).
S is the left ideal of S,(g, I) generated by {sPXPzB,SC,-~, sPs}. THEOREM 1. If A E A + such that A, > 1,) then 0: z C. Proof A general element in D is a linear combination of monomials S=Sp, . ..sPnu, u E U(h), where pi + ... + p,, = 0, because s commutes with h. If pn < 0, then s E 0 mod S . If p,, > 0, let k be the last index, for which ,uk< 0. Using Lemma 1 we can write s=a.s,;..s,,,,s,,
. . . s,” + terms of lower degree mod J,,
where a E R. Using induction on n and n - k we see that elements sP,,pi < 0 can be shifted to the left giving zero modulo S . It follows that any element of D is in @. f mod(t(U(g( 1,) + S). On the other hand Q$Dn(t(U(dZJ+S-). THEOREM 2. If 1 E A + such that A, = A,, then 0: z Z,. Proof: As in the proof of Theorem 1, we can show that any element of D is a polynomial of z modulo (t( U(g) IA) + S). Thus
0s; E GM(&)
n (t(U(d 1,) + S- 1.
Next we shall show that t(Z,) n S = 0. A general element s in S
is a
A ,-FINITE G,-MODULES
79
linear combination of monomials So,,... svm, where p, = --c( - j?. If s~t(Z,)nS~ we get pl+ . . + ~1,= 0, because s must commute with b. Using Lemma 1 we can show that s is of the form
Using induction it is easy to seethat all the highest order terms of s include the factor ez+B. On the other hand the same does not hold for z”. It follows that SE t(Z,) n S only when s =O. It is easy to see that t(Z,) n t( U(g) I;,) = 0. Thus 0: 1 t(Z,) z Z,. From Theorems 1 and 2 and Proposition 2 we now get COROLLARY
1. For each 1,E A + such that A, > I, there exists a unique
equivalence class in V.
qf irreducible
f-finite g-modules V such that V, is minimal
COROLLARY 2. For each A E A + such that AZ= A, and each cOE @ there exists a unique equivalence class of irreducible f-finite g-modules V such that V, is minimal in V and zO is represented by the scalar cO.
ACKNOWLEDGMENT I thank Professor Jouko Mickelsson
for helpful discussions.
REFERENCES 1. R. M. ASHEROVA, Yu. F. SMIRNOV, AND V. N. TOLSTOI, Projection operators for simple Lie groups, Teorer. Mar. Fiz. 8 (1971), 255-271; English transl. in Theoref. and Math. Phys. 8 (1971), 813-825. 2. R. M. ASHEROVA, Yu. F. SMIRNOV, AND V. N. TOLSTOI, Description of a class of projection operators for semisimple complex Lie algebras, Mar. Zametki 26 (1979), 15-25; English transl. in Math. Notes 26 (1979), 499-504. 3. A. VAN DEN HOMBERGH, “Harish-Chandra Modules and Representations of Step Algebras,” Ph.D. thesis, Department of Mathematics, Katholik University of Nijmegen, 1976. 4. J. MICKELSSON,Step algebras of semi-simple subalgebras of Lie algebras, Rep. Math. Phys. 4 (1973), 307-318. 5. J. MICKELSSON,On certain irreducible modules of the Lie algebra gl(4, C), Ann. Acad. Sci. Fenn. Ser. A I Math. 1 (1975), 285-296. 6. J. MICKELSSON,Discrete series of Lie superalgebras, Rep. Math. Phys. 18 (1980), 197-210. 7. M. F'ERROUD, On the irreducible representations of the Lie algebra chain G, 3 A,, J. Math. Phys. 17 (1976), 1998%2006.
80
PEKKA KEKiLiINEN
8. D. P. ZHELOBENKO, S-algebras and Verma modules over reductive Lie algebras, Dokl. Akad. Nauk SSSR 273 (1983), 785-788; English trans.]. in Souier Math. Dokl. 28 (1983) 696700. 9. D. P. ZHELOBENKO, Minimal K-types and classification of irreducible representations of reductive Lie groups, Funktsional. Anal. i Prilozhen. 18 (1984) 79-80; English transl. in Functional Anal. Appl. 18 (1984), 333-335. 10. D. P. ZHELOBENKO, S-algebras and Harish-Chandra modules over reductive Lie algebras, Dokl. Akad. Nuuk SSSR 283 (1985) 13061308; English transl. in Sooief Mufh. Dokl. 32 (1985). 328-331.