categories of nonstandard highest weight modules for...

Math. Z. 221, 193-209 (1996) Matllematis ZeitschriFt

© Springer-Verlag 1996

Categories of nonstandard highest weight modules for affine Lie algebras B. Cox l, V. Futorny 2'*'**, D. Melville 3

i Department of Mathematical Sciences, University of Montana, Missoula, MT 59812-1032, USA (e-mail: [email protected]) 2 Department of Mathematics, Kiev University, Kiev, 252617, Ukraine 3 Department of Mathematics, St. Lawrence University, Canton, NY 13617, USA (e-mail: [email protected] )

Received 21 October 1993; in final form 5 July 1994

0 Introduction

Modules of Verma type induced from nonstandard Borel subalgebras were studied extensively in [C1] and [FS]. The most significant development is made in the case when the central element c acts with a non-zero charge. In this case, the appropriate category ~ of modules for non-twisted affine algebras was introduced in [C2], and BGG duality was proved for the objects of dT~" 2 .

The present paper may be considered as a generalization of [C2]. For an arbitrary affine Lie algebra, we study nonstandard generalized Verma modules Mx, s(2) induced from a nonstandard parabolic subalgebra.

If 2(c)4:0, then the structure of Mx, s(2) is completely determined by the "finite" part (the sum of all finite-dimensional weight spaces) of Mx, s(2) (Proposition 4.5). When X is connected (5.1), we describe the submodules of Mx, s(2) (Theorem 5.2) and construct a generalized strong BGG resolution (Theorem 5.7).

In section 6 we define certain "truncated" categories dgx, s(2,q) which in- clude the category ~ as a particular case when S = 0 and q = 0. The central element c acts as a non-zero scalar 2(c) on any object of d~x,s(2,q). We show that the categories d~x,s(2,q) have enough projectives (Proposition 6.10) and prove the BGG duality theorem (Theorem 6.14). Here, nonstandard generalized Verma modules play the role which Verma modules originally played in the category C.

The crucial result is Theorem 6.8, which establishes the equivalence of the category Ox, s(2,q) and the category c S (2 ,q ) of modules for a certain affine Kac-Moody subalgebra. Many structural facts about the later category

* Current address: Department of Mathematics, Queen's University, Kingston ON K7M 3N6 (e-mail: [email protected]) ** NSERC International Fellow at Queen's University

194 B. Cox et al.

cS(A,q) were obtained in ([RW]) and we use them to our advantage to obtain information about (Px, s(2,q). In particular through the above equivalence of categories we obtain the BGG duality theorem and the generalized strong BGG resolution.

For a Lie algebra a , we denote by U(a ) the universal enveloping algebra of a . Let H be a Cartan subalgebra of a . An a-module V is called a weight module if V = ~),ze~/* V,t where V,I = {v E Vlhv = 2(h)v for all h E H}. Denote P(V) = {2 E H*IV~ 4=0 }. I f 2 E P(V), then 2 is called a weight of V and V~ is called a weight subspace. I f b c a , we set V b = {v E V[bv = 0}.

1 Modules of Verma type and their properties

In this section we review the construction and basic properties of Verma-type modules described in [C1] and [FS].

1.1. Let g be an affine Lie algebra of rank n + 1 with a Cartan subalgebra H, one-dimensional center C = ~ c C H and a root system A = A im U A re, where A re is the set of real roots and A/m = {k6 I k E 7/\ {0}} is the set of imaginary roots with indivisible imaginary root 6. Fix a basis n = {~0 . . . . . ~,} of A such that 3 n = Y]i=l kiwi with ko = 1 and either -~o + 6 E A or ½(-~0 + 6) E A.

For any 2 E H* and subset P c A which is closed under addition (i.e. if ~,/~ E P , and ~ + / ~ E A, then ~ + ] ~ E P ) , and such that P U - P = A and P M - P = 0, consider a subalgebra g(P) = ~ p g~ and a g-module

M(2 ,P) = U(g) @U(g(P)+H)

associated with P and 2, where C is viewed as a g(P)@ H-module under the action (x + h)l = 2(h)l for all x E g(P), h E H.

These modules are called Verma-type modules. Let W be the Weyl group of A, J = {1 . . . . . n}, X C J . Consider a linear

functional ~bx on H* defined by

EiEa \x~; - (EiEJ\xki)o~; if X =[=J, 4~x

t En=0 ~*, if X = J

where ~*(~j) = 3ij, i,j E {0, 1, . . . ,n}. Set P(X) = {~ E Al~bx(~) > 0} U {~ E Al~bx(~) = 0,q~j(~) > 0}. It was

shown in [JK] and [F1] that any P as defined above is W x {+l}-equivalent to P(X) for some X c o r.

Denote by Mx(2) := M(2,P(X)). Let Ax be a root subsystem of A generated by {~i + n6ti E X,n E 71}, A + = Ax 71P(X),Q = y~e~Z~,Qx = ~--]~ax Z~, Qx (resp. Q+) denote the monoid in H* generated by P(X) (resp. A +) and R(X) = P(X) \ Ax.

For any 2 E H*, denote 2 1 Q~c = {2 - #lP E Q~}. We will write v < 2 i f v E 2 $ Q +.

Highest weight modules for affine Lie algebras 195

1.2 Remarks. (i) If X = J , then Mj(2) = M(2) is a Verma module with highest weight 2.

(ii) If X:~J, then Mx(2) possesses both finite and infinite-dimensional weight spaces. Moreover, 0 < dimMx(2)u < oo if and only if # E 2 .[ Q+.

1.3. Let mx ~ = ~ n f c g+~ and m x = m~.@H®m x. Let ~ be the subalgebra of rrtx generated by all g~, 0~ E Ax fl A ~, and G = EkEZ\{0} ~k6.

1.4 Remarks. (i) If X = 0, then m~ = G @ H. (ii) If the Coxeter-Dynkin diagram corresponding to X is connected, then

Ox is the derived algebra of an affine Lie algebra of rank IXl + 1.

m X. 1.5. In general, assume that X = Ui=I i and the diagrams corresponding to each X/ are connected. Then fix = ~--]~"=1 g~), where [g~),g~)] = 0, i4:j, (']i%, g~) = Cc and g~) is the derived algebra of an affine Lie algebra of rank IX/I + 1 for each i. Set gx = fix + H. Then m x = gx @ g~ where g~m C G, g~c m O(gx fq G)- - -G and [g~m,~x ] ---0 (see IF2]). Note that ~iv m = g~m @Cc is a Heisenberg algebra.

1.6. The module Mx(2) has a unique irreducible quotient Lx(2). Denote Mf (2 ) = ~-']Mx(2)i, and Lf(2) = y"~Lx(2)u, for # E 2 J. Q+. Then Mf(2 ) and Lf(2) are rex-modules. Moreover, MT(2) has a unique maximal mx-submodule and Lf(2) is its unique irreducible quotient ([C1], [FS]).

Consider the subalgebras Ux ~ = ~--~R(x) ga:~, and Px = u~. G rex. Clearly g = p x ~ u x .

The key result about Verma-type modules is the following:

1.7 Theorem. ([FS], [C1]). Let 2 E H* and 2(c)4:0 .

(i) A g-module Mx(2) is irreducible if and only / fMf(2) is an irreducible mx -module.

(ii) I f N is a submodule of Mx(2) then N = U(g)®U(ox)(N M Mr(2) ) where N M MY(2) is a px-module with trivial action of u~.

(iii) Lx(2)-~ U(g)®V(px)Lf (2), where Lf(2) is an irreducible px-module with trivial action of uf~, which implies that Lx(2) ~ U(u x ) ®¢ Lf(2) as a vector space.

1.8, I f X = 0, then T = ~ t ~ z + M0(A)~-k6 is a G @ •c-submodule of M0(2) generated by a vacuum vector with eigenvalue 2(c) (see [K, Sect.9.13]). Hence, T is an irreducible G ~ Cc-module whenever 2(c)4=0 [K, Corollary 9.13] and we have the following

Corollary. (IF3]) I f 2 E H* and 2(c)4=0, then M0(2 ) is irreducible.

196 B. Cox et al.

1.9. Denote g± = g~m N mx ~. Let a E C \ {0} and M be a ~m-module such that for every m E M, cm = am and there exists n such that (g+)~v = 0. Then, by [K, Lemma 9.13b], M is isomorphic to a direct sum of copies of U(g~m)®vC~+ea:~)~ ~-- U(g-) , where C is regarded as a g+ ~3 ~c-module with c - 1 = a and trivial action of g+. This implies that M -~ U ( g - ) ®c M g+ as a vector space.

1.10. Furthermore, we will consider only Verma-type modules Mx(2) with 2(c)=~0. Due to Corollary 1.8, the case X = 0 is not interesting and from now on we will always assume that X ~ 0 .

2 "Truncated" categories of gx-modules

In this section, we follow [RW] and consider certain "truncated" categories of gx'modules.

2.1. Let Fix = {Gti[i E X } and S ~ Hx, t2s = ~-~es 7~ot, f2 + = f2s N Q+. Let rift = E~Eo~-\{0) g±~, mx, s = rt s O H @ rt~-, U~s = ( E ~ e ~ \ ~ + g±~)

fqgx, and Px, s = mx, s~)U+,s, so that gx = Px, s@U~s. In addition we let Hs denote the subalgebra ~--~Eo~-\(0}[g~,g-~]"

2.2 Remarks. (i) If X is not connected then dx need not have a basis. (For example consider the Dynkin diagram of s l (3 ,C) and take X be the set of two roots corresponding to two nodes that are not connected. A rather straight forward argument proves that dx does not have a basis. Nevertheless, since dx C Q, the algebra gx satisfies the conditions (T1)-(T2) of [RW, Sect. 1] with respect to Q (see also [C2, Remark 1.4]).

(ii) S satisfies the conditions (S1)-($6) of [RW, Sect. 3]. (iii) mx, s is a reductive Lie algebra. In particular, rex, ~ = H.

2.3. For q E Z+, define

qQ+ = {it = ~- 'm~ E Qf~lm~ E 7z+' ~-'m~ > q} ~E~

= ~# E H ' l # - 2 = ~-~m~ot E Qx, m~ E 7~, H(A,q) l rt E lt

(I t - ,~)+ = ~ m~t E Q+ \ qQ+ [ ,

%

mete]g+ J HS(2,q) = {It E Fl(2,q)[Its = #l~s is integral dominant}.

Naturally, there is one-to-one correspondence between elements of IIS(2,q) and irreducible finite-dimensional rex, s-modules Vs(It) with highest weight It ~ H(,~,q).


2.4. Let 2 E H*, q E Z+. Denote by cS(2 ,q) (resp. cS(2,q)) the full subcategory of the category of weight ~x-modules (resp. rex-modules) M such that:

(i) dim M u < c~ for all /2 E H*, (ii) M is a direct sum of Vs(/2)'s, /2 E IIS(2,q) (cf. [RW]).

The categories cS(2, q) and cS(2, q) are stable under the operations of taking submodules, quotients and finite direct sums. All the results of [RW, Sect. 4-6] can be applied to the categories cS(2 ,q) and cS(2,q) (see Remark 2.2, (ii)).

2.5. Through the rest of the paper we fix X c {1 . . . . . n}, X # O , S ~ H x , 2 E H*, q E Z+ and write / / = / / ( 2 , q ) , H s = HS(2,q), C s = cS(2,q) , ~ s = cS(2,q).

2.6. Let/2 E 17 s. Then we can view Vs(/2) as a Px, s-module with trivial action of ufc s and construct the generalized Verma ~x-module

M(Vs(/2)) = U(~x) ®u(p~,s) Vs(/2).

It is also clear that M(Vs(/2)) E C s. If S = 0, then M(Vo(/2)) = M(/2) is the usual Verma module for ~x with highest weight/2. For any S the module M(Vs(/2)) is a homomorphic image of M(/2). Denote by L(/2) the unique irreducible quotient of M(#). The modules L(#) exhaust all irreducible objects in C s [RW, Prop 3.3 (ii)].

Fix /2 E H s and set U(u+,s) ~ = • U(n+,s)~, where the sum is over all a E Qx such that v + a ~ / / for any v E P(Vs(/2)). Note that U(u~,s) has a natural Qx-gradation. Then

A(/2,S) = (U(u~ , s ) /~ (U~sY) ® Zs(/2)

is a weight pc, s-module where U+s acts on the left and mx, s by the tensor product action. Let

A-(/2,s)= E a(/2,s)v vEH*\H

and

Then

~i(/2,s) = A(/2,S)/U(px, s ),i(/2,S) .

PS(/2) = U(~x) ®uo, x ~ ~'(/2, s) is a projective object in C s.

There is a one-to-one correspondence between the irreducible objects in C s and the non-isomorphic indecomposable direct summands of the PS(p)'s, /2 E FIS(2,q) ([RW, Corollary 4.13]). Denote by IS(/2) the indecomposable projective cover for L(/2).

It follows from [RW, Corollary 4.10] that IS(/2), for any /2 E IIS(~.,q), has a generalized Verma composition series, i.e. there is a filtration IS(/2) =

198 B. Cox et al.

I ° D 11 D . . . D I l D I z+t = (0) o f submodules such that t i l l i+l "~ M ( V s ( I A i ) ) , Iti E I-lS(A,q), i = 0 . . . . . I. Let ( lS ( lo : M ( V s ( v ) ) ) be the number of indexes i in {0, 1 . . . . , l} such that v = gi.

2.7. Let a be a complex Lie algebra, b C a and a : a ~ a a linear involu- tive anti-automorphism such that b + a(b ) = a .

Let 2 : b ~ C be a 1-dimensional representation of b . Following [JK], we say an a -modu le V is a highest weight module o f highest weight 2 (with respect to b) , if there exists a vector v E V such that

U(a)v = V and xv = 2(x)v for x E b .

Let M be an a-module. A highest weight series (with respect to b) for M is a chain

(0) = M0 C M 1 C M 2 C . . .

o f submodules of M such that oo M (i) Ui=0 i = M and

(ii) Mi/Mi - l is a highest weight module (with respect to b ) for all i.

2.8. It follows from [RW, Lemma 5.1] that any M E ~ s (resp. M E C s) has a highest weight series {Mi} with respect to m~ @ H (resp. ~x M ( m E @ H ) ) such that, i f Mi/Mi-1 has highest weight 2i, then 2i < 2j implies i > j .

2.9. Any module M E C s has a local composition series ([MP]), i.e. for any # E H, there exists a sequence M = Mk D -- . D M0 = 0 o f modules in C s and a subset 1 C { 1 , . . . , k} such that

(i) I f i E I , then Mi/Mi-1 ~-- L(2i), 2i ~ //; (ii) I f i ~ I , then (Mi /Mi- l )v = 0 for all v > #.

Let [M : L(p)] denote the multiplicity o f L(#) in M, i.e. the number of i ' s in I such that # = 2i. []

2.10 Theorem [RWl. (i) L e t M E C s a n d # E l i s . Then

[M : L(#)] = dim H o m a x ( I S ( # ) , M ) .

(ii) Let #, v E I I s. Then

(pS(#) : M ( V s ( v ) ) = dim H o m ~ x ( P S ( # ) , M ( V s ( v ) ) ) .

(iii) ( B G G Duali ty) Let #, v E H s. Then

(zs(~) : M(gs(v))) = [M(gs(V)) : L(~)].


3 "Truncated" categories of rex-modules

Through the rest of this paper we will assume that 2(c) # 0. In this section we establish the equivalence of the categories C s and ~s.

The proof of the next Proposition follows the general lines of [C2, Theorem 2.4].

3.1 Proposition. Let 2(c)#:0. Then the categories C s and ~ s are equivalent.

Proof If M is an rex-module in ~s, then M g+ is a non-trivial ~x-module in C s due to the relation [gx, g+] = 0. Define a functor

Inv : ~ s __~ C s

such that Inv(M) = M g+, Inv( f ) = ftMg+ for all M , N E ~s , and f E Hommx(M,N).

Now let V E C s. Consider V as a ~x @ g+'module with trivial action of g+ and define the rex-module Ind(V) = U(mx)®U(gxeg+) V which is isomorphic to U ( g - ) ® ¢ V as a vector space.

We have a standard Z-gradation on U(g~m): U(g~ m) = ~bkezU(g~m)k. Let U(gx)k = U ( g - ) N U(g~z)k. It is clear that if Vs(ft), g E IIS(2,q), is a direct summand of V (as an rex.s-module) and 0 # y E U(g-)k, then y @ Vs(#) Vs(p + k6) is an mx, s-submodule of Ind(V). Now the Poincarb-Birkhoff-Witt theorem applied to U ( g - ) implies that Ind(V) E ~s.

For all f E Homax(M,N), M , N E C s, set Ind( f ) = 1 ® f . Then we have a functor

Ind : C s --~ ~ s .

Assume now that M E ~s. Then M --~ U ( g - ) ® ¢ M ,q+ as a vector space +

(by 1.8), and hence M - U(mx)®U(~x@g+)Mg which implies that

Ind o Inv __- Ics .

Conversely, let N E C s. Then

Ind(N) = U(mx) ®u(@~g+) N ~- U ( g - ) ®¢ N

as a vector space. Hence, we have (Ind(N)) ~+'~_ N, by 1.9, and therefore, Inv o Ind ~_ Ics. This completes the proof.

3.2. Corollary. (i) For any tt E 11 s, Ind(L(t0) is an irreducible module in ~ s (ii) I f V is an irreducible module in ~s , 2(c )#0 , then V g+~ - L(#) for

some tt E 1I s and V "~ Ind(V g+ ) ~- Ind(L(t0) .

4 Nonstandard generalized Verma modules

In this section, we define nonstandard generalized Verrna modules which are induced from a nonstandard parabolic subalgebra. This generalizes the construction of Verma-type modules.

200 B. Cox et al.

4.1. Recall that g(P(X)) = rt~- @ U~s @ g+ @ uf~. Denote T~s = U~s @ g± G u~x and ~x , s =mx, s @ Tf~s. Clearly, [~Arx, s, Tf~sl C Tf~ s. We call ,~x,s a nonstandard parabolic subalgebra of g.

Let # E H* such that #s is integral dominant. Then Vs(#) is an irreducible finite-dimensional rex, s-module with highest weight #. Consider Vs(#) as JV'x,s-module with trivial action of T~s and an induced g-module

Mx, s(~) = U(g) ®v(~x,s) Vs(#)

associated with X, S and #. The module Mx, s(#) is called a (nonstandard) generalized Verma module.

Clearly, Mx, s(#) is a weight g-module and Mx, s(#)~- U(Tf~s)®¢ Vs(#) as a vector space.

4.2 Remark. If S = 0, then Mx,~(#) = Mx(#) is the Verma-type module associated with P(X) and #.

4.3 Proposition. (i) Mx, s(#) is a homomorphic image of Mx(#). (ii) Mx, s(#) has a unique maximal submodule and Lx(lO is its unique

irreducible quotient. (iii) 0 < dimMx, s(#)~ < c~ ~ v E (# I Q~r)NP(Mx, s(#)) •

Proof (i) and (ii) are obvious, (iii) follows from Remark 1.2, (ii).

4.4. Denote Mfs(# ) = ~ M x , s(p)~, v E (# ~ Qfc)n P(Mx, s(#)). Clearly, Mx f, s(#) is an mx-submodule of Mx, s(P). The main properties of M f s(/~) are described in the following proposition.

Proposition. (i) M f s(P) ~ U(mx ) ®Ufpx, sag +) Vs(#) ~- Ind(g(Vs(#))), where Vs(#) is a Px, s @ g+-module with trivial action of n+s ® g+ and hence Mrs(#) ~- U(u~s ® g-) @¢ Vs(#) as a vector space.

(ii) As an rex, s-module, Mfs (# ) is a direct sum of Vs(v)' s and therefore Mfs (# ) E cS(#,q) for all q E 71.+.

(iii) Mfs(p ) has a unique maximal mx-submodule and Lfx(p) ~-- Ind(L(p)) is its unique irreducible quotient. Proof (i) simply follows from the definitions; (ii) follows from (i), and (iii) follows from 1.3, Proposition 4.3, (i), and Corollary 3.2, (ii).

4.5. The following statement shows that the structure of the g-module Mx, s(#) is completely determined by the structure of Mfs(#) in the case 2(c)#0.

Proposition. Let # E H*, p(c)~O and #s be dominant integral. I f N is a submodule of M~s(#), then N is generated by N f = N GMfs(#). Moreover,

N -~ U(g) ®V(px) Nf ,

where N f has a natural structure of px-module with trivial action of u +. In f particular, Mx, s(#) ~- U(g) ®e(px) M~,s(#).


P r o o f Follows from Proposition 4.3, (i) and Theorem 1.7, (iii).

4.6. Corollary. Let It, v E l I s. Then we have the fol lowing chain o f isomor- phisms:

Hom q(Mx, s(#) , Mx, s (v ) ) ~- Hommx (Mx f, s(it), M f s(V))

----- Hom~x (M(Vs(#)) , M ( V s ( v ) ) ) .

Proof. The first assumption follows from Proposition 4.5, while the second one follows from Proposition 3.1 and Proposition 4.4, (i).

4.7. It follows immediately from Proposition 4.5, Proposition 3.1 and 2.9 that the module Mx, s(it), p E 1I s, has a local composition series and, for v E H s, the multiplicity [Mx, s( i t ) : Lx(v)] of L x ( v ) in Mx, s( i t ) is well-defined. Moreover,

[Mx, s(It) : Lx(v)] = [M(Vs(I t ) ) : L(v) ] .

5 Nonstandard generalized Verma modules: The case of connected X

5.1. In this section we assume that the Coxeter-Dynkin diagram corresponding to X is connected (we will simply say that X is connected) and that S = ~. Then gx is a derived algebra of an affine Lie algebra and Mx, ~(#) = M x ( g ) .

L e t / I x be a basis o f Ax con ta in ing / /x and let Wx denote the Weyl group for gx, l be a length function and sfl a reflection with respect to f l E A ~ N A re.

+ r e For the elements w,w~E Wx, we write w ~-- w ~ if there exists a root fl E A x N A such that w = sflw ~ and l (w) = l (w ' ) + 1. The Bruhat order on Wx is given by: w <= w ' i f w = w ~ or if there are wl . . . . , Wr E Wx such that

W = W I + - - . . . +.-- W r .~ . W t "

For w E Wx and # E H*, define w . It = w(# + P x ) - Px where Px E H* is any fixed element such that Px(~) = 1 for all c¢E / Ix . Let P~c = {# E H*l(/~,~¢ ) > 0 for all ~ E / ) x } .

5.2. The following theorem is a generalization of [C2, Proposition 3.3, (iv)] for the case o f an arbitrary affine algebra.

Theorem. L e t X be connected, It E P+, # ( c ) ~ O , w , w ' E Wx. Then

(i) d i m H o m ~ ( M x ( w ' • I t ) , M x ( w • It)) --- 1 ¢* w' < w ¢* M x ( w • It) : Lx(w'. I t ) l , 0.

(ii) I f M x ( w p • It) C M x ( w • It) then there are wo = wP, wl . . . . . Wr-l ,Wr = W E Wx such that l(Wi+l) = l ( w i ) - l , i = 0 . . . . . r - l , a n d M ( w o • It) C . . . C M ( w r • It).

(iii) I f l ( w ) = l (w ' ) - 2, the number o f w" E Wx such that M x ( w ' • I t ) ~ M x ( w " • It) ~ M x ( w • It) is 0 or 2.

202 B. Cox et al.

Proof Observe that, for v, ~ E H*,

Homg(Mx(v),Mx( ~) ) ~-- Hom~x (M(v), M( ~) )

by Corollary 4.6, and

[Mx(v) : Lx(~)] = [M(v) : L(¢)]

by 4.7. Then the statements of the theorem follow from [RW, Lemma 8.14 and Theorem 8.15].

5.3. For any i E Z+, denote Wx (/) = {w E Wxll(w) = i}. Let /~ E P+. Set Ci = @w~w~)Mx(w •/z), i E Z+. Note that Co = Mx(l~). t f w', w E Wx and w ~ <_ w, we can fix an inclusion

i~,,w(#) : mx(w' • I~) --* mx(w • t~)

by Theorem 5.2, (i). By [RW, Lemma 9.6], to every pair (Wl,W2) E Wx x Wx, Wl ~-- w2, we

can associate a number c(wbw2) E {+1,--1} such that

C(W1, W2 )C(W2, W4)C(W1, W3 )C(W3, W4) = -- 1

for any square W 4 ~ W 3

1 l W 2 ~ W 1

Let dj : Cj ---* Cj-1, j > 1, defined by dj J " W(x j), = ®bwl,w21wl,w2(#), wl E w2 E W(x j - l ) , where j J bwl,w2 = C(Wl,W2) if Wl ~ w2 and bwl,w2 = 0 otherwise.

5.4. The following result is a generalization of [C2, Proposition 3.3, (v)] for the case of an arbitrary affine algebra.

Theorem (Strong BGG resolution). Let X be connected, # E P~ and t 1 : Mx(#) ~ Lx(lO be a canonical projection. Then the sequence

Cj Cj-1 ~ , Cl dr ¢ = • .. " " ,Mx(# ) Lx(#) ~ 0, ( j > 1)

is exact.

Proof Follows from Proposition 4.5, Proposition 3.1 and [RW, Theorem 9.7].

5.5. Let S ~ l I x , S connected, Pf~s = {# E H*l(#,ct) > 0 for any ~ E S} and Wx, s denote the subgroup o f Wx generated by {s/~lfl E S}. Let w S = {w E Wxlw-l(A + M t2s) C A+}. Set (wS) (j) = W s M W(x j), j E Z+.

Let I~, v E P~s and f : Mx(p) ~ Mx(v) be a non-zero map. By Corollary 4.6, f induces a non-zero ~x-homomorphism f : M(p) ~ M(v). Let f s : M(Vs(#)) ~ M(Vs(v)) be a standard map associated with f [RW, Definition


9.10]. Then, again by Corollary 4.6, f s induces a map f s : M.r,s(#) Mx, s(V). We will call the map f s a standard map associated with f .

5.6 Proposition. Assume that X is connected, # E P+ and w,w' E W s are such that l(w) = l(w') + 1. Then

.S Homg(Mx, s ( w . #),Mx, s ( w ' . # ) ) # 0 ~ w' ~---w ¢==~ tw, w , (# )#O,

where .s tw, w,(# ) is a standard map associated with i,~,w,(#).

Proof. Follows from Corollary 4.6 and [RW, Proposition 9.11].

5.7. Denote C s = @w~W~j)Mx, s ( w . #) and consider d s : C s ~ cS_l defined X,S

by t/s .s = @bJw,,w2tw,,w2(#), for wl E (wS) (j), w2 e (wS) (j-l), j => I.

Theorem (Generalized strong BGG resolution). Assume that X is connected, # E P+ and V1 : M~s (# ) ~ Lx(#) is a canonicalprojection. Then the sequence

. . . cS ds-~J cS_, , . . . - - - + C s aS, Mx, s (#)- - - -~Lx(#)- - -*O, ( j > l )

is exact.

Proof. Follows from Proposition 4.5, Proposition 3.1 and [RW, Theorem 9.12].

6 "Truncated" categories of g-modules

In this section we define certain categories of g-modules. The important objects in these categories are the (nonstandard) generalized Verma modules. We show that there are enough projectives in these categories and that BGG duality holds.

6.1. Let ) .(c)#0 and denote by (gx, s = d)x,s(2,q) the full subcategory of the category of weight o-modules M such that

(i) P(M) C U.~n(# .L Qx); (ii) The module M is generated by M y = ~'~#61"1M#;

(iii) dimM l, < oc for all # E H; ( iv) M y is a direct sum of Vs(#)'s, # e II s.

6.2 Remarks. (i) The particular case of the category (~x,s(2,q) with S = 0 and q = 0 for non-twisted affine algebras was studied in [C2].

(ii) If/~ E H s then the modules Mx, s(#) and Lx(#) are objects of the category d)x, s.

(iii) If M E (gx, s then M y 6 ~s.

204 B. Cox et al.

(iv) It is not obvious that any submodule of a module in (gx, s is an object of this category. We will prove that it is in Sect. 6.6. In fact, if X = 0 and 2(c) = 0, it is not always the case (see [F3] for examples).

6.3 Proposition. I f V is an irreducible module in (gx, s then V ~_ Lx(lZ) for some # E II s.

Proof. Let V be an irreducible module in (-gx, s. Then V f E ~ s (Remark 6.2, (iii)) and m o r e o v e r V f is an irreducible rex-module. Indeed, if N is a proper mx-submodule of v f , then U(ux)N is a proper g-submodule of V which contradicts the irreducibility. Hence, V f ,.v Ind(L(/z)) for some # E U s by Corollary 3.2, (ii), and V f is a highest weight module (with respect to m + @H) with highest weight #. However, II~V f = 0 by 6.1, (i). Therefore V is a highest-weight module (with respect to g(P(X) )@H) with highest weight #. We immediately conclude that V is a homomorphic image of Mx(#) and thus is isomorphic to Lx(l~).

6.4. In order to achieve our goal we establish the equivalence of the categories Ox, s and ~s.

Define an exact functor F : •x,s ~ ~s by F ( M ) = M f for M E (gx, s and F ( f ) = f l M f for any f E Homg(M,N) in (~x,s.

Let N E ~s. Then N is an rex-module and we can make it into a Px- module by letting u + act trivially. Denote Y(N) = U(g)®U(px)N and Y(g) = 1 ®g for any g E Hommx(N, Nr), N,N ~ E ~s. This defines a functor Y : ~s __~ (gx, s.

By Proposition 4.5, Y o F(Mx, s(2)) ~- Mx, s(2) and F o y ( M f s ( 2 ) ) ~_ Mfs(2) . We will show that F o Y and Y o F are both equivalent to identity functors and hence the categories d~x, s and ~s are equivalent.

6.5. The following is a key result.

Proposition. Let M E (gx, s. Then M has a highest weight series {Mi} (with respect to g(P(X))@ H) such that Mi/Mi-1 ~-Alx, s(#i)/Y(Ni) for some #i E 11 s and Ni E ~s .

Proof, The proof of the proposition follows the general lines of [C2, Propo- sition 2.2, (b)]. Since F ( M ) = M f E ~s, it has a highest weight series {Ki} (with respect to m + @ H ) of mx-submodules (see 2.8). Define Mi = U(g)Ki. Since u+Ki = 0, we have a surjective homomorphism ~bi : Y(Ki) ~ Mi for each i. Also, Ki/Ki-l is a highest weight module (with respect to m~. @ H ) in ~s with highest weight Pi and hence we have a surjective homomorphism

Mx, s(ILli) --~ Y(Ki/Ki_I ) ~_ Y(Ki) /Y(Ki_I )


which together with ~b i induces a surjective homomorphism

f i : Mx, s(Pi) ~ Mi/Mi-I .

Then F ( K e r f i ) E ds and K e r f i ~_ Y o F ( K e r f i ) for all i by Proposition 4.5. Set Ni = F(Ker f i). Observe that UMi = U(g)(UKi ) = U(g)F(M) = M by 6.1, (ii). This completes the proof.

6.6 Corollary. The category (gx, s is closed under the operations of taking submodules, quotients and finite direct sums.

Proof. That (gx, s is closed under finite direct sums is obvious. It is enough to show that if M E Ox, s and N C M, then N E (gx, s, i.e. we have to show that N satisfies 6.1, (i) -( iv) . The only non-trivial condition is 6.1, (ii). The module M has a highest weight series {Mi} by Proposition 6.5, and Mi/M,.-1 is a homomorphic image of Mx, s(#i) for some #i E H s and each i. Denote

oo N o~ M N i = N N M i . Then Ui=o i = N N ( U i = o i ) = N N M = N . Set N-i = NdN/-l. Then )Vi C Mi/Mi-1. We can assume that ,~i :t:0 for all i

(otherwise we just need to renumber the indexes). Then N f 4=0 and U(fl)/V f = Ni for all i by Proposition 4.5. Thus, induction on i and the Five Lemma imply that N f 4 : 0 and U(g)N f = Ni for all i. We conclude that N is generated by N f = Ui~0 Nf . Closure for quotients follows directly from closure for submodules and the corollary is proved.

6.7. We are now in a position to prove our main result which generalizes Theorem 2.3 in [C2].

Theorem. Let 2(c)4:0. Then F o Y (resp. Y o F ) is naturally equivalent to the identity functor I~s (resp. Ic~x, s) and thus the categories C?x, s and ~s are equivalent.

Proof Let N E ~s. Since Y(N) ~_ U(u x ) ®¢ N as a vector space, we obtain that F o Y(N) ~_ N.

Conversely, let M E d~x, s. By Proposition 6.5, there exists a highest weight M oo ___ series { i}i=0 such that Mi/Mi-1 Mx.s(2i)/Ni for some 2i E 1I s, Ni E ~s.

By Proposition 4.5, Y o F(Mi/Mi-1) ~-- Mi/Mi-1 for all i. Note also that Y o F(MI ) ~-- M1. Then induction on i and the Five Lemma imply that YaF(Mi) ~- Mi for all i. We have

(i 0) M = UMi ~- U Y o F(Mil = Y o F Mi = Y o F (M) i = 0 i=0

since tensoring commutes with direct limits ([R, Corollary 2.10]). Finally, for any M,M' E (?x,s and N,N' E ~ s we have canonical isomor-

phisms Uomg (M, M ') ~_ Hom,nx (F(M), F(M ~ ))

206 B. Cox et al.

and Hommx(N,N') ~-- Homg(Y(N), Y(N ' ) )

which imply that Y o F _--_ Icx, s and F o Y ~_ Ids. This completes the proof.

6.8 Theorem. Let &(c)4:0. Then the categories Ox, s and C s are equivalent.

Proof Follows from Theorem 6.7 and Proposition 3.1. The equivalence is defined by functors Y o Ind : C s --~ d~x, s and Inv o F : ~x, s ~ C s.

6.9. For any It E I I s, define

PSx(it ) = Y o Ind(ps( i t ) ) _ U(g) ®U(Dx, s) PS(it)

and IS (# ) = Y o Ind(IS(it)) ~ U(g)®U(Dx, s ) IS(g)

where Ox, s = gx E3 g+ @ u +, g+pS(#) = g+ i s (# ) = u+pS(#) = u+/S(it) = 0. Then pS( i t ) is a projective module in 6x, s and IS (# ) is an indecomposable direct summand o f p S ( # ) by Theorem 6.8.

6.10 Proposition. The module IS(#) is a projective cover for Lx(#).

Proof Since IS(v) is a projective cover for L(it) (see 2.6), and Lx(i t) ~- Y o Ind(L(#)) , the statement follows from Theorem 6.8.

6.11. A module M E 6x, s is said to have a generalized Verma composition series (GVCS) if there is a filtration of submodules

M = M ° D M 1 D . . . D M t D M t+l = O

such that M i / M TM ~- Mx, s(2~), 2i ~ 11 s, i = O, . . . , l . We denote by (M : Mx, s (#) ) the number of indexes i such that # = 2i.

I f M has a GVCS, then Inv o F ( M ) has a GVCS in C s and

(M : Mx, s (# ) ) = (Inv o F ( M ) : Inv o F(mx, s(i t))

= (Inv o F(M) : m ( g s ( i t ) ) ) .

Hence the number (M : Mx, s(it)) is independent o f the filtration.

6.12 Proposition. (i) The modules PSx(it ) and IS(it) have a GVCS. (ii) Let It, v E II s. Then

( PSx ( it ) : mx, s( v ) ) = dimHomg(/~x(it), Mx, s( V ) ) .

Highest weight modules for affme Lie algebras 207

Proof (i) follows from 6.9, 2.6 and Theorem 6.8; (ii) follows from Theorem 2.10, (ii) and Theorem 6.8.

6.13. It follows from Theorem 6.8 and 2.9 that any module M E Ox, s has a local composition series, i.e. for any 12 E H there exists a sequence M = Mk D " " D M0 = 0 of modules in Ox, s and a subset I C {1, . . . ,k} such that Mj/Mj-1 "~ Lx(12j), pj >= 12, j E I and (Mj/Mj-I)v = 0 for all v > 12 if j ~ l . Then the multiplicity [M : Lx(v)] of the module Lx(v) in M is well-defined and [114 : Lx(v)] = [Inv o F(M) : L(v)].

6.14. The discussion above and Theorem 2.10, (ii), (iii) together imply

Theorem. (i) Let M E Or, s, 12 E fis. Then

[M : Lx(#)] = dimHomg(IxS(#),M) •

(ii) (BGG Duality) Let #, v E H s. Then

(ISx(12) : Mx, s(V)) = [Mx, s(v) : Lx(t~)]

= (IS(#) :M(Vs(v)) )

= [MCVs(v)):L(12)].

Remark 6.15. If X is connected one can consider another version of the truncated categories. In this case, Ax has a basis / )x containing f ix . Let T ~ / ) x and

qQ+ = I 2 = ~ m~alm~ E Z+, ~ ma > q , ~EFI x ~Ellx

f /~r(2,q)= ~ # E H * I # - 2 = ~ m~ a E Q x , m ~ E 7 7 ,

l

-4) += E m~EQ+\q0+~, (. m~EZ+ J

/-)r(2,q) = {p E/)(&q)112r - 121~r is integral dominant}.

Then we can define analogously to 2.4 and 6.1 the categories C r(2, q) and (~x,r(2,q) which are equivalent if 2(c)4=0. Note that if S = T, then cS(2,q) (resp. Ox, s(2,q)) is a subcategory of Cr(2,q) (resp. (gx, r(2,q)). All the results of section 6 hold for the category (~x,r(2,q).

208

Table of

1.1

1.3 1.5 1.6

1.9 2.1

2.3 2.4 2.5 2.6

2.9 3.1 4.1 4.4 5.1 5.3

5.5

5.7 6.1 6.4 6.9 6.15

nonstandard notation

~(P), M(2,P), 4px, P(X), Mx(2), dx, A +, Q, Qx, Ox, Q;, R(x),

mff, mx, gx, G.

Lx(2), M~(X), txf(2), uff, px. g± . nx, s, ~s, aL

, m ,s, U s, P ,s, Us. qQ+, //(2,q), IIS(2,q), Vs(lO. cS(2,q), ~s(2,q). II, 11 s, C s, ~s. M(Vs(#) ), A(k,S), A(k,S), A'(k,S), pS(p), is(p), (is(p) : M(Vs(v))). [M : L(#)] Inv, Ind(V). g(e(x)), T~s, .hrx, s, Mx, s(p). Mrs(#). Wx, w ~-- w', w <= w', w • 12, px, P +. W~ ), C~, iw',w(P), c(wl,w2), dj, b/w,,w2 . P~s, Wx, s, W s, (WS) (j), fS , j~ fS. q , 4 ~x,s. F, Y. PSx(#), Is(u), Dx, s. fix, qQ+, I91(2,q), flr().,q), Cr(2,q), d3x, r(2,q).

B. Cox et al.

References

[CI] Cox, B.: Verma modules induced from nonstandard Borel subalgebras. Pac. J. Math. (to appear )

[C2] Cox, B.: Structure of the nonstandard category of highest weight modules. Modern Trends in Lie Algebra Representation Theory Futorny, V., Pollack, D. editors. Queen's University, 1994

[F1] Futorny, V.: Parabolic partitions of root systems and corresponding representations of affine Lie algebras. Ac. Sci. Ukraine, Inst. of Math. (preprint) 1990

[F2] Futomy, V.: The graded representations of affme Lie algebras. Suppl. ai Rendic. Cir- colo Matem di. Palermo, Sefie II N26, 156-161 (1991)

IF3] Futorny, V.: Imaginary Verma modules for a i d e Lie algebras. Canad. Math. Bull. 37, 213-218 (1994)

[FS] Futorny, V., Saifi, H.: Modules of Vcrma type and new irreducible representations for affme Lie algebras. Proceedings Series: 6th International Conference on Representa- tions of Algebras Ottawa, 1993


[JK] Jakobsen, H., Kac, V.: A new class of unitarizable highest weight representations of infinite dimensional Lie algebras. Lect. Notes in Physics, Vol. 226 Springer-Verlag, Berlin New York, 1985, pp. 1-20

[K] Kac, V.: Infinite dimensional Lie algebras. Cambridge Univ. Press, 1985 IMP] Moody, R., Pianzola, A.: Lie algebras with triangular decomposition. J. Wiley, 1994

[R] Rotman, J.J.: An introduction to homological algebra. Academic Press, 1979 [RW] Rocha-Caridi, A., Wallach, N.: Projective modules over graded Lie algebras. Math. Z.

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