annihilators of verma modules for kac-moody lie algebras

Invent. math. 81, 47-58 (1985) Inventiones mathematicae �9 Springer-Verlag 1985

Annihilators of Verma modules for Kac-Moody Lie algebras

Vyjayanthi Chari

School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India

Introduction

Let G(A) denote an infinite dimensional Kac-Moody Lie algebra associated to the indecomposable symmetrizable generalized Cartan matrix A. In this paper we study some primitive ideals in the universal enveloping algebra U(G(A)) of G(A). Also we show that the center of U(G(A)) is the enveloping algebra of the center of G(A). This was proved for Euclidean Lie algebras in [1].

Let p denote a finite dimensional semisimple Lie algebra. In [2] Duflo proved that the annihilator (in U(p)) of a Verma module is generated by a maximal ideal in the center of U(~). We extend this result to irreducible Verma modules over the infinite dimensional Lie algebra G(A). In particular, if A is nonsingular, the irreducible Verma modules are faithful representations of U(G(A)). The criterion for the irreducibility of Verma modules was obtained in [4].

The techniques used in this paper are different from those of Duflo's and work only for infinite-dimensional Lie algebras. The crucial result is the following: Let Wdenote the Weyl group of G(A) and let (cq . . . . . ~,) be the simple roots of G(A). There exists an element w in W of infinite order such that for every positive integer r and for all i = 1 . . . . . n, either wr~i or - w r ~ is a positive non-simple root. The proof of this result proceeds by an induction on the order of A and the induction step uses the hypotheses that A is symmetrizable and indecomposable.

w I. Preliminaries

k will denote an algebraically closed field of characteristic zero and 7Z the ring of integers in k.

Let A = ( a l ) be an n x n matrix such that: aije7Z, a , = 2 , a~j<0 if i#:j, a~j=0 iff aj~=0. Let G(A) denote the associated Kac Moody Lie algebra. G(A) is defined uniquely (upto an isomorphism) by the following properties:

48 V. Chari

(a) G(A) contains an abelian diagonalizable subalgebra H such that,

|

where, G~,={xeG(A): [h,x]=a(h) xVheH} and Go=H. (b) There exists a linearly independent set rc of functionals al . . . . . ct,,eH*

and elements e 1 .... ,e., f l .... ,f. in G(A) such that

(i) G~=ke i, G_~=kfl, i=1 . . . . ,n,

(ii) [e i,fj] -- 0 if i # j ,

(iii) {e 1 . . . . . e,, fl . . . . . fn} w H generates G(A) as a Lie algebra,

(iv) the elements h i = [e i,fl], 1 <_ i <_ n are linearly independent,

(v) ~i(hi) = ai~, 1 < i, j < n, (vi) if hell is such that ai(h)=0 for all i= 1 . . . . , n, then

he ~ kh i. i = 1

(c) Any ideal of G(A) which intersects H trivially is zero.

Let F denote the additive subgroup of H* generated by a s . . . . . c~,. Set F +

={ ~k'cq:k'ez'ki>O'i=l`=2 ... . . n } .Ane lemen tO#~eFi sca l l edaroo to fG(A)

if G,#{O}. Let A denote the set of roots of G(A). If aeA then ~ or -c~ is in F +. Set A + =dc~F +, A - = - A +. Set

N+= | N - = | a e A + c t e A -

N + and N - are subalgebras of G(A) and one has G ( A ) = N - O H @ N +. For

:eA, a= ~ kia i, set hta= ~ [kil. Clearly hta> 1 if aq~IIU(-H). Let C denote i = 1 i = 1

the center of G(A). Clearly C~_ s kh~, and d i m H = d i m C+n. i = 1

For a subalgebra B of G(A) let U(B) denote the universal enveloping algebra of B. By the Poincare-Birkhoff-Witt theorem one has

U(G(A)) = U(N-)| U(H)| U(N+). k k

Condition (c) in the definition of G(A) implies that ade~ and adf/(1 <=i<=n), are locally nilpotent endomorphisms of G(A) and hence define locally nilpotent derivations (also denoted by adei, adf/) of U(G(A)). Thus exp(adei) , exp(adf/) are well defined automorphisms of U(G(A)).

The matrix A is said to be symmetrizable if there exists a diagonal matrix D with positive rational diagonal entries such that DA is symmetric. From now on we shall assume that A is indecomposable and symmetrizable and that D is fixed say D=diag(d 1 . . . . . d,). In this case H admits a non degenerate bilinear form F satisfying F(hl, hj)=dj-xaii. Let ( , ) denote the

Annihilators of Verma nodules for Kac-Moody Lie algebras 49

corresponding form induced on H*. Then (cq, ej)=dlaij ([3(a)] Proposition 7; 2cq

[5(a)] Lemma 7). Set ~ = (~, ~)" Define reflections s i of H* by si(2)=A-(2,~i)cq, 2~H*. Let W be the

subgroup of AutH* generated by {si}x<i<,. W is defined by the relations: s~ ={e}, (slsj) . . . . 1 where mlj=2 , 3, 4, 6 or oo if ai2aji=O, 1, 2, 3 or aijaji>4 respectively [5(a)]. The form ( , ) on H* is W invariant, and W leaves the subset A of H* invariant. An element aeA is called a real root if it belongs to the W-orbit of an element ~6n. A root is called imaginary if it is not a real

2e root. For a real root g=wgi, set ~ : = - - s~=wsw -1. Then s~(2)=2 -(2, ~)~, V2eH*. (~' ~)'

Let weW. Set Aw={aeA+: waeA-}. A w is a finite subset of A +. Let l(w) denote the length of a reduced expression for w in terms of the reflections {sl}l_<~< .. Then l(w)= #A~. If aleA w for some i = 1 . . . . . n, then there exists an element w'e W such that w=w's~, l(w)= l(w')+ 1 and eiCAw,. The action of W on H* induces an action on H. We recall the following important fact about W.

Theorem [5(a), Theorem 2]. For weW,, there exists an automorphism O(w) of G(A) such that O(w)G~= Gw~ for all ~ A and O(w)ln=w.

The construction of O(w) is as follows: For w=sl, take O(si) =exp(adei)exp(adfl)exp(adel). For arbitrary w, fix a reduced expression w =si~...si,, in terms of the simple reflections { s i : l< i<n }. Take O(w) = 0 % ) . . . O(sl).

The matrix A is said to be of finite type if the Lie algebra G(A) is finite dimensional. In this case DA must be positive definite [3(a), 5(a)]. Throughout the paper we shall assume that A is an indecomposable, symmetrizable generalized Cartan matrix which is not of finite type. The Weyl group W is an infinite group and A is an infinite subset of F [3(a), 5(a)]. The matrix A is called an Euclidean Cartan matrix if A is singular and every principal submatrix is of finite type. A complete list of the Euclidean Cartan matrices may be found in [5(b), Table 1]. We may associate to A a Dynkin diagram, by drawing n vertices and joining the /th and jth vertices by a~ja~ lines. If A is indecomposable the Dynkin diagram is connected.

w 2. Statement of Results

For 2~H*, let 14 denote the left ideal of U(G(A)) generated by N+w {h-2(h) : h~H}. Set M(2)= U(G(A))/Iz and let v~eM(2) be the image of 1 in M(2). M(2) is called the Verma module for G(A) with highest weight 2, and v k is the highest weight vector with weight 2. Further

M(2)= @ M(2)u rFEF +

where, M(2),~={m~M(2): hm=(2-q)(h)mgh~H}. It is easy to see that M(2) is a free U(N-) module. Also dim M(2),=P(t/), where P(t/) is the Kostant partition function for G(A) [3(b)]. Denoty by AnnM(2) the set

{g~ U(G(A)): gm= 0Vm~M(2)}.

50 v. Chari

Fix an element peH* such that p(hi)=l, i=1 . . . . ,n. M(2) is irreducible iff2(2+p, c04=n(c~,c 0 for all non-negative integers n, and for all c~eA + ([41, Proposition 3.1). Let Z denote the center of U(G(A)). Then Z acts on M(2) through a homomorphism Zx: Z ~ k . Let Ja denote the two sided ideal of U(G(A)) generated by the kernel of Z~. Clearly Jx_AnnM(2) . We now state our main theorem.

(2.1) Theorem. Let A be an indecomposable symmetrizable Cartan matrix such that the Kac-Moody Lie algebra G(A) is infinite dimensional. Then

(i) Z = U(C), where C is the center of G(A). (ii) I f M(2) is an irreducible Verma module, then Ann M(2) is the two sided

ideal generated by the kernel of Zx, i.e. AnnM(2)=Jx.

(2.2) Remark. Part (ii) of the theorem holds in the case when G(A) is a finite dimensional semisimple Lie algebra. The theorem is due to M. Duflo [2] and holds for arbitrary Verma modules.

(2.3) Remark. Part (i) of Theorem (2.1) was proved in the case of Euclidean algebras in [13 by techniques different from the ones used in this paper.

Set U + = @ G~. Clearly U + is an ideal in H G N +. ~tEct +

h t a > l

(2.4) Proposition. Let gE U(N-OH) U(U+). Then (i) geZ implies ge U(C),

(ii) gEAnnM(2) only if geJx.

(2.5) Proposition. For every element gsU(G(A)), there exists an element WgeW such that

0(wg)ge U(N- OH) U(U+).

It is easy to see that Propositions (2.4) and (2.5) imply Theorem (2.1). Let gsZ. Choose w~ as in Proposition (2.5). Since exp(adel) and exp(adf~)

fix g one has g=0(wg)g and so by Proposition (2.4), geU(C). This proves Part (i) of Theorem (2.1).

For Part (ii), let geAnnM(2) and choose wg as in Proposition (2.5). Since AnnM(2) is a two sided ideal in U(G(A)) it follows that O(w~)gsAnnM(2). By Proposition (2.4), we can write,

(2.6) O(w~)g = ~ glc~ i = 1

where giEU(G(A)), c~U(C) and Z~(ci)=0. Applying O(w~) -1 to both sides of (2.6) one has the desired result.

In the next two sections we shall prove Propositions (2.4) and (2.5).

w 3. Proof of Proposition (2.4)

The algebra U(N*) has a F+-gradation

(3.1) U(N+)= @ U(N*). ~l~ F +


where, U(N+),={xsU(N+): [h,x]=tl(h)xVheH ]. Further dim U(N+),=P(tl), where P(r/) is the Kostant partition function for G(A). For t /eF +, set U(U+), = U(U+)c~ U(N+),. One has

(3.2) U(U+)= @ U(U+),. r /~F +

For r/~F +, r/= ~, slcq, si~2g, si>0, let e~ denote the element e 1 . . . . ...e~ of i = l

U(N +) and let V, be the subspace of U(N +) spanned by e,. Set V= @ V,. By r / eF +

the Poincare Birkhoff Witt theorem one has

/ U ( N + ) = V O U ( N +)

(3.3) [U(N+),=(~V~,| tf'~F + and t/ '+t/"=t/ .

Fix an ordering on F + by: t/<r/' iff r / ' - t /~F +. Let u~ . . . . . u r be linearly independent elements of U(U +) such that ui~U(U+),, for some r/i~F +, i= 1 .. . . ,r. Assume without loss of generality that r/1 is a minimal element in the set (r h . . . . . r/, ).

(3.4) Lemma. Let 2~H* be such that M(2) is irreducible, and let u 1 .... ,u,~U(U +) be as above. For every t/~F +, there exists

(i) 0 4=v,~M(2), such that uivn=O for all i= 1 ..... r, if ~ll 4=0, (ii) O~vn~M(2),+, 1 such that UlV,~O and uivn=O if i> 1. We shall need the following: Let fl: U(G(A))~U(H) be the projection

corresponding to the decomposition

U(G(A)) = U(H)O(N- U(G(A))+ U(G(A))N+ ).

(3.5) Lemma [[1], Lemma 3.13. Let ~.~H* be such that M(2) is irreducible. The bilinear pairing

Bn, a: M(2), x U(N+),~k

given by B,,z(yva, x)=2(fl(xy)), ye U(N-), is non degenerate. Thus, if x 1 ..... x, are linearly independent elements of U(N+), then, there

exists vl,.. . , v, eM(2)n such that xlv j = ~ov~.

Proof of Lemma (3.4). Let t /eF +, t/= ~ si~ i. If t h 4=0 the element e,eV, is i = l

linearly independent with the set {U(N+)ui: l<i<r} . By Lemma 3.5 we can choose 0 ~ vsM( 2 ) , such that (a) e,v=vx, (b) U(N+),_nuiv=O.

Clearly ujv=O if t / - r / j~F +. Suppose ulv4=O for some i with th<r/. By Lemma (3.5) there exists an element xeU(N+) ,_ , , with xu~v=vx contradicting (b) above. This proves Lemma (3.4)(i).

For Lemma (3.4)(ii) observe that we can choose veM(2),+,~ such that (a) enulv=v ~ and (b) U(N+)n+nl_.uiv=O for all i>1. Clearly UlV+O and we can prove as before that u~v=O for i> 1.

Proof of Proposition (2.4). Write H = C O H 1 where C={heH:ei(h)=OVi = 1 . . . . ,n} is the center of G(A), dim H~ =n. The subgroup F of H* is a Zariski

52 V. Chari

dense subset of H*. For 2el l* , let Tz: U(H)~U(H) be the automorphism extending Tx(h)= h-2(h), hell. If rice +, clearly T,[v(c)is identity on U(C).

Let geU(N-(gH)U(U+)c~Z. Write g as a sum

g = h + ~ PiUi i = l

where heU(H), PieU(N-@H), uleU(U+),,, rlieF +, tllq:O. Let t /eF + and 2e l l* be such that M(2) is irreducible. Choose 0 # v , eM(2),

as in Lemma (3.4)(i). Then

(3.6) 2(h) v, = g v, = (2 - 1/)(h) v,

where the first equality is because geZ. Since (3.7) holds for the Zariski dense subset consisting of 2e l l* with M(2) irreducible we have

(3.7) h = Tn(h) for all fIEF +.

Write h= ~ cik ~ where (c~)I<=~<=~EU(C) are linearly independent and i = 1

(ki)l<_.i<seU(H1). By (3.7) we have

elk,= ~ ciT, k i and so ki= T,(k,) for all t/~F +. i = 1 i = 1

Since F + may be regarded as a Zariski dense subset of H* it follows that k~ek

for all i=l,...,s. Thus heU(C) and ~ plui~Z. By Theorem (2.6)(0 of [1] it i = 1

follows that p~u~=O and (i) of Proposition 2.4 is proved.

For (ii), we shall first prove that the intersection

(3.8) U(N- OH 0 U (U +) c~ Ann M(2) = {0}.

Let g be any element of the intersection. Write

g = ~ PiUi i = 1

where pi~U(N-@HO, uieU(U+),, for some ~h~F +, and (U31<_i<_r are linearly independent. Assume that '/1 is minimal in (t/l, ...,t/r). By Lemma 3.4(ii) we can choose for every ~/eF +, an element 0#v,eM(2)~+, , such that u~v .#0 and u~v. = 0 for all i > 1. Then

O=gG=pl ua Gr

Write p : = ~ y~h~, where hieU(H1) and (yi)~=<, are linearly independent ele- @

i = 1

ments of U(N-). Since M(2) is U(N-) free the equality

0 = pl ul v. = y~ ( ,~- tt)(h~) yi(u~ v.)


implies that ( 2 - t / ) ( h l ) = 0 for all ~/eF + and all i = 1 . . . . . t. This forces h i = 0 for all i = 1 , . . . , t and so p~=0 . Repeat ing this a rgument we can conclude that p~=0 for all i = 1 . . . . . r and so g = 0 proving (3.7).

Let g be an element of U ( N - | By the Poincare Birkhoff Wit t theorem we can write g as a sum

g = ~ cigi i=1

where c i eU(C) and (gl)~_~i_~, are linearly independent elements of U ( N - O H i ) U(U+). If g ~ A n n M ( 2 ) one has

O = g v = ~ )~(ci)giv for all v~M(2).

Thus ~])~(c~)g~ is an element in the intersection Ann M(2)c~ U(N-@H1)U(U+),

and so by (3.8) we have ~ 2(ci)gi=O. This forces 2(ci)=0 for all i = 1 . . . . , r and i=1

Proposi t ion 2 4 follows.

w 4. Proof of Proposition (2.5)

We shall deduce Proposi t ion 2.5 from the following Proposi t ion and its Corol- lary.

(4.1) Proposition. There exists an element w in the Weyl group W(A) such that,

(i) {w': r~Z} is an infinite subgroup of W,,

(ii) AwrC__Awr+l for all r>=O (where Awr={c~6A +" w' c~EA-}), or equivalently

(ii)' Aw~_Aw,. for all r > 0 ,

(iii) htw%ti>l for all r > 0 , and i = 1 . . . . . n,

(iv) eti6Awr for some r > 0 , and some i= 1 . . . . . n iff cq~A w.

(4.2) Corollary. There exists a subset A i of A such that

(i) A i c ~ - A x = 0 , A = A I ~ ( - A i )

(ii) ~,3~Al,c~+ 3~A~c~+ 3~A 1

(iii) Let F be any finite subset of A a. There exists an element wF6 W(A) such that wv(F ) ~_ A + - n.

Proof of Corollary (4.2). Let w~W(A) satisfy the condit ions of Proposi t ion 4.1. Set A' = U A wr- A +. Take A l = (A + - A') w ( - A'). Clearly,

r>0 A l u ( - A O = A , A l c ~ ( - A 1 ) - - 0 .

No te that: c~A + -A '~wrc~eA + - A ' for all r > 0 . Let ~,/3~A~ be such that c~+/3~A. Then

(a) ~ A + - A ' , f lea + -A '~w ' ( e~+3)~A + - A ' for all r > 0 ~ c ~ + I ~ A + - A ' .

(b) ct~A + - A ' , /3e ( -A ' ) . Choose t > 0 such that w'[3EA +. Clearly wr(~+3)~A + for all r>t>O. Hence either ~ + f l E A + - A ' if c~+/3~A + or ~ + / ~ ( - A ' ) if c~+/~EA-. In any case ~+/3EA, .

54 V. Chari

(c) ~ , f l ~ - A ' . Choose t > 0 such that wtc~, w~fl~A +. Then wr(~+fl)~A + for all r>t>O and hence c~+flE-A' . This proves Corollary (4.2)(ii).

Let F be a finite subset of A x and t > 0 be such that F n ( - A ' ) ~ 0 A w~" By

Proposition (4.1), w~F~_A + for all s>t. Suppose that w~Fn~+-O for any s>t. Since F and rt are finite subsets of A, it follows that there exists ~eF and ~ie~ such that wS~=~i for infinitely many s>t. Chose p>q>t such that wPc~=wq~ =~i. Then wP-qc~i=~ contradicting condition (iii) of Proposition (4.1). This proves the Corollary.

Before proving Proposition (4.1) we shall deduce Proposition (2.5). Let A 1 - A be as in Corollary (4.2).

Set N + = ( ~ G , , N 1-= (~ G~. N + a n d N 1- are subalgebras of G(A) and one has ~A1 ~-A~

G(A) = N~ OH@N +, U(G(A)) = U(N~-)| U(H)| U(N+). k k

Recall the unique antiautomorphism ~: U(G(A))~U(G(A)) satisfying ~(ei) =f~, a(fl)=e~, a[u=id. Since ~G~=G_~ it follows that a maps N + onto N~-. Let w~W and let O(w) be as in Section [1]. The automorphism O(w)aO(w)-la leaves G~ invariant for all ~ A .

Let {x~,i~G,: eeA1, l_<i<dimG~} be a base for N +. Fix an ordering on the set {(~,i): c~A~, i = l . . . . . dimG~}. The set X={xl~ ,,~w..x~a~,~.): (flx, iO<...<(fl,,i,), rs~. +} together with 1 forms a base for U(N+). For any xeX, x=xta,,i,)...x~p,,~ ) let F~ be the subset {fl~}~<~<, of A~. Let X v be a finite subset of X.

Set F = U F~. By Corollary (4.2)(iii) we can choose w v - w s W such that X~XF

wFc_A + -~. Hence O(w)x~U(U +) for all x~X~. (Recall that O(w) is an automorphism of U(G(A)) mapping G~ to G~). Thus if x I . . . . . x~ are elements of U(N+), there exists an element we W(A) such that O(w)x~eU(U +) for all i = l , . . . , r .

Let ge U(G(A)), write g as a sum

g = ~ Yihjxk i , j ,k

where, yi~U(N~-), hFU(H ) and Xk~U(N+). Choose w~W such that O(w)x k, O(w)a(yi)eU(U +) for all i and j. Since O(w)trO(w) -~ a -~ leaves the root space G~ invariant it follows that aO(w)yi6U(U +) for all i, and so O(w)YieU(N- ). Hence 0(w)ge U ( N - 0 ) H ) U ( U § and Proposition (2.5) is proved.

Proof of Proposition (4.1). The proof proceeds by induction on n. If n=2 , the Weyl group W(A) is generated by the reflections s~ and s 2 with relations s 2 = s 2 =e. It is easy to check that the element w=sas 2 satisfies the conditions (i)- (iv) of Proposition (4.1).

Let A be an n • n, (n > 2) indecomposable symmetrizable generalized Cartan matrix such that G(A) is infinite dimensional. Clearly either,

Case (i). Every proper principal submatrix of A is the Cartan matrix of a finite dimensional semisimple Lie algebra, or


Case (ii). A has a principal ( n - l ) x ( n - l ) indecomposable submatrix B such that the associated K a c - M o o d y Lie algebra is infinite dimensional.

We shall prove that in either case there exists an element we W(A) with the requisite properties. In Case (ii) we assume by induction that the proposi t ion is true for B.

Proof of Case (i). It is fairly easy to show that: A must be Euclidean if n > 5 . If n = 3 or 4 then either A is Euclidean or A (or the transpose of A) has a Dynkin diagram of the following type [3(c), Excercise 4.1]:

? l ~ 3 . V n = 4 .

The fact that A is symmetrizable is impor tant in proving the above fact. (Recall that the Dynkin diagram of A r can be obtained from the diagram for A by reversing arrows and that W(A) and W(A r) are isomorphic.)

Suppose that A is an Euclidean Cartan matrix. In this case A(A) has a unique minimal positive imaginary root 7. Fur ther (7, cq)= 0 all i = 1 . . . . . n and ht 7>n. Let A~(A) denote the set of positive real roots of G(A). There exists a subset say (a z . . . . . c~,) of n such that A~(A) is contained in the union,

A~(A)~_ {c~+nT: c~eFz+,n >0} u { - ~ + n y : c~eFz+, n >0}

where F2+= nicti: nieZ, ni>=OVi=l . . . . . n}. Fur ther there exists a positive

integer k such that -eh+k7 is in A~(A) for all i = 2 . . . . . n. Take ti=s~sk~_~,, i = 2 . . . . ,n. A simple computa t ion shows that t~ej=ej

+ka~jT, i,j=2 ..... n. Choose non-negative integers (P2 . . . . . P,) such that n

y" p~a~j>O for all j = 2 . . . . . n. (This is possible since the matrix (a~j)2~,j__< . is i = 2

positive definite). Set w = f i tip ~.

i = 2

It is easy to check that w has all the required properties. The results on Euclidean Lie algebras used here may be found in [3 (b), 5 (b)].

N o w suppose that n = 3 or 4 and that A or A r has a Dynkin diagram as given before. We shall choose a root tleAR(A ), and a simple root e such that

(4.3) (t/, 4) ==_ - 2, and (c~, 0) __< - 2.

(4.4) (t/+ a, fl) =< 0 for any simple root ft. Fur ther (t/, fl) 4:0 if (a, fl) = 0.

Take w=s,s~. We shall prove that w satisfies all the conditions.

56 V. Chari

The following formulae hold for r > 0 and for some positive integers p and q depending on r, ~/ and ~.

(4.5) wrrl= pct+q~l q >p

w-rrl = --pet --qt I p > q,

(4.6) wrct = --pe --qtl q >p

w-rc~ = pct + qtl p > q,

(4.7) s~wrc~ = --pc~--qrl, p >q

s , w - ' e = pc~+qtl, p<q,

(4.8) s~wrrl= p~+qtl, p >q - r _ s~w t l - - p c t - q t l , p<q.

Let fl be a s imple root, fl 4: e. Then

(4.9) (e ,w~fl)<0 and (tl, S~Wr for all r > 0 .

For, (ct, w" f l)=(w-" e, fl)=(pct + qrl, fl)=((p - q ) ~ + q(rl + cO, fl) <O where the in- equali ty follows f rom (4.6) and (4.4). Similary one can show that (tl, s,w"fl)<O. Hence w~+lfleA + - ~ for all r__>0. Thus (4.5)(4.9) proves that w satisfies (i), (iii) and (iv) of Propos i t ion 4.1.

We now have to prove that Aw~_Awr for all r > 0 . Observe that dw={c~}u{s~fl:fl~A+,s~fleA-}. Clearly cteAwr for all r. Let y=s~fl be such that y e a w.

Since s, f l6A- it follows that (t/ ,fl)>0. Fur ther by (4.4) it follows that (17 + a, s, fl) __> 0. Hence (~, s, fl) >__ 0 and we have

(4.10) (ct, w's,fl)>=O, (tl, s~wrs~fl)>=O for all r > l .

This proves that YeAwr+, for all r > 1. Thus A,~C_Awr for all r > 0 and the p roof of Case (ii) is complete, modu lo the choice of t/.

n =4 . Let (as, ct 2, ct 3, ~4) be the simple roots of A(A). Assume (c~2, ~3, ~ ) form a root system of type B 3 and that ~2 is the long root. Take t / = a z + e 3 + a 4, a I =~-

n = 3 . Let (~1,~2,~3) be the simple roots. If (cq, a~)#:0 for all pairs (i,j) i,j =1 ,2 ,3 , take ~=~1 t /=~2+~3" If (c%,aj)=0 for some pair i,j assume without loss of generali ty tha t (e2, %) = 0. Then (el, ~2) 4=0, (c%, ct3) + 0. Take, t /= s as 2 aa,

F r o m now on we assume that A has a principal (n - 1) x (n - 1) indecomposable submatr ix B such that G(B) is infinite dimensional. One has the natural inclusions G(B)-~G(A), W(B)->W(A), A(B)~A(A). Assume without loss of generality that (a2 . . . . , ~,) are the simple roots of A (B) and that (a~, ~2)+0.

By induct ion there exists we W(B) such that (i)-(iv) of Proposi t ion (4.1) are satisfied for A(B). Clearly w'exsA+(A) for all r. If in addi t ion htwrct~ > 1 for all r > 0 then Propos i t ion (4.1) is p roved for A.

Assume therefore that h twra~=l for some r > 0 . Then w~a~=aa and we may assume without loss of generali ty that r = 1 (just replace w by w').


Let S~l...s~ be a reduced expression for w in terms of the simple reflections

Claim: (ea,c~i,)=0 for all r = l . . . . ,k and w belongs to the subgroup generated by the {si: 3 <_ i <_ n}.

Suppose (cq, c~J =~0. Then wcq=si~...si~_~(e ~ - (u~,~i~)ui~). Since s~.....si~_ cq and s~-.. . .s~ cq~ are positive roots and (~ , c%)<0 ,we 1 must be a non-simple positive root contradicting w ~ = c q . Hence ( c q , u J = 0 . A re- peated application of this argument proves the claim. The following assertions are clear.

(4.11) (a) W~zSA + for all r, l(w2w)=l(w)+l.

(c) rleAw~(~2,w'~)>O (d)(w~e2,~2)_-<l for all r>0 . If / > 2 then (c~2,w'ei)__>0 (resp.

(~2,w~c~i)<O if weieA- (resp. wc~ieA+). Suppose that w is such that (c~2,w'c~2)__<0 for all r>0 . The following

formulae are easily checked and together with the induction hypotheses on w prove that the element SzW of W(A) satisfies the conditions of Proposition (4.1). Let r>0 , ~eA +. There exist integers (a~)0~,_~, depending on r and c~ such that:

(4.12) (a) (S2W)ro~2 -=w~o~2 +ar_l W~-lo~2 + ... +al w~2 +ao~ 2, where ai=ai(r,c~2)>O for all i=0 ,1 . . . . . r - 1 .

(b) (S2W)r~I =~X 1 -t-ar_l(S2W)r-lo~2"~ ...-lf-als2wo~2-I-aoO~2, where ai=ai(r, cq)>O for all i=0 ,1 . . . . . r - 1 .

(c) If j > 2 , (S2w)ro~j=wro~j - l -ar_ l (S2W)r - io~2-1 - ...+alS2WO~2+aoO~2, where ai=ai(r,c~i)>O (resp. a i<0 ) if w~FA+ (resp. wc~i~A- ).

(d) Let /?~A~. Then

(S2 w)r fl = Wr fl-l- a r_ l (S2 W ) r - l O~2 -t- . . . -k- a l s2 W O~2 -k- ao O~ 2

where ag=a~(r,[3)<O. Since A,2w=Awu{w-~z} and Aw~_Aw~ it follows that A~2~_A~2w)~ for all r>0 .

If there exists r > 0 such that (~2, w~c%)>0, then by (4.11)(d) we may assume that (wr~2,0~2)=1. Without loss of generality we can take r = l , so that w~2-c~z=~l is a root. We claim that

(4.13) tlr for any r>0 , and hence ( ~ 2 , W r t ~ ) ~ 0 for all r>0 .

Suppose that the claim is false. Let t > 0 be the smallest integer such that

rl~Awr for all r>t. Note that wr~2=~2 + ~ wi-lq. The induction hypothesis i=0

implies that wi-lq is nonzero for all r>0 . Since wrtleA - for all r > t we can i=O r--1

choose r so large that ~ w~-~/does not belong to F + contradicting w'~2~A +. This proves (4.13). i= o

58 V. Chari

As be fo re we h a v e the fo l lowing fo rmu lae : Le t r > 0 , s e A +. T h e r e exists

in tegers (al)o<=i<=r_ 1 d e p e n d i n g on r a n d ~ such tha t

(4.14) (a) (Szw)ro~2=wr- lr l+ar_l wr- 2rl-k - ... +aarl+aoC~ 2 where ai>O , i = 0 , 1 . . . . . r - 1 .

(b) ( s2w) 'e a =c~ a + a r _ a ( S z W ) r - l a z + ... +aa(s2W)az+aoC~2 with ai>O , i = 0 , 1 . . . . , r - 1 .

Le t j > 2. T h e n

(C) (S2W)r(Zj=wro~j-[-ar_l(S2W)r-lo~2 +...-[-al(SzW)O~Z-[-ao0~2 with ai~O (resp. ai<O ), i = 0 , 1 . . . . , r - i if w~i~A+ (resp. wc~j~A-).

(d) Le t f l~A w. T h e n

(S2 W)r fl = wr fl-k- ar_ l (S2 W)r- l o~2 q-...-k-al(s2w)o~2 +ao0~ 2

wi th a i < 0 , i = 0 , 1 . . . . . r - 1 .

Th is impl ies tha t A s2 w - A~s2 ~)r~/r > 0 since A s2 w = A ~ w { w 1 ~e } a n d A w - A ~r for all r > 0 . I f wtr/ is n o t a s imple r o o t for any t > 0 , t hen (4.14) (a)-(d) shows

tha t the e l emen t s 2 w ~ W ( A ) satisfies the c o n d i t i o n s of P r o p o s i t i o n (4.1). If (w)tr/

is a s imp le r o o t for s o m e t > 0 then by the i n d u c t i o n hypo theses on w, t is u n i q u e a n d the element (s2w)t+2~. W(A) does the job .

Th is c o m p l e t e s the i n d u c t i o n s tep a n d P r o p o s i t i o n (4.1) fol lows.

Acknowledgements. I would like to thank Professor R. Parthasarathy, S. Ilangovan and S.E. Rao for discussions.

References

1. Chari, V., Ilangovan, S.: On the Harishchandra homomorphism for infinite dimensional Lie algebras. To appear in the Journal of Algebra

2. Duflo, M.: Construction of primitive ideals in an enveloping algebra, Lie groups and their representations. Summer School in Mathematics, edited by I.M. Gelfand, Bolyai-Janov Math. Soc., Budapest (1971)

3. Kac, V.G.: (a) Simple irreducible graded Lie algebras of finite growth. Math. USSR Izv. 2, 1271- 1311 (1968) (b) Infinite-dimensional algebras, Dedekinds q-function, classical Mobius function and the very strange formula. Advances in Mathematics 30, 85-136 (1978) (c) Infinite dimensional Lie Algebras. Progress in Mathematics 44. Boston: Birkhauser, 1983

4. Kac, V.G., Kazhdan, D.A.: Structure of representations with highest weight of infinite dimensional Lie algebras. Adv. in Maths. 34, 97-108 (1979)

5. R.V. Moody: (a) A new class of Lie Algebras. J. Algebra 10, 211-230 (1968) (b) Euclidean Lie algebras. Can. J. Math. 21, 1432 1454 (1969)

Oblatum 6-VI1-1984 & 28-I-1985

annihilators of verma modules for kac-moody lie algebras

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