ft · 6/4/1993 · A = C and B is finite dimensional as a C-vector space. 1.4. If g' is a Lie...

10URNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 6, Number 4, October 1993

TENSOR STRUcrURES ARISING FROM AFFINE LIE ALGEBRAS. I

D. KAZHDAN AND G. LUSZTIG

INTRODUCTION

Let g be a simple Lie algebra of type A, D, or E. There is a natural central extension of Lie algebras 0 -+ Cl -+ i -+ C((€)) ® g -+ 0 (see 1.3) which has been extensively studied by physicists and mathematicians. (1 is a distinguished element of the centre.) A i-module is said to have central charge K' ES if 1 acts on it as multiplication by K'. For any K E C* , we denote by 19'" the category of i-modules of finite length, with central charge K - h (where h is the C~eter number) which are integrable in the sense of Kac [K]. It is known that 19'" is semisimple; moreover, it is nonzero only if K is an integer ~ h. In the work of physicists (Belavin, Polyakov, and Zamolodchikov [BPZ], Knizhnik and Zam~odchikov [KZ], Moore and Seiberg [MS]) it has been realized that the category 19'" has an additional structure, namely, that of a rigid braided tensor category. This work of physicists has been put on a rigurous mathematical foundation by Tsuchiya and Kanie [TK] , Tsuchiya, Ueno, and Yamada [TUY], and independently by Beilinson and Feigin (unpublished; but see [BFM] for a discussion of the closely related ~irasoro case). However, the precise construction of the tensor structure on 19'" has not been given in the literature.

In t~s paper we are interested in a category 19'" of i-modules which is larger than 19',,: it consists of modules with central charge K - h, of finite length, whose composition factors are simple highest weight modules corresponding to weights which are dominant in the direction of g. (The simple modules in this category are in 1-1 correspondence with the simple finite-dimensional g-modules. )

We show that, in the case where K ft Q>o' this category has a natural structure of braided tensor category; the rigidity will be shown in a sequel to this paper. (There is no such structure on 19'" in the case where K E Q>o')

We will show in another paper how this tensor structure may be used to establish an equivalence of categories between 19'" and the category of finite-dimensional integrable representations of a quantum group with parameter e"r-::rx/" . (See the announcement in [KL].)

Received by the editors October 5, 1992. 1991 Mathematics Subject Classification. Primary 20G99. Both authors are supported in part by the National Science Foundation.

905

© 1993 American Mathematical Society 0894-0347/93 SI.oo + S.25 per page

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906 D. KAZHDAN AND G. LUSZTIG

The most interesting case is that where K E Q<o ; in this case, the category &K is not semisimple, just like that of finite-dimensional representations of a quantum group at a root of 1.

One important source of inspiration for our work was the work of Drin-feld [D1 in which he showed that the tensor category of the finite-dimensional representations of the quantized enveloping algebra corresponding to g, with formal parameter, is equivalent to a tensor category whose objects are the finite-dimensional representations of g and whose tensor structure is obtained from the Knizhnik-Zamolodchikov equations. (From our point of view, this last category is essentially &K for K in a formal neighbourhood of 00.)

We now describe the content of this paper in more detail. The category &K is studied in §§2 and 3; various equivalent definitions for it are given there, and the duality functor is introduced. In §§4-6 we give three constructions of the tensor product; the first two are equivalent, but the third one is the same as the first two only under suitable finiteness assumptions (as shown in §7). In §8 we study the effect of changing the ground ring in a tensor product.

CONTENTS

1. Preliminaries 2. The category &K 3. A characterization of &K 4. First definition of tensor product 5. Second definition of tensor product 6. Third definition of tensor product 7. Finiteness for tensor products 8. Change of rings in tensor products

1. PRELIMINARIES

1.1. Modules over a Lie algebra. Let A be a commutative C-algebra with 1. Let h be a Lie algebra over A or an A-Lie algebra; we denote by U(h) the enveloping algebra (over A) of g.

An h-module is, by definition, an A-module V together with a homomor-phism h --+ EndA(V) of A-Lie algebras; this is the same as a U(h)-module.

If A --+ B is a homomorphism of C-algebras with 1, we may regard B 0 A h naturally as a B-Lie algebra. If V is a h-module, then B 0 A V is naturally a B 0 A h-module.

1.2. The Lie algebra g. Let g be a semisimple Lie algebra over C with the following property: g is isomorphic to a direct sum of copies of 2f single simple Lie algebra of type A, D, or E .

Let h be the Coxeter number of any of the simple components of g. Let ( , ): g x g --+ C be the g-invariant bilinear form defined as (2h)-1

times the Killing form. We shall assume, as we may, that g is given in terms of the generators

ei , f;, hi (i E 1) and the Serre relations [ei , fj1 = 6ijhi ; [hi' e) = aije} , [hi' fj1 = -aijfj; [hi' h) = 0; rei' rei' e)1 = Lf;, Lf;, fj11 = 0 if aij = -1; and rei' ej1 = Lt;, fj1 = 0 if aij = O. Here (ai)i,JEI is the Cartan matrix.


TENSOR STRUcrURES ARISING FROM AFFINE LIE ALGEBRAS. I 907

The form ( , ) satisfies (hi' h) = aij and (ei , fj) = 0ij . Let g - be the Lie subalgebra of g generated by the elements 1; (i E/). Let Zl (resp. N 1 ) be the set of all maps I ~ Z (resp. I ~ N). Let (bi)

be the matrix inverse to (ai); for a, b E Zl, we set (a, b) = Ei ,j bijaU)bU) . For any a E N1 , let ~ = U(g-)/(Ei U(g-)f;(i)+I). Let Ya be the image

of 1 E U(g-) in ~. There is a unique g-module structure on ~ such that g- acts by left multiplication and eiYa = 0 and hi(ya ) = aU)Ya for all i E I. This g-module is simple and finite dimensional.

The g-module dual to ~ is isomorphic to ~ where a ~ a is an involution of N 1 • There is a well-defined involution i ~ I of I such that aU) = a(I) for all iEI. 1.3. The Lie algebras i A, i A • Let A be a commutative C-algebra with 1. Let f be an indeterminate. Let Al be the free A-module of rank 1 with basis element 1.

For J., J; E A((€)) we define {J., J;} E A to be the residue at f = 0 of the formal differential form J;d(J.). We have {J., J;} + {J;, J.} = 0 and {J.J;, .t;} + {J;.t;, J.} + {.t;f" J;} = 0 for all J., J;, .t; E A((€)). Hence

iA = A((€)) ® g EB Al

is an A-Lie algebra with bracket

[fe, t e'] = f t[e, e'] + {f, tHe, e')I, [fe, 1] = 0, [1, 1] = 0

for f, r E A((f)) and e, e' E g. (We shall often write fe instead of f ® c.) Let iA = A[f , f-I] ® g EB AI. This is naturally an A-sub-Lie algebra of i A. Its bracket can also be described by the formula

n m' n+m' , (a) [f e,€ e]=f [e,e]+on+m,on(e,e)I

for any n, m E Z and any e, e' E g, together with the requirement that 1 IS

in the centre. iA is called the affine Lie algebra over A. Let i~ be the A-Liesubalgebra A[[f]]®gEBAI ofiA. Let g~ be the A-Lie

subalgebra A[f] ® g EB Al of i A • In the case where A = C we shall often omit A in our notation; for example,

we write i, g instead of iA ,iA and ® instead of ® A • If A ~ B is a homomorphism of A into another commutative C-algebra

with 1, then the obvious homomorphism B ® A gA ~ gB is an isomorphism of B-Lie algebras. On the other hand, the obvious homomorphism B ® A iA ~ iB is not necessarily an isomorphism; one case when it is an isomorphism, is when A = C and B is finite dimensional as a C-vector space.

1.4. If g' is a Lie subalgebra of g which is a sum of some of the simple components of g, then we have an obvious imbedding of A-Lie algebras ~ ~ iA given by fe ~ fe, 1 ~ 1 for all f E A((f)) and e E g' . This restricts to an imbedding of A-Lie algebras i~ ~ iA .



1.5. Let ~ : KA - KA be the A-Lie algebra involution defined by (€nc)~ = (_€)-nc for all nEZ, c E g and by (1)~ = -1.

Let V be the KA-module defined by the Lie algebra homomorphism gA -EndA(V). Composing this with the involution ~ : KA - KA' we obtain a new Lie algebra homomorphism gA - EndA(V), and this defines a new KA-module structure on V, denoted V~.

1.6. If S is a finite set, then gS = gEB" 'EBg (summands in 1-1 correspondence with the elements of S) is naturally a semisimple Lie algebra of the kind con-sidered in 1.2. Hence all definitions and results given in this paper for g are automatically applicable to gS.

We shall write t, K~ instead of 2, ~, although t, g~ are not direct sums of copies of i A , gA'

For each S E S , the imbedding g c gS of the s-component gives rise, as in 1.4, to A-Lie algebra imbeddings t5s : iA - t and t5s : gA - g~.

Taking the sum over s E S we get surjective A-Lie algebra homomorphisms (iA)s - t and (gA)S - g~ with kernels consisting of the elements (dsl)sES with (ds) E AS such that Es ds = O.

We shall sometimes write the element Es t5s(~s) of t (where ~s E i A ) as (~s) .

We have a natural isomorphism A« €))s ® g EB Al ~ t ; it takes 1 to 1 and associates to g ® c the element (gsc) , where g = (gs) E A«€))s and c E g.

We shall sometimes write gc instead of (gsc).

1.7. For any integer N ~ 1, we define QN to be the A-submodule of U(gA) generated by the products (fe1)(fe2)'" (feN) with c1 ' C2 ' ••• ,cN in g. We also define Qo to be the A-submodule of U(gA) generated by the unit element 1. Let Q~ be the image of QN under the involution ~.

Lemma I.S. (a) fPC E Qp if p ~ 1 and c E g. (b) Qn(€-Pc) E E~:~(€-p+jg)Qn-j + gQn-p + Qn-p + 1Qn_p if n ~ p ~ 0

and c E g. (c) Let f E A[[€]] and c E g. We have (jc)Q: c E:,=o Q:, U(i~).

The proofs of (a), (b) are left to the reader. We prove (c). We argue by induction on t ~ O. The case where t = 0 is trivial; hence, we may assume that t> 1.

Let ~ E Q: ' and assume that x = (€ -1 C')X' where x' E Q:-l and c' E g. We have

(fc)x = (fC)(€-I CI)X' -1 I f I f -1 I I I -l d f) I =(€ c)( c)x +( € [c,c])x +(c,c)Reso(€ ()lx.

Consider the three terms in the last sum. By the induction hypothesis, the first term is in Q~ E:;;o Q~, U (i~) c E:, =0 Q~, U (i~) .


TENSOR STRUCTURES ARISING FROM AFFINE LIE ALGEBRAS. I 909

The second term is of the form f(O)(€-I[c, c'])x' + tllc, c'])x' (where f = f(O) + € J) and hence is in Q~ + E~~~ Q:, U (g~) (by the induction hypothesis).

The third term is in Q~_I U(g~). It follows that (fc)x E E~'=o Q:, U(i~), as required.

1.9. Given a KA-module V and an integer N we define an A-submodule V(N) of V as follows. If N ~ 0, we set V(N) = {x E VIQNX = O}. If N :::; 0, we set V(N) = {x E VIQ~NX = O}. For N = 0, the two definitions coincide: we have V(O) = O. We have V(O) c V(l) c V(2) c ... ; we set V(oo) = UN>o V(N). We have V(O) c V( -1) c V( -2) c ... ; we set V( -00) = UN~O V(N) -:- Clearly,

(a) V(oo) = VU(-oo).

For any N ~ 1 we denote by V[N] the set of all x E V such that (€nc)x = 0 for all n ~ N and all c E g. We have V[I] c V[2] c ... ; we set V[oo] = UN V[N].

The elements of V(oo) are called the smooth elements of V. We say that the i-module V is smooth if all its elements are smooth, i.e., if V = V(oo).

Lemma 1.10. (a) V(oo) is a gA-submodule of V; if N ~ 0, then V(N) is a K~-submodule of V.

(b) V(-oo) is a gA-submodule of v. (c) V(N) c V[N] for any N ~ 1; hence, V(oo) C V[oo]. (d) If N ~ 2, then we have a natural exact sequence

o ~ V(1) ~ V(N) ~ Homdg, V(N -1»; the last arrow associates to x E V (N) the linear map c t---> (€ c)x , '1:/ c E g.

(e) If A is Noetherian and V(I) is a finitely generated A-module, then V(N) is a finitely generated A-module for any N ~ 1 .

Assume that N ~ O. For any p ~ 0, we have QN(€P c) C U(gA)QN (see 1.8(a), (b»; hence, V(N) is stable under (€pc) : V -+ V. We also have QN1 = lQN; hence, V(N) is stable under 1 : V -+ V. Thus, V(N) is a ~-submodule. It also follows that V(oo) is a g~-submodule. Note that QN(€-PC) C E~=o U(KA)QN-J (see 1.8(b» if N ~ p. This implies that V(oo) is stable under (€-p c) : V -+ V and (a) is proved. N6w (b) follows from (a) using the involution U of KA .

If N ~ 1 and x E V(N) then (€pc)x = 0 for all c E g and all p ~ N by 1.8(a). This proves (c). Now (d) is obvious; (e) follows from (d) by induction on N. The lemma is proved.

1.11. Any smooth KA-module V (or, more generally, a gA-module V such that V = V[oo]) can be naturally extended to a gA-module as follows. Let x E V[N] and let f E A«€», c E g; we define (fc)x = U;c)x where J;. E A[€, €-I] is such that f - J;. E €N A[[€]]. The operators (fc) : V -+ V are independent of choices and, together with the operator 1 : V -+ V of the gA-module structure, define a iA-module structure on V.



1.12. Let K' E C. A gA-module V is said to have central charge K' if 1 EgA acts on V as multiplication by K' .

1.13. Let ~ (s E S) be gA-modules which have the same central charge K' E C. Then ®SES ~ (tensor product over A) has a unique g~-module structure such that for any ~ = (~s) E g~ and any family of elements xs' E~' (s' E S) we have ~(®S'ESXS') = L:sES(®S'ESXS,S') where xs,s' E ~, is equal to ~s(xs) if s' = s and to xs' if s' =1= s .

To see this, we must only check that, if ~s = dsl EgA with ds E A satisfying L:s ds = 0, then the definition above gives zero. But in this case, we have xs,s' = K'dsxs' if s' = sand L:sES(®S'ESXS,S') = K'L:sds(®S'ESXS') = 0, as required.

Note that the g~-module ®s ~ has central charge K'. It is clear that, if each ~ is smooth, then the g~-module ®s ~ is smooth. In that case, the g~-module ®s ~ can be regarded naturally as a t'-module, as in 1.11.

1.14. Let V be a gA-module such that V = V[oo] and such that V has central charge K - h where K E C is nonzero. (h is as in 1.2.) For any k E Z, the Sugawara operator L k : V -+ V is given by

1 . ~ 1 .~. (a) Lk(x) = 2K L L(E -] Cp)(E] cp)x + 2K L L(E] Cp)(E -] cp)x

i?,-kj2 p i<-kj2 p

where (cp) is a basis of g such that (cp ' cp,) = Jpp" This operator is well defined (only finitely many terms are nonzero for any fixed x) and independent of the choice of basis. We have (b) (Enc)(Lkx) - Lk((EnC)X) = n(En+kc)X

for any x E V , any C E g, and any n E Z and

(c) [Lo' L_ 1] = L_I '

(See [KR].)

1.15. If g is replaced by g. for some finite set ", and if V is as above (for g. ), then we can restrict the g! -module V to a gA-module, via JI : gA -+ g! , where t E ". The Sugawara operator Lk for this restriction is denoted L k ; I .

Clearly, the Sugawara operator of the g~ -module V is given by

Lk = LLk;r IE.

1.16. For any A-module V we set d V = Hom A (V, A); this is again an A-module. If V is, moreover, a gA-module, then we regard d V with the gA-module structure induced from that of V; thus, (~Cu))(x) = -.u(~(x)) for all ~ EgA' .u E d V , and x E V .

. For any gA-module V we define a gImodule D(V) = dV~(oo). Note that (d V)U = d (VU) as gA-modules; this justifies omitting the brackets.



Note that V f-+ D(V) is naturally a contravariant functor from iA-modules to iA-modules.

Assume that V is a smooth iA-module. We have an obvious A-linear map V ~ d (D( V)): to any x E V we associate the linear form f f-+ f(x) on D( V) . We may regard the same as a map V ~ d(D(V))ti; this is now a iA-module homomorphism. Hence it takes V(oo) into d(D(V))ti and hence defines a i A -

module homomorphism (a) V ~ D(D(V))

since V = V(oo). Let x E V and let fJ, E D(V). We have, using the definitions,

(b)

2. THE CATEGORY &K

In this section we will introduce a category &K of i-modules with central charge K - h, where K ¢. Q>o. (See 2.15.) We will give several alternative descriptions for it (see 2.22) and show that it is stable under the duality D.

2.1. We consider a module ./Y over the C-Lie algebra C[€] 0 g. We say that ./Y is a nil-module if

(a) dimc./Y < 00, and (b) there exists t ~ 1 such that the product in the enveloping algebra of

any t elements of €C[€] 0 g acts on ./Y as zero. Using Engel's theorem, we see that the condition (b) is equivalent to the

conjunction of the following two conditions: (bl) there exists t ~ 1 such that each element of €tc[€] 0 g acts on ./Y as

zero, and (b2) any element of €C[€] 0 g acts on ./Y as a nilpotent transformation.

2.2. From Engel's theorem we see also that for a nil-module ./Y , there exists a finite filtration ./Y = A(; :J ~ :J . .. by submodules such that on each successive quotient the Lie algebra €C[€] 0 g acts as zero. 2.3. We shall fix K E c* . Given a nil-module ./Y as above, we extend ./Y to a g+ -module by defining the action of 1 E i+ to be niultiplication by K - h . (Recall (1.2) that h is the Coxeter number.) Let

'/yK = U(g) 0 U(g+)./Y

be the induced i-module. We say that '/yK is a generalized Weyl module. Note that we have a canonical imbedding ./Y c'/yK as a i+ -module and its image generates '/yK as a i-module.

Clearly, the i-module '/yK has central charge K - h. Note that, for any i-module W with central charge K - h , composition with

the canonical imbedding defines an isomorphism

(a) H0Illg(./YK, W) ~ Homg[E](./Y' W).



2.4. We apply the previous definitions in the case where ./Y = ~ (see 1.2) for some a E N1 , regarded as a C[€] ® g-module with €C[€] ® g acting as zero. In this case, the i-module ./Y" is denoted by V: ; it is called a Weyl module. By definition, V: = U(i) ®U(g+) ~

where ~ is regarded as a g+ -module with €C[€] ® g acting as zero and 1 acting as multiplication by K - h .

Lemma 2.S. (a) For any generalized Weyl module ./Y" there exists a finite fil-tration by submodules such that the successive quotients are Weyl modules.

(b) Let V be a i-module with central charge K - h . Assume that there exists N ~ 1 such that dim V(N) < 00 and V(N) generates Vasa i-module. Then V is a quotient of a generalized Weyl module ~ .

(a) follows immediately from 2.2, using the exactness of induction (which in tum follows from the Poincare-Birkhoff-Witt theorem).

We now prove (b). By 1.lO(a), V(N) is a C[€] ® g-submodule of V; it is contained in V[N]

(see 1.10(c)), and, hence, (€NC[€]) ® g acts on V(N) as zero. We show that a product in U(i) of any N elements of (€C[E]) ® g acts on V(N) as zero. It suffices to show that any product of form (EtICl)(Et2C2)··· (EtNCN ) with t\ ~ I, t2 ~ 1, ... acts on V(N) as zero. But, by 1.8(a), this product is contained in Qt Qt ... Qt C Qt +t + ... +t and so it acts as zero on V(N) since tl + t2 + ... ~

I 2 N I 2 N N. Thus, the C[E]®g-module ./Y = V(N) is a nil-module. Then the generalized

Weyl module ./Y" is well defined and it has a canonical i-homomorphism into V which induces the identity map on ./Y = V (N). Its image contains V (N) ; hence, it is the whole V since V(N) generates V as a i-module. This proves (b).

2.6. The following result concerns the action of the Sugawara operator Lo on a Weyl module V: . For any A E C, we set A V: = {y E V:ILo(Y) = Ay}.

Proposition 2.7. (a) We have V: = EBAEC<A V:). In other words, the endomor-phism Lo: V: -+ V: is locally finite and semisimple.

(b) dim A V: < 00 for any A E C . (c) If A v: :;i: 0, then A = 2~ (a, a + 2) + n for some n EN. (d) If A = 2~ (a, a + 2) , then A v: is equal to ~ and, hence, is nonzero. (e) Any proper i-submodule V' of V: is contained in the subspace EB A (A V:)

where A runs over the set {2~ (a, a + 2) + nln = 1, 2, ... }. (f) We have V: = V: (00) .

Let (cp) be a basis of g such that (cp' cpl) = Oppl . Let y be a vector in the subspace ~ of V:. We have (EjCp)Y = 0 for all p and all j > 0 and, hence,

(g) Lo(Y) = 2~ L cpcp(Y) = 2~ (a, a + 2}y; p



the last equality is a well-known property of the Casimir operator on ~. The vector space V: is spanned by the vectors

(to -nlc~) ... (to -nrc~)(y)

for various c~, c; , ... ,c; E g and n1 , n2 , ••• ,n, EN. Such a vector is contained in AV: where A = 2~(a, a + 2) + n1 + ... + n, (by (g) and 1.14(b)). Hence (a)-(c) hold. We also see that ,tV: (for A = 2~ (a, a + 2) ) is spanned by the vectors above with n 1 = ... = nr = 0; hence, it is equal to ~. This proves (d).

In the setup of (e), V' is clearly stable under the endomorphism Lo : V: -+

V: ; hence, by (a), it is the sum of its intersections with the various ,tV: . It is enough to show that, for A = 2~ (a, a + 2) , the intersection V' n ,tV: = V' n ~ is zero. This intersection is clearly a g-submodule of ~. If it is nonzero it must be equal to ~ (which is simple); thus, ~ c V' . But ~ generates V: as a g-module; it follows that V' = V: ' a contradiction. This proves (e).

We now prove (f). Clearly, ~ c V:(l); hence, ~ c V:(oo). Since V:(oo) is a g-submodule of V: and ~ generates the g-module V:' it follows that V: = V: (00). The proposition is proved.

2.8. Part (e) of the previous proposition shows that for K i- 0 the g-module V: has a unique maximal submodule. The quotient of V: by this maximal submodule is denoted L:. This is a simple g-module. The natural surjective map v: -+ L: restricts to an injective map ~ -+ L: (by 2.7(e)); we use it to identify ~ with a subspace of L: .

I Lemma 2.9. For any a EN, we have

(a) L:(l) = ~.

In particular, if a, a' E N1 , the g-modules L:, L:, are isomorphic if and only if a = a'.

The inclusion ~ c L: (1) is obvious. Assume that this inclusion is strict. Using the definitions, we see that L: (1) is g-stable. The action of g on L: ( 1 ) is locally finite (it is locally finite on V: by 2.7(a), (b); hence, it is locally finite on the quotient L: of V: ' and hence it is locally finite on the subspace L: (1) of L:). By the complete reducibility of finite-dimensional g-modules, we can find a finite-dimensional simple g-submodule <y' of L: (1) , whose intersection with ~ is zero. Then, for some b E N1 , there is an isomorphism of g-modules 'Ph ~ <y' . Since <y' c L: (1) , there is a unique homomorphism of g-modules V~ -+ L: whose restriction to 'Ph is the isomorphism 'Ph ~ <y' just considered (see 2.3(a)). This homomorphism has image which is a simple g-module; hence, it factors through a homomorphism of g-modules L~ -+ L: (necessarily an isomorphism) which carries 'Ph onto <y'. By 2.7(c), we have that ~ is the A-eigenspace of Lo : L: -+ L: ' where A = 2~ (a, a + 2). By the same result for



b instead of a, we have that ~ is the ).' -eigenspace of Lo: L~ -+ L~ , where ).' = tc(b, b + 2). The isomorphism L~ ~ L: is compatible with the actions of Lo. Hence).' is an eigenValue of Lo: L: -+ L: (necessarily different from A, since r' n ~ = 0), and, similarly, A is an eigenvalue of Lo : L~ -+ L~ (necessarily different from A'). By 2.7(d), we then have).' - A E {I, 2, ... } ; by the same result for b instead of a, we have A -).' E {I , 2, ... }. This is a contradiction; (a) is proved. The lemma follows.

Lemma 2.10. Let W, W' be i-submodules of v: such that W C W' and W 1= W'. There exists a' E N1 such that:

(a) the 2~ (a' , a' + 2)-eigenspace of Lo : w' /W -+ w' /W is nonzero, and (b) 2~«(a', a' + 2) - (a, a + 2)) EN.

W, W' are stable under Lo : v: -+ v: and Lo induces semisimple oper-ators on W, W' ,and W' /W. (See 2.7(a).) By 2.7(c), we can find an eigen-value A E C of Lo : W' / W -+ w' / W with minimum possible real part. If x is a vector in the corresponding eigenspace, then for any c E g we have that Lo«Ec)x) = (u)(Lox) - (u)x = (A. - l)(u)x. By the choice of A. we then have (u)x = o. Thus we have x E (W' /W)(l). Thus the intersection of the A eigenspace of Lo : W' / W -+ w' / W with (W' / W)( 1) is nonzero. This intersection is a finite-dimensional g-submodule. It follows that there exists a' E N1 and a homomorphism of i-modules V:' -+ w' / W which maps the subspace ~, injectively into that intersection. Using now 2.7(d), it follows that A = tc(a' , a' + 2) .

Applying 2.7(c) to V:' we see that A = 2~ (a, a + 2) + n for some n EN. The lemma is proved.

Lemma 2.11. Let W' be a i-submodule of V: such that 01= Wi =f. v:. There exists a' E N1 such that 2~ «(a' , a' + 2) - (a, a + 2}) E {I , 2, 3, ... } .

Let a' be as in the previous lemma applied with W = 0 and Wi. Let A = 2~«a', a' + 2}. Applying 2.7(e) with V' = Wi (recall that W' =f. V:) we see that A - 2~ (a , a + 2) E {I , 2, 3, ... }. The lemma follows.

Proposition 2.12. (a) If K ¢ Q, then V: is an irreducible i-module for any a EN1 .

(b) If K E Q<o' then V: is an irreducible i-modulefor any a E N1 such that (a, a + 2) < -2K. In particular, Va is irreducible.

If v: is reducible, then from 2.11 we see that K is rational. If, in addition, we have K E Q<o' then 2.11 shows that (a, a+2}+2Kn = (a' , a' +2) for some integer n ;?: 1. Since (a', a' + 2) = Ebj,ja'(i)(a'(j) + 2) and the quantities bjj , a' (i) are ;?: 0, we have (a', a' + 2) ;?: o. Hence, (a, a + 2) + 2Kn ;?: O. Since K < 0 and n;?: 1, it follows that (a, a + 2) + 2K ;?: O. The proposition is proved.



2.13. In the following result we will use the following notation: for a E N1 , Fa denotes the finite set

I I I I Fa = {a EN I(a + 1, a + 1) ~ (a + 1, a + I)}.

Proposition 2.14. Assume that K ¢. Q>o. Let a E N1 . For any g-submodule W of v: we denote 6(W) = E). dim(W n). V:) E N where the sum is taken over all A. E C such that A. = 2~ (ai, a' + 2) for some a' E Fa .

(a) If W, Wi are g-submodules of V: such that We Wi and W =I- W', then 6(W) < 6(W').

(b) The i-module V: has finite length. All its composition factors are of the form L:, for various a' E Fa .

(c) All composition factors of the unique maximal submodule of V: are of the form L:, for various a' E N1 such that (a' + 1, a' + 1) < (a + 1, a + 1). In particular, the simple module L: appears exactly once in a composition series of V:.

In the setup of (a), we consider a' given by 2.10. Let A = 2~ (ai, a' +2). If we had (ai, a' + 2) - (a, a + 2) > 0, then from the inclusion

1 I I 2K ((a, a + 2) - (a, a + 2)) E N

(see 2.10(b)) we would deduce that 2~ E Q>o' contradicting our assumption on K. Thus, we must have the opposite ineqU"ality (ai, a' + 2) - (a, a + 2) ~ 0, which shows that a' E Fa .

By 2.1O(a), the A-eigenspace of Lo : Wi /W -+ Wi /W is nonzero. Hence, the dimension of the A-eigenspace of Lo : Wi -+ Wi is strictly bigger than the dimension of the A-eigenspace of Lo : W -+ W. Using the definition we deduce that 6(W) < 6(W') and (a) is proved.

From (a) it is clear that the g-module V: must be of finite length ~ 6(V:). Now let W C Wi be i-submodules of V: such that Wi /W is simple as

a i-module. By the proof of 2.10, there exists a nonzero homomorphism of i-modules V:' -+ Wi /W. Since Wi /W is simple, we must have W' /W ~ L:, and (b) is proved. (c) follows in the same way, using 2.7(d). The proposition is proved.

In the remainder of this paper it is assumed that K ¢. Q~o. Definition 2.1S. &'" is the full subcategory of the category of i-modules whose objects are the i-modules V with the following properties:

(a) V has central charge K - h, and (b) there exists a finite composition series of V with all subquotients of the

form L: for various a E N1 .

2.16. By 2.14(b), any Weyl module v: belongs to &'". Using 2.5(a), we deduce that any generalized Weyl module ./Y" belongs to &'" .

We have the following result.



Proposition 2.17. If V belongs to &", then V is smooth and dim V (N) < 00 for all N?I.

To prove that dim V(N) < 00 for N? 1, it is enough, by 1.1O(e), to prove this only for N = 1 .

A Weyl module V: is smooth, by 2.7(f); hence its quotient L: is smooth. Note also that L:(I) is finite dimensional by 2.9. Using the definition of &'", we are reduced to proving the following result.

Lemma 2.1S. Let 0 - V' - V - V" - 0 be an exact sequence of i-modules. (a) If V' = V'(oo) and V" = V" (00) , then V = V(oo). (b) We have an exact sequence 0 - V'(I) - V(I) - V"(I). Hence, if

dim V' (1) < 00 and dim V" (1) < 00 , then dim V ( 1) < 00 .

The proof of (b) is immediate. We now prove (a). Let B be a C-basis of g. Let Y E V and let y" be the image of y in V".

By assumption, we have y" E V"(N) for some N ? 1. Hence, the vectors (Eel)··· (fCN)(y) belong to V' for any sequence cI ' ... , cN in B. Since there are only finitely many such vectors (N is fixed), they are all contained in V' (N') for some N' ? 1 (by our assumption on V').

Thus, for any c;, ... ,C~f in B and any cI ' ... , cN in B we have

(fC~) ... (fC~f )(fCI) ... (EeN )(y) = 0

so that y E V(N + N'). This completes the proof of the lemma and, hence, that of 2.17.

2.19. Let V be an object of &". For any ..:t E C we denote by ). V the C-subspace of V consisting of all x E V such that x is in the kernel of some power of (Lo - U) : V - V where Lo is the Sugawara operator.

Lemma 2.20. (a) We have V = EB).Ed). V) and, for any ..:t E C, we have dim).V < 00.

(b) There exist II' ... ,1m in N such that

{..:t E CI). V oF O} c {~~ + N} U··· U {~~ + N}. The lemma is already known in the case where V is a Weyl module. (See

2.7.) If the lemma is true for V then it is clearly true for a quotient i-module of V. Therefore, it is true for the simple quotient of a Weyl module. If the lemma is true for a i-submodule of V and for the corresponding quotient, then it is clearly true for V itself. We apply this repeatedly to a composition series of V and the lemma follows.

We shall write !Rz for the real part of a complex number z. Proposition 2.21. Let V be an object of &'" .

(a) For any ..:t E C and any N? 0 we have Q~(). V) C .l.+N V . (b) For any real number t there exists N ? 1 such that EBm~t (). V) :J Q~ V .



(c) Consider the C-linear map ¢ J.. : g ® J.. V --+ J..+ 1 V given by c ® v 1-+ (€ -1 c)v (see (a)). There exists a real number t' such that ¢J.. is surjective for all A. E C such that !JU > t' .

(d) For any N ~ 1 there exists a real number t" such that E9!RJ..>t" (J.. V) c Q~V. -

(e) For any N ~ 0, the C-vector space V/Q~ V is finite dimensional.

In the case where N = 0, (a) is trivial. The case where N ~ 1 can be immediately reduced to the case where N = 1; in that case the result follows from the commutation formulas 1.14(b) for Sugawara operators.

We prove (b). Let t E R. Let 11' ... ,1m be as in 2.20. Let N ~ 1 be such that N ~ maxu (t - !R( 2~ K). Let A. be such that J.. V =f. 0; by the definition of the lu' we have A. = 2~/u + n for some n ~ 0 and some u.

By (a), we have Q~J..V C J..+NV, and it remains to show that !R(A. + N) ~ t or that !R( 2~ )Iu + n + N ~ t. This holds by the definition of N. Thus, (b) is proved.

We now prove (c). Assume that 0 --+ V' --+ V --+ V" --+ 0 is an exact sequence in &'". Then

from 2.20(a) we see that for any A. E C we have an induced exact sequence o --+ J.. V' --+ J.. V --+ J.. V --+ O. Tensoring with g we obtain an exact sequence that is the top row of the following commutative diagram with exact rows

o -----+ g ® J.. V' -----+ g ® J.. V -----+ g ® J.. V -----+ 0

1 1 1 I o -----+ J..+ 1 V -----+ J..+ 1 V -----+ J..+ 1 V -----+ 0

where the vertical maps are as in (c). From this we see that if (c) holds for V' and for V", then it also holds for

V; moreover, if (c) holds for V, then it also holds for V". Thus we are first reduced to the case where V is a simple object of &'" and

then we are further reduced to the case where V is a Weyl module V: . Let A.o = 2~ (a, a + 2). Then J..o V: = ~ (see 2.7(~).) By the definition of

Weyl modules we have EN>o Q~~ = V:. By (a) we have Q~~ C J.. +N V: ; - 0

this implies in view of the previous equality and 2.20(a) that Q~~ = J.. +N V: o

for all N ~ O. This shows that (c) holds with t' = !RA.o . This completes the proof of (c). By (c), for any N ~ 1 and any A. E C such that !RA. ~ t' + N, we have that

Q~J.._NV = J.. V . This clearly implies (d). We prove (e). Using (d) and 2.20(a) we see that, for any N, there exists a

real number t" such that the natural map E9!RJ..<t" (J.. V) --+ V / Q~ V is surjective. The last direct sum is a finite-dimensional C-vector space by 2.20. It follows that V / Q~ V is a finite-dimensional C-vector space. The proposition is proved.



Theorem 2.22. Let V be a i-module with central charge K - h. The following conditions are equivalent.

(a) V is in 19". (b) V is a quotient of a generalized Weyl module. (c) There exists N ~ 1 such that V(N) isfmite dimensional and generates

Vasa i-module.

If V satisfies (c), then it satisfies (b), by 2.5(b). If V satisfies (b), then it satisfies (a), by 2.16. (A quotient of an object in 19" is in 19".)

Now let V be an object in 19". We will show that V satisfies (c). Let M = E9!RA<t'+I(.1.V) where t' is as in 2.21(c). This is a finite-dimensional C-subspace of V by 2.20.

We show by induction on N ~ 0 that # ~ I I A V c QNM lor any A. E C such that t + N ~ lRA. < t + N + 1 .

For N = 0 this follows from the definition of M. Assume now that N > 1 and that our assertion is known for N - 1. If t' + N ~ lRA. < t' + N + 1, then by 2.21 (c) we have A V c Q~ (A-I V) and by the induction hypothesis we have Q~(A-I V) C Q~Q~_IM c Q~M. This proves our assertion by induction. It implies, using the definition of M and 2.20, that V = EN>o Q~M. Thus, M generates V as a i-module. Since V is smooth (2.17) and M is finite dimensional, we have M C V(N) for some N ~ 1. This V(N) will then generate V as a i-module; it is finite dimensional, by 2.17. We have proved that V satisfies (c). The theorem is proved.

2.23. We now stu4y the i-module D(V) (see 1.16) for V in 19". As a vector space, D(V) consists of those C-linear forms f: V --+ C such that f is zero on Q~V for some N ~ 1. By 2.21(c), (d), this condition on f is equivalent to the condition that f is zero on E9!RA>t(.l. V) for some t E R and, by 2.20, this is equivalent to the condition that f-is zero on all but finitely many of the subspaces A V of V. Thus, as a C-vector space, we have

(a) D(V) = EBd(AV). A

From the definition, D( V) is a smooth i-module with central charge K - h . Hence the Sugawara operators are well defined on D( V). From 1.16(b) we see that the subspace d (A V) of D( V) is precisely the set of all vectors in D( V) annihilated by some power of Lo - A. : D(V) --+ D(V); we denote it AD(V).

Proposition 2.24. If V is the i-module L~, then the i-module D( V) is iso-morphic to I4 where a is as in 1.2.

Assume that W is a i-submodule of D( V) distinct from 0 and D( V) . Then W is stable under Lo : D( V) --+ D( V) and the direct sum decom-position D(V) = E9A(AD(V)) (see 2.23) induces a direct sum decomposition W = E9A(W n AD(V)). Since W =f. 0 and W =f. D(V), we must have


TENSOR STRUcruRES ARISING FROM AFFINE LIE ALGEBRAS. I 919

(a)

for some A, t. Let Wl. = {x E Vlf(x) = 0 Vf E W}. Then Wl. is clearly a i-submodule

of V and from (a) we see that Wl. is distinct from 0 and V. This contradicts the fact that V is simple. Thus, D( V) is a simple i-module.

Let AO = 2~ (a, a + 2) . By 2.7, we have A V = 0 unless A - AO EN. By 2.23, we then have AD(V) = 0 unless A - AO EN. For any n ~ 1 and c E g, we have (€nc);. D(V) C A _nD(V) by 1.14(b) and, hence, (€nc)A D(V) = O. Thus,

o 0 0 the finite-dimensional g-stable subspace A D( V))s annihilated by (€C[ €]) ® g ;

o hence, there exists a i-module homomorphism V:' -+ D( V) that maps ~, injectively into A D( V). Since D( V) is simple, this must define an isomorphism

o of i-modules L:, ~ D( V). As a g-module, A D( V) is isomorphic to the dual

o of A V and, hence, to the dual of ~; in particular, it is isomorphic to ~. It

o follows that a' = a. The proposition is proved.

Proposition 2.25. (a) If 0 -+ V' -+ V -+ V" -+ 0 is an exact sequence in &'K' then the corresponding sequence of i-modules 0 -+ D( v") -+ D( V) -+ D( V') -+ o is exact.

(b) If V belongs to &'K' then the i-module D( V) belongs to &'K' We first prove (a). From 2.20 it follows that for any A E C we have an

induced exact sequence of C-vector spaces 0 -+ A V' -+ A V -+ A V -+ 0 . Taking dual spaces we obtain an exact sequence 0 -+ d (A V") -+ d (,t V) -+

d I (AV) -+ O. . Taking the direct sum over all A and using 2.23(a), we obtain an exact se-

quence 0 -+ D(V") -+ D(V) -+ D(V') -+ O. This proves (a). To prove (b) we see, by repeated application of (a), that we are reduced to

the case where V is simple. In that case, we may use 2.24. The proposition is proved.

Proposition 2.26. If V is in &'K' then the natural map V -+ D(D(V)) (see 1.16(a)) is an isomorphism of i-modules.

With the notation in 2.23 we have

(a) AD(V)~d(AV), Applying (a) to D(V) instead of V, we obtain

(b)

Taking the dual of (a) (in which the vector spaces involved are finite dimen-sional), we obtain

(c)

From (b) and (c), we obtain A V ~ AD(D(V)). Taking the direct sum over A, we obtain V ~ D(D(V)); it is clear that this coincides with the map 1.16(a). The proposition is proved.


920 D. KAZHDAN AND G. LUSZTlG

2.27. Now assume that we are given a set 0 with two elements S, S'. Let ~, ~, be objects of &". Then V = ~ ®~, is a ir::) -module as in 1.13.

Proposition 2.28. (a) V is in &" (relative to ir::)). (b) D(V) = D(~) ® D(~,).

(a) follows, for example, from the characterization 2.22(b) of &" . We prove (b). From the definitions we see that for any A. E C we have

). V = l:D (). ~ ®). ~,) W 1 2 ).1+).2=).

(finite sum). Taking dual spaces for both sides (which are finite dimensional) we obtain

d EB d d ). V = ( (). ~) ® ().~, )). 1 2

).1+).2=).

Taking now the direct sum over A., we obtain (b). Proposition 2.29. Let V, Vi be two objects of &".. The C-vector space Homt9' (V, Vi) is finite dimensional.

K

If V; -+ V is a surjective homomorphism of i-modules, we have a natural imbedding Homt9' (V, Vi) -+ Homt9' (V; , Vi). Thus our assertion for (V, Vi) follows from the Kassertion for (V;: Vi). By 2.22, we can find a surjective homomorphism of i-modules V; -+ V with V; a generalized Weyl module. Thus we are reduced to the case where V is a generalized Weyl module ./Y" . By 2.3(a), we are then reduced to showing that dim Homg[E]('/y , Vi) < 00. By the definition of a nil-module, there exists t ~ 1 such that the image of any g[t]-homomorphism ./Y -+ Vi is contained in Vi (t) . Thus, Homg[E](./Y' Vi) = Homg[E](./Y' Vi (t)). Since ./Y and Vi (t) are finite dimensional, the proposition follows. 2.30. Let V, Vi be two objects of &". Let W be the subspace of II).,).' Hom(). V ® ).' Vi ,C) consisting of all 1;.,).' such that

(a) 1;.,).' «tn c)x ® y) + I;.+n ,).' +n(x ® « _t)-n c)y) = 0

for all A., A.' E C, c E g, n E Z and all x E Mn V, Y E ).' Vi. Let »'; be the subspace of W consisting of those (I;. ).,) such that for any A. E C, we have I;. ).' = 0 for all but finitely many A.'. '

'Using 2.20 and 2.23, we see that Homt9' (V, D(V')) may be naturally iden-tified with the subspace WI of K

II Hom(). V, ).,D(V' )) = II Hom(). V ® ).' Vi , C) = W. ).,).' ).,).'

On the other hand, we may regard V ® Vi as a i.-module with zero action of 1 and with

(t n c)(x ® y) = «t n c)x) ® y + x ® « _t)-n c)(y) for all c E g, n E Z, x E V, Y E Vi. Using again 2.20, it is clear that the



vector space Homg( V ® V' ,C) (where C is taken with zero i-action) may be naturally identified with W. The imbedding W. c W may therefore be considered as a linear map

(b) Hom~K (V, D(V')) --+ Homg(V ® V', C).

Proposition 2.31. The linear map 2.30(b) is an isomorphism. We must prove that W = w.. Let A. E C. Since ). V is finite dimensional,

we can find N ~ 1 such that). V c V(N). By 2.21(d), we can find t > 0 such that for any A.' satisfying !RA.' ~ t we have ).' V' c Q~ V'. Consider y E ).' V' for such A.'. Then we can write y as a sum of elements of form (€-\) ... (€-ICN)Z with Z E ).'_NV' . For any x E). V we have

/;.,)., (x ®y) = L/;.,).I(X ® (€-ICI )··· (€-ICN)z)

= L/;'-N,).'_N((UN)··· (ul)x ® z)

where we have used N times 2.30(a) for n = 1. But the last sum is zero since x E V(N). Thus we have /;.,)., = 0 unless !RA.' < t. By 2.20, there are only finitely many eigenvalues A.' of Lo : V' --+ V' satisfying ru' < t. Thus we have WI = Wand the proposition is proved.

2.32. To V, V' in &'K we associate the C-vector space

(a) (V, V') = (V ® V')/i(V ® V')

where V ® V'is regarded as a i-module as in 2.30. (Thus this is not the standard tensor product of i-modules.)

It is clear that the C-linear isomorphism V ® V' ~ V' ® V given by x ® y I----t

y®x preserves the images ofthe i-module actions and induces an isomorphism of C-vector spaces

(b) (V, V') = (V', V).

The isomorphism 2.30(b) asserted in 2.31 can be regarded as an isomorphism

(c) Hom/9' (V, D(V')) ~ HomcC(V, V'}, C). K

This, together with 2.29, implies that dim(V, V'} < 00. Note that V, V' I----t

(V, V') is naturally a functor from &'K X&'K to finite-dimensional vector spaces.

Proposition 2.33. (a) Any nonzero i-submodule of D(V:) contains the image of D(L:) c D(V:) (see 2.25(a)).

(b) D(V:)( 1) is isomorphic to ~ as a g-module.

(a) is dual to the statement that V: has a unique maximal submodule, namely the kernel of v: --+ L: .

We now prove (b). Let r' be an irreducible g-submodule of D(V:)(I). As in the proof of 2.9, we can find a homomorphism of i-modules rp : V~ --+ D(V:) that carries ~ isomorphicallyonto r'. Since L~ is a quotient


922' D. KAZHDAN AND G. LUSZTIG

of if>(V~), it is a composition factor of D(V~). By duality, L~ is a composition factor of V~ (see 2.24); hence,

(c) 2~((b,b+2)-(a,a+2))EN (see 2.7). By (a), if>(V~) contains the submodule D(L~). Assume that it con-tains it strictly. Then D(L~) is a composition factor of the maximal submodule of V~ ; hence, again by 2.24 and 2.7, we have

1 2K ((a, a + 2) - (b, b + 2)) E {I , 2, 3, ... , }.

This contradicts (c). Thus, we must have if>(~) = D(L~). This implies that r' c D(L~)(I). We now use 2.9; the proposition follows.

3. A CHARACTERIZATION OF &"

3.1. In this section we will give a criterion for a i-module to be in &" by a condition which is weaker than that in 2.22(c). This criterion will be crucial in our study of tensor products.

Theorem 3.2. Let V be a i-module with central charge K - h. The following conditions are equivalent.

(a) V is in &". (b) V is smooth and dim V(I) < 00.

The proof will occupy the rest of this section. One of the main ingredients in the proof is the use of projective objects in (a truncation of) &". Soergel [So] has proved an abstract version of the Brauer (or BGG) reciprocity which is well adapted to our needs. In the subsections 3.3-3.7 we will show that Soergel's axioms hold in the case of interest to us. (Alternatively, one could use [RW].)

3.3. Let t be an integer ~ 1. We consider the full subcategory &: of &" whose objects are the V in &" with the following property: all composition factors of V are of form L~ for some a in the finite set

pt = {a E N11(a, a + 2):5 t}.

We have &:' c &: if t' :5 t. Clearly, any V in &" belongs to &: for some t.

We regard pt as a partially ordered set: we say that a' :5 a if either a' = a or (a', a' + 2) < (a, a + 2).

The following properties are obvious.

(a)

(b)

&: is closed under extensions in the category of i-modules with central charge K - h .

If V is in &:' then D(V) is in &:. Lemma 3.4. If a E pt , then V~ is in &:. Moreover, all composition factors if V~ other than L~ are of form L~, with a' E pt, a' < a .

This follows from 2.14(c).



Lemma 3.S. If a is a maximal element of Ft for ~, then the canonical map V: -+ L: is a projective cover for L: in ~:.

If f: V: -+ X is a morphism in ~: ' then f restricts to a g-morphism from ~ into the (generalized) 2~ (a, a + 2}-eigenspace Wx of Lo : X -+ X. Then 2~ (a, a + 2) - 1 is not an eigenvalue of Lo: X -+ X (by the choice of a). An argument in the proof of 2.10 shows that Wx is contained in X(l). Hence, restriction defines a bijection between g-morphisms v: -+ X and g-morphisms from ~ to Wx .

Now let p : X' -+ X be a sUljective morphism in ~: and let f: V: -+ X be any morphism in ~:. We want to lift f to a morphism I : V: -+ X' such that pI = f. Our morphisms restrict to g-morphisms PI : Wx' -+ Wx and It : ~ -+ Wx with PI surjective. But in the category of finite-dimensional g-modules, all objects are projective. Hence, we can find a g-morphism .r: : ~ -+ Wx' such that pl.r: = It. By the earlier argument, .r: extends uniquely to a i-module homomorphism I : V: -+ X'. This clearly has the required property. Thus we have proved that V: is projective in ~:. It is clearly an indecomposable object. The lemma is proved.

Lemma 3.6. For any a E N1 , the endomorphisms of the i-module V = L: are just the scalars in C.

Indeed such an endomorphism is completely determined by its restriction to V(l) = ~, and then one uses Schur's lemma for simple g-modules.

3.7. Let C?f (resp. C?f+, C?fo) be the abelian category whose objects are C-vector spaces V with a given vector space decomposition V = €a .l.Ed.l. V) and with a given i-module (resp. g+-module, g EB IC-module) structure such that such that:

(a) fnC(.l. V) C .l.-n V for all c E g and all n E Z (resp. n EN, n = 0), (b) I acts as multiplication by K - h , (c) the action of g on each .l. V is locally finite. We have a natural imbedding ~" C C?f as a full subcategory, defined by

attaching to a g-module V in &'" the direct sum decomposition defined by the generalized eigenspaces of Lo: V -+ V . The following holds.

I I s~~ ~~ (d) For any a,a EN and any s~ 1 wehaveExt~(l'a,D(l'a'))=O.

We will sketch a proof following Polo [P]. The objects of C?fo are both projective and injective. Using objects induced

or "co-induced" from C?fo to C?f, C?f+ , we see that C?f, C?f+ have enough projec-tive and injective objects. We can regard ~ as an object of C?f+ so that the corresponding induced object in C?f is v: . Since induction from C?f+ to C?f is exact and takes projectives to projectives, we see that

(e)



Next one verifies that ~ can be regarded as an object of ~o so that the corresponding "co-induced" object in ~+ is the restriction of D(V~,). It fol-lows that D(~,) is an injective object of ~+. Hence, the right-hand side of (e) is zero and (d) follows. 3.8. From 3.3(a),(b), 3.4, 3.5, 3,6, 3.7(a), 2.24 we see that Soergel's axioms [So] for his abstract Brauer (or BGG) reciprocity are satisfied for ~:. Thus we can apply Soergel's result in our case and obtain the following result. Proposition 3.9. For each a E Ft there is a projective object Pa of~: such that:

(a) dim Hom/9: (Pa , L~,) = daa, for a, a' EFt, and K

(b) Pa admits afinitefiltration with subquotients ofform V~ (b EFt) and the number of occurences of ~ is equal to [~ : L~l .

Here we have used the following notation: if V is in ~IC' we denote by [V : L~] the number of subquotients in a composition series of V which are isomorphic to L~ .

To state the following result we need some further notation. For a finite-dimensional g-module or we denote by [or : ~lg the number of times ~ appears in a decomposition of or as a direct sum of simple g-modules. Corollary 3.10. For any V in ~IC and any a E N1 , we have

[V: L:l :5 I)~ : L:UV(l) : ~lg. b

(Here b runs over N1 ; but only finitely many terms in the right-hand side are nonzero since dim V(l) < 00.)

We choose t such that V is in ~:. Since Pa is projective in ~:, the number dim Homi9' (Pa , Vi) is additive

with respect to short exact sequences in Vi E ~: . It follo~s that

dim Hom/9: (Pa , V) = L dim Hom/9: (Pa , L:, )[V : L:, 1 K K

a'

a'

Moreover, if 0 -+ X' -+ X -+ X" -+ 0 is a short exact sequence in ~:, then we have

dim Hom/9: (X, V) :5 dim Hom/9: (X', V) + dim Hom/9: (X", V). KKK

Applying this repeatedly to a filtration of Pa as in 3.9(b), we obtain

dim Hom/9: (Pa , V) :5 L[~ : L:ldimHom/9: (~, V) K K

and the corollary follows.

b

= L[~: L:ldimHomg(~' V(l)), b


TENSOR STRUcruRES ARISING FROM AFFINE LIE ALGEBRAS. I 925

3.11. Proof of Theorem 3.2. Let V be a i-module as in 3.2(b). For any N ~ 1, V(N) is finite dimensional (see 1.1O(e)) and the i-module V{N} of V generated by V(N) satisfies 2.22(c); hence, it is in &". Therefore, 3.10 may be applied to it. Since V{N}(I) = V(I), we see from 3.10 that

a,b a,b

The last expression is finite and independent of N. It follows that the ascending sequence of i-submodules V {I} c V {2} c ...

must be stationary. The union of the terms of this sequence is equal to V (since already the union of the V(N) is equal to V). It follows that we must have V {N} = V for some N ~ 1 . Since V {N} is in &", the same must hold for V.

Conversely, any i-module in &" is as in 3.2(b), by 2.17. The theorem is proved.

4. FIRST DEFINITION OF TENSOR PRODUCT

4.1. Assume that we are given n smooth g-modules with the same central charge K - h. The usual tensor product of these modules is a g-module with central charge n(K - h); moreover, it does not have reasonable finiteness prop-erties when the factors do.

Our present aim is to give a modified construction of a tensor product which leads to a module with the same central charge K - h .

Unlike the usual tensor product, the modified tensor product is defined in terms of a choice, namely, of local parameters (up-to second order) at (n + 1) distinct points of a curve.

4.2. The setup for our definition is the following one. We assume given a finite nonempty set S with a given nonempty subset \;? We assume that for each So E \;? we are given a subset [so] of S containing So such that, when So runs through \;?, the subsets [so] form a partition of S. Thus any s E S belongs to [so] for a unique So E\;? Let • = S - \;? We assume that each subset [so] contains at least two elements (thUS it contains, besides so' at least one element in .).

We assume given a smooth projective curve Cover C, such that any con-nected component of C is isomorphic to pi and such that the set of connected components of C is in a given bijection (so +-+ Cs ) with the set \;?

o We assume that C has specified distinct points Ps (s E S) such that Ps E

Cs for all s E [so] . o We also assume that for each s E S we are given a morphism Ys : pi ---- C

such that Ys defines an isomorphism pi ~ Cs (where s E [so]) and such that o

Ys(O) = Ps ' (These morphisms are called charts.)

4.3. Let ~ (s E .) be a collection of smooth i-modules with the same central charge K - h , indexed by the set •.



Our definition of tensor product will attach to the g-modules ~ and to the system of charts (ys) , a gl;:) -module T( W)U .

The most important case for applications is that where C is connected (so o has a single element) and • consists of two elements; thus, we are given two i-modules ~.

However, for technical reasons, it is necessary to consider the more general setup described above.

4.4. An outline of the definition of T(W)U is as follows. We will define a natural filtration of the usual tensor product of C-vector spaces W = ®SE. ~ , and we will form the completion W of W with respect to this filtration. We will show that this completion is naturally a gl;:) -module. We restrict this to a il;:) -module, apply to it U, and in the resulting gl;:) -module we take the subspace of smooth vectors. This subspace will be T(W)U.

4.5. Let C' be the affine open curve obtained from C by removing the points Ps (s E S). Let R be the algebra of all regular functions C' ---+ C.

For any S E S , we have an algebra homomorphism R ---+ C((€»

defined as follows:

(a) Sf is the power series expansion at 0 of the rational function fys on pl.

4.6. The Lie algebra r. For 1; ,h in R, let {1;, h} be the sum of residues of the differential form hd(1;) on C' at the points Ps (s E 0). The bilinear pairing { , }: R x R ---+ C satisfies

{1; , h} + {h ' 1;} = 0, {1; h ' i;} + {hi; , 1;} + {i;1; , h} = 0 for all 1;, h ' f3 E R .

Hence, r = (R ® g) EB Cl has a natural structure of C-Lie algebra with 1 in the centre and with lfe, Ie'] = fIle, e'] + {f, IHe, e')1 for all f, I E R and all e, e' E g. (We write f e instead of f ® e .)

We have natural Lie algebra homomorphisms

(a)

(b) The map (a) is given by

the map (b) is given by

r ~. ---+ g .

f e t-+ I: 0/ f e and 1 t-+ 1 ; sEI;:)

fe t-+ I: 0/ fe and 1 t-+ -1. sE.

The fact that (a) is a Lie algebra homomorphism is obvious. The fact that (b) is a Lie algebra homomorphism is a consequence of the following fact: if



1;. , 1; are functions in R, then the sum of the residues at Ps (s E .) of the differential form 1;d (1;.) on C' is equal to minus the sum of residues of that form at Ps (s E 0). (This follows from the residue theorem since that differential form is regular on C'.)

4.7. r-module structure on W. Now W is naturally a i·-module since the ~ have the same central charge. (See 1.13.) By restriction, W is therefore also a r-module. Note that 1 E r acts on W as (-K + h) times the identity.

4.8. Definition of a decreasing filtration of W. For any n E Z, let Rn be the subspace of R consisting of those f such that So f E €nq[€]]} for all So EO. We have··· eRn c R n_ l C .... Not that RnRm c Rn+m for all n, m.

For any integer N ~ 1, let G N be the C-subspace of U (r) spanned by all products (1;.cl )(1;c2)··· (lNCN) with 1;.,1;, ... , fN E RI and cI ' c2' ... , cN E g. It is convenient to define Go as the set of all C-multiples of the unit element in U (r) .

The vector space W has the following natural decreasing filtration

W:::>G I W:::>G2W:::>··· where G N W is defined using the U (r)-module structure on W.

This gives rise to a projective system of vector spaces: WjG I W f- WjG2W f- WjG3W f- •••

the corresponding projective limit is denoted W. 4.9. i-module structure on W. We state the following three facts ((a) is left to the reader and (b),(c) will be verified in 4.12,4.13):

(a) (Approximation property.) Assume that for'each So E 0 we are given a power series W sEC( (€ )). Then, for any N ~ 0, there exists g E R such that

o So g _ Ws E €N q[€]] for all So EO.

o (b) Let N~ 1 andlet gERN , cEg. Then gCEGN .

(c) Let t ~ r ~ O. Let C E g and f E R_r . We have (fc)Gt C Gt_rU(r). Let W = (WS)SEC/ E C((€))C/ and let c E g. We define a C-linear map (wc) : W ---+ W as follows. We choose a sequence

gl' g2' ... of elements of R such that S gn - Ws E €nq[€]] for n = 1, 2, ... and all s EO. (See (a).) We also choose q ~ 0 such that Ws E €-qq[€]] for all s EO. We represent an element of W by a sequence (Xl' X 2 ' ••• ) of elements in W such that xn+l - xn E Gn W for n = 1, 2, ....

We have \gn+l - gn) E €nq[€]] for all s EO; hence, by (b),

(d) (gn+l - gn)c E Gn· We have gn+l E R_q by the choice of q; hence, by (c),

(e) (gn+lc)Gq+n We Gn W. We set

(f)


928 D. KAZHDAN AND G. LUSZTlG

(Here, (gnc)xq+n is defined by the r-module structure of W.) The right-hand side of (f) defines an element of W; indeed,

(gn+lc)xq+n+1 - (gnc)xq+n = (gn+lC)(xq+n+1 - xq+n) + ((gn+l - gn)c)xq+n E (gn+lC)Gq+nW + GnW C GnW + GnW C GnW.

(We have used (d), (e).) We show that the map (wc) is well defined. Let g:, g~, ... be another

sequence of elements in R such that S g~ - Ws E €nC[[€]] for n = 1,2, ... , i and for all s EO; let q ~ 0 be such that Ws E €- C[[€]] for all S EO.

We assume that q' ~ q. Let (x~, x~, ... ) be a sequence of elements in W such that x~ -xn E Gn W

for all n> 1. We ha~ g~ - gn ERn; hence, by (b),

(g) (g~ - gn)C E Gn·

We have gn E R_q by the choice of q and xi+n - Xq+n E Gq+n W (by the definition of (xn) and by the assumption q' ~ q); hence, by (c),

(h)

We have g~ E R_q, by the choice of q' and x~'+n -xq'+n E Gq'+n W; hence, by (c),

(i) (g~c)(x~'+n - xq'+n) E Gn W.

Using (g), (h), (i), we see that

(g~c)x~'+n - (gnc)xq+n = ((g~ - gn)c)xq'+n

+ (g~c)(x~'+n - xq'+n) + (gnc)(xq'+n - xq+n) E Gn W. In other words, we have

((glC)Xq+1 , (g2c)xq+2, ... ) = ((g;C)X~'+l' (g~C)X~'+2' ... )

as elements of W. This shows that the map (wc) is well defined.

Lemma 4.10. The endomorphisms (wc) define a t' -module structure on W with 1 acting as multiplication by -K + h .

Let W = (WS)SE<::1 E C((€))<::1 and let C E g. We choose a sequence gl' g2' ... of elements of R such that Sgn - Ws E

€nC[[€]] for n = I, 2, ... and for all s EO. (See 4.9(a).) We also choose t ~ 0 such that Ws E €-tc[[€]] for all s EO.

Let ww = (WSWS)SE<::1 E C((€))<::1; then (ww[c, C)) : W - W is well defined. Let hn = gn+tgn+q. Then shn - wsws E €nC[[€]] for n = 1,2, ... and for

allsEO.



From the definitions, the commutator ofthe endomorphisms (wc) and (w' c') applied to (Xl' X 2 , .•• ) gives the element of W represented by (Y l '

Y2' ... ) where

Yn = (gnc)(in+i)Xq+t+n - (inC)(gn+lc)xq+l+n = (in+qC)(gnc)Xq+l+n - (gnC)(gn+lc)xq+l+n + (gngn+q[c, C]xq+l+n + (-K + h){gn' gn+q}(c, C)xq+l+n·

On the other hand, the endomorphism

(ww[c, C]) + (-K + h) L {ws' ws}(c, C)Idw sE<::>

applied to (Xl' X2' ... ) gives the element of W represented by (Zl' z2' ... ) where

Zn = (hn[c, C])xq+l+n + (-K + h) L {ws' ws}(c, C)xn· sE<::>

We must prove that (Y l ' Y2' ... ) and (Zl' z2' ... ) represent the same ele-ment of W.

It is enough to show that Yn -zn E Gn_tW for all n > t. (Then for all n;::: 1 we have Yn+l- zn+l E GnW, and, since Yn+l - Yn E GnW, zn+l- zn E GnW, it would follow that Yn - zn E GnW for all n.)

It is therefore enough to show that for all n > t we have

(a)

(b)

and

(c) {gn' gn+q}(c, C)xq+t+n - L{ws ' ws}(c, C)xn E Gn_tw. sE<::>

Since fqWs ' fnWs ' fnr;, rs are in C[[f]] (recall that n > t), we see that the last three residues are zero; hence, {gn' gn+q} = 2:sE<::>{WS ' ws}. This together with Xq+t+n - xn E Gn W implies (c).

We have gn - gMt E Rn and gn+q E R_t ; hence, (gn - gn+t)gn+q E R n- t so that (b) follows from 4.9(b) (recall that n > t).

The expression in (a) is equal to the sum of four terms:

«gn+q - gn)C)(gnc)xq+l+n ' «gn - gn+t)c)(gnC)xq+l+n ' Cin(gn - gn+l)[c, c])xq+l+n' (-K + h){gn' gn - gn+l}(c, c)xq+l+n·



It is enough to show that each of these four terms is in Gn_tW. We have 'in+q - 'in ERn; hence, by 4.9(b), the first term is in GnW hence in Gn_tW.

Similarly, we have gn - gn+t E Rn; hence, by 4.9(b), the second term is in G n Wand, hence, in G n-t W .

This shows also that 'in(gn - gn+t) E R n_t (recall that 'in E R_t ). Using again 4.9(b), we see that the third term is in Gn_tW.

To deal with the fourth term it is enough to show that {'in' gn - gn+t} = O. This follows from 'in E R_t' gn - gn+t ERn and from n > t.

This completes the proof of the lemma.

4.11. The definition of T( W)". We restrict the gO -module W to gO using the obvious imbedding gO C gO. We now consider the gO -submodule T( W) = W(-oo) of the gO-module W (see 1.9). Hence the gO-module T(W)" is defined (see 1.5). It has central charge K - h..

We get the same gO -module if we first apply U to the gO -module Wand then take the set of smooth vectors: (W") (00) .

4.12. Proof of 4.9(b). We argue by induction on N. The case where N = 1 is trivial. Now assume that N> 1. If g E RN , we can write g as a sum of products g' gil where g' ERN' , gil E R Nil and N' + Nil = N, N' 2: 1, Nil 2: 1 . Also we can write c as a sum of commutators [c', c"] in g. Clearly, we may assume that both sums have a single term: g = g' gil and c = [c' , c"]. We have gc = [g' c' , gil c"] in r; hence, gc = (g' C')(g" c") - (gil c")(g' c') in U(r).

By the induction hypothesis, this is contained in G N' G Nil + G Nil G N' C G N as required. 4.13. Proof of 4.9(c). We argue by induction on t. The case where t = r is trivial. We now assume that t > r. Let 1;,1;, ... ,1; be elements of Rl and let c1 ' c2 ' ••• , ct be elements of g. Let e = U;c2) ••• (1;ct ) •

The product (f c) (1; c l)e is equal to the sum of three terms:

(1; c1)(fc)e + (f 1;[c, cd)e + {f, 1;}(c, c1)le . The first term is in G1 Gt- r- 1 U(r) (by the induction hypothesis). The second term is in Gt_rU(r) (trivially, if r = 0, and by the induction hypothesis, if r 2: 1). The third term is in Gt_rU(r) (trivially, if r 2: 1, and since {f, 1;} = 0, if r = 0). This completes the proof.

4.14. Remarks. For any s E S with s E [so] (so E c::?), let fs E R be the function defined by

1 (a) fs(P) = y;l(p) for p E CSo and fs = 0 on C - Cso '

This function is regular on C - {ps} and satisfies s fs = t. A system of charts (ys) on C is said to be special if, for any SEc::?, the

point ys(oo) belongs to the set {ps,ls' E .}. An equivalent condition is that fs does not vanish on C' if SEc::? In this

case, the function fs-n belongs to R for any n E Z and we have s (fs-n) = En ;


TENSOR STRUCTURES ARISING FROM AFFINE LIE AWEBRAS. I 931

therefore, we have a simple formula for the action of ~s€n C E g<' (with C E

g, nEZ, S E 0 on W (or W(-oo)), namely,

(a) (~s€nC)(YI' Y2' ... ) = ((fs-nC)Yq+l, (fs- nC)Yq+2' ... )

where q E N is large enough. If (Ys ) is not necessarily special, the formula (a) remains valid provided

that n ~ 0 (we can take q ~ -n); however, for n > 0, the action of ~s€n C must be defined by the more complicated procedure explained earlier, involving approximations.

4.15. Relation to coinvariants. Let X be a smooth gO -module with central charge (K - h). Then X extends naturally to a gO -module (see 1.11) which then can be restricted to a r-module using the homomorphism of Lie algebras r -+ gO (see 4.6(a)).

The tensor product W ® X of the r-modules W, X is a r-module in the standard way; it is clear that 1 acts as zero on this r-module.

On the other hand, the tensor product T( W) ® X of the gO -modules T( W) , X is a gO -module in the standard way; again it is clear that 1 acts as zero on this gO -module.

We show that

(a) ifxEGtW,VEX(N), andt~N, thenx®vEG1(W®X).

(In the last expression we use the restriction of the r-module structure to the Lie subalgebra G 1 of r.)

We can assume that x = C;I .. 'C;tW with C;I' ..• ,C;t E G1 and W E W. By definition, C;I ((C;2'" C;tw ) ® v) = (C;1C;2" 'C;tw ) ® v + (C;2 ... C;tw ) ® C;IV ; hence, (C;1C;2" 'C;tw)®v is equalto -(C;2" 'C;tW)®C;IV modulo G1(W®X). Similarly, (C;2C;3"'C;tW)®v is equal to -(C;3"'C;tw)®(C;2C;IV) modulo G1(W®X).

Continuing in this fashion, we see that (C;1C;2" 'C;tw) ® v is equal to ±w ® (C;t" 'C;2C;IV) modulo G1(W ®X). Since t ~ N, we have C;t" 'C;2C;IV = 0 by the definition of X(N). Thus (a) is proved.

4.16. We will define a canonical C-linear map

(a)

If t ~ N , the C-linear map W ® X (N) -+ W ® X , induced by the obvious inclusion X(N) -+ X, takes the image of Gt W ® X(N) -+ W ® X(N) into the subspace G1 (W ® X) of W ® X (see 4.15(a)); hence, it induces by passage to quotients a C-linear map

(b)

We consider the C-linearmap W®X(N) -+ (W/GtW)®X(N) (thecanonical map W -+ W / Gt W on the first factor and the identity map on the X (N)-factor) for some t ~ N; its composition with (b) is a C-linear map

(c) W ® X(N) -+ (W ® X)/G1 (W ® X).



From the definitions it is clear that the map (C) is independent of the choice of t. It is also clear that if N' ~ N , the maps (c) defined in terms of Nand N' are compatible with the linear map W ® X(N) -+ W ® X(N') induced by the canonical imbedding X(N) c X(N'). Hence the maps (c) define a linear map lim (W®X(N)) -+ (W®X)/GI(W®X) or, equivalently, W®X-+

--+N>I (W ® X) / G~ (W ® X). (We use our assumption that X is smooth.) Composing the last linear map with the linear map T(W) ® X -+ W ® X induced by T(W) c W, we obtain the linear map (a). 4.17. The map v in 4.16(a) has the following property:

(a) vCg~ (T(W) ® X)) is contained in r(W ® X)/GI (W ® X).

Indeed, let -r E T(W), U EX, S E <::?, a E Z and C E g. We must show that v(os€ac)(-r ® u)) E r(W ® X)/GI (W ® X). We represent -r = (XI' x2' ... ). Assume first that a ~ O. We have

(os€a c)(-r ® u) = (crs-a c)xl _a, U;-a c)x2_a, ., .) ® u + -r ® (os€a c)u

and v«Os€aC)(-r ® u)) = U;-aC)Xt ® u + Xt ® 0s(€aC)u = U;-aC)(Xt ® u)

for sufficiently large t, and this is in the image of the r-action, as required. Assume next that a > O. Choose a sequence gl' g2' ... in R such that

s gn - €a E €nC[[€]] for all n and all S E <::? We have

and (os€ac)(-r ® u) = «glc)xI , (gIC)X2, ... ) ® u + r ® (Os€ac)u

v«Os€aC)(-r ® u)) = (gtc)xt ® r + xt ® (os€ac)u,

v«Os€a c)(-r ® u)) = (gtc)(xt ® r) + Xt ® « _s gt + €a)c)u for sufficiently large t. The first term is in the image of the r-action; the second term is zero for large t, by the definition of gt'

This proves (a). 4.18. From 4.17(a) we see that the linear map 4.16(a) induces a linear map (a) (T(W) ® X)g<:> -+ (W ® X)r'

5. SECOND DEFINITION OF TENSOR PRODUCT

The results in this section will not be used elsewhere in the paper. 5.1. We preserve the setup of 4.2 and the notation of §4. We will give an alternative definition of T(W) (see 4.11) as a lim lim ( ).

--+ +---

5.2. Let HI be the subspace of r spanned by the elements fsc (s E <::? , C E g) , where fs is as in 4.14.

For any integer r ~ 1, let H, be the subspace of U(r) spanned by the products (fs cl)(fs c2)··· (fs c,) for various cI ' c2' ... , c, E g and SI' S2' ... ,

1 2 r S, E <::?


TENSOR STRUCTURES ARISING FROM AFFINE LIE ALGEBRAS. I

Since Is ,Is ' ... E R_1 ' we have (by repeated application of 4.9(c» 1 2

(a) HrGt C Gt_rU(r) for all t ~ r ~ 1.

For t ~ r ~ 1 we can consider the following C-subspace of W:

~,r = {x E Wlex E Gt_rW "fie E Hr}

and its quotient space Wt,r = ~,r/Gtw.

933

(We have GtW C ~,r by (a).) The inclusions ~,1 C ~,2 C ... C ~,t induce by passage to quotients imbeddings (b)

The inclusions ... C w,.+2,r C w,.+l,r C w,.,r

and ... C Gr+2W C Gr+1 We GrW

induce by passage to quotients C-linear maps (c) ... - W r+2,r - W r+1,r - Wr,r·

For any integer r ~ 1, we consider the C-subspace w,. of the product TIt>r W t r consisting of families of elements x t r E W t r (for our fixed r) whIch are compatible with each other under the maps (c).' Equivalently,

w,. = ~ Wt,r· t

The imbeddings (b) define natural imbeddings WI C ~ C Jf; C .... We define W = Ur >l w,.. Thus, W = lim lim W t r.

- --+r -+-t '

5.3. We will construct a natural isomorphism of C-vector spaces

T(W) = W(-oo) ~ W. We consider an element x E W(-oo). We represent it by a sequence

(Xl' X2' ... ) of elements in W such that xn+1 -xn E GnW for n = 1, 2, .... By the definition of W( -00), there exists an r~ 1 such that (t5s € -\ )(t5s € -1 C2 )

1 2

... (t5s €-l cr )X = 0 for all C1, C2 ' ... , cr in g and Sl' S2' ... , sr E <::?

In ~ew of 4.14, this means that Hrxr+l E G1 W, Hrxr+2 E G2 W, .... Thus, Xr+ 1 E w,.+ 1 , r' Xr+2 E w,.+2, r' ... , and we have a sequence of elements

in W r+ 1 , r' W r+2, r' ... . This sequence defines an element YEw,. C W. It is easy to check that x t-+ Y is a well-defined map W ( -00) - W.

A map in the opposite direction is constructed as follows. If t ~ max(r, N) ~ 1, we have a map

(a)

induced by the inclusion ~, r C W .



The composition of the canonical map Jt:. --+ W t , r with the map (a) is a map Jt:. --+ W / G N W which is clearly independent of the choice of t. These maps are compatible with the imbeddings Jt:. c Jt:.+ I and, hence, give rise to a C-linear map lim Jt:. --+ W / G N W or equivalently W --+ W / G N W .

---+ r The maps W --+ W/GNW (for various N) are compatible with the maps

W/GN+I W --+ W/GNW and, hence, define a C-linear map W --+ W whose image may be checked to be in W( -00). Thus we have obtained a C-linear map W --+ W( -00). One can verify that this is the inverse of the map W ( -00) --+ W constructed earlier.

6. THIRD DEFINITION OF TENSOR PRODUCT

6.1. We preserve the setup of 4.2 and the notation of §4. We shall give another definition of the tensor product. It involves something like a double dual. We will show in §7 that it coincides with the first definition of tensor product under certain finiteness conditions.

6.2. We consider the C-vector space Z = HomcCW, C) with the r-module structure inherited from W. For any N 2: 1 , we denote by ZN the C-subspace of Z consisting of all elements A E Z such that <!"A = 0 for all <!" E G N' In other words, ZN is the annihilator of GNW c W. We have ZI C Z2 C ...

and denote by ZOO the union of all ZN. Note that ZOO is a r-submodule of Z. (Indeed, let fER, C E g, and let

A E ZN . We can find t 2: 0 so that f E R_t . By 4.9(c), we have (fC)A E ZN+t . Thus (fc): Z --+ Z maps ZOO into itself.)

6.3. We define a g'Y -module structure on ZOO C((€»'Y and let C E g so that

(a) ~'Y WC Eg .

Let A E ZOO . We can find N > 1 such that A E ZN. We choose a function g E R such that S g - Ws E €Nq~]] for all s E CV. (See 4.9(a).)

We define (WC}A = (gc);' E Zoo. (Here (gC)A is given by the r-module structure of ZOO .) This is independent of the choice of Nand g, by 4.9(b). It is easy to verify that the endomorphisms (wc) define a gO -module structure on ZOO with 1 acting as multiplication by K - h .

The restriction of the gO -module ZOO to a gO -module is denoted T' (W) .

Lemma 6.4. (a) The gO-module T'(W) is smooth. (b) The g'Y -module ZOO is the gO -module attached to the smooth gO -module

T' ( W) as in 1.11.

Let A E ZN with N 2: 2. Let cI ' ... , cN be elements of g, and let Sl' S2'

••• ,SN be a sequence in CV. Let w(i)cj = 0sfCj E gO. We can find g E R

such that s g - € E €N q[€]] for all S E CV. I



By definition, we have (w(N)cN)). = (gcN)).. Moreover, (gcN) E Gl and hence GN_l(gcN)). C GN). = O. Thus, (gcN)). E ZN-l . Applying the previous argument to (gc N)). instead of )., we obtain

(w(N - 1)CN_ l )(w(N)cN)). = (gcN_l)(gcN))..

Continuing in this way we see that (w(1)cl )(w(2)c2 )··· (w(N)cN))' = (gcl )··· (gcN)).·

The last expression is zero since ). E ZN . It follows that (w(1)cl )(w(2)c2 )··· (w(N)cN))' = O.

This shows that). E T' (W)(N) and (a) is proved. If ). E ZN and wc as in 6.3(a) satisfies Ws E EN C[[E]] for all S E S, then,

using 4.9(b), we see that (wc)). = O. This shows that (b) holds. The next result follows immediately from the definitions.

Lemma 6.5. We have T'(W)(1) = Zl .

One can also easily show that T'(W)(N) = ZN for all N 2': 1.

6.6. The relationship between the gO -modules T( W), T' (W) can be given by two natural maps:

(a) a homomorphism of gO-modules T(W)ti --- D(T'(W)) , (b) a homomorphism of gO-modules T'(W) ___ D(T(W)ti).

These will be constructed in the following two subsections.

6.7. By definition, we have ZN = d(W/GNW). Taking again duals, we obtain d(ZN) = d(d(W/GNW)). Hence the obvious linear map

(a) W/GNW ___ d(d(W/GNW))

may be identified with a linear map d N

(b) W /GNW --- (Z ). This passes to projective limits and gives a linear map

(c) lim W/GNW ___ lim d(ZN), +-- +--N N

but

(d) ~ W/GNW= Wand N

hence, we have obtained a linear map

~ d(ZN) = d(Zoo); N

(e) W ___ dT'(W).

A routine verification using the definitions shows that this is compatible with the gO -module structures where d T' (W) is given the gO -module structure in-herited from that of T' (W) .



It is clear that (e) restricts to a homomorphism of g'V -modules --- d, T(W) = W(-oo) -+ ( T (W))(-oo).

Applying U, we get the map 6.6(a).

6.8. Remark. If it were known that dime W/GNW < 00 for every N, so that 6.7(a) is an isomorphism for every N, then the previous argument shows that each of the maps 6.7(b), (c), (e) and 6.6(a) would be an isomorphism.

6.9. Let A E T'(W). Then A is a C-linear form W -+ C which is zero on G N W for sufficiently large N. Hence, it defines a linear form AN: W / G N W -+ C for sufficiently large N. The composition of W -+ W / G N W with AN is a linear form i : W -+ B which is independent of N. Let i' : T( W) -+ C be the restriction of i to T( W). Then A 1-+ i' is a C-linear map T'(W) -+ d(T(W)). One can verify that it is a homomor-phism of g'V -modules. Hence, it restricts to a homomorphism of g'V -modules T'(W)(oo) -+ (dT(W))(oo). Since T'(W)(oo) = T'(W) (see 6.4), this gives a homomorphism

T'(W) -+ (dT(W))(oo).

This is, by definition, the map 6.6(b).

6.10. Now let X be a smooth g'V-module with central charge K - h. We can regard X as a g'V -module as in 1.11 and as a r -module via the homomorphism r -+ g'V given in 4.6(a).

We will define natural isomorphisms

(a) HOIllg<:> (X , T'(W)) ~ Homr(W ® X, C),

(b)

Let p : X -+ Z = HomcCW, C) be a C-linear map. We associate to p the C-linear map p' : W ® X -+ C given by p'(w ® v) = p(v)(w). It is clear that p 1-+ p' is an isomorphism

(c)

The condition that p above is compatible with the r-actions on X and Z is that p(Cjc)v)(w) = -p(v)«jc)w) for all c E g, all S E Yl and all j E R; this is the same as the condition that the corresponding p' is compatible with the actions of r.

Hence, (c) restricts to an isomorphism (d)

If q : X -+ ZOO is a homomorphism of g'V -modules, then the composition of q with the inclusion ZOO C Z is a homomorphism q' : X -+ Z of r-modules. The correspondence q 1-+ q' is then an injective C-linear map (e) HOIllg<:> (X, Zoo) -+ Homr(X, Z).



We show that it is surjective and, hence, an isomorphism. Let q' : X -+ Z be a homomorphism of r-modules.

Since X is smooth, for any x E X there exists an N ~ 1 such that x E X(N). Then we have also GNx = o.

Since q' commutes with the action of r, it follows that q' (x) E ZN . Thus, the image of q' is contained in ZOO • Thus, there is a unique homomorphism of r-modules q : X -+ ZOO such that q' is the composition of q with the inclusion ZOO c Z. By the density of r in t' (see 4.9(a)), q is automatically a homomorphism of gO -modules. This proves that (e) is an isomorphism. Composing it with (d) we get the isomorphism (a).

We now essentially repeat the previous arguments with W, r replaced by --r::J T(W), g .

The natural homomorphism analogous to (c):

(cl) HomdX, d T(W)) ~ HomdT(W) ® X, C)

restricts to an isomorphism

(dl) d Homg<:> (X , T(W)) ~ Ho~(T(W) ® X, C).

If r : X -+ (dT(W))(oo) = D(T(W)~) is a homomorphism of gO-modules, then the composition of q with the inclusion (d T( W))( 00) C d T( W) is a homomorphism r' : X -+ d T( W) of gO -modules. The correspondence r 1-+ r' is then an isomorphism

(el)

Composing it with (dl) we get the isomorphism (b).

6.11. The homomorphism T'(W) -+ D(T(W)~) (see 6.6(b)) gives rise to a linear map

(a) HOIllg<:> (X , D(T(W)~)) -+ HOIllg<:> (X , T' (W)).

Combining this with the isomorphisms 6.1O(a),(b), we obtain a linear map

(b) Homr(W ® X, C) -+ Homg<:> (T(W) ® X, C)

or, equivalently, a linear map of coinvariants

(c) Homd(W ® X)r' C) -+ HOIIlc((T(W) ® X)gO , C).

The last map can easily be identified with the transpose of the map 4.18(a).

6.12. Remark. If the linear map 6.6(b) were known to be an isomorphism, then, by construction, the maps 6.11(a), (b), (c) in the previous subsection are also isomorphisms.

7. FINITENESS FOR TENSOR PRODUCTS

7.1. We preserve the setup of 4.2 and the notation of §§4 and 6. We shall assume that for any S E • the g-module ~ has central charge



K - s and that we can find Ns 2: 1 such that the subspace ~(Ns) is finite dimensional and generates ~ as a g-module.

[Note that, under our hypothesis that K i. Q>o' the assumption above is equivalent to the assumption that the ~ belong to &'K (see 2.22); however, the hypothesis on K is not used in the proofs of 7.4, 7.5.]

The main result of this section is that, when applied to objects of &'K ' the constructions of the tensor product given in §§4 and 6 lead to the same object and that, moreover, this object is in &'K. (See Theorem 7.9.)

7.2. For S E ., with s E [so] (so E C:::» , let gs E R be the function defined by

1 (a) gs(p) = l/y;l(p) _ l/y;l(ys(O)) for p E CSo ' and gs = 0 on C - Cso ·

o 0

We have s 2 gs E f. + f. C[[f.]],

s' gs E C[[ f.]] if s' =1= s , S' E • ,

s' gs = 0 if s' i. [so], s -I' * , gs = rf. + r for some r E C , r E C.

7.3. We choose, for any So E C:::>, a function g(so) E R such that So g(so) E

f. +f. 2C[[f.]] and g(so) = 0 on C - Cso . (For example, we could take g(so) = gs for some S E [so] - {so}; see 7.2.)

For any integer N 2: 1 we denote by X N the C-subspace of U(r) spanned by the products (g(SI)CI )(g(S2)C2)··· (g(SN)CN) with cI ' c2' ... , cN in g and SI,S2, ... ,SN in c:::>. Clearly, XNcGN ·

We shall prove the following result:

Proposition 7.4. Let ~ be the subspace ®sE. ~(Ns) 0/ W = ®sE. ~, where Ns are as in 7.1. For any M 2: 1 we have

(a) M-I

W = WI + L XI WI + G M W. 1=1

Corollary 7.5. W/GMW is a finite-dimensional C-vector space/or any M 2: 1.

Indeed, the proposition shows that

M-I dim(W/GMW):::; L(IC:::>ldimg/ II dim V(Ns )·

1=0

7.6. We begin the proof of the proposition by showing, by induction on M 2: 1 , that

(a) GM C X M + L GM, . M':M'>M



Assume first that M = 1. Let e E g and I E R be such that for some So E \? we have So 1- f E f2C[[f]] and 1=0 on C - Cs • Then, by the choice

o of g(so)' we have 1= g(so) + g' where g' E R2 .

By 4.9(b), we have g' e E G2 • By definition, we have g(so)e E XI. Hence, Ie = g(so)e + g'e E XI + G2 • Since the elements Ie as above span GI as a vector space, we have GI C XI + G2 , as required.

Next we assume that M ~ 2. Using (a) with M replaced by 1 and by M - 1 and the inclusion X NeG N ' we have

GIGM _ I C GI(XM_I + L GM,) M'?,M

C XIXM_I + G2XM_I + L XIGM, + L G2GM, M'?,M

( a) is proved. 7.7. Assume that M ~ 2 and that Proposition 7.4 is known to be true whenever M is replaced by M' where 1 ~ M' < M. We show that it is then also true for M.

By our assumption, we have W = ~ + GI W. It follows that GM_I W C GM_I~ +GM_IGIW.

Using now 7.6(a), we deduce GM_I We XM_I WI + GMW. Combining this with the induction hypothesis W = ~ + E~~2 Xt WI +

GM _ I W we see that M-2

W C ~ + L XtWI +XM_I~ +GMW, t=O

as required. 7.8. We are thus reduced to proving the proposition in the special case where M = 1. The general case can immediately be reduced to the case where C is connected (hence \? has a single element). For simplicity we shall carry out the proof for C connected, assuming also that • consists of two elements S I ' S2 • The proof for general • is the same, but the notation is more complicated.

By our assumption, we have ~ = Et>o Q:V(Ns ) (notation of 1.7) for s = SI' S2· We shall write ~, l'2, NI , N2 instead of ~ , ~ , Ns ' Ns . It is then

1 2 1 2

enough to verify the following statement: For any t l , t2 ~ 0, any el E Q: ' 1 e2 E Q: ' and any XI E ~(NI)' X2 E l'2(N2), we have

2

W ~~0~~E~+~W We will prove (a) by induction on tl + t2 ~ o. When tl + t2 = 0, (a) is

trivial. Hence, we may assume that at least one of t l , 12 (say, t l ) is ~ 1. We may assume that e l = (f-Ie)e~ for some e~ E Q: -I and some e E g.

1



By the definition of the r-module structure of W, we have

(b)

We have S2gs E C[[€)) (see 7.2); hence, using 1.8(c), •

(2 gs.c) c;2 x2 E L Q:' ~(N2)' t' : t' :<:,t2

Using the induction hypothesis for (t, - 1 , t') where t' :5 t2 , we deduce that

(c)

Recall now that s. g = r€-' + r' for some r, r' E C with r =1= 0 (see 7.2). s. Using again 1.8(c) we see that

(r' c)c;~ x, E L Q:, V, (N,) , t'9.-'

and using, as before, the induction hypothesis, we see that

This together with (c) and (b) shows that r(€-'c)c;~x, ® c;2X2 E W, + G, W. (The left-hand side of (b) is in G, W since gs c E G, ' see 7.2). Since r =1= 0,

• it follows that C;,X, ® c;2X2 E »'; + G, Wand the induction step is established. Proposition 7.4 is proved.

Theorem 7.9. (a) The ";-modules T(W)U and T'(W) belong to &". (b) The natural map T(W)U --+ D(T(W)) (see 6.6(a)) is an isomorphism of v: -modules. (c) The natural map T'(W) --+ D(T(Wh (see 6.6(b)) is an isomorphism of

jr::) -modules.

By 7.5, we have dim W/GNW < 00 for any N ~ 1. By Remark 6.8, we see that (b) holds.

Since T'(W) is a smooth jr::)-module (see 6.4(a)) with central charge K - h and since T(W)(l) = Z' = d(W/G, W) is finite dimensional, we see from Theorem 3.2 that T' (W) belongs to &". Since D takes &" into itself (see 2.25), we deduce that D(T'(W)) belongs to &,,; using (b), it follows that (a) holds. Now (c) follows from (a), (b), since D is involutory on &" (see 2.26).

Proposition 7.10. Let X be a smooth jr::) -module on which 1 acts as multipli-cation by K - h. We regard X as a gr::) -module as in 1.11 and as a r-module via the homomorphism r --+ gr::) given in 4.6(a). If the C-vector spaces of coin-variants (T(W) ® X)g and (W ® X)r are finite dimensional, then the natural


TENSOR STRUCTURES ARISING FROM AFFINE LIE ALGEBRAS. I

linear map

(T(W) 18) X)i ~ (W 18) X)r

(see 4. 18(a)) is an isomorphism.

941

From (e) and Remark 6.12, we deduce that the natural map Homd(W 18) X)r' C) ~ Homd(T(W) 18) X)g, C) (the transpose of the map 4.18(a)) is an isomorphism. The proposition follows.

7 .11. We now assume that e has precisely two connected components or, equivalently, that 0 consists of two elements s', s". Let S' = [s') and S" = [s"). Then S', S" form a partition of S. Let 0' = {s'}, 0" = {s"} , , , {' """} ,,, • = S - s}, and. = S - {s . Let e ,e be the connected components of e corresponding to s', s" respectively.

Note that e', e" inherit systems of charts (YS)SES" (YS)SES" from e. Thus (S' , , e' ) d (S" "." e" ) d f h k· d , 0 , ., , .. . an ,0" , ... are ata 0 t e same m as (S, 0, ., e, ... ) in 4.2. We can define the Lie algebras r', r" in terms of these data in the same way as r was defined in terms of the datum in 4.2. Similarly, the subspaces G~ c U(r') and G~ c U(r") are defined like the subspaces G N c U (r) .

As in 4.8, W' = ®SE.' ~ (resp. W" = ®SE." ~ ) has a natural filtration " " " .• -, ( -II ) GNW (resp. GNW ) and correspondmg completIon W resp. W . As

in 4.11, we get from this a well-defined g-module T(W') (resp. a g-module T(W")). These modules have the same central charge; hence, T(W') 18) T(W") is naturally a t::>-module (as in 1.13).

We have the following result.

Proposition 7.12. We have a natural isomorphism of gO -modules

T(W') 18) T(W") ~ T(W).

Let (XI' x2' ... ) and (YI ' Y2' ... ) be two sequences of elements in W', ". , , G" "£, 11 W respectIvely such that X'+I - x, E G,W and Y'+I - Y, E , W lor a

r ~ 1. Define a sequence (zl' Z2' ... ) of elements in W = W' 18) W" by ZI = XI 18) YI ' Z2 = x2 18) Y2' .... We have

Z'+I - z, = (X,+I - x,) 18) Y,+I + x, 18) (Y,+I - Y,) E G,W.

Moreover, if x, E G~W' for all r ~ 1 or if Y, E G~W" for all r ~ 1, then z,EG,W for all r~ 1.

Thus ((XI' x2' ... ), (YI ' Y2' ... )) 1-+ (zi ' z2' ... ) defines a C-linear map

(a) --, ---" .-W 18)W ~W.

We want to construct a map in the opposite direction. For any object X m &" (relative to gO) we have



Homd(W ® X)r' C) = Homd(T(W) ® X}g" , C) d = Homg" (T(W), X)

~ d ~ = Homg,,(T(W) , X)

= Homg"(T(W)~ , d X"(oo))

= Homg" (T( W) , D(X)").

The first equality is by 6.12, which is applicable in view of 7.9(c); the second and third are obvious; the fourth is because T( W)" is smooth; the fifth is obvious.

We apply this with X = D(T(W' )") ® D(T(W")~) (see 2.28(a)). Using X = D(T(W')# ® T(W")~) (see 2.28(b)) and D(X) = T(W')~ ® T(W")" (see 2.26), we obtain

(b)

Homd(W ® (D(T(W' )") ® D(T(W")"))r' C)

= Homg" (T(W), T(W') ® T(W")).

Similarly, we have ( ) (( I I ~ I I C Home W ®D(T(W) )r" C) = Homg(T(W), T(W))

and (d) ",," "" Homd(W ® D(T(W ) )r'" C) = Homg(T(W ), T(W )).

To the identity map of T(W') there corresponds under (c) a r'-linear map b' : W' ® D(T(W' )" ---+ C; to the identity map of T(W") there corresponds under (d) a r" -linear map b" : W" ® D(T(W")" ---+ C.

The tensor product b' ® b" is a r-linear map

W ® (D(T(W' )") ® D(T(W")") ---+ C,

and this corresponds under (b) to a g~-linear map T(W) ---+ T(W') ® T(W"). One can verify that this is the inverse of the map (a). The proposition follows.

8. CHANGE OF RINGS IN TENSOR PRODUCTS

8.1. In this section we assume that we are given a commutative C-algebra B with 1.

Most of the definitions and results of §§4, 6, 7 extend with only minor changes to the case where C is replaced by B. We will review those definitions and results in this more general context, and we will point out the places where special care should be taken. 8.2. We place ourselves in the setup of 4.2. For any S E S we tensor the homomorphism R ---+ q(€)) (given by f I---> sf, see 4.5(a)) by B, and we obtain a B-algebra homomorphism B ® R ---+ B ® q (€)). Composing this last homomorphism with the obvious B -algebra homomorphism B ® q (€ )) ---+

B((€)), we obtain a B-algebra homomorphism (a) B®R---+B((€)) (/1---> sf).



We consider the B-Lie algebra r B = B i&I r. We have natural B-Lie algebra homomorphisms

(b)

(c)

defined in terms of the maps (a), just as 4.6(a), (b).

8.3. We assume that we are given a collection of iB-modules Vs (s E .) with the same central charge K - h , indexed by the set •.

We regard W = ®SE. VS (tensor product over B) as a ~-module, as in 1.13 and as a rB-module via the homomorphism 8.2(b). We have a natural homomorphism of C-Lie algebras r -. r B (x ....... 1 i&I B x); we identify an element of r with the corrresponding element of r B. Via this homomorphism, we may regard W also as a r-module. Hence the B-submodules GNW of W are well defined; they form a descending filtration of Wand we may form, as in 4.8, the projective limit W = ~ N W /GNW.

8.4. Let W = (WS)SEI:;) E B((E»I:;) , and let C E g. Then wc E g~ . We choose a sequence gl' g2' ... of elements of B i&I R such that S gn - Ws E En B[[E]] for n = 1, 2, ... and all S EO. (Compare 4.9(a).) We also choose q 2': 0 such that Ws E E-qB[[E]] for all S EO.

We define a B-linear map (wc): W -. W by setting

for any element of W represented by a sequence (Xl' X 2 ' ••• ) of elements in W such that xn+l -xn E GnW for n = 1,2, .... (Here, (gnc)xq+n is defined by the r B-module structure of W.)

As in 4.9, 4.10, we see that the B-linear maps (wc) : W -. Ware well defined and that they give a g~ -module structure on W in which 1 acts as multiplication by -K + h.

We now restrict the g~ -module W to i~ using the obvious imbedding g~ c i~ and we consider the i~-submodu1e T(W) = W(-oo) of the i~-

-- I:;) # module W (see 4.11). iB-module T(W) (see 1.5) has central charge K - h.

8.5. Now let X be a smooth i~-module on which 1 acts as multiplication by (K-h). Then X extends naturally to a i~-module which then can be restricted to arB-module using the homomorphism of B-Lie algebras r B -. i~ (see 8.2(b».

The tensor product W i&I B X of the r B-modules W, X is then arB-module in the standard way; it is clear that 1 acts as zero on this r B-module.



On the other hand, the tensor product T(W) 0 B X of the g~-modules T(W) , X is a g~-module in the standard way; again it is clear that 1 acts as zero on this g; -module.

Just as in 4.16, there is a canonical B-linear map v: T(W) 0 B X ~ (W 0 B X)/GI (W 0 B X)

such that V((XI , x 2 ' ••• ) 0 B z) = x t 0 B z

where (XI' x2 ' ••• ) E T(W), Z E X(N) , and t ~ N. Just as in 4.18, this induces a B-linear map

(a) (T(W) 0 B X)-gC::> ~ (W 0 B X)r . B B

8.6. As in 6.2, we now consider the B-module Z = HomB(W, B) with the rB-module structure inherited from W. For any N ~ 1, we denote by ZN the B-submodule of Z consisting of all elements A. E Z such that c;A. = 0 for all c; E GN . Equivalently, ZN is the annihilator of GNW c W. We have ZI C Z2 C ... and denote by ZOO the union of all ZN.

As in 6.2, ZOO is a r B-submodule of Z . As in 6.3, there is a natural g~-module structure on Zoo, defined as follows.

Let W = (WS)SEO E B((€))o and let c E g so that wc E g~. Let A. E ZN. We choose g E B ° R such that S g - Ws E €N B[[€]] for all

S E O. We define (wc)A. = (gc)A. E Zoo. (Here (gc)A. is given by the r B-module

structure of ZOO .) This gives the required g~-Iilodule structure on Zoo; 1 acts as mUltiplication by K - h .

The restriction of the g~-module ZOO to a g~-module is denoted T'(W). As in 6.4, the g~-module T'(W) is smooth and the g~-module ZOO is the g~ -module attached to the g~ -module T' (W) as in 1.11.

It is clear that T' (W) (1) = Z I .

8.7. As in 6.6, we have natural homomorphisms of g~-modules

(a) T(W)U ~ D(r'(W)) ,

(b) T'(W) ~ D(T(W)U).

Moreover, as in 6.8, the map (a) is an isomorphism, provided that for any N ~ 1 the B-module W /GNW is reflexive; that is, the natural map W /GNW ~ d (d (W / G N W)) is an isomorphism. 8.8. Let X be as in 8.5.

As in 6.12, we see that, if the B-linear map 8.7(b) were known to be an isomorphism, then the B-linear map (a)

given by the transpose of 8.5(a) would be an isomorphism.



8.9. In the remainder of this section we assume that ~ are objects of &'K indexed by s E. and such that, for all s E ., V s = B ® ~ as gn-modules.

By the right exactness of tensor product, we have for all N 2: 1

(a)

where W is as in 4.4; since dim(WjGNW) < 00 (see 7.5), it follows that W / G N W is a finitely generated free B-module, and in particular it is reflexive. Hence, by 8.7, we have a natural isomorphism of g~-modules

(b) T(W)" s:! DCr' (W».

8.10. We show that we have a natural isomorphism of g~-modules

(a) B ® T'(W) s:! T'(W).

Taking duals in 8.9(a), we obtain

Homn(W /GNW, B) = B ® HomdW/GN W , C)

(since W/GNW is finite dimensional) or, equivalently, ZN =B®ZN. Since in-ductive limits commute with base change, it follows that lim ZN = B ® lim ZN or, equivalently, T'(W) = B ® T'(W) , as re---N>l- --N>l -quired. -

8.11. We have a natural isomorphism of g~-modules

(a) B ® D(T' (W» s:! D(B ® T' (W».

This follows from Lemma 8.16, which is applicable since T' (W) is in &'K.


T(W)" s:! B ® T(W)".

This is obtained as a composition

T(W)" = D(T' (W» = D(B ® T' (W» = B ® D(T' (W» = B ® T(W)"

where the first equality comes from 8.9(b), the second from 8.10(a), the third from 8.11(a), and the fourth follows by tensoring with B from 6.6(c).


(a) T' (W) s:! D(T(M)")

which coincides with the map 8.7(b). This is obtained as the composition

T' (W) = D(D(T' (W») = D(T(W)").

The first equality comes from 8.10(a) and 2.26; the second comes from 8.9(b).



8.14. Let X be as in 8.5. As we have seen in 8.13, the map 8.7(b) is an isomorphism. This implies, by Remark 8.8, that the map 8.8(a) given by the transpose of the map 8.5(a) is an isomorphism.

Hence if X is such that both

(W 0 B X)r and (T(W) 0 B X)-g\? B B

are reflexive B-modules, then the B-linear map

(T(W) 0 B X)-<:>g -+ (W 0 B X)r B B

(see 8.5(a)) is an isomorphism.

8.15. We conclude this section with a result relating duality with change of rings. (This result has been used in 8.11.)

Let V be an object of &'1(' Then B 0 V is naturally a iB-module. Let

(a) B 0D(V) -+ D(B 0 V)

be the B-linear map which associates to b ° Ji, (with Ji, E D(V) and b E B) the B-linear form B 0 V -+ B given by b' ° X 1--+ Ji,(x)bb'. Clearly, (a) is a homomorphism of iB-modules.

Lemma 8.16. The map 8.15(a) is an isomorphism of iB-modules.

We have B 0 V = EB,1.EdB 0 A. V); this follows from the corresponding statement (2.20) over C. Consider the following two families of submod-ules of B 0 V: the first family consists of the submodules Q~(B 0 V) for N = 1 , 2, ... ; the second family consists of the submodules EBlR,1.>t B 0 A. V for t running through the real numbers. Then -

any submodule in the first family contains some sub module in (b) the second family; any submodule in the second family contains

some submodule in the first family.

This follows from the corresponding property over C, asserted in 2.21. By definition, a B-linear form B 0 V -+ B is in D(B 0 V) precisely when it vanishes on some submodule in the first family. By (b), this is equivalent to the condition that the linear form vanishes on some submodule in the second family; by 2.20, this is also equivalent to the condition that the linear form vanishes on all but finitely many submodules B 0 A. V .

Thus we have naturally D(B 0 V) = EB,1. (d (B 0 A. V)) . On the other hand, we have B 0 D(V) = EB,1.EdB 0,1.D(V)) since D(V) is

in &'1(' It is then enough to prove that for each A, the natural map

B 0 ,1.D(V) -+ d(B 0,1.V)

is an isomorphism. But this map is the composition of the isomorphism B 0 ,1.D(V)~(d(B0,1.V)) (induced by the isomorphism ,1.D(V)-+d(,1.V) (see 2.23) with the natural isomorphism B 0 d (A. V) ~ d (B 0 A. V) (which comes from the fact that A. V is finite dimensional over C). The lemma is proved.



REFERENCES

[BPZ] A. A. Belavin, A. N. Polyakov, and A. B. Zamolodchikov, Infinite conformal symmetries in two-dimensional quantum field theory, Nuclear Phys. B 241 (1984), 333-380.

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A. Beilinson, B. Feigin, and B. Mazur, Introduction to algebraic field theory on curves, preprint. v. G. Drinfeld, On quasitriangular quasi-Hopf algebras closely related to Gal(Q/Q) , Alge-bra Anal. 2 (1990),149-181. v. G. Kac, Infinite dimensional Lie algebras, Birkhiiuser, Boston, MA, 1983. V. G. Kac and A. K. Raina, Bombay lectures on highest weight representations of infinite dimensional Lie algebras, World Scientific, Singapore and Teaneck, NJ, 1988. D. Kazhdan and G. Lusztig, Affine Lie algebras and quantum groups, Internat. Math. Res. Notes 2 (1991), 21-29. V. G. Knizhnik and A. B. Zamolodchikov, Current algebra and Wess-Zumino models in two dimensions, Nuclear Phys. B 247 (1984),83-103. G. Moore and N. Seiberg, Classical and conformal field theory, Comm. Math. Phys. 123 (1989), 177-254. P. Polo, Projective versus injective modules over graded Lie algebras and a particular parabolic category 0 for affine Kac-Moody algebras, preprint. A. Rocha-Caridi and N. R. Wallach, Projective modules over graded Lie algebras, Math. Z. 180 (1982),151":'177. W. Soergel, Construction of projectives and reciprocity in an abstract setting, preprint. A. Tsuchiya and Y. Kanie, Vertex operators in conformal field theory on PI and mon-odromy representations of braid groups, Advanced Stud. Pure Math., vol. 16, Academic Press, Boston, MA, 1988, pp. 297-372.

[TUY] A. Tsuchiya, K. Ueno, and Y. Yamada, Conformal field theory on universal family of stable curves with gauge symmetries, Advanced Stud. Pure Math., vol. 19, Academic Press, Boston, MA, 1989, pp. 459-565.

DEPARTMENT OF MATHEMATICS, HARVARD UNIVERSITY, CAMBRIDGE, MASSACHUSETTS 02138 E-mail address: [email protected]

DEPARTMENT OF MATHEMATICS, MASSACHUSETTS INSTITUTE OF TECHNOLOGY, CAMBRIDGE, MASSACHUSETTS 02139

E-mail address: [email protected]


ft · 6/4/1993 · A = C and B is finite dimensional as a C-vector space. 1.4. If g' is a Lie...

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