Part 2. Quantum affine algebras

Transcription

1 Part 2 Quantum affine algebras

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3 CHAPTER 9 Affine Lie algebras reminder 1. The central extension of a loop algebra A standard textbook of affine Lie algebras is, of course, [Kac]. We also recommend [ ]. But we follow the notation of [Beck-N], which follow [Ka-IV] in turn. type affine Dynkin graph entries of δ entries of c A (1) n (n 1) B n (1) (n 3) C n (1) (n 2) D n (1) (n 4) E (1) E (1) E (1) F (1) G (1) Table 1. Affine Dynkin diagrams and coefficients of δ, c In these lectures, we only treat untwisted affine Lie algebras (type X n (1) ). But all results remained true for twisted cases. More precisely, we take the 0-vertex as a long simple root contrary to the convention in [Kac] for A (2) 2n. Let g 0 : a finite dimensional complex simple Lie algebra of type ABCDEFG, (, ) : a normalized nondegenerate symmetric invariant bilinear form with the length of long roots = 2). An affine Lie algebra g has two presentations: (1) an example of Kac-Moody Lie algebras, (2) a central extension of the loop Lie algebra of g 0. It is a nontrivial fact that two presentations give rise an isomorphic Lie algebra. We first give the presentation (2): Lg 0 = g 0 C C[t, t 1 ] g = Lg 0 Cc c is a central element 59

4 60 9. AFFINE LIE ALGEBRAS REMINDER [X t m, Y t n ] = [X, Y ] t m+n + mδ m+n,0 (X, Y )c g = g Cd (d is called the degree operator) [d, X t m ] = mx t m The difference of two versions g and g of an affine Lie algebra is hard to guess at the first sight. It will become clear when we consider the corresponding representation theories. At this moment, just note that they have different root data below: g has smaller one, g larger one. The two choices are possible since the Cartan matrix is not invertible. Let g 0 = h 0 α 0 (g 0 ) α be the root space decomposition of the finite dimensional Lie algebra g 0. Then ( ) ( ) ( ) g = h 0 Cc Cd (g 0 ) α t m h 0 t m }{{} def. = h α 0 m Z m Z\{0} is a direct sum decomposition. Let us define the Cartan subalgebra of g by h = h 0 Cc Cd. Then the last two summands are root spaces with roots α + mδ, mδ respectively. Here δ h is defined by two conditions δ, d = 1, δ, h 0 Cc = 0. We put the values of α 0 for c, d as 0 and consider it as an element of h. The root of an affine Lie algbra is separated as = {α + mδ α 0, m Z} {mδ m Z \ {0}} into two types. A root in the former (resp. latter) class is called real (resp. imaginary). A real root has multplicity 1, while the imaginary root has #I. If + 0 denotes the set of positive roots of g 0, we define the set + of positive roots of g by + = {α + mδ α + 0, m Z 0} {α + mδ α + 0, m Z >0} Z >0 δ. Note that = + +. We will see that + is the set of positive roots as a Kac-Moody Lie algebra. Next we give the presentation of g as a Kac-Moody Lie algebra. Since a positive root is defined as above, simple roots are automatically determined: α i : a simple root of g 0 α 0 def. = δ θ : θ is the highest root of g 0 Let I 0 be the index set of simple roots of g 0. Then the set of simple roots of g is I = I 0 {0}. The Dynkin diagram of g is obtained by adding a vertex corresponding to 0. In Table 1 the 0-vertex is denoted by. If it is removed, one get the Dynkin diagram of g 0. We take a vector E 0 from the root space (g 0 ) θ and define F 0 by F 0 = ω 0 (E 0 ), where ω 0 is the Chevalley involution for g 0. We normalize E 0, F 0 by (E 0, F 0 ) = 1. Then we define the corresponding Chevalley generators by e 0 def. = E 0 t, f 0 def. = F 0 t 1, h 0 def. = [e 0, f 0 ] = [E 0, F 0 ] 1 + c. Other Chevalley generators are given by e i 1, f i 1 (i I 0 ), where e i, f i are Chevalley generators for g 0. The affine Cartan matrix is defined by ( h i, α j ) i,j I.

5 2. AFFINE WEYL GROUP 61 We have the following: Theorem 9.1 (Kac, Moody). The affine Lie algebra g is given by the following defining relations of a Kac-Moody Lie algebra: [h, h ] = 0 (h, h h), [e i, f j ] = δ ij h i, [h, e i ] = α i, h e i, [h, f i ] = α i, h f i Serre relation (ad e i ) 1 aij e j = 0, (ad f i ) 1 aij f j = 0 (i j). We define a i by δ = a i α i. i I Their explicit values are given in the second column of Table 1. We define a i by c = a i h i. i I There explicit values are given in the third columon of Table 1. A Cartan datum on g consists of the set of simple roots together with a symmetric bilinear (, ) form on Qα i with certain properties. We take it as the restricition of the unique bilinear form on h which coincides with the normalized bilinear form (, ) on h 0, extended to h so that We have (c, h i ) = 0 (i I 0 ), (c, c) = 0, (c, d) = 1, (d, h i ) = 0 (i I 0 ), (d, d) = 0. (α i, α i ) = 2a i a i, and it takes values 1, 1/2 or 1/3 from the table. Therefore d appeared in the definition of the quantum enveloping algebra is 1, 2 or 3 respectively. Since this bilinear form is non-degenerate, we have an isomorphism ν : h h. Then we have ν(h i ) = α i def. = 2α i, ν(c) = δ. (α i, α i ) 2. Affine Weyl group Next we describe the affine Weyl group. For each i I, we define an involution s i on h by s i (λ) = λ λ, h i α i. The group generated by s i i I GL(h ) is called the Weyl group, and denoted by W. This is the same definition as the Weyl group W 0 (appeared already many times) for the finite dimensional Lie algebra g 0. It is well-known (and used many times already) that the set of roots is the union of Weyl group orbits of simple roots: 0 = i I 0 W 0 α i. This is no longer true for affine Lie algebras. We only get real roots: re = i I Wα i. In fact, we have Wδ = δ by definition, and hence no way to produce δ.

6 62 9. AFFINE LIE ALGEBRAS REMINDER Next we describe the relation between W and W 0. We first relate the duals h, h 0 of Cartan subalgebras by the map cl: h 0 def. = {λ h λ, c = 0} h 0 defined by cl(α i ) = α i (i I 0 ), cl(δ) = 0. It induces an isomorphism h 0 /Cδ = h 0. The subspace h 0 is invariant under the Weyl group action. And δ is fixed by W. Therefore we have an induced group homomorphism W W 0. From its definition, s i W (i I 0 ) is mapped to s i W 0, and hence it is surjective. Moreover it is known that we have an exact sequence of groups: 1 Q cl def. = i I 0 Zα i t W W 0 1. Here the inclusion Q t cl W is given by the restriction of t: h 0 /Cδ GL(h ) defined by { } (ξ, ξ) t(ξ)(λ) = λ + (λ, δ)ξ (λ, ξ) + (λ, δ) δ. 2 (Q cl is the root lattice of the Langlands dual of g 0, the Lie algebra defined for the transpose of the Cartan matrix.) We see the appearance of Q cl naturally from the formula: s αi nδs αi = t(nα i ). When we restrict t to λ h 0, the above formula is simplified as (9.2) t(ξ)(λ) = λ (λ, ξ)δ. For a later purpose, we define the extended affine Weyl group W. Let P cl def. = {λ mod Cδ h 0 /Cδ (λ, α i ) = α i, ν(λ) Z i I 0 }. (This is the weight lattice of the Langlands dual of g 0, and we have Pcl Q cl.) def. Then we define W = Pcl W 0. If we set T def. = {g W g( + ) = + }, we have W = T W. The group T is a subgroup of the Dynkin diagram automorphisms of g. To define a quantum enveloping algebra, we need a root datum. There are two version. The first one is Hom(P, Z) = i I Zh i Zd. The corresponding quantum enveloping algebra is denoted by U q as before. Another one is Pcl 0 = cl(p h 0 ). The corresponding quantum enveloping algebra is denoted by U q. (Kashiwara simply took P cl = cl(p). Since we only consider representation on which c acts by 0, the difference is not essential.)

7 CHAPTER 10 A PBW base of a quantum affine algebra 1. A recipe of a construction Now we define a PBW base of a quantum affine algebra. In order to generalize the construction for finite type case, we need to overcome the following difficulties: Since the Weyl group is not a finite group, there is no longest element. What is an analog of a reduced expression of the longest? Root vectors for imaginary roots cannot be constructed by the braid group operators. How do we define them? We first review a general construction in [Lusztig, 40] to overcome the first difficulty. Let w W be a Weyl group element, and let w = s i1 s i2... s im be its reduced expression. Then we consider the subspace spanned by elements of form f (c1) i 1 T i1 (f (c2) ) (T i1 T i2 T im 1 )(f (cm) i 2 i m ) with various c 1,..., c m. They are products of root vectors correspoding to positive roots in + w 1 ( ). The previous arguments show The above elements are linearly independent. The subspace depends only on w, not on the reduced expression. We denote it by U q (w). U q (w) is closed under the product. Therefore it is a subalgebra. (See Proposition 3.6.) The same is true for the integral form A U q (w). (See Lemma 3.12.) It is natural to take longer and longer w, and consider the limit of the above construction. So we take the doubly-infinite sequence h = (..., i 2, i 1, i 0, i 1,...) such that any finite part is a reduced expression. We also fix p Z. The following construction depends on h, p, but we do not include it in the notation. For (c +p,c p ) Z Z 0 = ZZ p, we define 0 ZZ>p 0 L(c +p ) = f (cp) i p T ip (f (cp 1) i p 1 ), L(c p ) = T 1 i p+1 (f (cp+2) i p+2 )f (cp+1) i p+1. The former type of elements span the limit of U q (w) for w = s i p s ip 1 s ip 2. We denote it by U q (+ p). Similarly we denote the subspace corresponding to the latter type elements by U q ( p ). The former consists of those elements which will be outside of U q after sufficiently many applications of T 1 i p, T 1 i p 1, etc. The latter elements are preserved in U q these operators. But if we apply T ip+1, T ip+2 and so 63

8 A PBW BASE OF A QUANTUM AFFINE ALGEBRA on, they will be outside of U q. Therefore they are not contained in the subspace U q (0 p ) def. = U q ( T ip (U q ) T ip T ip 1 (U q ) ) ( T 1 i p+1 (U q ) T 1 i p+1 T 1 i p+2 (U q ) ). By the orthogonality of the inner product, one see that U q (+ p) U q (0 p) U q ( p) U q is injective. But we do not know whether this is surjective in general. The above construction depends on the choice of the infinite sequence h. The following should be a minimum requirement: Take an element w W so that l(w n ) = nl(w). We take a reduced expression of w and repeat it periodically to get h. Then we can hope some kind of periodicity in the construction, otherwise no hope to understand U q (0 p). For such a w, it is naturl to require the followings: It is sufficiently small (i.e., a period is short). Sufficiently many positive roots appear in n Z + w n ( ). So far, two such choices are known: (1) One which was used in [Beck-N]. (It was first used in [Be94].) Let w = t(ξ), where ξ is the sum of fundamental coweights i (i I 0 ) of Pcl : ξ = n Pcl. (2) w is the Coxeter element, i.e., the product of simple reflections for each vertex. This is suitable with the representation theory of an affine quiver. Let us study what positive roots are missing in n Z + w n ( ). In the first case, the positive roots corresponding to U q (+) (when p = 0) are {α + mδ α + 0, m Z 0}, while those to U q ( ) are {α + mδ α + 0, m Z >0}. These follow from (9.2). Therefore U q (0) should be given by root vectors for imaginary roots. Here U q (+ p ), etc with p = 0 are simply denoted by U q (+), etc. On the other hand, in the second case, those corresponding to U q (+) are roots for preprojective representations of the affine quiver, and those to U q ( ) are preinjective representations. Unless g = ŝl 2, those sets of roots are different from ones in the first case. In fact, there are finitely many real positive roots α such that w(α) α and w p (α) = α (p > 1). The corresponding indecomposable representations are in the tube. The category of such representations is equivalent to the category of nilpotent representations of a cyclic quiver with p-vertices. Therefore the constrution of U q (0) is very different. 2. Imaginry root vectors Let ξ be the sum of fundamental coweights of P cl as above: ξ = n P cl. This is in the extended Weyl group W, but not in W in general. (For example, for g = ŝl 2, we have s 0 s 1 = t(2ξ).) We decompose t(ξ) to the W-part and the Dynkin automorphism part as t(ξ) = s i1 s i2 s in τ. Then we extend the sequence to the both direction so that i k+n = τ(i k ).

9 2. IMAGINRY ROOT VECTORS 65 Then we define L(c +p ), L(c p ) as above. When p = 0, we denote the root vector for a positive root β by F β. Important root vectors are F kδ±αi, which correspond to e i t k, f i t k in the classical limit q s = 1. They have the following simple expressions ([Be94]): (10.1) F kδ+αi = T e i(f k i ), F kδ αi = T i T k (f e i i ). The lower one contains T i, but the reduced expression of t( i) ends with s i, so the first part of T k and T e i i cansel out. These vectors are so-called Drinfeld generators, and one can present U q as a q-analog of the loop algebra. (Drinfeld presentation. See [Be94].) The following lemma is important in many calculations, though we do not directly use it in these lectures: Lemma f 1 f i, f 0 F δ αi give an embedding U q i (ŝl 2) U q. For g = A (2) 2n, we do not have this lemma in general. We need to replace U q (ŝl i 2) by U q i (A (2) 2 ), for which a detailed calculation was performed in [Ak02]. We now come to a point where we consider the second difficulty mentioned at the beginning of the previous section. We construct imaginary root vectors by hand: def. ψ i,k = F kδ αi f i qi 2 f i F kδ αi. It has weight kδ, and equals to h i t k at the classical limit q s = 1. Lemma ψi,k s are commuting. The PBW base should be almost orthonormal, but powers of ψ i,k are not. We instead calculate r( ψ i,k ) and find that it is similar to the coproduct on symmetric functions applied to power sums. Then it is natural to replace ψ i,k by analog of Schur functions. Thus we follow [BCP99] to define P i,k inductively by P i,k = 1 [k] i k s=1 q s k i ψ i,s Pi,k s. Then identifying P i,k with elementary symmetric functions, and define Schur functions S i,y corresponding to a Young diagram Y by using the determinantal identities. For an I 0 -tuple of Young diagrams c 0 = (Y i ) i I0, we define S c0 = i I 0 S i,y i. Finally we set L(c, p) = L(c +p ) (T ip+1 T ip+2... T i0 )(S c0 ) L(c p ). The p = 0 case is most important and we denote the corresponding element by L(c). But for the actual proofs, we need to consider all L(c, p) simultaneously. In fact, to increase p by 1 corresponds to the replacement of h = (i 1,..., i ν ) by h = (i 2, i 3,..., i ν, ι(i 1 )). (Recall the proof of Theorem 5.1.) Therefore not only S c0, but also its images under the braid group operators T w are important. Roughly speaking, their study are nothing but the fact that S c0 is an extremal vector. We introduce an order p on c by c p c c +p c + p and c p c p holds, at least one of is strict.

10 A PBW BASE OF A QUANTUM AFFINE ALGEBRA Here is the lexicographic order reading from the left. Note that c 0 is not related to the order. For example, if c ±p = 0, the above just means that either of c ±p is nonzero. (If we also consider weights, we conclude both are nonzero.) The followings are proved by calculation: Theorem (1) L(c, p) A U q. (2) (L(c, p), L(d, p)) δ c,d mod q s A 0. In particular, {L(c, p) c} is linearly independent. By a comparison with dimensions of weight spaces, they form a Q(q s )- vector space base of U q. In particular, for each c there exist a canonical base element b(c, p) such that L(c, p) ±b(c, p) mod q s L ( ). (3) For g = A (1), D (1), E (1) or A (2) 2, {L(c, p)} is an A-base of AU q. By (2) we get a parametrization of the canonical base. We use it for a study of the structure of extremal weight modules in next section, and then using it in turn, we remove the sign umbiguity above. Properties of S c0 must be studied differently from other root vectors. We use the following: Lemma We have P i,k = F (k) δ α i f (k) i + a c L(c, 0), c where a c is in q s A A 0 = q s Z[q s ], and if a c 0, the corresponding c satisfies 0 0 c, i.e., if c = (c +,c 0,c ), both c +, c are nonzero. Moreover S c0 contains only P i,t with 0 < t < k. By using this lemma, the upper-triangular property of the bar involution with respect to the PBW base {L(c, p)} is proved as follows: (1) We first show that the upper triangular property for the real root parts L(c ±p ). The proof is the same as the finite type case. (2) Second we show P i,k = P i,k + 0 a 0c cl(c, 0) by the above lemma and (1). (3) We show the upper triangularity for S c0 by (2) and the exchange relation on real root vectors. (4) We study the general cases. Once (3) is established, the remaining part is same as one for the finite type case. Together with Theorem 10.4(3), these allow us to construct the canonical base from {L(c, p)} for g = A (1), D (1), E (1) or A (2) 2 without invoking Kashiwara s theory. But we cannot remove the sign umbiguity. It, a priori, depends on h and p.

11 CHAPTER 11 Level 0 fundamental representations 1. The convexity for level 0 representations For each i I 0, we define the level 0 fundamental weight i by i = Λ i a i Λ 0. Since a 0 = 1, we have i, c = a i a i a 0 = 0, hence it is of level 0. By a Weyl group element any level 0 weight element can be mapped uniquely to that of form λ = i 0 λ i i, λ i Z 0. If we identify i with the usual fundamental weight of g 0, this is nothing but a dominant weight. We call the weight λ of above form level 0 dominant. Theorem 11.1 ([Ka-IV, Th. 5.1]). Let V (λ) be a level 0 extremal weight module. (1) A weight µ of an extremal vector of V (λ) satisfies cl(µ) cl(wλ) = W 0 cl(λ). (2) Weights of V (λ) are contained in the convex hull of Wλ. Once (1) is shown, (2) follows as any component of B(λ) contains an extremal vector (Theorem 8.13). We prove a stronger result by using the PBW base. Once the following is proved, we get the above immediate as wt(s c0 v λ ) λ + Zδ. Theorem An extremal vector of B(λ) is in a Weyl group orbit of ±S c0 v λ for some c 0. We cannot remove the sign umbiguity at this moment, but we remove ± from the notation for brevity. Proof. We consider B(λ) as a subset of B(U q a λ ), and determine elements b which are both extremal and -extremal (i.e., b is extremal. It is enough to show that b can be mapped to S c0 v λ by applying Weyl group operators S w and Sw. Let us write b = b 1 t λ b 2. By a property on ((8.11)), we may assume b 2 = u. Next we take a Weyl group element w so that w wtb = w(λ + wt b 1 ) is level 0 dominant. By [AK97, Lem. 1.4] we can write w = s jl s j1 so that h jk, s jk 1 s j1 λ > 0. Then by the tensor product rule for crystals, S w b is still of the form b 1 t λ u. Thus it is enough to determine b 1 B( ) under the further assumption that wtb is level 0 dominant. 67 f max i

12 LEVEL 0 FUNDAMENTAL REPRESENTATIONS By using the PBW base, we write b 1 L(c, 0) = L(c + )S c0 L(c ) mod q s L ( ). The extremality of b and the tensor product rule give that max(0, h i, wtb ) = ε i (b) = max(ε i (b 1 ), h i, λ + wtb 1 ). Take i 0. Since wtb is level 0 dominant, we must have ε i (b) = 0, and hence ε i (b 1 ) = 0. Remember that root vectors L(c + ) = f (c0) i 0 T i0 (f (c 1) i 1 ) in U q (+) are in {α + mδ α + 0, m 0}. In particular, i 0 0 and hence 0 = ε i0 (b 1 ) = c 0 from the above. Next consider S i0 b = b 1 t λ u and note b 1 T 1 i 0 L(c + ) = f (c 1) i 1. Noticing h i 1, wt(s i0 b) = s i0 h i 1, λ + wtb 1 0, we use the same as above to get ε i 1 (b 1 ) = c 1 = 0. Repeating this, we get c + = 0. Next we use that b is extremal. Noticing wtb = λ, we use the assumption that λ is level 0 dominant to show c = 0 as above. Therefore b 1 S c0 for some c 0. The above theorem does not tell us which c 0 appear as above, we will determine them by several steps. Lemma Suppose λ is level 0 dominant. (1) We have P i,k v λ = F (k) δ α i f (k) i v λ, and the right hand side vanishes if k > λ i. (2) If S c0 v λ = i I 0 S i,y iv λ is nonzero, the number of rows of the Young diagram Y i is less than or equal to λ i. Proof. (2) is immediate from (1). Let us show (1). By Lemma 10.5, we have P i,k v λ = F (k) δ α i f (k) i v λ + a c L(c, 0)v λ, where c appearing in has c 0. Since the root vectors in U q ( ) are in Z >0δ, cl wt(l(c )v λ ) is outside of cl(λ) Q + cl. Therefore the convexity in Theorem 11.1 implies L(c )v λ = 0. This is the first claim. The integrality implies f λi+1 i v λ = 0. Therefore the first term also vanishes if k > λ i. We rephrase the above condition on Young diagrams. Recall that Young diagrams with numbers of rows are less than or equal to λ i correspond to irreducible polynomial representations of GL λ i(c). Therefore c 0 = (Y i ) i I0 with each Y i has λ i or less rows is bijective to polynomial representations of def. G λ = GL λ i(c). i I 0 Next we consider the maximum nontrivial k, i.e., k = λ i. The corresponding GL λ i(c)-representation is the determinant representation, and hence is an easy 1- dimensional one. One can hope the above is also easy. It turns out to be true by the following: Lemma For t(α i ) = s α i δs αi, we have P i,λi v λ = F (λi) δ α i f (λi) i v λ = S t(α i )v λ.

13 2. LEVEL 0 FUNDAMENTAL REPRESENTATIONS 69 T 1 e i Proof. The first equality was proved in the previous lemma. We prove the second equality by calculation. By (10.1), we note F (λi) δ α i = T i T 1 (f e i i ). Write T i T 1 = T 1. Then F (λi) e i e i δ α i = ). Next note that f (λi) v λ = S i v λ. Therefore we get (f (λi) i F (λi) δ α i f (λi) i v λ = T 1 i (f (λi) e i i ) S i v λ = T 1 (f (λi) e i i T e i S i v λ ) = T 1 (S i T e i S i v λ ). Now we make a calculation in the Weyl group and make precise the normalization factors from the difference between T w and S w. (See Lemma 8.5.) From this calculation P i,λi v λ is a canonical base element. Since λ i can be arbitrary, we can remove the sign mbiguity of P i,k in the PBW base. It must be Level 0 fundamental representations We study the case λ = i in this section. We have G λ = C and its polynomial representation is given by t t n for some n 0. Theorem 11.5 ([Ka-IV, Th. 5.3]). The extremal vectors of V ( i) are {S w v i w W }. In particular, B( i) is connected. Proof. As in the proof of Theorem 11.2, we determine b = b 1 t i u which is extremal and -extremal. We may assume b 1 S c0 as before. By Lemma 11.3, S c0 cannot contain P j,k (j i) or P i,k (k 2). Therefore S c0 is a power of P i,1. Hence Pi,1 n = S v i t(nα i )v i Y i has a single row with n boxes, other S c0 v i = Y j s are empty, 0 otherwise. Since the Weyl group operators S w are invertible, vectors in the above case are nonzero. The latter assertion follows since S w are compositions of Kashiwara operators on the crystal. Corollary V ( i) i is 1-dimensional. Proof. By convexity, a vector with weight i is automatically extremal. Therefore it is of the form S w v i by the above theorem. By a comparison of weights, we must have w i = i. But it implies S w v i = v i. (See [Ka-IV, Lem. 5.7]) Next we understand S t(α i ) in view of the Weyl group symmetry of extremal weight modules (Theorem 8.15(4)). First note that wt(s t(α i )v i) = t(α i ) i = i + (α i, i)δ = i + δ. Recall that U q is the subquotient of U q, where q d is setted to be 1. Since i and i + δ are equal modulo Zδ, we have a homomorphism of U q -representations V ( i) V ( i) sending v i to S t(αi)v i by the universality. Moreover, the associated crystal homomorphism is S t(αi), and hence an isomorphism. Therefore the above itself is an e i

14 LEVEL 0 FUNDAMENTAL REPRESENTATIONS isomorphism. We denote this by z, and consider it as a U q-representation isomorphism. It adds weights by δ. We define W( i) def. = V ( i)/z, and call level 0 fundamental representation. Theorem 11.7 ([Ka-IV, Th. 5.17]). (1) W( i) has the canonical base induced from that of V ( i). We denote it by B(W( )). (2) W( i) is a finite dimensional irreducible U q-representation. (3) V ( i) is isomorphic to the affinization Q(q)[z, z 1 ] W( i) of W( i). For the proof of finite dimensionality, see the original paper.

15 CHAPTER 12 Structures of extremal weight modules of affine types In this chapter, we first construct canonical bases on tensor products of level 0 fundamental representations. We follow [Ka-IV, 8]. Then we study its relation to extremal weight modules following [Beck-N]. It will determine structures of extremal weight modules. 1. Tensor products of level 0 fundamental representations Here for brevity of notation, we only consider tensor products of two level 0 fundamental representations V ( i) V ( j). But all results here can be generalized to the case of tensor products of more than two level 0 fundamental representations. The following is [AK97, Lem. 1.6], which is the converse of Lemma Lemma B 1, B 2 is a finite normal U q -crystal. If b 1 b 2 B 1 B 2 is extremal, then both b 1 and b 2 are extremal and their weights are in the same Weyl chamber. Let z i, z j denote the automorphisms of U q-representations on V ( i), V ( j) constructed in Chapter 11. We need to introduce a bar involution on V ( i) V ( j). As we saw in Chapter 7 in the case of a tensor product of highest and lowest weight representations, we need to use a quasi R-matrix to twist the product of the bar involutions on factors. But the problem here is that the quasi R-matrix Θ exists only in the completion and does not act on V ( i) V ( j). This is a crucial difference of this case from a tensor product of highest and lowest weight representations. We introduce V ( i) V ( j) def. = Q(q s )[[z i /z j ]] Q(qs)[z i/z j] (V ( i) V ( j)). Proposition V ( i) V ( j) has the unique bar involution v i v j = v i v j, it commutes with z i, z j. We explain the sketch of the proof. satisfying (1) We first check that ( ) Θ: V ( i) V ( j) V ( i) V ( j) is welldefined: Recall the definition of Θ, and use the observation that W( i) has only finitely many weight spaces to it is enough to complete by formal power series in z i /z j. 71

16 STRUCTURES OF EXTREMAL WEIGHT MODULES OF AFFINE TYPES (2) (V ( i) V ( j)) i+ j is 1-dimensional as a Q(q s )[[z i /z j ]]-vector space. (Lemma 12.1) Therefore there exists ϕ(z i /z j ) Q(q s )[[z i /z j ]] such that ( ) Θ(v ωi v ωj ) = ϕ(z i /z j )u. From the definition of Θ we have ϕ(0) = 1, and hence we can define = ϕ(z 1 /z 2 ) 1 ( ) Θ. It has the required properties. (3) The uniqueness follows from the irreducibility of W( i) W( j), which is easy to show. The is upper-triangular with respect to the tensor product of the canonical bases of V ( i) and V ( j), as in the case of a tensor product of highest/lowest weight representations, we can prove the following in the same way: (though V ( i) V ( j) is infinite dimensional, it is finite dimensional over Q(q s )[[z i /z j ]] Q(qs)[z i/z j] Q(q s )[z i ±, z± j ], and the base is defined over there.) Proposition For each pair b B( i), b B( i), there exists the unique element G(b b ) V ( i) V ( j) satisfying the following two properties: (A) G(b b ) = G(b b ). (B) It is written as G(b b ) = b b + a b1,b b 1 1 b 1 (a b1,b q 1 sz[q s ]) b 1:wt(b 1)>wt(b) b 1 :wt(b 1 )<wt(b ) by B( i) B( j). Here we may have infinitely many b 1 b 1. Let B(V ( i) V ( j)) def. = {G(b b ) b B( i), b B( j)}. Next we study extremal vectors. Lemma Any extremal vector in B(V ( i) V ( j)) is contained in the usual tensor product V ( i) V ( j). Proof. First note that v i v j, which is in B(V ( i) V ( j)), is contained in the usual tensor product V ( i) V ( j). An extremal vector in B(V ( i) V ( j)), after the Weyl group action, has the weight in i + j + Zδ. (Theorem 11.2) Then it is of a form zi m z n v i j v j. In particular, it is contained in V ( i) V ( j). Therefore any extremal vector is contained in V ( i) V ( j). Now we cut out a finite part: V ( i) V ( j) def. = U q [z ± i, z± j ]v i v j V ( i) V ( j) V ( i) V ( j), L (V ( i) V ( j)) def. = (L ( i) L ( j)) ( V ( i) V ( j) ), B(V ( i) V ( j)) def. = B(V ( i) V ( j)), where L ( i), L ( j) are A 0 B( i), A 0 B( j) respectively. The followings are clear: V ( i) V ( j) is preserved under the bar involution. L (V ( i) V ( j))/q s L (V ( i) V ( j)) L ( i) L ( j)/q s (L ( i) L ( j))

17 2. EXTREMAL WEIGHT MODULES AND TENSOR PRODUCTS 73 Let us show that the inclusion in the second line is, in fact, =. For a weight µ W ( i + j) + Zδ, we have (V ( i) V ( j)) µ = (V ( i) V ( j)) µ thanks to the previous lemma. Therefore L (V ( i) V ( j)) µ = (L ( i) L ( j)) µ. Any vector in B(V ( i) V ( j)) is mapped to an extremal vector by a successive application of Kashiwara operators. The result is contained in the space L (V ( i) V ( j))/q s L (V ( i) V ( j)) by the above remark. Therefore the claim is proved. We still need to show that B(V ( i) V ( j)) is a subset of V ( i) V ( j). We only know that extremal vectors are in V ( i) V ( j) at this stage. We check the assertion for general vectors, by a reduction to the extremal case. See the original paper for the proof of this part. 2. extremal weight modules and tensor products Let λ be a level 0 dominant weight. We decompose it as λ = i I 0 λ i i. We consider the tensor product Ṽ (λ) = i I 0 V ( i) λi of level 0 fundamental representations. We fix an order on I 0 so that the above tensor product is taken in that order. Since the results below are true for any choice of the order, we do not include the choice in the notation. Let z i,ν (1 ν λ i ) be the automorphism of the U q-representation on the νth V ( i) in Chapter 11. They are commuting each other on Ṽ (λ). Therefore Ṽ (λ) is a module over the Laurent polynomial ring Q(q s )[z ± i,1,...,z± i,λ i ]. i I 0 Let ṽ λ Ṽ (λ) be the tensor product of v i in V ( i). We further define V (λ) def. = U q [z ± i,1,..., z± i,λ i ] i I0 ṽ λ Ṽ (λ). By the results in the previous section, it has the canonical base B( V (λ)). The corresponding A 0 -form is L ( i) λi V (λ). We denote it by L (λ). Let B 0 ( V (λ)) be the connected component of B( V (λ)) containing ṽ λ. Since ṽ λ is an extremal vector with weight λ, we have the unique U q -homomorphism Φ λ : V (λ) V (λ); v λ ṽ λ by the universality Theorem 8.15(1). The set of I 0 -tuples of Young diagrams c 0 = (Y i ) i I0 such that the numbers of rows of Y i are less than or equal to λ i is denoted by PolIrr(G λ ). It is identified with the set of irreducible polynomial representations of G λ. Proposition Let c 0 PolIrr(G λ ). We consider the corresponding pure imaginary element S c0 U q and s c0 (z), the Schur function whose variables are substituted by the above automorphisms z i,ν. Then we have Φ λ (S c0 v λ ) = s c0 (z)ṽ λ.

18 STRUCTURES OF EXTREMAL WEIGHT MODULES OF AFFINE TYPES Here all higher variables are set to be zero: z i,λi+1 = z i,λi+2 = = 0. Note that by the assumption on c 0, Schur functions are nonvanishing even after this substitution. For example, for λ i = 2, we have but z 1 z 2 + z 1 z 3 + z 1 z 2, z 1 z 2 z The proof is given as follows: By applying Lemma 11.4 to a level 0 fundamental representation, the assertion is true for P i,1. The coproduct P i,k is almost equal to the usual coproduct on the symmetric function. (This is proved by calculation.) The difference vanishes when they act on the extremal vector. Corollary (1) There is no sign umbiguity for S c0, i.e., S c0 B( ) + q s L ( ). (2) The set of the extremal vectors of V (λ) is exactly the union of Weyl group orbits in S c0 v λ with c 0 PolIrr(G λ ). Proof. (1) For brevity of the notation, we suppose that all Young diagrams except one on i I 0 are. We prove the assertion on the induction on the number of columns. There is nothing to prove for c 0 =. Let k be the length of the longest column. Consider V (λ) and Ṽ (λ) for λ = k i. The k th elementary symmetric function is just a monomial z 1 z 2 z k. It is a crystal automorphism, and we can divide s c0 (z)ṽ λ by z 1 z k. The corresponding element is s c 0 (z)ṽ λ = Φ λ (S c 0 u λ ), where c 0 is obtained from c 0 by removing the column of length k. Since S c 0 B( ) + q s L ( ) by the induction hypothesis, the same is true for S c0. (2) By Theorem 11.2 we already know that extremal vectors are in the Weyl group orbits of S c0 u λ. If c 0 / PolIrr(G λ ), we have S c0 v λ = 0 by Lemma The above remark also shows that S c0 v λ 0 if c 0 PolIrr(G λ ). Theorem There is no sign umbiguity for arbitrary element L(c, p) in the PBW base. Proof. Once the assertion was established for the above pure imaginary elements, the assertion follows from Lusztig s compatibility result [Lu-IV] for the braid group and the canonical base: Let i π: U q U q [i], π i : U q U q [i] be as in Chapter 5. Then the image of B( ) under the projection i π, π i, after removing 0, are bases of U q [i], U q [i] respectively. Suppose that b B( ) satisfies i π(b) 0. Then there exists b B( ) such that T i ( i π(b)) = π i (b ). The correspondence b b gives a bijection between subsets which are mapped to nonzero under i π and π i. Theorem Φ λ is injective and maps B(λ) to { } s c0 (z)b c PolyIrr(Gλ ), b B 0 ( V (λ)) bijectively.

19 3. BILINEAR FORM 75 Sketch of proof. Since Φ λ is a homomorphism of representations, it commutes with ẽ i, f i. By Proposition 12.5 and the fact that z i,ν preserves the crystal structure, we know that the images of extremal vectors are belonged to L (λ). Therefore we have Φ λ (L (λ)) L (λ). We consider the induced linear map Φ λ qs=0 : L (λ)/q sl (λ) L (λ)/q s L (λ). Again by Proposition 12.5, B(λ) is mapped to the image of the set in the assertion or 0. If we have an element b mapped to 0, it is so on the connected component containing b. But by Proposition 12.5 no extremal vector is mapped to 0. Since any component contains an extremal vector, this is a contradiction. Now both surjectivity and injectivity of Φ λ qs=0 becomes clear. It is clear that Φ λ commutes with, and maps the A-form to the A-form. Therefore the canoical base elements are mapped to canonical base elements. Therefore Φ λ is injective. From this result, we see that V (λ) has a structure of a module of i I 0 Q(q s )[z ± i,1,...,z± i,λ i ] Sλi, where S λi is the symmetric group of λ i letters acting the Laurent polynomial ring Z[z i,1 ±1,..., z±1 i,λ i ] by exchanging variables. This structure will be seen in a geometry approach to quantum loop algebras via quiver varieties. (See 14.8.) 3. Bilinear form In this section we review results on a bilinear form on the extremal weight module, and a characterization of the canonical base (up to sign) similar to Theorem 5.9. These results were obtained in [Na04, 4] by generalizing results for the special cases of level 0 fundamental representations [VV-III]. The proofs will be only sketched, so consult the original papers for the detail. First we show Proposition 12.9 (Kashiwara (see [Na04, 4.1]). The extremal weight module V (λ) has a unique bilinear form (, ) satisfying { 1 if b = v λ, (v λ, b) = 0 otherwise (xu, v) = (u, ψ(x)v) for x U q, u, v V (λ). Here ψ is an involutive anti-automorphism of U q given by ψ(e i ) = qi 1 ti 1 f i, ψ(f i ) = qi 1 t i e i, ψ(q h ) = q h. This is proved as follows. Use ψ, we define a U q -module structure on Hom(V (λ), Q(q s )). Then by the convexity of weights of V (λ), one shows that the vector u λ Hom(V (λ), Q(q s )) given by v λ, b = δ b,vλ is extremal. Then we have a homomorphism V (λ) V (λ) sending u λ to u λ by the universality. It gives the bilinear form. For the special case when λ is a level 0 fundamental weight i, we have Lemma (zu, zv) = (u, v) for u, v V ( i).

20 STRUCTURES OF EXTREMAL WEIGHT MODULES OF AFFINE TYPES From the uniqueness of the bilinear form it is enough to check (zu, zv) satisfies the above properties. The second one is clear. To show the first one, we calculate (zv i, zv i). This can be done first by studying the relation between the bilinear form and the braid group operators, and then use zu i = S t(α i )v i and the relation between T w and S w (Lemma 8.5). We define a Q(q)[z ± ]-valued bilinear form ((, )) on V ( i) by { z m (z m u, v) if wt(u) = wt(v) + mδ for m Z, ((u, v)) = 0 otherwise. We then define a Q(q)[z i,ν ± ] i I,ν=1,...,λ i -valued bilinear form ((, )) on Ṽ (λ) by ((u, v)) def. = ((u i,ν, v i,ν )), i,ν where u i,ν, v i,ν is the ν th V ( i)-factor of u, v Ṽ (λ). We define a bilinear form (, ) on Ṽ (λ) by (u, v) def. = 1 ((u, v)) z i,µ zi,ν m i! i I µ ν(1 1 ), where [f] 1 denote the constant term in f. Then by the uniqueness of the bilinear form and the computation of (s c0 (z)ṽ λ, s c 0 (z)ṽ λ ), we can show Proposition Let u, v V (λ). Then (u, v) = (Φ λ (u), Φ λ (v)). Using this, we obtain the following characterization: Theorem (1) B(λ) is an almost orthonormal base for (, ), that is, (b, b ) δ bb mod q s Z[q s ]. (2) {±b b B(λ)} = {u A V (λ) u = u, (u, u) 1 mod q s Z[q s ]}. (2) follows from (1) as in Theorem 5.9. For (1) we use Kashiwara operators ẽ i, fi to reduce the proof for the case of extremal vectors. Then the previous proposition gives the assertion. 1

21 CHAPTER 13 Structures of quantum affine algebras We consider the modified quantum enveloping algebra Ũq associated with Pcl 0. It is more reasonable to denote this by Ũ q in comparison with our previous notation U q and U q. But we do not consider the modified quantum enveloping algebra for U q, so we choose a simpler notation. 1. Affine analog of the refined Peter-Weyl theorem In this section we review a structural result on the canonical base of the modified quantum enveloping algebra Ũq proved in [Beck-N, 6.1]. This result is an affine analog of the result by [Lusztig, Ch. 29] (see also [Lu-III, 4] for further treatements), called a refinement of the Peter-Weyl theorem. We introduce a family of subquotients Ũq[λ] of Ũq, and then show that they are all compatible with the canonical base. Moreover, their canonical bases are completely understood by the canonical bases of V (λ), something like B(λ) B( λ), but with an essential modification. Let # def. =. The canonical base B(Ũq) is invariant under #. Let B[λ] def. = {β B(Ũq) β is connected to an element in B(λ) # by ẽ i, f i.} If λ and µ are in the same W 0 -orbit, we have B[λ] = B[µ], and B[λ] B[µ] = otherwise. Therefore we have a decomposition B(Ũq) = B[λ], λ is level 0 dominant which is compatible with the bi-crystal structure, i.e., with both crystal and - crystal (or more naturally #-crystal) structure. We identify Pcl 0 with the weight lattice of the finite dimensional Lie algebra g 0, and introduce the dominance order. Let Pcl,+ 0 be the semigroup of level 0 dominant weights. The following definition is an affine analog of a statement in [Lusztig, ]. Definition Let Ũq[ λ] (resp. Ũ q [ > λ]) be the two sided ideal of Ũ q consisting of all elements x Ũq acting on V (λ ) by 0 for any λ λ (resp. λ λ). Let Ũq[λ] = Ũq[ λ]/ũq[ > λ]. We define A Ũ q [ λ], A Ũ q [ > λ] and A Ũ q [λ] in an analogous manner. In the finite type case, Ũ q [λ] is compatible with B[λ], i.e., Ũ q [ λ] (resp. Ũ q [ > λ]) is the subspace spanned by λ λ B[λ ] (resp. λ >λ B[λ ]) and B[λ] defines a base of Ũq[λ] ([Lusztig, Ch. 29]). Using this result, Lusztig showed that for β B[λ] there exists b 1, b 2 B(λ) such that β b 1 a λ b # 2 modulo Ũq[ > λ], and b (b 1, b 2 ) gives a bi-crystal isomorphism between B[λ] and B(λ) B(λ) 77

22 STRUCTURES OF QUANTUM AFFINE ALGEBRAS ([Lu-III, 4.4]). This completely determines the bi-crystal structure of B(Ũq). We will state a similar result (Theorem 13.3 below) for the affine case. We first explain what is the main difference between finite and affine cases: the middle term a λ in β b 1 a λ b # 2 is replaced by a canonical base element corresponding to Schur functions in a pure imaginary root vectors. Let us say this a little bit more precise. Let Irr(G λ ) be the set of all irreducible representations of G λ. They corresponds to Schur functions possibly multiplied by negative powers of the determinant z i,1 z i,λi. If c 0 is a dual of a polynomial representation c 0, we define S c0 as an element in U + q as S # c Then we consider Irr(G 0. λ ) as a subset of B(Ũq) by identifying c 0 with the canonical base element corresponding to the PBW base element S c0. We replace {a λ } in the finite type case to Irr(G λ ) in the affine case. Let B W (λ) be the finite crystal i I 0 B(W( i)) λi. By the discussion in the previou section, we have a bijection (13.2) B(λ) = Irr(G λ ) B W (λ). Note that this decomposition is not compatible with the crystal structure. Theorem 13.3 ([Beck-N, 6]). (1) β B(Ũq) is in B[λ] if and only if β Ũq[ λ] and β 0 in Ũq[λ]. (2) B[λ] defines a base on Ũq[λ]. (3) For β B[λ] we have This correspondence β b 1 sb # 2 mod Ũq[ > λ], b 1, b 2 B W (λ), s IrrG λ. B[λ] β (b 1, s, b 2 ) B W (λ) Irr(G λ ) B W (λ) is bijection, and Kashiwara operators ẽ i, f i are given by considering (b 1, s) as an element of B(λ) by (13.2). (4) B[λ] is invariant under #. It is given by (b 1, s, b 2 ) (b 2, s #, b 1 ) under the bijection in (3). Here s # denote the dual representation of s. Let us explain the main points of the proof. Let us return back to Definition For example, the projector a λ acts on V (λ ) by a nonzero map, λ is a weight of V (λ ). Therefore we have λ λ. This means that a λ Ũq[ λ]. More generally U q a λ is contained in Ũq[ λ]. It is also easy to see that b B(λ) is nonzero in Ũq[λ]. Then we observe that b # mod Ũq[ > λ] is an extremal vector in the integrable representation Ũ q [λ]. Therefore we have the unique U q -homomorphism Ψ b : V (λ) Ũq[λ] sending a λ to b #. This homomorphism is similar to Φ λ : V (λ) V (λ). We can show compatibility of various structures of V (λ) and Ũq[λ]. For example, it commutes with ẽ i, fi. Then it is not difficult to show that it sends L (λ) to L [λ] def. = L (Ũq) Ũq[ λ]/l (Ũq) Ũq[ > λ] if b B W (λ). Thus it induces a homomorphism L (λ)/q s L (λ) L [λ]/q s L [λ]. Then one can show that it gives the crystal embedding B(λ) B[λ]. Since Ψ b2 (b) = bb # 2 mod Ũq[ > λ] by definition, and is easy to show that bb # 2 b 1sb # 2 mod Ũq[ > λ] if b = (b 1, s) under (13.2), we

23 2. CELLS 79 obtain (3) at q s = 0. Then we use the bar involution to the actual (3) for generic q s. 2. Cells In this section we review results in [Beck-N, 6], where Lusztig s conjecture in [Lu-III] was proved. All the proofs will be omitted Cell structure of Ũq. We recall [Lusztig, 29.4] the definition of cells in Ũq with respect to the canonical base B(Ũq). Definition Let c β ββ (q s ) A be the structure constant with respect to the canonical base B(Ũq): ββ = β c β ββ (q s )β. (1) For β, β B(Ũq) we say β L β (resp. R ) if there is a sequence β 1 = β, β 2,..., β N = β in B(Ũq) and a sequence γ 1,..., γ N 1 B(Ũq) such that c βs+1 γ s,β s 0 (resp. c βs+1 β s,γ s 0) for s = 1,...,N 1. We write β β if either of the above structure constants is non zero for all β s, γ s, β s+1, s = 1,...,N 1. (2) For β, β B(Ũq), we say that β β if β β and β β. The equivalence classes of are called two sided cells. (2) Similarly, by considering L or R, we define the equivalence classes, and call them left cells or right cells respectively. In the following theorem, we identify β B[λ] with (b 1, s, b 2 ) B W (λ) IrrG λ B W (λ) by Theorem 13.3(3). Theorem (1) B[λ] is a two sided cell of B(Ũq). Thus we have a bijection between Pcl,+ 0 and the set of two sided cells of B(Ũq). (2) For any b 2 B W (λ), {(b 1, s, b 2 ) B[λ] s IrrG λ, b 1 B W (λ)} is a left cell. (3) For any b 1 B W (λ), {(b 1, s, b 2 ) B[λ] s IrrG λ, b 2 B W (λ)} is a right cell. In particular, (1) There exist only finitely many left and right cells in each two sided cell. Both of their numbers are equal to #B W (λ). (2) Any left cell and any right cell intersect. The intersection is bijective to IrrG λ. By the definition of B[λ], we have λ µ if B[λ] B[µ]. It is natural to expect the converse is also true. Thus the bijection Pcl,+ 0 {two sided cells of B(Ũq)} in the above theorem respects the two natural orderings. Let β B(Ũq). We have a bi-module, called the cell module / Q(q s )β Q(q s )β β β β β,β β This depends only on the two sided cell B[λ] containing β. It has a basis B[λ]. (This construction is general for based rings.) The above result says that the based

24 STRUCTURES OF QUANTUM AFFINE ALGEBRAS bi-module is isomorphic to Ũq[λ] with the base B[λ]. Similarly we have a left Ũ q -module associated with a left cell. Theorem 13.5(2) shows that this module is isomorphic to V (λ). In particular, its isomorphism class depends only on the two sided cell containing the left cell The function a. Let β = (b 1, s, b 2 ) B[λ]. Define a(β) = (wt(b 2 ), 2λ+ wt(b 2 ))/2, where wt is a weight considered as an element in Ũq. We have Theorem (1) Let L [λ] be the A 0 -submodule of Ũq[λ] as before. Then we have (q a(β) β)l [λ] L [λ] and for each β, a(β) 1 d Z 0 is the smallest number with this property. This gives a characterization of a. (2) a(β) < for any β. (This is obvious from the definition. But not obvious from the above characterization.) (3) For any two sided cell B[λ] and any λ 1 Pcl,+ 0, the restriction of a to B[λ] Ũqa λ1 is constant The asymptotic algebra. Let c β β,β be the structure constant of the multiplication in Ũq, i.e., ββ = β c β β,β β. For β B[λ] define ˆβ = q a(β) β. Then the Z[q s ] submodule Ũq[λ] of Ũq[λ] generated by {ˆβ β B[λ]} is a Z[q s ] subalgebra of Ũq[λ] by Theorem We define the asymptotic algebra Ũq[λ] 0 by Ũ q [λ] /q s Ũ q [λ]. Define t β to be the image of ˆβ in Ũq[λ] 0. Then Ũq[λ] 0 is a ring with a Z basis {t β β B[λ]}. Its structure constant γ β β,β is equal to the constant term of q a(β) c β β,β. We define Ũ0 = Ũ q [λ] 0 to be the direct sum of these rings. Theorem (1) Let us define a finite subset D B[λ] of B[λ] by {(b, 1, b) b B W (λ)}. Then D B[λ] is a generalized unit of Ũq[λ] 0 compatible with the basis {t β }, i.e., dd = δ d,d d and any β B[λ] is contained in dũq[λ] 0 d. (2) For d D B[λ], we have d # = d and d Ũq(0)a λ1 for some λ 1 P 0 cl. Here Ũ q (0) denote the subalgebra generated by a µ Ũ q a µ (µ P 0 cl ). (3) Let (, ) denote the bilinear form on Ũq defined in [Lusztig, Chap. 26]. Let λ 1 P 0 cl. If β Ũq(0)a λ1 B[λ], then q a(β) (a λ1, G(β)) 1 mod q s Z[q s ] if β D B[λ] and 0 otherwise. In particular, this gives us a characterization of D B[λ]. We call an element of D B[λ] distinguished involutions. We have (1) Any left cell and any right cell contain exactly one distinguished involution. (2) The number of distinguished involutions in B[λ] is D B[λ] A homomorphism Φ: Ũq Ũq[λ] 0. Fix a two sided cell B[λ]. We define a Q(q s ) linear map Φ: Ũq Q(q s ) Ũq[λ] 0 by Φ(β) = c β β,d (q s)t β, (β B(Ũq)) d D B[λ],β B[λ] which is well-defined since D B[λ] is finite and for a fixed β, d there are only finitely many c β β,d 0.

25 2. CELLS 81 Theorem (1) Φ is an algebra homomorphism. (2) Let P(B[λ]) be the set of λ P such that a λ d = d for some d D B[λ]. Then Φ( λ P(B[λ]) a λ) = 1, and Φ(a λ ) = 0 for λ / P(B[λ]) Structure of the asymptotic algebra. Next we describe an explicit realization of the ring structure of Ũq[λ] 0. Let J λ be the matrix algebra over the representation ring R(G λ ) of G λ where the size of matrices is #B W (λ). We denote by E(b 1, b 2, s) the matrix whose (b 1, b 2 )-entry is s IrrG λ and other entries are 0. (We identify the index set of matrices with B W (λ).) Then {E(b 1, b 2, s) b, b B W (λ), s IrrG λ } is a Z-basis of J λ. Theorem There exists a ring isomorphism Ũq[λ] 0 Jλ which gives a bijection between {t β β B[λ]} and {E(b 1, b 2, s) b, b B W (λ), s IrrG λ }. The bijection β (b, s, b ) is given by Theorem 13.3(3). The multiplication is calculated by using the almost orthonormality (still to be written).

26