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1 TitleTwo-sided BGG resolutions of admiss Author(s) Arakawa, Tomoyuki Citation Representation Theory of the Americ (014), 18(7): 183- Issue Date URL First published in 'Representation by the American Mathematical Societ Rightpublished version. Please cite only 論文は出版社版でありません 引用の際には出版社版をご確認ご利用ください Type Journal Article Textversion author Kyoto University

2 TWO-SIDED BGG RESOLUTIONS OF ADMISSIBLE REPRESENTATIONS TOMOYUKI ARAKAWA Abstract. We prove the conjecture of Frenkel, Kac and Wakimoto [FKW] on the existence of two-sided BGG resolutions of G-integrable admissible representations of affine Kac-Moody algebras at fractional levels. As an application we establish the semi-infinite analogue of the generalized Borel-Weil theorem [Kos] for minimal parabolic subalgebras which enables an inductive study of admissible representations. 1. Introduction Wakimoto modules are representations of non-twisted affine Kac-Moody algebras introduced by Wakimoto [Wak] in the case of ŝl and by Feigin and Frenkel [FF1] in the general case. Wakimoto modules have useful applications in representation theory and conformal field theory. In these applications it is important to have a resolution of an irreducible highest weight representation L(λ) of an affine Kac- Moody algebra g in terms of Wakimoto modules, that is, a complex C (λ) : C i 1 (λ) d i 1 C i (λ) d i C i+1 (λ)... with a differential d i which is a g-module homomorphism such that C i (λ) is a direct sum of Wakimoto modules and H i (C L(λ) if i = 0, (λ)) = 0 otherwise. The existence of such a resolution has been proved by Feigin and Frenkel [FF] for any integrable representations over arbitrary g and by Bernard and Felder [BF] and Feigin and Frenkel [FF] for any admissible representation [KW] over ŝl. In their study of W -algebras Frenkel, Kac and Wakimoto [FKW, Conjecture 3.5.1] conjectured the existence of such a resolution for any principle admissible representations over arbitrary g. In this paper we prove the existence of a two-sided resolution in terms of Wakimoto modules for any g-integrable admissible representations over arbitrary g (Theorem 6.11), where g is the classical part of g. For a general principal admissible representation of g we obtain the two-sided resolution in terms of twisted Wakimoto modules (Theorem 6.15). Let us sketch the proof of our result briefly. By Fiebig s equivalence [Fie] the block of the category O of g containing an admissible representation L(λ) is equivalent to the block containing an integrable representation 1. Therefore an admissible This work is partially supported by JSPS KAKENHI Grant Number No and No In the case L(λ) is a non-principal G-integrable admissible representation this is a block of another Kac-Moody algebra. 1

3 TOMOYUKI ARAKAWA representation admits a usual BGG type resolution in terms of Verma modules by the result of [GL, RCW]. Hence the idea of Arkhipov [Ark1] is applicable in our situation: One can obtain a twisted BGG resolution of L(λ) in terms of twisted Verma modules by applying the twisting functor T w [Ark1] to the BGG resolution of L(λ) as we have the Borel-Weil-Bott vanishing property [AS] L i T w L(λ) L(λ) if i = l(w), = 0 otherwise for w W(λ), where W(λ) is the integral Weyl group of λ and l : W(λ) Z 0 is the length function, see Theorem 5.1. It remains to show that one can construct an inductive system of twisted BGG resolutions B w(λ)} of L(λ) such that the complex lim w B w(λ) gives the required two-sided resolution of L(λ), see 6 for the details. We note that by applying the (generalized) quantum Drinfeld-Sokolov reduction functor [FKW, KRW] to the (duals of the) two-sided BGG resolutions of admissible representations we obtain resolutions of some of simple modules over W -algebras in terms of free field realizations due to the vanishing of the associated BRST cohomology [A1, A, A3, A4, A5]. In particular we obtain two-sided resolutions of all the minimal series representations [FKW, A7] of the W -algebras associated with principal nilpotent elements in terms of free bosonic realizations. As an application of the existence of two-sided BGG resolution for admissible representations we prove a semi-infinite analogue of the generalized Borel-Weil theorem [Kos] for minimal parabolic subalgebras (Theorem 7.7). This result is important since it enable an inductive study of admissible representations, see our subsequent paper [A6]. This paper is organized as follows. In we collect and prove some basic results about semi-infinite cohomology [Feĭ] and semi-regular bimodules [Vor1] which are needed for later use. In particular we establish an important property of semiregular bimodules in Proposition.1. In we collect basic results on the semiinfinite Bruhat ordering (or the generic Bruhat ordering) of an affine Weyl group defined by Lusztig [Lus] and study the semi-infinite analogue of parabolic subgroups. Semi-infinite Bruhat ordering is important for us since it (conjecturally) describes the space of homomorphisms between Wakimoto modules, see Proposition 4.10 and Conjecture The semi-infinite analogue of the minimal (or maximal) length representatives (Theorem 3.3) is important for describing the semi-infinite restriction functors studied in 7. In 4 we define Wakimoto modules and twisted Verma modules following [Vor] and study some of their basic properties. In particular we prove the uniqueness of Wakimoto modules which was stated in [FF] without a proof (Theorem 4.7). In 5 we generalize the Borel-Weil-Bott vanishing property of the twisting functor established in [AS] to the affine Kac-Moody algebra cases. In 6 we state and prove the main results of this paper. In 7 we study the semi-infinite restriction functor and establish the semi-infinite analogue of the generalized Borel-Weil theorem [Kos] for minimal parabolic subalgebras. This is a non-trivial fact since admissible representations are not unitarizable unless they are integrable. Acknowledgments. Some part of this work was done while the author was visiting Weizmann Institute, Israel, in May 011, Emmy Noether Center in Erlangen,

4 TWO-SIDED BGG RESOLUTIONS OF ADMISSIBLE REPRESENTATIONS 3 Germany in June 011, Isaac Newton Institute for Mathematical Sciences, UK, in 011, The University of Manchester, University of Birmingham, The University of Edinburgh, Lancaster University, York University, UK, in November 011, Academia Sinica, Taiwan, in December 011. He is grateful to those institutes for their hospitality.. Semi-regular bimodules and semi-infinite cohomology.1. Some notation. For Z-graded vector spaces M = n Z M n, N = n Z N n with finite-dimensional homogeneous components let Hom C (M, N) = n Z Hom C (M, N) n, Hom C (M, N) n = f Hom C (M, N); f(m i ) N i+n }, End C (M) =Hom C (M, M). We denote by M = n Z (M ) n the space Hom C (M, C), where C is considered as a graded vector space concentrated in the degree 0 component. If M, N are module over an algebra A we denote by Hom A (M, N) the space of all A-homomorphisms in Hom C (M, N)... Semi-infinite structure. Let g be a complex Lie algebra. A semi-infinite structure [Vor1] of g is is the following data: (i) a Z-grading g = n Z g n of g with finite-dimensional homogeneous components, dim C g n < for all n, (ii) a semi-infinite 1-cochain γ : g C. Here by a semi-infinite 1-cochain we mean the following: Decompose g into the direct sum of two subalgebras (1) () g = g + g, g + = g i, g = g i. i 0 i<0 A linear map γ : g C is called a semi-infinite 1-cochain if γ satisfies γ([x, y]) = tr((ad x) + (ad y) + (ad y) + (ad x) + ) for x, y g, ad x where (ad x) ± denotes the composition g g projection g ±. In the rest of this section we assume that g is equipped with a semi-infinite structure such that γ( i 0 g) = 0. We denote by U, U, U +, the enveloping algebras of g, g +, g by respectively. These algebras inherit a Z-grading from the corresponding Lie algebras. Let Õg be the category of Z-graded g-modules M = n Z M n with dim M n < for all m on which j>0 g + acts locally nilpotently and g 0 acts locally finitely..3. Semi-infinite cohomology. Choose a basis x i ; i Z} of g such that x i ; i 0} and x i ; i < 0} are bases of g + and g, respectively, and let c k ij } be the structure constant: [x i, x j ] = k ck ij x k.

5 4 TOMOYUKI ARAKAWA Denote by Cl(g) the Clifford algebra associated with g g, which has the following generators and relations: generators: ψ i, ψ i for i Z, relations: ψ i, ψ j } = δ i,j, ψ i, ψ j } = ψ i, ψ j } = 0. Here X, Y } = XY + Y X. The space of the semi-infinite forms + (g) of g is by definition the irreducible representation of Cl(g) generated by the vector 1 satisfying ψ i 1 = 0 for i 0, ψ i 1 = 0 for i > 0. It is graded by deg 1 = 0, deg ψ i = 1 and deg ψ i = 1: + (g) = p Z For A End C ( + (g)) of degree n set ψ k A if k < 0, (3) : ψ k A := ( 1) n : ψka := Aψ k if k 0, The following defines a g-module structure on + (g): ψk A if k 0, ( 1) n Aψk if k > 0. +p (g). (4) where x i : ad(x i ) : +γ(x i ), : ad x i := j,k c k ij : ψ k ψ j :. For M Õg, define d End(M + (g)) by d = x i ψi 1 1 c k ij : ψi (: ψj ψ k :) : +1 γ(x i )ψi i Z i,j,k Z i Z Then d = 0, d(m +p (g)) M +p+1 (g). The cohomology of the complex (M + (g), d) is called the semi-infinite g- cohomology with coefficients in M and denoted by H + (g, M) ([Feĭ, Vor1])..4. Semi-regular bimodules. We consider the full dual space Hom C (U, C) of U as a U-bimodule by (Xf)(u) = f(ux), (fx)(u) = f(xu) for X g, f Hom C (M, C), u U. The graded duals U ± of U ± are g ± -submodule of Hom C (U, C). By abuse of notation we denote by U the image of the embedding U + C U Hom C (U, C), f + f (u u + f + (u + )f (u )), f ± U ±, u ± U. Then U is a U-bisubmodule of Hom C (U, C) and coincides with the image of the embedding U C U + Hom C (U, C), f f + (u + u f + (u + )f (u )). Following [Vor] define US(g) = H +0 (g, U C U), where g is given the opposite semi-infinite structure and the semi-infinite g-cohomology is taken with respect to the diagonal left action on U C U. Here by the opposite semi-infinite structure we mean the one obtained by replacing g ± with g and γ

6 TWO-SIDED BGG RESOLUTIONS OF ADMISSIBLE REPRESENTATIONS 5 with γ. The space US(g) inherits the U-bimodule structure from U U defined by X(f u) = (fx) u, (f u)x = f (ux) for X g, U, u U. The U-bimodule US(g) is called the semi-regular bimodule of g. One has (5) US(g) = U + U+ U as left g + -modules and right g-modules, and the left g-module structure of US(g) is defined through the isomorphism (6) U + U U = Hom C (U +, U) = Hom U (U, U C C γ ) ([Vor1, Soe, Vor]). Let M be a g-module and consider the following four left g-module structures on US(g) C M: (7) (8) π 1 (X)(s m) = (sx) m + s Xm,, π (X)(s m) = (Xs) m, π 1(X)(s m) = (sx) m, π (X)(s m) = (Xs) m + s (Xm), for X g, s US(g), m M. Clearly, the two actions π 1 and π (resp. π 1 and π ) commute. Proposition.1 (cf. [AG, 6.4]). For M Õg the two U-bimodule structures (π 1, π ) and (π 1, π ) on US(g) C M are equivalent. Namely there exists a linear isomorphism Φ : US(g) C M US(g) C M such that Φ π i (X) = π i(x) Φ for i = 1,, X g. Proof. Define the linear isomorphism Φ 1 : U C U C M U C U C M by Φ 1 (f u m) = f ( (u)(1 m)) for f U, u U, m M, where : U U C U is the coproduct. We have Φ 1 π,l (X) = (π,l (X) + π 3,L (X)) Φ 1 Φ 1 (π,r (X) + π 3,R (X)) = π,r (X) Φ 1, where π i,l (resp. π i,r ) denotes the left action (resp. the right action) of g on the i-th factor of U U M, and M is considered as a right g-module by the action mx = xm for m M, x g. Next consider the graded dual M = n (M ) n as a right module by the action (fx)(m) = f(xm). Let Ψ : U C M U C M be the linear isomorphism defined by Ψ(f m)(u g) = (f m)((1 g) (u)) for f U, m M, u U, g M, where M is identified with (M ). Extend this to the linear isomorphism Φ : U C U C M U C U C M

7 6 TOMOYUKI ARAKAWA by setting Φ (f u m) = i f i u m i if Ψ(f m) = i f i m i with f i U, m i M. Then Φ π 1,R (X) = (π 1,R (X) + π 3,R (X)) Φ, Φ (π 1,L (X) + π 3,L (X)) = π 1,L (X) Φ. Set Then (9) (10) (11) Φ = Φ Φ 1 : U C U C M U C U C M. Φ (π 1,L (X) + π,l (X)) = Φ (π 1,L (X) + π,l (X) + π 3,L (X)) Φ 1 = (π 1,L (X) + π,l (X)) Φ, Φ (π,r (X) + π 3,R (X)) = Φ π,r (X) Φ 1 = π,r (X) Φ, Φ π 1,R (X) = Φ π 1,R (X) Φ 1 = (π 1,R (X) + π 3,R (X)) Φ. By (9) and the definition of US(g), Φ gives rise to a linear isomorphism Φ : US(g) C M US(g) C M. Moreover Φ satisfies the required properties by (10) and (11)..5. Semi-infinite induction. Let h = n Z h n be a graded Lie subalgebra of g such that γ h is a semi-infinite 1-cochain of h. Following [Vor] we define the semi-induced g-module S-ind g h M as S-ind g h M := H +0 (h, US(g) C M), where US(g) C M is considered as an h-module by the action π 1 defined in (7). The space S-ind g h M inherits the structure of a g-module from the action π defined in (7). Lemma.. The assignment S-ind g h : M S-indg h M defines an exact functor from Õh to Õg. Proof. Clearly S-ind M is an object of Õg since US(g) C M is. By Proposition.1 we may replace the actions of π 1 and π on US(g) C M with π 1 and π, simultaneously. It follows that (1) H + (h, US(g) C M) = H + (h, US(g)) C M. Since US(g) is free over h and cofree over h +, H +i (h, US(g)) = 0 for i 0 by [Vor1, Theorem.1]. (Note that the spectral sequence on [Vor1] converges since the complex US(g) + (h) is a direct sum of finite-dimensional subcomplexes consisting of homogeneous vectors.) We have shown that the functor S-ind g h is exact. In the case that h = g and γ 0 = γ, we have the following assertion. Proposition.3 ([Vor, (1.9)]). The functor S-ind g g : Õ g Õg is isomorphic to the identify functor. Proof. As H +0 (g, US(g)) is isomorphic to the trivial representation C of g ([Vor1, Theorem.1]), (1) gives the g-module isomorphism S-ind g g M = M as required.

8 TWO-SIDED BGG RESOLUTIONS OF ADMISSIBLE REPRESENTATIONS 7.6. Suppose that g admits a decomposition g = a ā with graded subalgebras a and ā such that the restrictions γ a and γ ā of γ are semi-infinite 1-cochains of a and ā, respectively. Lemma.4. US(g) = US(a) C US(ā) as left a-modules and right ā-modules. Proof. We have U + = U(a + ) C U(ā + ) as left a + -modules and right ā + -modules. Consider the composition US(a) C US(ā) (U(a ) C U(a + ) ) C (U(ā + ) C U(ā )) U(a + ) C U+ C U(ā + ) US(g), where the last map is the multiplication map. From the description (5), (6) of the g-bimodule structure of semi-regular bimodules one sees that the image of the above map is stable under the left and the right action of g on US(g). Hence the image must coincides with US(g) since it contains U+. By the equality of the graded dimensions it follows that above map is an isomorphism. Lemma.5. For M Õā, S-ind ḡ a M = US(a) C M as a-modules, where a acts only on the first factor US(a) of US(a) C M. Proof. We have S-ind ḡ a(m) = H +0 (ā, US(a) C US(ā) C M) = US(a) C S-indāā(M) = US(a) C M by Lemmas.3 and Semi-infinite Bruhat ordering 3.1. Affine Kac-Moody algebras and affine Weyl groups. We first fix some notation which are used for the rest of the paper. Let g be a finite-dimensional complex simple Lie algebra, and fix a Cartan subalgebra h of g. Let h be the set of roots of g. Choose a subset + h of positive roots and the set Π = α i ; i I} +, I = 1,,... l}, of simple roots. Let θ be the highest root, θ s the highest short root, = +, ρ = 1 α. α + Let Q = Zα h, the root lattice of g, Set n = g α, n = g α, where α α + α g α is the root space of g with root α. We have the triangular decomposition g = n h n. Let ( ) be the normalized invariant bilinear form of g. We identify h with h using ( ). Let = α ; α }, the set of coroots, Q = Zα h = h, the coroot lattice of g, ρ = 1 α, where α = α/(α α). α + α

9 8 TOMOYUKI ARAKAWA Let W GL( h ) be the Weyl group of g, s α W be the reflection corresponding to α : s α (λ) = λ λ(α )α. Let g be the affine Kac-Moody algebra associated with g: g = g[t, t 1 ] CK CD. The commutation relations of g are given by [xt m, yt n ] = [x, y]t m+n + mδ m+n,0 (x y)k, [K, g] = 0, [D, xt n ] = nxt n. We consider g as a subalgebra of g by the natural embedding g g, x xt 0. Let h = h CK CD, the Cartan subalgebra of g. The bilinear form ( ) from h to h by letting (K h) = (D h) = (K K) = (D D) = 0 and (D K) = 1. We identify h with the subspace of h consisting of elements which vanishes on CK CD. Thus, h = h CΛ 0 Cδ, where Λ 0 and δ are defined by Λ 0 (K) = δ(d) = 1, Λ 0 ( h Cδ) = δ( h CK) = 0. The number λ, K is called the level of λ. Let re + = + α + nδ; α, n N}, the set of positive real roots of g, re = re +, re = re + re the set of real roots, Π = α 0 = θ + δ, α 1,..., α l } the set of simple roots. Let W be the Weyl group of g, or the affine Weyl group of W. We have W = W Q. The extended affine Weyl group W e of g is the semidirect product W e = W P where P = λ h; α, λ Z for all α }, the coweight lattice of g. We have W e = W e + W, where W e + subgroup of W e consisting of elements which fix the set Π. We denote by t α or simply by α for the element of W e corresponding to α P. The reflection s α corresponding α = ᾱ + nδ re is given by s α = t nᾱ sᾱ. We set s i = s αi for i I := 0, 1,..., l}, so that W = s i ; i I. The action of W on h is extended to the action of W e on h by w(λ 0 ) = Λ 0, w(δ) = δ w W, t α (λ) = λ + Λ, K α ( λ, α + (α α) λ, K )δ, λ h. For λ h let λ h be its restriction to h.

10 TWO-SIDED BGG RESOLUTIONS OF ADMISSIBLE REPRESENTATIONS Twisted Bruhat ordering. Let l : W e Z 0 the length function: l(w) = ( re + w( re )). We have (13) l(t µ y) = (α µ) + 1 (α µ) α + y( + ) α + y( ) for µ P, y W. The twisted length function [Ark1] l y : W e Z with the twist y W e is defined by l y (w) = ( re + w( re ) y( re + )) ( re + w( re ) y( re )). Lemma 3.1. Let w, y W e. (i) Suppose that l(ys i ) = l(y) + 1 for i I. Then l y (w) if w 1 y(α l ysi i ) re +, (w) = l y (w) if w 1 y(α i ) re. (ii) l y (w) = l(y 1 w) l(y 1 ). Proof. (i) The assertion follows from the definition and the fact that re + ys i ( re ) = re + y( re ) y(α i )} if l(ys i ) = l(y) + 1. (ii) We prove by induction on l(y). If l(y) = 0 then l y (w) = l(w) = l(y 1 w). Suppose that l(ys i ) = l(y) + 1. If w 1 y(α i ) re + then l(s i y 1 w) = l(y 1 w) + 1. Hence by (i) and induction hypothesis, l ysi (w) = l y (w) = l(y 1 w) l(y 1 ) = l(s i y 1 w) l(s i y 1 ). If w 1 y(α i ) re then l(s i y 1 w) = l(y 1 w) 1. Again by (i) and induction hypothesis, l ys i (w) = l y (w) = l(y 1 w) l(y 1 ) = l(s i y 1 w) l(s i y 1 ). This completes the proof. For w 1, w, y W and γ re, write w 1 γ y w if w 1 = s γ w and l y (w 1 ) > l y (w ). Below, we shall often omit the symbol γ above the arrow. Also, we shall omit the symbol y under the arrow if y = 1. By Lemma 3.1 (ii) we have (14) w 1 y(γ) y w y 1 w 1 γ y 1 w. In particular for β = y(α i ) re +, α i Π, and w 1, w W such that l y (w ) l y (w 1 ) = 1 we have the equivalence (15) w 1 by [BGG, Lemma 11.3]. s β w 1 s β w 1 y y s β w y y w w

11 10 TOMOYUKI ARAKAWA Define w y w if there exists a sequence w 1, w,..., w k of elements of W such that Note that (16) w y w 1 y w y... y w k y w. w y w y 1 w y 1 w, by (14), where = 1, the usual Bruhat ordering of W. It follows that y defines a partial ordering of W. We will use the symbol w y w to denote a covering in the twisted Bruhat order y. Thus w y w means that w y w and l y (w) = l y (w ) Semi-infinite Bruhat ordering. Define the semi-infinite length [FF] l (w) of w W e by We have (17) l (w) = α re + w( re ); ᾱ + } α re + w( re ); ᾱ }. for λ P, w W. Set l (tλ y) = l(y) ( ρ λ) P+ = λ P ; α(λ) 0 for all α + }, We say that λ P + is sufficiently large if α(λ) if sufficiently large for all α +. By (13) and (17) we have the following assertion. Lemma 3.. l (w) = l λ (w) = l λ (w) for a sufficiently large λ P +. We write γ w 1 w for w 1, w W and γ re if w 1 = w s γ and l (w 1 ) < l (w ). (We shall often omit the symbol γ above the arrow.) Define w w if there exists a sequence w 1, w,..., w k of elements of W such that By Lemma 3. w w 1 w... w k w. w w tλ w for a sufficiently large λ P+, w t λ w for a sufficiently large λ P +. It follows that defines a partial ordering of W. Following Arkhipov [Ark1], we call it the semi-infinite Bruhat ordering on W. By [Soe1, Claim 4.14] the semi-infinite Bruhat ordering coincides with the generic Bruhat ordering defined by Lusztig [Lus]. We will use the symbol w w to denote a covering in the twisted Bruhat order. Thus w w means that w w and l (w) = l (w ) 1.

12 TWO-SIDED BGG RESOLUTIONS OF ADMISSIBLE REPRESENTATIONS Semi-infinite analogue of parabolic subgroups and minimal (maximal) length representatives. Let S be a subset of Π, S the subroot system of generated by α i S, S = r S,i the decomposition into the simple subroot i=1 systems 1,S,..., r,s. Let θ i be the longest root of S,i. Set S = α + nδ re ; α S, n Z}, W S = s α ; α S W. Then S is a subroot system of re isomorphic to the affine root system associated with S. Put S,+ = S re +, the set of positive root of S. Then Π S = S θ 1 +δ,..., θ s +δ} is a set of simple roots of S. We have W S = W S t Q S, where Q S = Zα. By (17), the restriction of the semi-infinite length function α S to W S coincides with the semi-infinite length function of the affine Weyl group W S. Define W S = w W; w 1 ( S,+ ) re + }. Theorem 3.3 ([Pet]). The multiplication map W S W S W, (u, v) uv, is a bijection. Moreover, we have l (uv) = l (u) + l (v) for u WS, v W S. Proof. First, we show the injectivity of the multiplication map. Suppose that u 1 v 1 = u v with u i W S, v i W S. Then v 1 = uv with u = u 1 1 u W S. If u 1 then there exists α S,+ such that u 1 (α) S,+. But then v W S implies that v1 1 (α) = v 1 u 1 (α) re, and this contradicts that v 1 W S. Hence u 1 = u, and so v 1 = v. Second, we show that the multiplication map W S W S W is surjective. We will prove by induction on (w 1 ( S,+ ) re ) that there exists u W S such that u 1 w W S. If (w 1 ( S,+ ) re ) = 0, w W S there is nothing to show. Suppose that (w 1 ( S,+ ) re ) > 0. Then there exists β Π S such that w 1 (β) re. Indeed, any element α S,+ is expressed as α = β Π S n β β with n β Z 0. Thus w 1 (α) = β Π S n β w 1 (β) re implies that one of w 1 (β) must belong to re. Now because (s β w) 1 ( S,+ ) = w 1 s β ( S,+ ) = w 1 ( S,+ \β} β}) = w 1 ( S,+ )\w 1 (β)} w 1 (β)}, (s β w) 1 ( S,+ ) re = w 1 ( S,+ ) re \w 1 (β)}. Hence by applying the induction hypothesis to s β w we find an element u W S such that u 1 s β w W S. Finally, we prove the equality of the semi-infinite length. By (17), we have l (t µ w) = l (t µ ) + l (w) for any µ Q. Hence we may assume that u W S. We will prove by induction on the length l(u) of u W S that l (uv) = l (u) + l (v) for any v W S. Suppose that l(u) = 1, so that u = s i for some α i S. Let

13 1 TOMOYUKI ARAKAWA v = t µ y W S with µ Q, y W. Note that v W S is equivalent to that if α 0 if y 1 (α) +, (18) S,+ then α(µ) = 1 if y 1 (α). Since l (si t µ y) = l(t si(µ)s i y) = l(s i y) (ρ µ α i (µ)α i ) = l(s i y) (ρ µ) + α i (µ), (18) implies that l (s i v) = l (v) + 1. Next let u = s i u 1 W S with u 1 W S, α i S, l(u) = l(u 1 ) + 1, so that u 1 1 (α i) +. Let v = t µ y W S as above. We have l (uv) = l (tsi u 1 (µ)s i u 1 y) = l(s i u 1 y) (ρ s i u 1 (µ)). If l(s i u 1 y) = l(u 1 y)+1, then + (u 1 y) 1 (α i ) = y 1 (u 1 1 (α i)). Hence (µ u 1 1 (α i)) = 0 by (18), which means s i u 1 (µ) = u 1 (µ). By the induction hypothesis, this proves that l (uv) = l (u) + l (v). If l(s i u 1 y) = l(u 1 y) 1, then (u 1 y) 1 (α i ) = y 1 (u 1 1 (α i)). So (18) gives (µ u 1 1 (α i)) = 1, which means s i u 1 (µ) = u 1 (µ) αi. By the induction hypothesis, this proves that l (uv) = l (u)+l (v) as required. 4. Wakimoto modules and twisted Verma modules 4.1. The category O of g. For any h-module M we set M µ = m M; hm = µ(h)m for all h h}. Let O g be the full subcategory of Õ g consisting of modules on which h acts semisimply. The formal character of M O g is defined by ch M = µ h (dim C M µ )e µ. Let O g k be the full subcategory of Og consisting of objects of level k, where a g-module M is said to be of level k if K acts as the multiplication by k. 4.. Twisting functors and twisted Verma modules. By abuse of notation we denote also by w a Tits lifting of w W e to Aut(g). For each w W the twisting functor T w : O g O g is defined as follows ([Ark1]): Let n w = n w 1 (n + ) and set N w = U(n w ). Put S w = U Nw N w. The space S w has a U-bimodule structure, which is described as follows: Let f n \0}, and set U (f) = U C[f] C[f, f 1 ]. Then U (f) is an associative algebra which contains U as a subalgebra. We set S f = U (f) /U. Choose a filtration n w = F 0 F 1 F r 0, r = l(w), consisting of ideals F p n w of codimension p. If f p F p 1 \F p we have an isomorphism of U-bimodules (19) We have (0) S w = S f1 U S f U... U S fr. S w = N w Nw U

14 TWO-SIDED BGG RESOLUTIONS OF ADMISSIBLE REPRESENTATIONS 13 as right U-modules and left N w -modules. Put For M O g define 1 w = f f... fr S w. T w (M) = φ w (S w U(g) M), where φ w means that the action of g is twisted by the automorphism w of g. This define a right exact functor T w : O g O g such that (1) T wsi = Tw T i if α i Π and l(ws i ) = l(w) + 1, where T i = T si. The functor T w admits a right adjoint functor G w in the category O g ([AS, 4]): G w (M) = Hom U (S w, φ 1 w (M)). It is straightforward to extend the definition of T w and G w to w W e ([A1]). The following assertion follows in the same manner as [Soe, Theorem.1]. Lemma 4.1. Let M O g, w W e (i) Suppose that M is free over n w. Then M = G w T w (M). (ii) Suppose that M is cofree over w(n w ). Then M = T w G w (M). For λ h, let M(λ) be the Verma module of g with highest weight λ. Set M w (λ) = T w M(w 1 λ). The g-module M w (λ) O g is called the twisted Verma module M w (λ) with highest weight λ and twist w W e. Note that by (0) we have () M w (λ) µ = φw (Nw Nw U(n )) µ λ = (U(w(n ) n + ) C U(w(n ) n )) µ λ as h-modules. Hence In particular M w (λ) is an object of O g. By Lemma 4.1 (1) we have ch M w (λ) = ch M(λ). M(µ) = G w M w (w µ). Hence the functor T w gives the isomorphism (3) Hom g (M(λ), M(µ)) Hom g (M w (w λ), M w (w µ)) for λ, µ h. We have [AL, Proposition 6.3] (4) M w (λ) = M(λ) if λ + ρ, α N for all α re + w( re ).

15 14 TOMOYUKI ARAKAWA 4.3. Hom spaces between twisted Verma modules. For λ h let (λ) and W(λ) be its integral root system and integral Weyl group, respectively: (λ) = α re ; λ + ρ, α Z}, W(λ) = s α ; α (λ) W. Let (λ) + = (λ) re + the set of positive roots of (λ), Π(λ) (λ) + the set of simple roots of (λ), l : W(λ) Z 0 the length function. For y W(λ) the twisted length function l y and the twisted Bruhat ordering λ,y are defined for W(λ). We will use the symbol w λ,y w to denote a covering in the twisted Bruhat order λ,y. Recall that a weight λ h is called regular dominant if λ+ρ, α 0, 1,,... } for all α re +. It is called regular anti-dominant if λ + ρ, α 0, 1,,... } for all α re +. Theorem 4.. Let w, w, y W(λ). (i) If λ is regular dominant then dim C Hom g (M y (w λ), M y (w λ)) = (ii) If λ is regular anti-dominant then dim C Hom g (M y (w λ), M y (w λ)) = 1 if w λ,y w, 0 otherwise. 1 if w λ,y w, 0 otherwise. Proof. (i) By (3) the assertion follows from (16) and [KT, Proposition.5.5 (ii)]. Proof of (ii) is similar Wakimoto modules. Let g, h be as in 3.1, and let us consider the Z-grading of g with g 0 = h, g 1 = α Π g α, where g α is the root space of g of root α. Let ρ = ρ + h Λ 0 h, where h is the dual Coxeter number of g. Then ρ, α = 1 for all α Π and ρ define a semi-infinite 1-cochain of g [Ark]. Let L n, L n, a and ā be graded subalgebras of g defined by L n = n[t, t 1 ], L n = n [t, t 1 ], a = L n h[t 1 ]t 1, ā = L n h[t] CK CD. Then 0 = ρ Ln = ρ Ln = ρ a gives semi-infinite 1-cochains of Ln, Ln, a, and ρ ā gives a semi-infinite 1-cochain of ā. Following [Vor] we define the Wakimoto module W (λ) of g with highest weight λ h by W (λ) = S-ind ḡ a C λ, where C λ is the one-dimensional representation of h corresponding to λ regarded as a ā-module by the natural projection ā h. By Lemma.5 we have (5) W (λ) = US(a) as a-modules,

16 TWO-SIDED BGG RESOLUTIONS OF ADMISSIBLE REPRESENTATIONS 15 and hence (6) (7) H +i (a, W (λ)) = ch W (λ) = ch M(λ). C λ if i = 0, 0 otherwise as h-modules, In particular W (λ) is an object of O g. Theorem 4.7 below shows that the above definition of Wakimoto module coincides with that of Feigin and Frenkel [FF, Fre] Wakimoto modules as inductive limits of twisted Verma modules. Let y, w, u W such that w = yu and l(w) = l(y) + l(u). Then T w = T y T u and S w = Sy U φ y (S u ). Let j w,y : S y S w be the homomorphism of left U-modules which maps s S y to s 1 u S y U φ y (S u ) = S w. Define ν λ w,y Hom g (M y (λ), M w (λ)) by ν λ w,y(s v y 1 λ) = j w,y (s) v w 1 λ for s S y, where v µ denotes the highest weight vector of M(µ) for µ h. Then by (3). We have (8) Hom g (M y (λ), M w (λ)) = Cν λ w,y ν λ w 3,w ν λ w,w 1 = ν λ w 3,w 1 if w 3 = w u, w = w 1 u 1 with l(w 1 ) = l(w ) + l(u ), l(w ) = l(w 1 ) + l(u 1 ). Let γ 1, γ,... } be a sequence in P + such that γ i γ i 1 P + and lim n α(γ n) = for all α +. Then t γi+1 = t γi t (γi+1 γ i ) with l(t γi+1 ) = l(t γi ) + l(t (γi+1 γ i )) for all i. It follows that M γn (λ) : ν λ γ m, γ n } forms an inductive system of g-modules. Proposition 4.3 ([Ark1, Lemma 6.1.7]). There is an isomorphism of g-modules W (λ) = lim M γn (λ). n Proof. For the reader s convenience we shall give a proof of Proposition 4.3 here. Set W (λ) = lim M γn (λ). First note that n t γi (n γi ) =t γi (n ) n + = span C x α t n ; α +, 0 n < α(γ i )}, t γi (n ) n = ( h n)[t 1 ]t 1 span C x α t n ; α +, n > α(γ i )}, where x α is a root vector of g of root α. Thus we have t γ1 (n γ1 ) t γ (n γ ) a + and a + = i 1 t γ i (n γi ). The map j γi, γ j : S γi S γj restricts to the embedding j γi, γ j : N γ i N γ j for i < j, and we have U(a + ) = lim φ γi (N γ i ) i as left a + -modules. Let j γi : φ γi (N γ i ) U(a + ) be the embedding of left φ γi (N γi )-modules under the above identification.

17 16 TOMOYUKI ARAKAWA Since t γi (n γi ) = span C x α t n ; α +, 0 < n α(γ i )} a, by Lemma 4.1 (ii). Hence W (λ) = T γi G γi (W (λ)) Hom g (M γi (λ), W (λ)) = Hom g (M(t γi λ), G γi (W (λ))). As ch G γi (W (λ)) = ch M(t γi λ), there exists a unique g-module homomorphism ψ i : M(t γi λ) G γi (M) which sends v tγi λ to w i, a vector of G γi (W (λ)) of weight t γi λ. Up to a non-zero constant multiplication, w i equals to the the to j γi (f) 1 λ US(a) C λ = W (λ). The corresponding homomorphism T γi (ψ i ) : M γ i (λ) W (λ) is given by element of G γi (W (λ)) = Hom N γi (N γ i, φ 1 γ i (W (λ))) which sends f N t γi (9) T γi (ψ i )(f v tγi λ) = j γi (f) 1 λ for f N γ i. It follows that T γi (ψ j ) ν λ γ j,γ i = T γi (ψ i ) for i < j, and the sequence T γi (ψ j )} yields a g-module homomorphism Φ : W (λ) = lim M γ i (λ) W (λ). i Fix µ h. Since W (λ) = US(a) as an a-module, it follows from () that T γi restricts to the isomorphism M γi (λ) µ W (λ) µ for a sufficiently large i. This completes the proof Endmorphisms of Wakimoto modules. Proposition 4.4. Let α P +, λ h. (i) T α W (λ) = W (t α λ). (ii) G α W (λ) = W (t α λ). Proof. (i) Let γ 1, γ,... } be a sequence in P + such that γ i γ i 1 P + and lim β(γ n) = for all β +. Set γ n i = γ i + α. Then the sequence γ 1, γ,... } satisfies the same property. Hence by Proposition 4.3 and the fact that a homology functor commutes with inductive limits we have T α W (λ) = T α (lim M γ i (λ)) = lim T α M γ i (λ) = lim T α T γi M(t γi λ) = lim T γ i M(t γi λ) = lim M γ i (tα λ) = W (t α λ). (ii) Since n t α a, W (λ) is free over n t α. Hence W (t α λ) = G α T α W (t α λ) = G α W (λ) by Lemma 4.1 and (i). Corollary 4.5. Let α P +. The functor G α gives the isomorphism for λ, µ h. Hom g (W (λ), W (µ)) = Hom g (W (t α λ), W (t α µ)). Proposition 4.6. For λ h we have End g (W (λ)) = C. Proof. Let γ 1, γ,..., } be in Subsection 4.5. Then End g (W (λ)) = Hom g (lim M γi (λ), W (λ)) (by Proposition 4.3) i = lim i Hom g (M γ i (λ), W (λ)) = lim Hom g (M(t γi λ), G γi W (λ)) i Hom g (M(t γi λ), W (t γi λ)) (by Proposition 4.4). = lim i

18 TWO-SIDED BGG RESOLUTIONS OF ADMISSIBLE REPRESENTATIONS 17 As we have seen in the proof of Proposition 4.3, the space Hom g (M(t γi λ), W (t γi λ)) is one-dimensional and ν γ λ m,γ n induces the isomorphism Hom g (M γm (λ), W (λ)) Hom g (M γn (λ), W (λ)). This completes the proof Uniqueness of Wakimoto modules. A finite filtration 0 = M 0 M 1 M M r = M of a g-module M is called awakimoto flag if each successive quotient M i /M i 1 is isomorphic to W (λ i ) for some λ i. Theorem 4.7. Suppose that k is non-critical, that is, k h. For an object M of O g k the following conditions are equivalent. (i) M admits a Wakimoto flag. (ii) H +i (a, M) = 0 for i 0 and H +0 (a, M) is finite-dimensional. If this is the case the multiplicity (M : W (λ)) of W (λ) in a Wakimoto flag of M equals to dim H +0 (a, M) λ. In particular if H +i (a, M) C λ if i = 0, = 0 otherwise as h-modules, M is isomorphic to W (λ). The proof of Theorem 4.7 will be given in Subsection 4.8. We put on record some of consequences of Theorem 4.7: Proposition 4.8. A tilting module in O g at a non-critical level admits a Wakimoto flag. Proof. By definition a tilting module M admits both a Verma flag and a dual Verma flag. It follows that M is free over n and cofree over n +. In particular M is free over n[t 1 ]t 1 and cofree over n[t]. Hence by [Vor1, Theorem.1], we have H +i (a, M) = 0 for i 0. The fact that H +0 (a, M) is finite-dimensional follows from the Euler-Poincaré principle. Proposition 4.9. Suppose that λ + ρ, K Q 0. Then W (t α λ) = M(t α λ) for a sufficiently large α P +. Proof. Let α be sufficiently large. By the hypothesis t α (λ + ρ), β N for all β re + such that β +. It follows from [A1, Theorem 3.1] that M(t α λ) is cofree over n[t] = a +. Because M(t α λ) is obviously free over a we have H +i (a, M(t α λ)) C tα λ for i = 0, = 0 otherwise. The following assertion follows from Proposition 4.9 and Corollary 4.5. Proposition Let λ, µ h be of level k, and suppose that k + h Q 0. Then Hom g (W (λ), W (µ)) = Hom g (M(t α λ), M(t α µ))

19 18 TOMOYUKI ARAKAWA for a sufficiently large α P +. In particular if λ h is integral, regular antidominant, then 1 if w dim C Hom g (W (w λ), W (y λ)) = y 0 else for w, y W. Conjecture Let λ h be integral, regular dominant. Then 1 if w dim C Hom g (W (w λ), W (y λ)) = y 0 else for w, y W. In Theorem 6.11 below we prove Conjecture 4.11 in the case that w y (in a slightly more general setting) Proof of Theorem 4.7. Let H = h[t, t 1 ] CK g, the Heisenberg subalgebra. Denote by π λ the irreducible representation of H with highest weight λ. We have π λ = U( h[t 1 ]t 1 ) as a module over h[t 1 ]t 1 H provided that λ(k) 0. For M O g k one knows that H + (L n, M) is naturally an H-module of level k + h ([FF]). Lemma 4.1. Let M be an object of O g k with k h. Then the following conditions are equivalent: (i) H +i (a, M) = 0 for i 0; (ii) H +i (L n, M) = 0 for i 0. Proof. The assumption that k h implies that H + (L n, M) is semi-simple as an H-module and is a direct sum of π µ s. Consider the Hochschild-Serre spectral sequence for the ideal Ln a to compute H + (a, M). By definition, we have E p,q H = p ( h[t 1 ]t 1, H +q (Ln, M)) for p 0, 0 for p > 0. By the above mentioned fact H +q (Ln, M) is free over U( h[t 1 ]t 1 ). Hence E p,q H = +q (Ln, M))/ h[t 1 ]t 1 (H +q (Ln, M))) for p = 0. 0 for p 0. Therefore the spectral sequence collapses at E = E, and H +i (a, M) = 0 for i 0 if and only if H +i (L n, M) = 0 for i 0. This completes the proof. Proposition Let M be an object of O k at a non-critical level k such that H +i (a, M) = 0 for i 0. Then M = US(a) C H +0 (a, M) as a-modules and h-modules, where a acts only on the first factor US(a) and h acts as h(s m) = ad(h)(s) m + s hm.

20 TWO-SIDED BGG RESOLUTIONS OF ADMISSIBLE REPRESENTATIONS 19 Proof. By Proposition.3 it suffices to show that S-ind a a M = US(a) C H +0 (a, M). As in the proof of Lemma 4.1, we shall consider the Hochschild-Serre spectral sequence for the ideal L n a to compute H + (a, US(a) M). By definition we have (30) E,q 1 = H +q (L n, US(a) C M) C ( h[t 1 ]t 1 ), (31) E p,q = H p ( h[t 1 ]t 1, H +q (Ln, US(a) C M)). To compute the E 1 -term set F p US(a) = US(a) µ, µ, ρ p where US(a) is considered as an h-module by the adjoint action. Then US(a) = F 0 US(a) F 1 US(a)..., F p US(a) = 0, F p US(a) L n F p+1 US(a). Define the filtration F (US(a) C M C + (L n)) by setting F p (US(a) C M C + (L n)) = F p US(a) C M C + (L n). This defines a decreasing, weight-wise regular filtration of the complex. Consider the associated spectral sequence E r H + (L n, US(a) C M). Because the associated graded space gr US(a) with respect to this filtration is a trivial L n-module the E 1 -term of the spectral sequence E r is isomorphic to US(a) C H + (Ln, M). Hence by the hypothesis and Lemma 4.1 the spectral sequence E r collapses at E 1 = E and we obtain the isomorphism of h-modules H +i (Ln, US(a) C M) US(a) C H +0 (Ln, M) for i = 0, (3) = 0 for i 0. This is also an isomorphism of a-modules since US(a) = gr US(a) as left a-modules, where x α t n a is considered as an operator on gr US(a) = p F p US(a)/F p+1 US(a) which maps F p US(a)/F p+1 US(a) to F p+α( ρ ) US(a)/F p+α( ρ )+1 US(a). We have computed the E 1 -term (30): E,q 1 It follows that (33) = US(a) C H +0 (L n, M) C ( h[t 1 ]t 1 ) for q = 0, 0 for q 0. E p,q = US(a) C H +0 (a, M) for p = q = 0, 0 otherwise as h-modules and a-modules, see the proof of Lemma 4.1. The spectral sequence collapses at E = E and we obtain the required isomorphism. Set Q,+ = Z 0α + Z 0δ h α re ᾱ,

21 0 TOMOYUKI ARAKAWA and define the partial ordering on h by µ λ λ µ Q,+. Note that µ λ if and only if t α µ t α λ for a sufficiently large α Q. Theorem 4.7. Since The direction (i) (ii) in Theorem 4.7 is obvious by (6), we shall prove that (ii) implies (i). Let λ 1,..., λ r } be the set of weights of H +0 (a, M) with multiplicities counted, so that (34) M = r US(a) C C λi i=1 as a-modules and h-modules by Proposition We may assume that if λ i λ j then j < i. Set λ = λ 1. We shall show that there is a g-module embedding W (λ) M. Let γ 1, γ,... } be a sequence in P + such that γ i γ i 1 P + and lim n α(γ n) = for all α +, so that W (λ) = lim M γn (λ) by Proposition 4.3. By Lemma 4.1 (ii) n we have M = T γi G γi (M), and hence, Hom g (M γ i (λ), M) = Hom g (M(t γi λ), G γi (M)). By (34), ch G γi (M) = r i=1 ch M(t γ i λ). Let i be sufficiently large so that t γi λ is maximal in G γi (M). Denote by Φ i the g-module homomorphism ψ i : M(t γi λ) G γi (M) which sends v tγi λ to a vector of G γi (M) of weight t γi λ. As in the proof of Proposition 4.3 T γi (ψ i ) : M γ i (λ) M} yield an injective g-module homomorphism Φ : W (λ) = lim M γ i (λ) M. i The map Φ induces the homomorphism H +0 (a, W (λ)) = C λ H +0 (a, M) which is certainly injective. It follows from the long exact sequence associated Φ with the exact sequence 0 W (λ) M M/W (λ) 0 we obtain that H +i (a, M/W (λ)) = 0 for i 0 and dim H +0 (a, M/W (λ)) = dim H +0 (a, M) 1. Theorem 4.7 follows by the induction on dim H +0 (a, M) Twisted Wakimoto modules. For w W we have the decomposition g = w(a) w(ā), and ρ defines a semi-infinite 1-cochain of the graded subalgebra w(ā). Hence we can define the twisted Wakimoto module W w (λ) with highest weight λ and twist w W by W w (λ) = S-ind g w(ā) C λ, where C λ is the one-dimensional representation of h corresponding to λ regarded as a ā-module by the projection ā h. We have W w (λ) = US(w(a)) as w(a)-modules and ch W w (λ) = ch M(λ), H +i (w(a), W w (λ)) C λ for i = 0, = as h-modules. 0 otherwise,

22 TWO-SIDED BGG RESOLUTIONS OF ADMISSIBLE REPRESENTATIONS 1 Let γ 1, γ,... } be a sequence in P + such that γ i γ i 1 P + and lim n α(γ n) = for all α +. The following assertion can be proved in the same manner as Proposition 4.3. Proposition Let λ h, w W. There is an isomorphism of g-modules W w (λ) = lim M w(γn) (λ). n The following assertion can be proved in the same manner as Theorem 4.7. Theorem Let λ h be non-critical, w W. Let M be an object of O g such that H +i (w(a), M) C λ if i = 0, = 0 otherwise, as h-modules. Then M is isomorphic to W w (λ). 5. Borel-Weil-Bott vanishing property of Twisting functors 5.1. Left derived functors of twisting functors. The functor T w, w W e, admits the left derived functor L T w in the category O g since it is a Lie algebra homology functor: L i T w (M) = φ w (H i (g, S w C M)), where g acts on N w C M by X(f m) = fx m + f Xm. Because (35) L i T w (M) = φ w (H i (n w, N w C M)) as w(n w )-modules, we have the following assertion. Lemma 5.1. Suppose M O g is free over n w. Then L i T w (M) = 0 for i 1. Let e i, h i, f i ; i I}, e i g αi, f i g αi, be the Chevalley generators of g. For i I, let sl (i) denote the copy of sl in g spanned by e i, h i, f i } Proposition 5.. Let M O g, i I. Denote by N the largest sl (i) -integrable submodule of M. Then T i (M) = T i (M/N), ch L 1 T i (M) = ch N and L p T i (M) = 0 for p. Proof. Let T (i) i denote the twisting functor for sl (i) corresponding to the reflection s αi. Because T i (M) = T (i) i (M) as sl (i) -modules and h-modules, we have (36) L p T i (M) = L p T (i) i (M) as sl (i) -modules and h-modules. In particular L p T i (M) = 0 for p. It follows that the exact sequence yields the long exact sequence 0 N M M/N 0 0 L 1 T i (N) L 1 T i (M) L 1 T i (M/N) T i (N) T i (M) T i (M/N) 0. Since M/N is free as C[f i ]-module L 1 T i (M/N) = 0 by Lemma 5.1. Also, T i (N) = 0 and L 1 T i (N) = N as h-modules by [AS, Theorem 6.1] and (36). This completes the proof.

23 TOMOYUKI ARAKAWA Let L(λ) O g be the irreducible highest weight representation of g with highest weight λ h. Theorem 5.3 ([AS, Theorem 6.1]). Let λ h and suppose that λ, αi Z 0 with i I. Then L p T i (L(λ)) L(λ) if p = 1, = 0 if p 1. Proof. The hypothesis implies that L(λ) is sl (i) -integrable. Therefore L pt i (L(λ)) = 0 for p 1 and ch L 1 T i (L(λ)) = ch L(λ) by Proposition Twisting functors associated with integral Weyl group. Lemma 5.4. Let λ h, α Π(λ). There exists x W and α i Π such that s α = xs i x 1, l(s α ) = l(x) + 1 and re + x( re ) (λ) =. Proof. Let s α = s jl s jl 1... s j1 be a reduced expression of s α in W. Then re + s α ( re ) = α 1, s j1 (α j ),..., s j1... s jl 1 (α jl )} Since l λ (α) = 1, re + s α ( re ) (λ) = α}. Thus there exists r such that α = s j1... s jr 1 (α jr ). Set x = s j1... s jr 1, i = j r. Then s α = s x(αi) = xs i x 1. It follows that s jl... s jr+1 = x and l(s α ) = l(x)+1. Also re + s α ( re ) (λ) = α} implies that re + x( re ) (λ) =. Note that if λ, α, α i, x are as in Lemma 5.4 then T α = T x T i T x 1. Let O g [λ] be the block of O g corresponding to λ, that is, the full subcategory of O g consisting of objects M such that [M : L(µ)] 0 µ W(λ) µ, where [M : L(µ)] is the multiplicity of L(µ) in the local composition factor of M. Lemma 5.5. Let λ h, y W, and suppose that λ + ρ, α Z for all α re + y 1 ( re ). Then T y M(w λ) = M(yw λ), T y L(w λ) = L(yw λ) for w W(λ). Moreover T w gives an equivalence of categories O g [λ] O g [w λ]. The same is true for G w. Proof. First note that the assumption implies that W(y λ) = yw(λ)y 1. We prove by induction on l(y). Let l(y) = 1, so that y = s i for i I. Then the fact that T i M(wλ) = M(s i w λ) with w W(λ) follow from (4). By [A1, Theorems 3.1, 3.] any object of O g [λ] and Og [s i λ] is free over C[f i] and cofree over C[e i ]. Hence by Lemma 4.1 T i gives an equivalence of categories O g [λ] O g [s i λ] with a quasi-inverse G i. It follows that T i L(λ) is a simple g-module which is a quotient of T i M(λ) = M(s i λ), and hence is isomorphic to L(s i λ). Next let y = s i z with z W, l(y) = l(z) + 1. Then re + y 1 ( re ) = z 1 (α i )} ( re + z 1 re ). The assertion follows from the induction hypothesis. Corollary 5.6. Let λ, α, α i, x be as in Lemma 5.4. Then T x give an equivalence of categories O g [x 1 λ] O g [λ] such that T xm(µ) = M(x µ), T x L(µ) = M(x µ) for µ W(x 1 λ) x 1 λ = x 1 W(λ) λ. Lemma 5.7. Let λ h, α i Π such that λ + ρ, α i Z. Then T im w (λ) = M s iws i (s i λ) for w W(λ).

24 TWO-SIDED BGG RESOLUTIONS OF ADMISSIBLE REPRESENTATIONS 3 Proof. By Lemma 5.5, T i M w (λ) = T i T w M(w 1 λ) = T i T w T i M(s i w 1 λ) = T s iws i M(s i w 1 s i s i λ). Lemma 5.8. Let λ h, α i Π such that λ + ρ, αi Z. Then T i is isomorphic to the identity functor, and so is G i : Og [λ] Og [λ]. : O g [λ] Og [λ] Proof. By Lemma 5.5 Ti induces an auto-equivalence of the category O g [λ] such that Ti M(w λ) = M(w λ) and Ti (L(w λ)) = L(w λ) for all w W(λ). The standard argument shows that such a functor must be isomorphic to the identify functor. Corollary 5.9. Let λ h, w = s α y W(λ), α Π(λ), y W(λ), l λ (w) = l λ (y) + 1. Then T w : O g [λ] Og [w λ] is isomorphic to the functor T s α T y : O g [λ] O g [w λ]. Proposition Let λ h, w W(λ), α Π(λ) and suppose that w(λ + ρ), α N. Then the following sequence is exact: 0 M(s α w λ) ϕ1 M(w λ) ϕ M s α (w λ) ϕ3 M s α (s α w λ) 0, where ϕ 1, ϕ, ϕ 3 are any non-trivial g-homomorphisms. Proof. First observe that Hom g (M(s α w λ), M(w λ)), Hom g (M(w λ), M s α (w λ)) and Hom g (M s α (w λ), M s α (s α w λ)) are all one-dimensional. (The first and the third are one-dimensional by Theorem 4..) By Lemma 5.4 there exists x W and α i Π such that s α = xs i x 1, l(s α ) = l(x) + 1, and re + x( re ) (λ) =. We have M(y λ) = T x M(x 1 y λ), M s α (y λ) = T x T i T x 1M(xs i x 1 y λ) = T x T i M(s i x 1 y λ) = T x M s i (x 1 y λ) for y W(λ) by Lemma 5.5. Since x 1 w(λ + ρ), α i = w(λ + ρ), α N there is an exact sequence 0 M(s i x 1 w λ) M(x 1 w λ) M s i (x 1 w λ) M s i (s i x 1 w λ) 0 by [AL, Propostion 6.]. The required exact sequence is obtained by applying the exact functor T x : O g [x 1 λ] Og [λ] to the above. Proposition Let λ h, α Π(λ), M O g [λ]. Take α i Π, x W such that α = x(α i ) and x 1 (λ) + re + as in Lemma 5.4. Let N be the largest sl (i) -integrable submodule of T x 1(M) and set N = T x(n ) M. Then T α (M) = T sα (M/N), ch L 1 T sα (M) = ch N and L p T sα (M) = 0 for p. Proof. We have T α = T x T i T x 1 and T x 1 : O g [λ] Og [x 1 λ], T x : O g x 1 λ Og [λ] are exact functors by Corollary 5.6. Therefore (37) Hence Proposition 5. gives that L p T sα (M) = T x (L p T i (T x 1M)). T sα (M) = T x T i T x 1(M) = T x T i (T x 1(M)/N ) = T x T i T x 1(M/N) = T sα (M/N), ch L 1 T sα (M) = ch T x T x 1(N) = ch N, L p T sα (M) = 0 for p 0. This completes the proof.

25 4 TOMOYUKI ARAKAWA Theorem 5.1. Let λ h be regular dominant weight, w W(λ). Then L p T w (L(λ)) L(λ) if p = l(w), = 0 otherwise. Proof. Let α Π(λ). Since T x 1L(λ) = L(x 1 λ) and x 1 λ + ρ, αi = λ + ρ, α N, T x 1L(λ) is sl (i) -integrable. Thus, L p T i T x 1L(λ) T x 1L(λ) if p = 1, = 0 if p 0 by Theorem 5.3. It follows from (37) that L p T sα (L(λ)) L(λ) if p = 1, (38) = 0 otherwise. Finally the assertion follows in the same manner as in [AS, Corollary 6.] by Corollary Two-sided BGG resolutions of admissible representations 6.1. Admissible representations. A weight λ h is called admissible if it is regular dominant and Q (λ) = Q re. The irreducible representation L(λ) is called admissible if λ is admissible. A complex number k is called an admissible number for g if the weight kλ 0 is admissible. Let r be the lacing number of g, that is, the maximal number of the edges of the Dynkin digram of g. Also, let h be the Coxeter number of g. Proposition 6.1 ([KW, KW3]). A complex number k is admissible if and only if k + h = p h if (r, q) = 1, (39) with p, q N, (p, q) = 1, p q h if (r, q) = r. A complex number k of the form (39) is called an admissible number with denominator q. For an an admissible number k with denominator q, we have (kλ 0 ) = α + nqδ; α, n Z} = re and W(kΛ 0 ) = W if (r, q) = 1, (kλ 0 ) = α + nqδ; α, n Z} = L re and W(kΛ 0 ) = L W if (r, q) = r, where (λ) = α ; α (λ)} and L re and L W are the real root system and the Weyl group of the non-twisted affine Kac-Moody algebra L g associated with the Langlands dual L g of g, respectively. Set α 0 = θ + qδ if (r, q) = 1, θ s + q r δ if (r, q) = r. Then Π(kΛ 0 ) = α 1,..., α l, α 0 }. Put ṡ 0 = s α0 W(kΛ 0 ), so that W(kΛ 0 ) = s 1,..., s l, ṡ 0. For an admissible number k let P r + k be the set of admissible weights λ of level k such that λ(α ) Z 0 for all α +. Then L(λ); λ P r + k } is the set of

26 TWO-SIDED BGG RESOLUTIONS OF ADMISSIBLE REPRESENTATIONS 5 irreducible admissible representations of level k which are integrable over g g. We have (λ) = (kλ 0 ) for λ P r + k. For an admissible number k denote by P r k the set of admissible weights λ of level k such that (λ) = (kλ 0 ) as root systems. Then [KW] (40) P r k = P r k,y, P r k,y = y P r + k. Note that (41) y W e y( (kλ 0 ) re + W(λ) = yw(kλ 0 )y 1 for λ P r k,y. For λ P r k, let l λ (?) be the semi-infinite length function of the affine Weyl group W(λ). The semi-infinite Bruhat ordering λ, are also defined for W(λ). We will use the symbol w λ, w to denote a covering in the twisted Bruhat order λ,. Remark 6.. The admissible weight λ P r k is called the principal admissible weight [KW] if (λ) = re, that is, if the denominator q of k is prime to r. 6.. Fiebig s equivalence and BGG resolution of admissible representations. The following theorem is the special case of a result of Fiebig [Fie, Theorem 11]. Theorem 6.3 ([Fie]). Let λ be regular dominant. Suppose that there exists a symmetrizable Kac-Moody algebra g whose Weyl group W is isomorphic to W(λ). Let λ be an integral dominant weight of g, O g [λ ] the block of O g containing the irreducible highest weight representation L g (λ ) of g with highest weight λ. Then there is an equivalence of categories O g [λ] = O g [λ ] which maps M(w λ) and L(w λ), w W(λ), to M g (φ(w) λ ) and L g (φ(w) λ ), respectively. Here M g (λ ) is the Verma module of g with highest weight λ and φ : W(λ) W is the isomorphism. Let k be an admissible number with denominator q, λ P r k. By Theorem 6.3 the block O g [λ] is equivalent to a block of the category O of g or L g containing an integrable representation. In particular the existence of a BGG resolution of an integrable representation of an affine Kac-Moody algebra [GL, RCW] implies the existence of a BGG resolution for L(λ): Theorem 6.4. Let k be an admissible number, λ P r k. complex B (λ) : d3 B (λ) d B 1 (λ) d1 B 0 (λ) d0 0 of the form B i (λ) = M(w λ), d i = w W(λ) l λ (w)=i λ), M(w λ)), such that H i (B (λ)) = Then there exists a d w,w, d w,w Hom g (M(w w,w W(λ) l λ (w)=i, w λ w L(λ) if i = 0, 0 otherwise.