Characters of the Negative Level Highest-Weight Modules for Affine Lie Algebras

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IMRNInternational Mathematics Research Notices 1994, No. 3

Characters of the Negative Level Highest-Weight Modules for Affine Lie Algebras

Masaki Kashiwara and Toshiyuki Tanisaki

0 Introduction

Our main result in this paper is the following theorem conjectured by G. Lusztig [L].

Letgbe an affine Lie algebra with Cartan subalgebrahand simple coroots{hi}i∈I. Forλ ∈ h^∗ letM(λ) (resp.L(λ)) be the Verma module (resp. the irreducible module) with highest weightλ.

Theorem. Forλ∈h^∗such that (λ+ρ)(hi)∈Z<0for anyi∈I, we have

chL

w(λ+ρ)−ρ

=

yw

(−1)^l(w)−^l(y)Py,w(1) chM

y(λ+ρ)−ρ

(0.1)

for anyw∈ W. HereWis the Weyl group,ρ ∈h^∗is such thatρ(hi)=1 for anyi∈I,lis the length function,is the Bruhat order,P_y,wis the Kazhdan-Lusztig polynomial, and ch denotes the character.

This type of character formula was first conjectured by Kazhdan and Lusztig [KL]

for finite-dimensional semisimple Lie algebras and was proved by Beilinson-Bernstein [BB] and Brylinski-Kashiwara [BK]. Then its generalization to symmetrizable Kac-Moody Lie algebras concerning the dominant integral weights λ was given by Kashiwara (-Tanisaki) [K2], [KT] and Casian [C1]. This paper is concerned with a different version in the case of the antidominant integral weights for affine Lie algebras.

The schemes of the proofs of those character formulas are all similar. The g- modules correspond toD-modules on the flag manifold, and theD-modules correspond to perverse sheaves by the Riemann-Hilbert correspondence. Since the perverse sheaf corresponding to a dual Verma module (resp. irreducible highest-weight module) is the zero (resp. minimal) extension of the constant sheaf on a Schubert cell, the proof is

Received 17 January 1994.

at Kyoto University Library on August 9, 2012http://imrn.oxfordjournals.org/Downloaded from

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152 Kashiwara and Tanisaki

reduced to the calculation of the local intersection cohomology groups of the Schubert varieties. This last step is now standard by the theory of Weil sheaves (cf. [BBD]) or the theory of Hodge modules (cf. [S]).

However, there are some differences between [K2], [KT] and our case. The natural setting in our case is of the rightD-modules supported on the finite-dimensional Schubert varieties, while in [K2], [KT] we used leftD-modules supported on the finite- codimensional Schubert varieties. The category of leftD-modules and the one of right D-modules are equivalent on finite-dimensional manifolds. The flag manifold in our case is infinite-dimensional, and those two categories are not equivalent. The leftD-modules behave well under the pull-back while the rightD-modules behave well under the push- forward. This is the reason why we use rightD-modules.

For a technical reason, we do not directly treat the rightD-modules on the flag manifold itself. Instead, considering the fact that the flag manifold is locally isomorphic to the projective limit of finite-dimensional smooth varieties, we use a “projective limit”

of rightD-modules on these finite-dimensional varieties as a substitute.

In this paper we will give descriptions of the category of those projective limits of right D-modules, and the functor from this category to the category of g-modules.

Our main result follows from several properties of this functor. Details of the proof will appear elsewhere.

The same result is claimed in the preprint of Casian [C2]. Our method is different, since a functor in the opposite direction is used in [C2].

1 The Kac-Moody Lie algebra

We recall basic facts concerning the Kac-Moody Lie algebra.

Lethbe a finite-dimensional vector space overC, and let{αi}i∈I,{hi}i∈Ibe linearly independent vectors ofh^∗ andhrespectively, such that ( hi, αj)i, j∈I is a symmetrizable generalized Cartan matrix. The Kac-Moody Lie algebra associated to (h,{αi},{hi}) is the Lie algebraggenerated by the vector spacehand the elementse_i,f_i(i∈I) satisfying the following fundamental relations:

[h, h]=0 forh, h∈h, (1.1)

[h, ei]=α_i(h)ei forh∈h, i∈I, (1.2)

[h, fi]= −αi(h)fi forh∈h, i∈I, (1.3)

[ei, f_j]=δ_{i j}h_i fori, j∈I, (1.4)

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Affine Lie Algebras 153

ad(ei)¹⁻^α^j^(hⁱ⁾(ej)=0 fori, j∈Iwithi= j, (1.5) ad(fi)¹⁻^α^j^(hⁱ⁾(fj)=0 fori, j∈Iwithi= j. (1.6) Fori∈I, lets_ibe the linear automorphism ofh^∗ given by

s_i(λ)=λ−λ(hi)αi forλ∈h^∗. (1.7)

The Weyl groupW is by definition the subgroup ofGL(h^∗) generated by {si}i∈I. ThenW is a Coxeter group with a canonical system of generators {s_i}i∈I. We denote the length function byland the Bruhat order ofWby.

Set

gα= {x∈g|[h, x]=α(h)xforh∈h} forα∈h^∗, (1.8)

∆=

α∈h^∗|gα= {0}

− {0}, (1.9)

∆⁺ =∆∩

i∈I

Z₀α_i, (1.10)

∆re=

i∈I

W(αi), ∆⁺_re=∆⁺∩∆re, (1.11)

and letn,n⁻,b,b⁻be the subalgebras ofggenerated by{e_i}i∈I,{f_i}i∈I,{e_i}i∈I∪h,{f_i}i∈I∪h, respectively. We then have

∆=∆⁺∪(−∆⁺), W(∆)=∆, (1.12)

g=n⁺⊕h⊕n⁻, (1.13)

n=

α∈∆+

gα, n⁻=

α∈∆+

g₋α, (1.14)

b=h⊕n⁺, b⁻=h⊕n⁻. (1.15)

Forλ∈h^∗letM(λ),N(λ) be theg-modules defined by

M(λ)=U(g)



h∈h

U(g)

h−λ(h) +

i∈I

U(g)ei



, (1.16)

N(λ)=U(g)



h∈h

U(g)

h−λ(h) +

i∈I

U(g)fi



, (1.17)

whereU(g) denotes the enveloping algebra ofg. LetM^∗(λ) be theg-module consisting of h-finite vectors in Hom_C

N(−λ),C

. The irreducibleg-moduleL(λ) with highest weightλis naturally isomorphic to the image of the unique nonzero homomorphismM(λ)→M^∗(λ).

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2 The flag manifold

We fix aZ-latticePofh^∗satisfying

α_i∈P, h_i, P ∈Z fori∈I. (2.1)

Forα=

i∈Im_iα_i∈∆⁺set|α| =

i∈Im_i, and let nk=

α∈∆⁺

|α|k

gα, n⁻_k =

α∈∆⁺

|α|k

g₋α fork∈Z0. (2.2)

Define group schemes by

T =Spec(C[P]), (2.3)

U=lim

←−k

exp(n/nk), (2.4)

U⁻=lim←−

k

exp(n⁻/n⁻_k), (2.5)

B=(the semidirect product ofT andU), (2.6)

B⁻=(the semidirect product ofT andU⁻). (2.7)

Here, for a finite-dimensional nilpotent Lie algebraa, exp(a) denotes the unipotent alge- braic group withaas its Lie algebra. We have natural isomorphisms

exp :

α∈∆+

gα−→U, exp :

α∈∆+

g₋α−→U⁻. (2.8)

In [K1] the first-named author constructed the flag variety of (g,h, P) as the quo- tient

X=G/B, (2.9)

whereGis a scheme with locally free right action ofB.

Letλ∈P. We denote the composite of the homomorphisms B−→T −→^λ Gm

byb→b^λ. Letπ: G→ Xbe the canonical morphism. An invertibleOX-moduleOX(λ) is

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defined by

Γ

V;O^X(λ)

=

ϕ∈Γ(π⁻¹V;O^G)|ϕ(gb)=ϕ(g)b⁻^λfor (g, b)∈π⁻¹V×B

(2.10)

for any open subsetVofX.

We have natural left actions of the group schemeB⁻and the braid groupWofW onX. LetY be aT-stable subset ofX. Forw∈W, the setwY depends only on the image w∈Wofwunder the canonical homomorphismW→W, and hence we simply denote it bywY.

LetB₀be the subgroup ofBgenerated by T and the elements of the form exp(x) withx∈gα,α∈∆⁺_re. Note thatB₀is not a scheme but an inductive limit of schemes. We have a natural left action ofB₀on Xcompatible with those ofB⁻ andW. Letx₀ ∈Xbe 1 modB. Thenx₀is a uniqueB₀-fixed point. We set

X^w=B⁻wx₀, X_w=B₀wx₀ forw∈W. (2.11)

Proposition 1 [K1]. (i)X^wis a locally closed subscheme ofXwith codimensionl(w), and ifgis not of finite type, it is isomorphic to the infinite-dimensional affine spaceA^∞.

(ii)X= w∈WX^w. (iii)X^w = ywX^y.

Moreover we have the following results.

Proposition 2. (i) Xw is a locally closed subscheme ofXisomorphic to the affine space A^l(w).

(ii)w∈WXwis naturally isomorphic to the “flag manifold” treated in [KP], [T], and others.

(iii)X_w = ywX_y.

3 D-modules

For a finite subsetFofWset

X^F=

w∈F

X^w, X_F=

w∈F

X_w. (3.1)

By Propositions 1 and 2, the setX^F(resp.XF) is open (resp. closed) if and only ifFsatisfies

w∈F, y∈W, yw⇒y∈F. (3.2)

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A subsetY(resp.Z) ofXis called an admissible open (resp. closed) subset ifY =X^F(resp.

Z=X_F) for a finite subsetFofWsatisfying (3.2).

LetZ be an admissible closed subset ofX. In this caseZis projective. Take an admissible open subsetY ofXcontainingZ. We can take suchY since we haveX^F ⊃X_F ifF satisfies (3.2). Fork0 setU⁻_k =exp(

α∈∆+

|α|k

g₋α)⊂U⁻. Ifkis sufficiently large, then U⁻_k acts onY locally freely, and hence the quotientU⁻_k\Y is a finite-dimensional smooth variety. We do not know ifU⁻_k\Y is separated or not. Ifk is large enough, the natural morphismZ→U⁻_k\Yis a closed immersion. Fix suchZ,Y, andk. Forlkset

Yl=U⁻_l\Y (3.3)

and let

πl:Y−→Yl (3.4)

p_l:Y_l₊₁−→Y_l (3.5)

il:Z−→Yl (3.6)

be the natural morphisms. Forλ ∈ P, we can naturally define an invertibleOY_l-module OY_l(λ) satisfying

π^∗_lOY_l(λ)∼=OX(λ)|Y and (3.7)

p^∗_lOY_l(λ)∼=OY_l₊₁(λ). (3.8)

LetDY_l be the sheaf of differential operators onY_l, and set DY_l₊₁→Y_l =OY_l₊₁⊗_p−1

l O_Yl p⁻_l¹DY_l, (3.9)

DY_l(λ)=OY_l(−λ)⊗_O_YlDY_l⊗_O_YlOY_l(λ), (3.10) DY_l₊₁→Y_l(λ)=OY_l₊₁(−λ)⊗O_Yl₊₁ DY_l₊₁→Y_l⊗_p−1

l O_Ylp⁻_l¹OY_l(λ). (3.11) ThenDY_l(λ) is a ring acting onOY_l(−λ) from the left, and on Ω_Y_l⊗OY_l(λ) from the right.

HereΩ_Y_l is the sheaf of differential forms of degree dimY_l. We have a natural (DY_l₊₁(λ), p⁻_l¹DY_l(λ))-bimodule structure onDY_l₊₁→Y_l(λ).

LetHlbe the abelian category of right holonomicDY_l(λ)-modules supported inZ.

ForM∈Hlwe defineM^∗ ∈Hlby M^∗ =Ω_Y_l⊗OY_l(2λ)⊗Ext^dimY_D ^l

Yl(λ)

M,DY_l(λ)

. (3.12)

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This defines a contravariant exact functor∗:Hl→Hlsuch that∗∗ =id. ForM∈Hl+1we define

p_lM∈Hlby

p_l

M=(pl)_∗

M⊗D_Yl₊₁₍λ)DY_l₊₁→Y_l(λ)

. (3.13)

Sincei_l₊₁(Z)→^∼ i_l(Z),

p_l induces an equivalenceHl+1→∼ Hl.

We define an abelian categoryH = H(Z, λ, Y, k) as follows. An object is a family M=(Ml)lk∈

lkOb(Hl) together with isomorphisms r_l:

p_l

Ml+1−→∼ Ml, (3.14)

and morphisms are given by Hom_H(M,N)

=



(ϕl)∈

lk

Hom_H_l(Ml,Nl)ϕl◦rl=rl◦

p_l

ϕl+1

forlk



. (3.15)

Note thatHis equivalent toHl. The duality functor∗:H→His defined by

M^∗ =(M^∗l)lk. (3.16)

Since

p_l commutes with the duality, this is well defined. It is a contravariant exact functor such that∗∗ =id. Note that the categoryH(Z, λ, Y, k) does not depend onYandk.

Moreover forλ,λ∈P,H(Z, λ, Y, k) andH(Z, λ, Y, k) are equivalent byM^l→M^l⊗O^Yl(λ−λ).

For M ∈ H the homomorphism OY_l₊₁ → DY_l₊₁→Y_l(λ) induces a homomorphism pl∗Ml+1→Ml, which gives a homomorphism

Hⁱ(Yl+1;Ml+1)−→Hⁱ(Yl;Ml) fori0. (3.17) Set

Hⁱ(Y;M)=lim←−

l

Hⁱ(Yl;Ml) fori0. (3.18)

Proposition 3. ForM∈H,Hⁱ(Y;M) carries a natural structure of a g-module.

There is no homomorphism fromgtoDY_l(λ) because there is no action ofgonY_l. However, we can construct a section ofDY_l₊_m→Y_l(λ) corresponding toA∈gifm 0. This induces an action ofgonHⁱ(Y;M).

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Forw∈Wsuch thatX_w⊂Z, we defineBw,Mw,Lw∈Ob(H) by (Bw)l=Hi^codim_l(Xw)ⁱ^l^(X^w⁾

Ω_Y_l⊗O_YlOY_l(λ)

, (3.19)

Mw =B^∗w, (3.20)

Lw =the image of the natural morphismMw→Bw. (3.21)

ThenLw is a simple object ofHsatisfyingL^∗w∼=Lw.

Let H0 = H0(Z, λ, Y, k) be the full subcategory ofH consisting ofM ∈ Hwhose composition factors are isomorphic toLw for somew∈WsatisfyingXw ⊂Z.

4 The main result

We fixρ∈h^∗satisfying

hi, ρ =1 for anyi∈I. (4.1)

Define a shifted action of the Weyl groupWonPby

w◦λ=w(λ+ρ)−ρ. (4.2)

Assume thatλ∈Psatisfies the following:

hi, λ<−1 for anyi∈I; (4.3)

the Verma moduleM(λ) is an irreducibleg-module. (4.4) We keep the notation of Section 3. Let F ⊂ W be such thatZ = XF. Then we have the following theorem.

Theorem.

(i) Hⁿ(Y;M)=0 for anyM∈H0andn >0;

(ii) H⁰(Y; ∗) defines an exact functor fromH0to the category ofU(g)-modules;

(iii) H⁰(Y;Bw)= ˆM^∗(w◦λ) for anyw∈F;

(iv) H⁰(Y;M^w)= ˆM(w◦λ) for anyw∈F;

(v) H⁰(Y;Lw)= ˆL(w◦λ) for anyw∈F.

HereMˆ^∗(w◦λ) is the completion ofM^∗(w◦λ), etc.

By [KK] the condition (4.4) is satisfied ifgis an affine Lie algebra and ifλsatisfies (4.3). The conjecture of Lusztig is easily derived from this theorem along with the standard arguments.

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The proof of this theorem is accomplished by showing the following statements by induction on the dimension ofZ.

ForM∈H0,µ∈P, andn∈Z, the weight spaceHⁿ(Yl;Ml)µ is

constant forl k. (4.5)

This implies that Hⁿ(Y;M) is isomorphic to

µ

lim←−

l

Hⁿ(Yl;Ml)µ

. We denote

⊕µ

lim←−

l

Hⁿ(Yl;Ml)

µbyH¯ⁿ(Y;M). HenceH¯ⁿ(Y;M) is the set ofh-finite vectors ofHⁿ(Y;M).

H¯ⁿ(Y;M)=0 forM∈H0andn >0. (4.6)

Hom_H₀(M,N)#Homg

H¯⁰(Y;M),H¯⁰(Y;N)

forM,N∈H0. (4.7)

H¯⁰(Y;M)⊗D(λ)=M forM∈H0. (4.8)

LetM∈H0and letNbe aU(g)-submodule ofH¯⁰(Y;M).

IfN⊗D(λ)=M, thenN= ¯H⁰(Y;M). (4.9)

H¯⁰(Y;Bw)=M^∗(w◦λ) forw∈F. (4.10)

H¯⁰(Y;Mw)=M(w◦λ) forw∈F. (4.11)

H¯⁰(Y;Lw)=L(w◦λ) forw∈F. (4.12)

In (4.8) and (4.9), for a U(g)-submodule N of H¯⁰(Y;M), N⊗D(λ) denotes the subobject of M“generated by”N. We remark that we do not know how to construct the functor M→M⊗D(λ) from a category ofg-modules toH0, but there is no problem in constructing N⊗D(λ)⊂MforN⊂ ¯H⁰(Y;M).

References

[BB] A. Beilinson and J. Bernstein,Localisation deg-modules, C. R. Acad. Sci. Paris S ´er. I Math.

292(1981), 15–18.

[BBD] A. Beilinson, J. Bernstein, and P. Deligne, “Faisceaux pervers” inAnalyse et topologie sur les espaces singuliers, Ast ´erisque100, Soc. Math. France, Paris, 1983, 5–171.

[BK] J.-L. Brylinski and M. Kashiwara, Kazhdan-Lusztig conjecture and holonomic systems, Invent. Math.64(1981), 387–410.

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[C1] L. Casian,Kazhdan-Lusztig multiplicity formulas for Kac-Moody algebras, C. R. Acad. Sci.

Paris S ´er. I Math.310(1990), 333–337.

[C2] ,Kazhdan-Lusztig conjecture in the negative level case (Kac-Moody algebras of affine type), preprint.

[KK] V. Kac and D. Kazhdan,Structure of representations with highest weight of infinite dimen- sional Lie algebras, Adv. Math.34(1979), 97–108.

[KP] V. Kac and D. Peterson,Infinite flag varieties and conjugacy theorems, Proc. Nat. Acad. Sci.

USA80(1982), 1778–1782.

[K1] M. Kashiwara, “The flag manifold of Kac-Moody Lie algebra” inAlgebraic Analysis, Geometry, and Number Theory, Johns Hopkins Univ. Press, Baltimore, 1989.

[K2] , “Kazhdan-Lusztig conjecture for symmetrizable Kac-Moody Lie algebras” in The Grothendieck Festschrift,Vol. 2, Progr. Math.87, Birkh ¨auser, Boston, 1991, 407–433.

[KT] M. Kashiwara and T. Tanisaki, “Kazhdan-Lusztig conjecture for symmetrizable Kac-Moody Lie algebras II” inOperator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory, Progr. Math.92, Birkh ¨auser, Boston, 1990, 159–195.

[KL] D. Kazhdan and G. Lusztig,Representations of Coxeter groups and Hecke algebras, Invent.

Math.53(1979), 165–184.

[L] G. Lusztig,On quantum groups, J. Algebra131(1990), 466–475.

[S] M. Saito,Mixed Hodge modules, Publ. Res. Inst. Math. Sci.26(1989), 221–333.

[T] J. Tits, “Groups and group functors attached to Kac-Moody data” in Arbeitstagung Bonn, 1984, Lecture Notes in Math.1111, Springer-Verlag, Berlin, 1985.

Kashiwara: Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606, Japan Tanisaki: Department of Mathematics, Hiroshima University, Higashi-Hiroshima 724, Japan