Kazhdan–Lusztig Polynomials and Character Formulae for The

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Advances in Mathematics 182 (2004) 28–77 http://www.elsevier.com/locate/aim

Kazhdan–Lusztig polynomials and character formulae for the Lie superalgebra qðnÞ$ Jonathan Brundan Department of Mathematics, University of Oregon, Eugene, OR 97403, USA

Received 8 August 2002; accepted 19 November2002

Communicated by Pavel Etingof

Abstract

We compute the characters of the ﬁnite dimensional irreducible representations of the Lie superalgebra qðnÞ and formulate a precise conjecture for the other irreducible characters in category O: The guiding principle is that the representation theory of qðnÞ should be described by Lusztig’s canonical basis for‘‘tensorspace’’ overthe quantum groupof type bN: r 2003 Elsevier Science (USA). All rights reserved.

1. Introduction

The problem of computing the characters of the ﬁnite-dimensional irreducible representations for the classical Lie superalgebras was posed originally by Kac in 1977 [K1,K2,K3]. Forthe family qðnÞ; various special cases were treated by Sergeev [S2] (polynomial representations) and Penkov [P1,P3] (typical then generic representations), culminating in the complete solution of Kac’ problem for qðnÞ in the work of Penkov and Serganova [PS2,PS3] in 1996. In this article, we will explain a different approach to computing the characters of the irreducible ‘‘integrable’’ representations of qðnÞ; i.e., the representations that lift to the supergroup QðnÞ: The strategy followed runs parallel to our recent work [B3] on representations of GLðmjnÞ; and is in keeping with the Lascoux–Leclerc–Thibon philosophy [LLT].We ﬁrst study the canonical basis of the representation ^ n Fn :¼ V;

$Work partially supported by the NSF (Grant DMS-0139019). E-mail address: [email protected].

J. Brundan / Advances in Mathematics 182 (2004) 28–77 29 where V denotes the natural representation of the quantized enveloping algebra UqðbNÞ: This provides a natural Lie theoretic framework for the combinatorics associated to the representation theory of QðnÞ: The idea that bN should be relevant here is already apparent from [LT,BK1]. Our main theorem shows that the transition matrix between the canonical basis and the natural monomial basis of Fn at q ¼ 1is transpose to the transition matrix between the bases for the Grothendieck group of ﬁnite-dimensional representations of QðnÞ given by certain Euler characteristics and by the irreducible representations. In order to deﬁne the canonical basis of Fn; we start by considering the tensor space n Tn :¼ # V:

Work of Lusztig [L2, Chapter27] shows how to construct a canonical basis for Tn: We then pass from there to the space Fn; which we realize as a quotient of Tn following [JMO]. The entries of the transition matrix between the canonical basis and the natural monomial basis of Tn should be viewed as the combinatorial analogues for the Lie superalgebra qðnÞ of the Kazhdan–Lusztig polynomials of [KL]. We conjecture that these polynomials evaluated at q ¼ 1 compute the composition multiplicities of the Verma modules in the analogue of category O for qðnÞ; see Section 4.8 fora precise statement. n We now state our main result precisely. Let Zþ denote the set of all tuples l ¼ n ðl1; y; lnÞAZ such that l1X?Xln and moreover lr ¼ lrþ1 implies lr ¼ 0 foreach y A n y r ¼ 1; ; n À 1: For l Zþ; let zðlÞ denote the numberof lr ðr ¼ 1; ; nÞ that equal zero. Also let dr denote the n-tuple with rth entry equal to 1 and all other entries A n equal to zero. Given l Zþ; there is an irreducible representation LðlÞ of QðnÞ of highest weight l; unique up to isomorphism. Let Ll denote the character of LðlÞ; giving us a canonical basis fLlg A n for the character ring of QðnÞ; see Section 4.1. l Zþ There is another basis denoted fEmg A n which arises naturally from certain Euler m Zþ characteristics, see Section 4.2. We can write X Em ¼ dm;lLl A n l Zþ forcoefﬁcients dm;lAZ; where dm;m ¼ 1 and dm;l ¼ 0 for l4/ m in the dominance ordering. The Em’s are explicitly known: they are multiples of the symmetric functions known as Schur’sP-functions. So the problem of computing Ll foreach A n l Zþ is equivalent to determining the decomposition numbers dm;l foreach A n l; m Zþ:

A n Main Theorem. Let l ¼ðl1; y; lnÞ Zþ: Choose p to be maximal such that there ? ? y exist 1pr1o orpospo os1pn with lrq þ lsq ¼ 0 for all q ¼ 1; ; p: Let I0 ¼fjl1j; y; jlnjg: For q ¼ 1; y; p; deﬁne Iq and kq inductively according to the ARTICLE IN PRESS

30 J. Brundan / Advances in Mathematics 182 (2004) 28–77 following rules: e (1) if lrq 40; let kq be the smallest positive integer with lrq þ kq IqÀ1; and set

Iq ¼ IqÀ1,flrq þ kqg; 0 0 e 0 (2) if lrq ¼ 0; let kq and kq be the smallest positive integers with kq; kq IqÀ1; kqokq 0 0 if zðlÞ is even and kq4kq if zðlÞ is odd, and set Iq ¼ IqÀ1,fkq; kqg: y A p n Finally, for each y ¼ðy1P; ; ypÞ f0; 1g ; let RyðlÞ denote the unique element of Zþ p that is conjugate to l þ q¼1 yqkqðdrq À dsq Þ: Then, ( ðzðlÞÀzðmÞÞ=2 p 2 if m ¼ RyðlÞ for some y ¼ðy1; y; ypÞAf0; 1g ; dm;l ¼ 0 otherwise:

The remainder of the article is organized as follows. In Section 2, we introduce the n quantum group of type bN; and construct the canonical basis of the tensor space T : In Section 3, we pass from Tn to the quotient Fn; and study its canonical basis. This time it turns out to be quite easy to compute explicitly. Finally in Section 4, we prove the characterformulae. Note thereis one difﬁcult place in ourproof when we need to appeal to the existence of certain homomorphisms between Verma modules, see Lemma 4.36. For this we appeal to the earlier work of Penkov and Serganova [PS2, Proposition 2.1], which in turn relies upon a special case of Penkov’s generic character formula [P3, Corollary 2.2]. It would be nice to ﬁnd an independent proof of this fact.

Notation. Generally speaking, indices i; j; k will run over the natural numbers N ¼ f0; 1; 2; yg; indices a; b; c will run over all of Z; and indices r; s; t will run over the set f1; 2; y; ng where n is a ﬁxed positive integer.

2. Tensor algebra

2.1. Quantum group of type bN

Let P be the free abelian group on basis e1; e2; e3; y; equipped with a symmetric bilinearform ð:; :Þ deﬁned by

ðei; ejÞ¼2di;j forall i; jX1: Inside P; we have the root system f7ei; 7ei7ej j 1piojg of type bN: We use the following labeling forthe Dynkin diagram:

We take the simple roots a0; a1; y deﬁned from

a0 ¼Àe1; ai ¼ ei À eiþ1 ðiX1Þ: ARTICLE IN PRESS

J. Brundan / Advances in Mathematics 182 (2004) 28–77 31

This choice induces a dominance ordering p on P; namely, bpg if g À b is a sum of simple roots. The Cartan matrix A ¼ðai;jÞi;jX0 is defined by ai;j ¼ 2ðai; ajÞ=ðai; aiÞ: We will work over the ground field QðqÞ; where q is an indeterminate. Let qi ¼ ðai;aiÞ=2 2 q ; i.e. q0 ¼ q and qi ¼ q for i40: Define the quantum integer

n Àn qi À qi ½ni ¼ À1 qi À qi

! and the associated quantum factorial ½ni ¼½ni½n À 1iy½1i: There is a ﬁeld automorphism À : QðqÞ-QðqÞ with q% ¼ qÀ1: We will call an additive map f : V-W between QðqÞ-vectorspaces antilinear if f ðcvÞ¼cf% ðvÞ forall cAQðqÞ; vAV: The quantum group U ¼ UqðbNÞ is the QðqÞ-algebra generated by elements Ei; Fi; Ki ðiX0Þ subject to the relations

À1 À1 KiKi ¼ Ki Ki ¼ 1; KiKj ¼ KjKi;

À1 ðai;aj Þ À1 Àðai;aj Þ KiEjKi ¼ q Ej; KiFjKi ¼ q Fj;

À1 Ki À Ki EiFj À FjEi ¼ di;j À1 ; qi À qi 1XÀai;j 1XÀai;j k ðkÞ ð1Àai;j ÀkÞ k ðkÞ ð1Àai;j ÀkÞ a ðÀ1Þ Ei EjEi ¼ ðÀ1Þ Fi FjFi ¼ 0 ði jÞ; k¼0 k¼0

ðrÞ r ! ðrÞ r ! where Ei :¼ Ei =½ri and Fi :¼ Fi =½ri: Also let "# Yr 1Às À1 sÀ1 Ki K q À K q ¼ i i i i : qs qÀs r s¼1 i À i foreach rX1: We regard U as a Hopf algebra with comultiplication D : U-U#U deﬁned on generators by

# # À1 DðEiÞ¼1 Ei þ Ei Ki ;

DðFiÞ¼Ki#Fi þ Fi#1;

DðKiÞ¼Ki#Ki:

This is the comultiplication used by Kashiwara [Ka2], which is different from the one in Lusztig’s book [L2]. ARTICLE IN PRESS

32 J. Brundan / Advances in Mathematics 182 (2004) 28–77

Let us introduce various (anti)automorphisms of U: First, we have the bar involution À : U-U; the unique antilinear algebra automorphism such that

À1 Ei ¼ Ei; Fi ¼ Fi; Ki ¼ Ki : ð2:1Þ

We will also need the linearalgebraantiautomorphisms s; t : U-U and the linear algebra automorphism o : U-U deﬁned by

À1 sðEiÞ¼Ei; sðFiÞ¼Fi; sðKiÞ¼Ki ; ð2:2Þ

À1 À1 tðEiÞ¼qiFiKi ; tðFiÞ¼qi KiEi; tðKiÞ¼Ki; ð2:3Þ

À1 oðEiÞ¼Fi; oðFiÞ¼Ei; oðKiÞ¼Ki : ð2:4Þ

Lemma 2.5. The maps t and À3s are coalgebra automorphisms, i.e. we have that DðjðxÞÞ ¼ ðj#jÞðDðxÞÞ for all xAU and either j ¼ t or j ¼À3s: The map o is a coalgebra antiautomorphism, i.e. DðoðxÞÞ ¼ Pððo#oÞðDðxÞÞÞ where P is the twist x#y/y#x:

À1 We will occasionally need Lusztig’s Z½q; q -form UZ½q;qÀ1 for U: We recall from À1 [L1, Section 6] thatÂÃ this is the Z½q; q -subalgebra of U generated by the elements r r 7 ð Þ ð Þ 1 Ki X X Ei ; Fi ; Ki and r forall i 0; r 1: It inherits from U the structure of a Hopf algebra over Z½q; qÀ1:

2.2. Tensor space

The natural representation of U is the QðqÞ-vectorspace V on basis fvagaAZ with action deﬁned by

À1 E0va ¼ da;0ðq þ q ÞvÀ1 þ da;1v0; Eiva ¼ da;iþ1vi þ da;ÀivÀiÀ1;

À1 F0va ¼ da;0ðq þ q Þv1 þ da;À1v0; Fiva ¼ da;iviþ1 þ da;ÀiÀ1vÀi;

2da;À1À2da;1 2da;iÀ2da;iþ1þ2da;ÀiÀ1À2da;Ài K0va ¼ q va; Kiva ¼ q va;

" n forall aAZ and iX1; see forexample [J2, Section 5A.2]. Let T ¼ nX0 T be the tensoralgebraof V; so Tn ¼ V#?#V (n times) viewed as a U-module in the natural way. n Let Z denote the set of all n-tuples l ¼ðl1; y; lnÞ of integers. The symmetric n group Sn acts on Z by the rule wl ¼ðlwÀ11; y; lwÀ1nÞ: We will always denote the ARTICLE IN PRESS

J. Brundan / Advances in Mathematics 182 (2004) 28–77 33

n longest element of Sn by w0; so w0l ¼ðln; y; l1Þ: Given lAZ ; let #?# A n Nl ¼ vl1 vln T : ð2:6Þ n A n The vectors fNlglAZn obviously give a basis for T : For l Z ; let zðlÞ denote the n numberof lr ðr ¼ 1; y; nÞ that equal zero. We get another basis fMlglAZn for T by deﬁning

À1 ÀzðlÞ Ml ¼ðq þ q Þ Nl ð2:7Þ foreach lAZn: Let ð:; :Þ be the symmetric bilinear form on Tn such that

zðlÞ ðMl; NmÞ¼q dl;m ð2:8Þ forall l; mAZn: Also deﬁne an antilinearautomorphism s : Tn-Tn and a linear automorphism o : Tn-Tn by

ÀzðlÞ sðNlÞ¼q NÀl; ð2:9Þ

oðNlÞ¼NÀw0l ð2:10Þ forall lAZn: The following lemma is checked by reducing to the case n ¼ 1 using Lemma 2.5.

Lemma 2.11. The following hold for all xAU and u; vATn: (i) sðxvÞ¼tðsðxÞÞsðvÞ; (ii) ðxu; vÞ¼ðu; tðxÞvÞ; (iii) oðxuÞ¼oðxÞoðuÞ:

2.3. Bruhat ordering

ða ;gÞ Recall that a vector v in a U-module is said to be of weight gAP if Kiv ¼ q i v for each iX0: The weight of the basis vector vi of V is ei; where we write e0 ¼ 0 and eÀi ¼Àei: Hence the vectors Nl and Ml have weight ? A wtðlÞ :¼ el1 þ þ eln P: ð2:12Þ

The degree of atypicality of lAZn is deﬁned by 1 X #l :¼ n À jðwtðlÞ; eiÞj; ð2:13Þ 2 iX1 recalling that ðei; ejÞ¼2di;j: Forexample, if n ¼ 7 and l ¼ð1; 0; 4; 1; 0; À1; À2Þ; then n wtðlÞ¼e1 À e2 þ e4 and #l ¼ 4: We will say that lAZ is typical if #lp1: ARTICLE IN PRESS

34 J. Brundan / Advances in Mathematics 182 (2004) 28–77

Equivalently, l is typical if lr þ lsa0 forall 1 prospn: We record the following obvious but useful lemma.

n Lemma 2.14. If l; mAZ are typical and wtðlÞ¼wtðmÞ; then there exists wASn such that l ¼ wm:

We next deﬁne an important partial ordering % on Zn; which we call the Bruhat ordering. Recall that p denotes the dominance ordering on the weight lattice P: For each r ¼ 1; y; n; let

? A wtrðlÞ :¼ elr þ þ eln P;

n and declare for l; mAZ that l%m if wtrðlÞpwtrðmÞ foreach r ¼ 1; y; n with equality holding when r ¼ 1: P P % r r y Lemma 2.15. If l m then s¼1 lsp s¼1 ms for each r ¼ 1; ; n with equality holding when r ¼ n:

Proof. For lAZn; r ¼ 1; y; n and iX1; introduce the shorthand

#ðl; i; rÞ :¼ #fs ¼ 1; y; r j lsXigÀ#fs ¼ 1; y; r j lsp À ig:

TheP dominanceP ordering p on P satisﬁes bpg if and only if jXi ðb; ejÞX jXi ðg; ejÞ forevery i ¼ 1; 2; y : Using this it is easy to expand the above deﬁnition of the Bruhat ordering to show that l%m if and only if #ðl; i; rÞp#ðm; i; rÞ foreach iX1andr ¼ 1; yP; n with equalityP holding when r ¼ n: r The lemma follows at once on observing that s¼1 ls ¼ iX1 #ðl; i; rÞ: &

Remark 2.16. Here is an equivalent description of the Bruhat ordering %: Since I have been unable to ﬁnd a proof of this observation, it will not be used in the remainder of the article. Write lkm if one of the following holds:

(1) forsome 1 prospn such that lr4ls; we have that mr ¼ ls; ms ¼ lr and mt ¼ lt forall tar; s; (2) forsome 1 prospn such that lr þ ls ¼ 0; we have that mr ¼ lr À 1; a ms ¼ ls þ 1 and mt ¼ lt forall t r; s: n Then, it appears to be the case that m%l if and only if there exist n1; y; nkAZ such that l ¼ n1kn2kyknk ¼ m:

2.4. Bar involution

n cn n We next deﬁne a barinvolution on T (actually on some completion T of T ) compatible with the barinvolution on U; following [L2, Section 27.3]. To begin with, ARTICLE IN PRESS

J. Brundan / Advances in Mathematics 182 (2004) 28–77 35 let us recall the deﬁnition of Lusztig’s quasi-R-matrix Y; translated suitably since we are working with a different comultiplication. Write U ¼ UÀU0Uþ ¼ UþU0UÀ forthe usual triangulardecompositions of U; so À 0 71 þ U is generated by all Fi; U is generated by all Ki and U is generated by all Ei: þ À þ À Fora weight nX0; let Un (resp. Un ) denote the subspace of U (resp. U ) spanned y y ? by all monomials Ei1 Eir (resp. Fi1 Fir ) with ai1 þ þ air ¼ n; so

þ " þ À " À U ¼ Un ; U ¼ Un : nX0 nX0 P P # 4 Deﬁne tr n :¼ iX0 ai if n ¼ iX0 aiai: Let ðU UÞ denote the completion of the # # vectorspace U U with respect to the descending ﬁltration ððU UÞd ÞdAN where X # À 0 þ# # þ 0 À ðU UÞd ¼ ðU U Un U þ U U U Un Þ: tr nXd

Exactly as in [L2, Section 4.1.1], we embed U#U into ðU#UÞ4 in the obvious way, then extend the QðqÞ-algebra structure on U#U to ðU#UÞ4 by continuity. The barinvolution on U#U is deﬁned by x#y :¼ x%#y%; and also extends by continuity to ðU#UÞ4: Finally, the antiautomorphism s#s : U#U-U#U and the automorphism P3ðo#oÞ : U#U-U#U (where P is the twist x#y/y#x) extend to the completion too.

There is a unique family of elements A þ# À such that and LemmaP 2.17. Yn Un Un Y0 ¼ 1 A # 4 % A # 4 Y :¼ n Yn ðU UÞ satisﬁes DðuÞY ¼ YDðuÞ for all u U (identity in ðU UÞ ). # Moreover, each Yn belongs to UZ½q;qÀ1 Z½q;qÀ1UZ½q;qÀ1:

Proof. The ﬁrst part of the lemma is [L2, Theorem 4.1.2(a)]. The second part about integrality follows using [L2, Corollary 24.1.6]. Actually [L2, Corollary 24.1.6] applies only to ﬁnite type root systems but one can pass from bn to bN by a limiting argument. &

Lemma 2.18. The following equalities hold in ðU#UÞ4: (i) YY% ¼ YY% ¼ 1#1; (ii) ðs#sÞðYÞ¼Y; (iii) ðs#sÞðY% Þ¼Y% ; (iv) ðP3ðo#oÞÞðYÞ¼Y:

Proof. (i) This is [L2, Corollary 4.1.3]. (ii) Recall by Lemma 2.5 that the map À3s ¼ s3À is a coalgebra automorphism. Using this one checks that

DðuÞ¼ðs#sÞðDðsðu%ÞÞÞ; Dðu%Þ¼ðs#sÞðDðsðuÞÞÞ: ARTICLE IN PRESS

36 J. Brundan / Advances in Mathematics 182 (2004) 28–77

Now apply the antiautomorphism s#s to the equality DðsðuÞÞY ¼ YDðsðu%ÞÞ to get that ðs#sÞðYÞDðu%Þ¼DðuÞðs#sÞðYÞ: Hence, ðs#sÞðYÞ¼Y by the uniqueness in Lemma 2.17. (iii) Combine (i) and (ii). (iv) Follows easily from the uniqueness and Lemma 2.5. &

cn n Let T denote the completion of the vectorspace T with respect to the descending filtration ð n Þ ; where n is the subspace of n spanned by fN g for P Td dAZ Td T l A n n X n cn l Z with r¼1 rlr d: We embed T into T in the natural way. We note using cn Lemma 2.15 that T contains all vectors of the form Nl þ ðÃÞ where ðÃÞ is an infinite n linearcombination of Nm’s with m!l: The action of U on T extends by continuity cn n n to T ; as does the map o : T -T : cn Now we are ready to inductively define the bar involution on T : It will turn out to satisfy the following properties:

cn cn (1) À : T -T is a continuous, antilinearinvolution; cn (2) xv ¼ x%v% forall xA ; vA ; P U T P (3) A c À1 A c À1 A n Nl Nl þ m!l Z½q; q Nm and Ml Ml þ m!lZ½q; q Mm forall l Z ; cn (4) oðvÞ¼oðv%Þ forall vAT :

If n ¼ 1; the barinvolution is deﬁned by setting va ¼ va foreach aAZ; then extending by antilinearity then continuity. Properties (1)–(4) in this case are easy to check directly. Now suppose that n41; write n ¼ n1 þ n2 forsome n1; n2X1; and assume cn cn we have already constructed bar involutions on T 1 and T 2 satisfying properties cn (1)–(4). Because of the way the completion T is deﬁned, multiplication by Y gives a linearmap

cn1 #cn2 -cn Yn1;n2 : T T T :

A n1 A n2 # %# % Given v T ; w T ; let v w :¼ Yn1;n2 ðv wÞ; deﬁning an antilinearmap n cn À : T -T : The explicit form of Y from Lemma 2.17 combined with the inductive hypothesis implies that

Nl ¼ Nl þða possibly infinite QðqÞ-linearcombination of Nm’s with m!lÞ;

Ml ¼ Ml þða possibly infinite QðqÞ-linearcombination of Mm’s with m!lÞ: # Each Yn belongs to UZ½q;qÀ1 Z½q;qÀ1UZ½q;qÀ1 by Lemma 2.17, and UZ½q;qÀ1 leaves the À1 cn Z½q; q -lattices in T generated by either the Nl’s orthe Ml’s invariant. Hence, the coefﬁcients actually all lie in Z½q; qÀ1; so property (3) holds. Now property (3) immediately implies that baris continuous, so it extends uniquely to a continuous cn cn antilinearmap À : T -T still satisfying (3). The argument in [L2, Lemma 24.1.2] ARTICLE IN PRESS

J. Brundan / Advances in Mathematics 182 (2004) 28–77 37 shows that property (2) is satisfied. Lemma 2.18(i) gives that bar is an involution, hence property (1) holds, while property (4) follows from Lemma 2.18(iv). Note finally that as in [L2, Section 27.3.6], the definition is independent of the initial choices of n1; n2:

c2 Example 2.19. The barinvolution on T is uniquely determined by the following formulae:

va#vb ¼ va#vb ðapb; a þ ba0Þ;

2 À2 va#vb ¼ va#vb þðq À q Þvb#va ða4b; a þ ba0Þ; X À1 2 À2 2 bþ1 v0#v0 ¼ v0#v0 þ ðq þ q Þðq À q ÞðÀq Þ vb#vÀb; bo0 X 2 À2 2 aþ1Àb vÀa#va ¼ vÀa#va þ ðq À q ÞðÀq Þ vÀb#vb ðaX1Þ; b4a

2 2 À2 va#vÀa ¼ va#vÀa þ q ðq À q ÞvÀa#va ðaX1Þ X 2 À2 2 bþ1Àa þ ðq À q ÞðÀq Þ vb#vÀb 0oboa À1 2 1Àa þðq À q ÞðÀq Þ v0#v0 X 2 2 À2 2 bþ1Àa þ q ðq À q ÞðÀq Þ vb#vÀb: bo0

cn Now that the barinvolution has been deﬁned on T ; we can deﬁne a new bilinear cn form /:; :S on T by setting

/u; vS ¼ðu; sðv%ÞÞ ð2:20Þ

cn forall u; vAT ; where ð:; :Þ and s are as in Lemma 2.11. Note that this makes sense even though the map s and the form ð:; :Þ are not deﬁned on the completion.

Lemma 2.21. /:; :S is a symmetric bilinear form with /xu; vS ¼ /u; sðxÞvS for all cn xAU; u; vAT :

Proof. Forthe second partof the lemma, we calculate using Lemma 2.11 to get that

/xu; vS ¼ðxu; sðv%ÞÞ ¼ ðu; tðxÞsðv%ÞÞ ¼ ðu; tðsðsðxÞÞÞsðv%ÞÞ

¼ðu; sðsðxÞv%ÞÞ ¼ ðu; sðsðxÞvÞÞ ¼ /u; sðxÞvS: ARTICLE IN PRESS

38 J. Brundan / Advances in Mathematics 182 (2004) 28–77

Now let us show by induction on n that /:; :S is a symmetric bilinear form, this being obvious in case n ¼ 1: For n4P1; write n ¼ n1 þ n2 for n1; n2X1: Take # # A n1 # n2 # A # 4 u1 u2; v1 v2 T T : Write Y ¼ iAI xi yi ðU UÞ : Recall by Lemma 2.18(iii) that X X sðxiÞ#sðyiÞ¼ xi#yi: iAI iAI

Combining this with the inductive hypothesis, we calculate from the deﬁnition of the cn barinvolution on T : X / # # S # # # # u1 u2; v1 v2 ¼ðu1 u2; sðYn1;n2 ðv1 v2ÞÞÞ ¼ ðu1 u2; sðxiv1 yiv2ÞÞ X iAXI ¼ ðu1; sðxiv1ÞÞðu2; sðyiv2ÞÞ ¼ /u1; xiv1S/u2; yiv2S XiAI XiAI ¼ /xiv1; u1S/yiv2; u2S ¼ /v1; sðxiÞu1S/v2; sðyiÞu2S XiAI iAI ¼ /v1; xiu1S/v2; yiu2S ¼ /v1#v2; u1#u2S: iAI

Hence /:; :S is symmetric. &

cn 2.5. Canonical basis of T

cn Now that we have constructed the bar involution on T satisfying properties (1)– (4) above, we get the following theorem by general principles, cf. the proof of [B3, Theorem 2.17].

cn Theorem 2.22. There exist unique topological bases fTlglAZn and fLlglAZn for T such that:

(i) T ¼ T and L ¼ L ; l l P l l P (ii) A c A c À1 À1 Tl Nl þ mAZn qZ½qNm and Ll Ml þ mAZn q Z½q Mm; P n c for all lAZ : Actually, we have that TlANl þ qZ½qNm and that LlAMl þ P m!l c À1 À1 m!l q Z½q Mm: Also, oðTlÞ¼TÀw0l and oðLlÞ¼LÀw0l:

Example 2.23. Suppose that n ¼ 2: Using Example 2.19, one checks: a Tða;bÞ ¼ Nða;bÞ ðapb; a þ b 0Þ; 2 a Tða;bÞ ¼ Nða;bÞ þ q Nðb;aÞ ða4b; a þ b 0Þ; ARTICLE IN PRESS

J. Brundan / Advances in Mathematics 182 (2004) 28–77 39

2 TðÀa;aÞ ¼ NðÀa;aÞ þ q NðÀaÀ1;aþ1Þ ðaX1Þ;

2 4 Tða;ÀaÞ ¼ Nða;ÀaÞ þ q ðNðaÀ1;1ÀaÞ þ Nð1Àa;aÀ1ÞÞþq NðÀa;aÞ ðaX2Þ;

3 Tð0;0Þ ¼ Nð0;0Þ þðq þ q ÞNðÀ1;1Þ;

4 Tð1;À1Þ ¼ Nð1;À1Þ þ qNð0;0Þ þ q NðÀ1;1Þ:

Note that in this example that each Tl is a ﬁnite sum of Nm’s. I conjecture that this is true in general. On the other hand, the Ll’s need not be ﬁnite sums of Mm’s even for n ¼ 2:

cn We call the topological basis fTlglAZn the canonical basis of T and fLlglAZn the dual canonical basis. Let us introduce notation for the coefﬁcients: let X X Tl ¼ tm;lðqÞNm; Ll ¼ lm;lðqÞMm ð2:24Þ mAZn mAZn

À1 forpolynomials tm;lðqÞAZ½q and lm;lðqÞAZ½q : We know that tm;lðqÞ¼lm;lðqÞ¼0 unless m%l; and that tl;lðqÞ¼ll;lðqÞ¼1:

n Lemma 2.25. For l; mAZ ; /Ll; TÀmS ¼ dl;m:

Proof. A calculation using the deﬁnition of /:; :S shows that X À1 /Ll; TÀmS ¼ ln;lðqÞtÀn;Àmðq Þ; ð2:26Þ m%n%l

X À1 /TÀm; LlS ¼ ln;lðq ÞtÀn;ÀmðqÞ: ð2:27Þ m%n%l

À1 À1 Hence, /Ll; TÀmS equals 1 if l ¼ m and belongs to q Z½q if lam: Similarly, /TÀm; LlS equals 1 if l ¼ m and belongs to qZ½q if lam: But /Ll; TÀmS ¼ /TÀm; LlS by Lemma 2.21. & P A n À1 Corollary 2.28. For l Z ; Ml ¼ mAZn tÀl;Àmðq ÞLm: P / S Proof. By Lemma 2.25, we can write Ml ¼ mAZn Ml; TÀm Lm: Now a calculation À1 from the deﬁnition of the form /:; :S gives that /Ml; TÀmS ¼ tÀl;Àmðq Þ: & ARTICLE IN PRESS

40 J. Brundan / Advances in Mathematics 182 (2004) 28–77

Example 2.29. Suppose that n ¼ 2: Using Example 2.23, one checks: a Mða;bÞ ¼ Lða;bÞ ðapb; a þ b 0Þ;

À2 a Mða;bÞ ¼ Lða;bÞ þ q Lðb;aÞ ða4b; a þ b 0Þ;

À2 MðÀa;aÞ ¼ LðÀa;aÞ þ q LðÀaÀ1;aþ1Þ ðaX1Þ;

À2 À4 Mða;ÀaÞ ¼ Lða;ÀaÞ þ q ðLðaÀ1;1ÀaÞ þ LðÀaÀ1;aþ1ÞÞþq LðÀa;aÞ ðaX2Þ;

À1 Mð0;0Þ ¼ Lð0;0Þ þ q LðÀ1;1Þ;

À1 À3 À2 À4 Mð1;À1Þ ¼ Lð1;À1Þ þðq þ q ÞLð0;0Þ þ q LðÀ2;2Þ þ q LðÀ1;1Þ:

I conjecture for arbitrary n that each Ml is always a ﬁnite linearcombination of Lm’s.

2.6. Crystal structure

Now we describe the crystal structure underlying the module Tn: The basic reference followed here is [Ka2]. Let A be the subring of QðqÞ consisting of rational functions having no pole at q ¼ 0: Evaluation at q ¼ 0 induces an isomorphism A=qA-Q: Let VA be the A-lattice in V spanned by the va’s. Then, VA togetherwith the basis of the Q-vectorspace VA=qVA given by the images of the va’s is a lower crystal base for V at q ¼ 0 in the sense of [Ka2, 4.1]. Write E˜i; F˜i forthe corresponding crystal operators. Rather than view these as operators on the crystal base fva þ qVAgaAZ; we will view them simply as operators on the set Z parameterizing the crystal base. Then, the crystal graph is as follows:

F˜ F˜ F˜ F˜ F˜ F˜ ?- À 3 !2 À2 !1 À1 !0 0 !0 1 !1 2 !2 3-? :

Thus, F˜iðaÞ equals a þ 1ifa ¼ i or a ¼Ài À 1; | otherwise, and E˜iðaÞ equals a À 1if | - a ¼ i þ 1ora ¼Ài; otherwise. The maps ei; ji : Z N deﬁned by ˜k a| ˜k a| eiðaÞ¼maxfkX0 j Ei a g; jiðaÞ¼maxfkX0 j Fi a g only evertake the values 0 ; 1 or 2 (the last possibility occurring only if i ¼ 0). Since Tn is a tensorproduct of n copies of V; it has an induced crystal structure. n The crystal lattice is TA; namely, the A-span of the basis fNlglAZn ; and the images n n of the Nl’s in TA=qTA give the crystal base. Like in the previous paragraph, we ARTICLE IN PRESS

J. Brundan / Advances in Mathematics 182 (2004) 28–77 41 will view the crystal operators as maps on the underlying set Zn parameterizing the ˜0 ˜ 0 crystal base. However, we will denote them by Ei; F i; since we want to reserve the unprimed symbols for something else later on. In order to describe them explicitly, we introduce a little more combinatorial notation. Given lAZn and iX0; let ðs1; y; snÞ be the i-signature of l; namely, the sequence deﬁned by 8 > þ if ia0 and l ¼ i or À i À 1; > r > a > À if i 0 and li ¼ i þ 1or À i; <> þþ if i ¼ 0 and lr ¼À1; sr ¼ ð2:30Þ > Àþ if i ¼ 0 and l ¼ 0; > r > > ÀÀ if i ¼ 0 and lr ¼ 1; :> 0 otherwise:

Form the reduced i-signature by successively replacing subwords of ðs1; y; snÞ of the form þÀ (possibly separated by 0’s in between) with 0’s until we are left with a sequence of the form ðs* 1; y; s* nÞ in which no À appears after a +. For r ¼ 1; y; n; let dr be the n-tuple with a 1 in the rth position and 0’s elsewhere. Then, ( | if there are no À ’s in the reduced i-signature; ˜0 EiðlÞ¼ l À dr otherwise; where the rightmost À occurs in s* r;

( | if there are no þ ’s in the reduced i-signature; ˜0 FiðlÞ¼ l þ dr otherwise; where the leftmost þ occurs in s* r;

0 eiðlÞ¼the total numberof À ’s in the reduced i-signature;

0 jiðlÞ¼the total numberof þ ’s in the reduced i-signature:

n ˜0 ˜ 0 0 0 n We have now described the crystal ðZ ; Ei; F i; ei; ji; wtÞ associated to the module T purely combinatorially.

Example 2.31. Consider l ¼ð1; 2; 0; À3; À2; À1; 0; 1ÞAZ8: The 1-signature is ðþ; À; 0; 0; þ; À; 0; þÞ: Hence the reduced 1-signature is ð0; 0; 0; 0; 0; 0; 0; þÞ; so we ˜0 | ˜ 0 get that E1l ¼ and F 1l ¼ l þ d8: On the otherhand, the 0-signatureis ˜0 ðÀÀ; 0; Àþ; 0; 0; þþ; Àþ; ÀÀÞ; which reduces to ðÀÀ; 0; Àþ; 0; 0; 0; 0; 0Þ; so E0l ¼ ˜ 0 l À d3; F 0l ¼ l þ d3:

The following lemma is a general property of canonical bases/lower global crystal bases. It follows ultimately from [Ka1, Proposition 5.3.1]. See [B3, Theorem 2.31] for a similarsituation. ARTICLE IN PRESS

42 J. Brundan / Advances in Mathematics 182 (2004) 28–77

Lemma 2.32. Let lAZn and iX0:

P 0 0 i i 1ÀjiðmÞ (i) EiTl ¼½j ðlÞþ1 T ˜0 þ A n u Tm where u Aqq Z½q is zero unless i i EiðlÞ m Z m;l m;l i 0 0 ejðmÞXejðlÞ for all jX0: P 0 0 i i 1ÀeiðmÞ (ii) FiTl ¼½e ðlÞþ1 T ˜0 þ A n v Tm where v Aqq Z½q is zero unless i i FiðlÞ m Z m;l m;l i 0 0 jjðmÞXjjðlÞ for all jX0:

˜0 (In (i) resp. (ii), the ﬁrst term on the right-hand side should be omitted if EiðlÞ resp. ˜0 | FiðlÞ is :)

Motivated by Lemmas 2.21 and 2.25, we also introduce the dual crystal operators deﬁned by ˜Ã ˜0 ˜Ã ˜0 Ei ðlÞ¼ÀFiðÀlÞ; Fi ðlÞ¼ÀEiðÀlÞ; ð2:33Þ

Ã 0 Ã 0 ei ðlÞ¼jiðÀlÞ; ji ðlÞ¼eiðÀlÞ: ð2:34Þ

These can be described explicitly in a similar way to the above: for ﬁxed iX0 and n lAZ ; let ðs1; y; snÞ be the i-signature of l deﬁned according to (2.30). First, replace all sr that equal Àþ with þÀ: Now form the dual reduced i-signature ðs* 1; y; s* nÞ from this by repeatedly replacing subwords of ðs1; y; snÞ of the form Àþ (possibly separated by 0’s) by 0’s, until no þ appears after a À: Finally, ( 0 if there are no À ’s in the dual reduced i-signature; ˜Ã Ei ðlÞ¼ l À dr otherwise; where the leftmost À occurs in s* r; ( 0 if there are no þ ’s in the dual reduced a-signature; ˜Ã Fi ðf Þ¼ l þ dr otherwise; where the rightmost þ occurs in s* r;

Ã ei ðlÞ¼the total numberof À ’s in the dual reduced i-signature;

Ã ji ðlÞ¼the total numberof þ ’s in the dual reduced i-signature:

n ˜Ã ˜Ã Ã Ã In this way, we obtain the dual crystal structure ðZ ; Ei ; Fi ; ei ; ji ; wtÞ:

Lemma 2.35. Let lAZn and iX0:

P Ã Ã i i 1Àei ðlÞ (i) EiLl ¼½e ðlÞ L ˜Ã þ A n w Lm where w Aqq Z½q is zero unless i i Ei ðlÞ m Z m;l m;l i Ã Ã jj ðmÞpjj ðlÞ for all jX0: P Ã Ã i i 1Àji ðlÞ (ii) FiLl ¼½j ðlÞ L ˜Ã þ A n x Lm where x Aqq Z½q is zero unless i i Fi ðlÞ m Z m;l m;l i Ã Ã ej ðmÞpej ðlÞ for all jX0:

Proof. Dualize Lemma 2.32 using Lemmas 2.21 and 2.25. & ARTICLE IN PRESS

J. Brundan / Advances in Mathematics 182 (2004) 28–77 43

Remark 2.36. (i) Suppose we are given ei; jiAN forall iX0: One can show from the Ã Ã combinatorial description of the maps ei ; ji above that there exist only finitely many n Ã Ã lAZ with ei ðlÞ¼ei and ji ðlÞ¼ji forall iX0: (ii) Using (i), one deduces easily that all but finitely many terms of the sums À1 occurring in Lemma 2.35 are zero, i.e. both EiLl and FiLl are finite Z½q; q -linear combinations of Lm’s. We will not make use of this observation in the remainder of the article. (iii) If the finiteness conjecture made in Example 2.23 holds, then it is also the case that all but finitely many terms of the sums occurring in Lemma 2.32 are zero, i.e. À1 both EiTl and FiTl are finite Z½q; q -linearcombinations of Tm’s, cf. the argument in the last paragraph of the proof of Lemma 3.30 below.

3. Exterior algebra

3.1. Exterior powers

Deﬁne K to be the two-sided ideal of the tensoralgebra T generated by the vectors

va#va ðaa0Þ;

2 va#vb þ q vb#va ða4b; a þ ba0Þ;

2 4 va#vÀa þ q ðvaÀ1#v1Àa þ v1Àa#vaÀ1Þþq vÀa#va ðaX2Þ;

4 v1#vÀ1 þ qv0#v0 þ q vÀ1#v1; forall admissible a; bAZ: These relations are a limiting case of the relations in [JMO, Proposition 2.3] (with q replaced by q2), hence K is invariant under the n action of U: Let F :¼ T=K: Since K ¼ "nX0 K is a homogeneous ideal " n n n n of T; F is also graded as F ¼ nX0 F ; where FV ¼ T =K : We view the n n space F as a quantum analogue of the exterior power V in type bN: As usual, n we will write u14?4un forthe image of u1#?#unAT underthe quotient map p : Tn-Fn: n A n As in the introduction, let Zþ denote the set of all tuples l Z such that lr4lrþ1 if a A n lr 0; lrXlrþ1 if lr ¼ 0; foreach r ¼ 1; y; n À 1: For l Zþ; let

? A n Fl ¼ pðNw0lÞ¼vln 4 4vl1 F : ð3:1Þ

The following lemma follows from the deﬁning relations for Kn: ARTICLE IN PRESS

44 J. Brundan / Advances in Mathematics 182 (2004) 28–77

A n A þ Lemma 3.2. For l Z ; pðNw0lÞ equals Fl if l Zn ; otherwise pðNw0lÞ is a qZ½q-linear A n k combination of Fm’s for m Zþ with m l:

n This shows that the elements fFlg A n span F : In fact, one can check routinely l Zþ using Bergman’s diamond lemma [Bg, 1.2] that:

n Lemma 3.3. The vectors fFlg A n give a basis for F : l Zþ

cn n cn cn cn cn Let K be the closure of K in T and F :¼ T =K ; giving a completion of the n cn vectorspace F : The vectors fFlg A n give a topological basis for F : Note that l Zþ s : Tn-Tn leaves Kn invariant, hence induces s : Fn-Fn: Similarly the cn cn cn continuous automorphism o : T -T leaves K invariant, so induces cn-cn o : F F with oðFlÞ¼FÀw0l:

cn 3.2. Canonical basis of F

cn Now we construct the canonical basis of F : To start with, we need a bar involution.

cn cn Lemma 3.4. The bar involution on T leaves K invariant, hence induces a continuous cn cn antilinear involution À : F -F such that

cn (1) xv ¼ x% v% forP all xAU; vAF ; A À1 A n (2) Fl Fl þ mgl Z½q; q Fm for all l Zþ; cn (3) oðvÞ¼oðv%Þ for all vAF :

Proof. In the case n ¼ 2; all the generators of K2 are bar invariant by Example 2.23, hence K2 is bar invariant. In general, Kn is spanned by vectors of the form v#k#w n1 2 n2 for vAT ; kAK ; wAT and some n1; n2X0 with n1 þ n2 þ 2 ¼ n: By the deﬁnition of the barinvolution,

# # %# %# % v k w ¼ Yn1þ2;n2 ðv Y2;n2 ðk wÞÞ:

We have already shown that k%AK2; and K2 is U-invariant, hence this belongs to cn n cn cn K : This shows that K CK ; hence K itself is barinvariantby continuity. Now properties (1) and (3) are immediate from the analogous properties of the bar cn A n involution on T : For property (2), take l Zþ: We know Nw0l equals Nw0l plus a À1 ! g Z½q; q -linearcombination of Nw0m’s with w0m w0l; orequivalently, m l: By ARTICLE IN PRESS

J. Brundan / Advances in Mathematics 182 (2004) 28–77 45

À1 k Lemma 3.2, pðNw0mÞ is a Z½q; q -linearcombination of Fn’s with n m: Hence (2) holds. &

We get the following theorem by general principles, just as in Theorem 2.22 before.

cn Theorem 3.5. There exists a unique topological basis fUlglAZn for F such that P þ c n Ul ¼ Ul and UlAFl þ A n qZ½qFm; for all lAZ : Actually, we have that UlAFl þ m Zþ þ Pc mgl qZ½qFm: Also, oðUlÞ¼UÀw0l:

Example 3.6. For n ¼ 2; one deduces from Example 2.23 and Lemma 3.8 below that: a Uða;bÞ ¼ Fða;bÞ ða4b; a þ b 0Þ;

2 Uða;ÀaÞ ¼ Fða;ÀaÞ þ q Fðaþ1;ÀaÀ1Þ ðaX1Þ;

3 Uð0;0Þ ¼ Fð0;0Þ þðq þ q ÞFð1;À1Þ:

cn We call the topological basis fUlg A n the canonical basis of F : Write l Zþ X Ul ¼ um;lðqÞFm ð3:7Þ A n m Zþ forpolynomials um;lðqÞAZ½q: We know that um;lðqÞ¼0 unless mkl; and that ul;lðqÞ¼1: The following lemma explains the relationship between Ul and the cn canonical basis element Tl of T constructed earlier.

Lemma 3.8. For lAZn; we have that ( n Ul if lAZ ; pðT Þ¼ þ w0l e n 0 if l Zþ:

A n Proof. Suppose ﬁrst that l Zþ: We know that Tw0l equals Nw0l plus a qZ½q-linear g combination of Nw0m’s with m l: Moreover, by Lemma 3.2, pðNw0mÞ is a Z½q-linear combination of Fn’s with nkm: Hence, g pðTw0lÞ¼Fl þða qZ½q-linearcombination of Fm’s with m lÞ: ARTICLE IN PRESS

46 J. Brundan / Advances in Mathematics 182 (2004) 28–77

Since it is automatically barinvariant, it must equal Ul by the uniqueness in e n Theorem 3.5. An entirely similar argument in case l Zþ shows that g pðTw0lÞ¼ða qZ½q-linearcombination of Fm’s with m lÞ

¼ða qZ½q-linearcombination of Um’s with mglÞ:

Since it is bar invariant, it must be zero. &

cn Corollary 3.9. The vectors fTw lg A n n form a topological basis for K : 0 l Z ÀZþ 3.3. Dual canonical basis

A n For l Zþ; deﬁne X À1 El ¼ uÀw0l;Àw0mðq ÞLm: ð3:10Þ A n m Zþ

n cn Let E be the subspace of T spanned by the fElg A n : Since there are only ﬁnitely l Zþ A n % many m Zþ with m l; we see that El is a ﬁnite linearcombination of Lm’s, and vice n versa. So the vectors fLlg A n also form a basis for E : l Zþ

Example 3.11. For n ¼ 2; we have by Example 3.6 that: a Eða;bÞ ¼ Lða;bÞ ða4b; a þ b 0Þ;

À2 Eða;ÀaÞ ¼ Lða;ÀaÞ þ q LðaÀ1;Àaþ1Þ ðaX2Þ;

À1 À3 Eð1;À1Þ ¼ Lð1;À1Þ þðq þ q ÞLð0;0Þ;

Eð0;0Þ ¼ Lð0;0Þ:

bn n cn bn cn Let E be the closure of E in T : By Corollary 3.9 and Lemma 2.25, E and K are orthogonal with respect to the bilinear form /:; :S: Hence we get induced a well- bn cn deﬁned pairing /:; :S between E and F : By Lemmas 3.8 and 2.25, we have that / S LÀw0l; Um ¼ dl;m ð3:12Þ

A n forall l; m Zþ:

bn cn Lemma 3.13. E is a U-submodule of T : ARTICLE IN PRESS

J. Brundan / Advances in Mathematics 182 (2004) 28–77 47

bn A n Proof. It sufﬁces toP show that EiLl and FiLl both belong to E foreach l Zþ and / S iX0: Write EiLl ¼ mAZm bm;lðqÞLm: Apply :; TÀm to both sides and use Lemma / S e n cn cn 2.25 to get bm;lðqÞ¼ Ll; EiTÀm : For m Zþ; TÀm belongs to K : But K is U- cn e n invariant, hence EiTÀm belongs to K : So we get that bm;lðqÞ¼0 forall m Zþ by bn Corollary 3.9. Hence EiLl belongs to E still, and similarly for FiLl: &

The bilinearform ð:; :Þ on Tn also induces a pairing ð:; :Þ between En and Fn; actually by (2.20) we have that

ðu; vÞ¼/u; sðvÞS ð3:14Þ foreach uAEn; vAFn:

A n zðlÞ Lemma 3.15. For all l; m Zþ; we have that ðEl; NmÞ¼q dl;m: P P Proof. Let Fm ¼ A n vg;mðqÞUg; so A n um;gðqÞvg;nðqÞ¼dm;n: Now calculate: g Zþ g Zþ

/ S ðEÀw0l; NÀw0mÞ¼ EÀw0l; sðNÀw0mÞ

zðmÞ/ S zðmÞ/ S ¼ q EÀw0l; Nw0m ¼ q EÀw0l; Fm X zðmÞ À1 À1 / S zðmÞ ¼ q ul;nðq Þvg;mðq Þ LÀw0n; Ug ¼ q dl;m: & A n n;g Zþ

Now we get the following characterization of the basis fLlg A n in terms of the l Zþ bn restriction of the bar involution to E :

n bn Theorem 3.16. For lAZ ; L is the unique element of E such that L ¼ L and P þ l l l c À1 À1 LlAEl þ A n q Z½q Em: Moreover, m Zþ X Ll ¼ lm;lðqÞEm; ð3:17Þ A n m Zþ where lm;lðqÞ is as in (2.24).

A n zðmÞ Proof. ForeachP m Zþ; ðLl; NmÞ¼q lm;lðqÞ: NowP apply Lemma 3.15 to deduce 0 À1 À1 that Ll ¼ A n lm;lðqÞEm: Finally, if L AEl þ A n q Z½q Em is anotherbar m Zþ l m Zþ bn 0 invariant element of E ; then Ll À Ll is bar invariant and can be expressed as a À1 À1 q Z½q -linearcombination of Ln’s. Hence it must be zero. & ARTICLE IN PRESS

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A A n A n To state the next lemma, we deﬁne polynomials al;mðqÞ Z½q foreach l Zþ; m Z by X

pðNw0mÞ¼ al;mðqÞFl: ð3:18Þ A n l Zþ

Recalling Lemma 3.2, we have that al;mðqÞ¼0 unless lkm: P n À1 A n Lemma 3.19. For each l Zþ; El ¼ mAZ aÀw0l;Àw0mðq ÞMm:

Proof. Applying the antilinearmap s to (3.18) gives that X zðmÞÀzðlÞ À1 pðNmÞ¼ q aÀw0l;Àw0mðq ÞpðNlÞ A n l Zþ

A n zðmÞ À1 foreach m Z : Hence, invoking Lemma 3.15, ðEl; NmÞ¼q aÀw0l;Àw0mðq Þ for each mAZn: The lemma follows. &

n Finally in this subsection, we describe the action of U on the basis fElg A n of E l Zþ n cn explicitly. In particular, this shows that E itself is a U-submodule of T ; as could also be proved using Lemma 3.13 and Remark 2.36(ii).

A n X y Lemma 3.20. Let l Zþ and i 0; and let ðs1; ; snÞ be the i-signature of l deﬁned according to (2.30). Then, X a ;e ? e Àð i lrþ1 þ þ ln Þ EiEl ¼ q cl;rðqÞElÀdr ; A n r with lÀdr Zþ; sr¼À;Àþ or ÀÀ X a ;e ? e ð i l1 þ þ lrÀ1 Þ FiEl ¼ q cl;rðqÞElþdr ; A n r with lþdr Zþ; sr¼þ;Àþ or þþ where ( P À1 zðlÞ À2 s ðq þ q Þ s¼0 ðÀq Þ if sr ¼ÀÀ or þþ; cl;rðqÞ¼ 1 otherwise:

Proof. WeP sketch the proof for Ei: By Lemmas 3.13, 3.15 and 2.11(ii), we may write EiEl ¼ A n cm;lðqÞEm where m Zþ

ÀzðmÞ ÀzðmÞ À1 cm;lðqÞ¼q ðEiEl; NmÞ¼q ðEl; qiFiKi NmÞ:

The right-hand side is computed using Lemma 3.15 and the fact that El is orthogonal to Kn: & ARTICLE IN PRESS

J. Brundan / Advances in Mathematics 182 (2004) 28–77 49

3.4. Crystal structure

cn The crystal structure underlying the canonical basis fUlg A n of F is easily l Zþ deduced from the results of Section 2.6 and Lemma 3.8. Let us denote the resulting n cn ˜ ˜ crystal operators on the index set Zþ parameterizing the bases of F by Ei; Fi; ei; ji: By deﬁnition,

˜ ˜0 ˜ ˜0 EiðlÞ¼w0Eiðw0lÞ; FiðlÞ¼w0Fiðw0lÞ; ð3:21Þ

0 0 eiðlÞ¼eiðw0lÞ; jiðlÞ¼jiðw0lÞ; ð3:22Þ

˜0 ˜0 0 0 where Ei; Fi; ei and ji are as in Section 2.6. By properties of the automorphism o (or ˜ ˜ by directly checking all of the cases listed below), the operators Ei; Fi; ei; ji are the n ˜Ã ˜Ã Ã Ã same as the restrictions to Zþ of the dual crystal operators Ei ; Fi ; ei ; ji deﬁned in Section 2.6, so we will not need the latternotation again. In fact, there are now so few possibilities that we can describe the crystal graph explicitly. First suppose that i ¼ 0: Then the possible i-strings in the crystal graph are as follows:

(1) ð?Þ; F˜ F˜ (2) ðy; 0r; À1; yÞ !0 ðy; 0rþ1; yÞ !0 ðy; 1; 0r; yÞ; (3) ðy; 1; 0r; À1; yÞ: Here, y denotes some ﬁxed entries different from 1; 0; À1 and rX0: Similarly for i40; the possible i-strings in the crystal are as follows: (1) ð?Þ; F˜ (2) ðy; i; yÞ !i ðy; i þ 1; yÞ;

F˜ (3) ðy; Ài À 1; yÞ !i ðy; Ài; yÞ;

F˜ F˜ (4) ðy; i; y; Ài À 1; yÞ !i ðy; i; y; Ài; yÞ !i ðy; i þ 1; y; Ài; yÞ; (5) ðy; i þ 1; y; Ài À 1; yÞ; (6) ðy; Ài; Ài À 1; yÞ; (7) ðy; i þ 1; i; yÞ; F˜ (8) ðy; i þ 1; i; y; Ài À 1; yÞ !i ðy; i þ 1; i; y; Ài; yÞ;

F˜ (9) ðy; i; y; Ài; Ài À 1; yÞ !i ðy; i þ 1; y; Ài; Ài À 1; yÞ; (10) ðy; i þ 1; i; y; Ài; Ài À 1; yÞ; where again y denotes ﬁxed entries different from i; i þ 1; Ài; Ài À 1: A crucial observation deduced from this analysis is that all i-strings are of length p2: ARTICLE IN PRESS

50 J. Brundan / Advances in Mathematics 182 (2004) 28–77

A n X Lemma 3.23. Let l Zþ and i 0: P i i 1ÀeiðlÞ (i) EiLl ¼½eiðlÞ L ˜ þ A n w Lm where w Aqq Z½q is zero unless i EiðlÞ m Zþ m;l m;l i j ðmÞpj ðlÞ for all jX0: j j P i i 1ÀjiðlÞ (ii) FiLl ¼½j ðlÞ L ˜ þ A n x Lm where x Aqq Z½q is zero unless i i FiðlÞ m Zþ m;l m;l i ejðmÞpejðlÞ for all jX0:

˜ ˜Ã ˜ ˜Ã Proof. This is a special case of Lemma 2.35, since Ei ¼ Ei ; Fi ¼ Fi ; y : &

A n X Lemma 3.24. Let l Zþ and i 0: P i i 1ÀjiðmÞ (i) EiUl ¼½j ðlÞþ1 U ˜ þ A n y Um where y Aqq Z½q is zero unless i i EiðlÞ m Zþ m;l m;l i e ðmÞXe ðlÞ for all jX0: j j P i i 1ÀeiðmÞ (ii) FiUl ¼½eiðlÞþ1 U ˜ þ A n z Um where z Aqq Z½q is zero unless i FiðlÞ m Zþ m;l m;l i jjðmÞXjjðlÞ for all jX0:

(In (i) resp. (ii), the ﬁrst term on the right-hand side should be omitted if E˜iðlÞ resp. F˜iðlÞ is |:)

Proof. Dualize Lemma 3.23. &

A n X Corollary 3.25. Let l Zþ and i 0: (i) If e l 40 then E U j l 1 U : ið Þ i l ¼½ ið Þþ i E˜iðlÞ (ii) If j l 40 then F U e l 1 U : ið Þ i l ¼½ ið Þþ i F˜iðlÞ

Proof. We prove (i), (ii) being similar. Lemma 3.24 gives us that EiUl ¼½j ðlÞþ P i i i 1ÀjiðmÞ 1 U ˜ þ A n y Um where y belongs to qq Z½q and is zero unless i EiðlÞ m Zþ m;l m;l i i a ejðmÞXejðmÞ forall jX0: Suppose that ym;l 0 forsome m: By assumption, a i A eiðmÞXeiðlÞX1; so jiðmÞp1 since all i-strings are of length p2: So 0 ym;l qZ½q: i But ym;l is bar invariant, so this is a contradiction. &

3.5. Computation of Ul’s

Now we explain a simple algorithm to compute Ul: Recall the deﬁnition of the degree of atypicality of l from (2.13). We will need the following: ARTICLE IN PRESS

J. Brundan / Advances in Mathematics 182 (2004) 28–77 51

A n X A n Procedure 3.26. Suppose we are given l Zþ with #l 2: Compute m Zþ and an operator XiAfEi; FigiX0 by following the instructions below starting at step (0).

(0) Choose the minimal rAf1; y; ng such that lr þ ls ¼ 0 forsome s4r: Go to step (1). (1) If r41 and lr ¼ lrÀ1 À 1; replace r by ðr À 1Þ and repeat step (1). Otherwise, go to step (2). 0 (2) If lr þ ls þ 1 ¼ 0 forsome (necessarily unique) sAf1; y; ng go to step ð1Þ :

Otherwise, set Xi ¼ Elr and m ¼ l þ dr: Stop. 0 0 ð1Þ If son and ls ¼ lsþ1 þ 1; replace s by ðs þ 1Þ and repeat step ð1Þ : Otherwise, go to step ð2Þ0: 0 ð2Þ If lr þ ls À 1 ¼ 0 forsome (necessarilyunique) rAf1; y; ng go to step (1).

Otherwise, set Xi ¼ FÀls and m ¼ l À ds: Stop.

The following lemma follows immediately from the nature of the above procedure and Corollary 3.25.

A n X A n Lemma 3.27. Take l Zþ with #l 2: Apply Procedure 3.26 to get m Zþ and XiAfEi; FigiX0: Then, #mp#l and XiUm ¼ Ul: Moreover, after at most ðn À 1Þ repetitions of the procedure, the atypicality must get strictly smaller. Hence after ﬁnitely many recursions, the procedure reduces l to a typical weight.

Lemma 3.27 implies the following algorithm for computing Ul: If l is typical then A n Ul ¼ Fl; since by Lemma 2.14 there are no other m Zþ with wtðmÞ¼wtðlÞ: A n A Otherwise, apply Procedure 3.26 to get m Zþ and Xi fEi; FigiX0: Since the procedure always reduces l to a typical weight in ﬁnitely many steps, we may assume Um is known recursively. Then Ul ¼ XiUm:

Example 3.28. Let us compute Uð5;3;2;1;0;0;À1;À4;À6Þ: Apply Procedure 3.26 repeatedly to get the following sequence of weights:

ð5; 3; 2; 1; 0; 0; À1; À4; À7Þ; ð6; 3; 2; 1; 0; 0; À1; À4; À7Þ;

ð6; 3; 2; 1; 0; 0; À1; À5; À7Þ; ð6; 4; 2; 1; 0; 0; À1; À5; À7Þ;

ð6; 4; 3; 1; 0; 0; À1; À5; À7Þ; ð6; 4; 3; 2; 0; 0; À1; À5; À8Þ;

ð7; 4; 3; 2; 0; 0; À1; À6; À8Þ; ð7; 5; 3; 2; 0; 0; À1; À6; À8Þ;

ð7; 5; 4; 2; 0; 0; À1; À6; À8Þ; ð7; 5; 4; 3; 0; 0; À1; À6; À8Þ;

ð7; 5; 4; 3; 0; 0; À2; À6; À8Þ; ð7; 5; 4; 3; 1; 0; À2; À6; À8Þ: ARTICLE IN PRESS

52 J. Brundan / Advances in Mathematics 182 (2004) 28–77

Hence,

Uð5;3;2;1;0;0;À1;À4;À6Þ ¼ F6E5F4E3E2E1F7E6F5E4E3E2F1E0Fð7;5;4;3;1;0;À2;À6;À8Þ

2 ¼ Fð5;3;2;1;0;0;À1;À4;À6Þ þ q Fð7;5;3;2;0;0;À4;À6;À7Þ

3 þðq þ q ÞFð8;5;3;2;1;À1;À4;À6;À8Þ

3 5 þðq þ q ÞFð8;7;5;3;2;À4;À6;À7;À8Þ:

Example 3.29. Using the algorithm, one computes for n ¼ 3 that:

a Uða;b;cÞ ¼ Fða;b;cÞ ða þ b; a þ c; b þ c 0Þ;

2 a Uða;b;ÀbÞ ¼ Fða;b;ÀbÞ þ q Fða;bþ1;ÀbÀ1Þ ðaXb þ 2; b 0Þ;

2 a Uðbþ1;b;ÀbÞ ¼ Fðbþ1;b;ÀbÞ þ q Fðbþ2;bþ1;ÀbÀ2Þ ðb 0Þ;

2 a Uða;Àa;ÀbÞ ¼ Fða;Àa;ÀbÞ þ q Fðaþ1;ÀaÀ1;ÀbÞ ðbXa þ 2; a 0Þ;

2 a Uða;Àa;ÀaÀ1Þ ¼ Fða;Àa;ÀaÀ1Þ þ q Fðaþ2;ÀaÀ1;ÀaÀ2Þ ða 0Þ;

2 Uða;b;ÀaÞ ¼ Fða;b;ÀaÞ þ q Fðaþ1;b;ÀaÀ1Þ ða4b4 À aÞ;

3 Uða;0;0Þ ¼ Fða;0;0Þ þðq þ q ÞFða;1;À1Þ ðaX2Þ;

3 Uð1;0;0Þ ¼ Fð1;0;0Þ þðq þ q ÞFð2;1;À2Þ;

3 Uð0;0;ÀbÞ ¼ Fð0;0;ÀbÞ þðq þ q ÞFð1;À1;ÀbÞ ðbX2Þ;

3 Uð0;0;À1Þ ¼ Fð0;0;À1Þ þðq þ q ÞFð2;À1;À2Þ;

5 3 Uð0;0;0Þ ¼ Fð0;0;0Þ þðq À q ÞFð1;0;À1Þ þðq þ q ÞFð2;0;À2Þ: ARTICLE IN PRESS

J. Brundan / Advances in Mathematics 182 (2004) 28–77 53

3.6. Specialization

When we specialize at q ¼ 1; various things become simpler to compute. To formalize the specialization process, we will work with the Z½q; qÀ1-lattices X X n À1 À1 EZ½q;qÀ1 ¼ Z½q; q Ll ¼ Z½q; q El; A n A n l Zþ l Zþ

X X n À1 C À1 FZ½q;qÀ1 ¼ Z½q; q Ul Z½q; q Fl: A n A n l Zþ l Zþ

cn n cn Also let FZ½q;qÀ1 denote the completion of FZ½q;qÀ1; i.e. its closure in F : We need A n n this because the element Fl F need not belong to the lattice FZ½q;qÀ1; though cn it always belongs to F À1 : Indeed, the fFlg A n form a topological basis Z½q;q l Zþ cn for FZ½q;qÀ1:

n n cn Lemma 3.30. EZ½q;qÀ1; FZ½q;qÀ1 and FZ½q;qÀ1 are modules over UZ½q;qÀ1:

À1 cn Proof. The Z½q; q -lattice in T generated by fMlglAZn is invariant under UZ½q;qÀ1: ðrÞ Combining this with Corollary 2.28 and Lemma 3.19, we deduce that each Ei El À1 can be expressed as a (possibly infinite) Z½q; q -linearcombination of Em’s. ðrÞ Actually, it is a finite linearcombination by Lemma 3.20. Similarly,each Fi El is a À1 n finite Z½q; q -linearcombination of Em’s. So EZ½q;qÀ1 is a UZ½q;qÀ1-module. À1 cn Considering instead the Z½q; q -lattice in T generated by fNlglAZn and passing cn ðrÞ ðrÞ to the quotient F ; we also get easily that each Ei Fl and each Fi Fl is a finite À1 cn Z½q; q -linearcombination of Fm’s, hence FZ½q;qÀ1 is a UZ½q;qÀ1-module. This means ðrÞ ðrÞ À1 in particular that each Ei Ul and each Fi Ul is a (possibly infinite) Z½q; q -linear n combination of Um’s. Finally to show that FZ½q;qÀ1 is a UZ½q;qÀ1-module we need to ðrÞ ðrÞ À1 show Ei Ul and Fi Ul are actually finite Z½q; q -linearcombination of Um’s. We ðrÞ explain the argument for Ei Ul only. By the algorithm described in the previous subsection, Ul is a finite linear ðrÞ combination of Fm’s, hence Ei Ul is a barinvariant,finite linearcombination of Fm’s. Say X ðrÞ Ei Ul ¼ fmðqÞFm mAZn

À1 forpolynomials fmðqÞAZ½q; q : Let d be minimal such that all fmðqÞ belong to Àd q Z½q: If do0 then the right-hand side is a qZ½q-linearcombination of Fm’s, hence ARTICLE IN PRESS

54 J. Brundan / Advances in Mathematics 182 (2004) 28–77 since it is also a barinvariant combination of Um’s it must be zero. Otherwise, define gmðqÞ to beP the unique barinvariant polynomial with gmðqÞfmðqÞðmod qZ½qÞ and subtract m gmðqÞUm from the right-hand side, summing over the finitely many m 1Àd such that deg gmðqÞ¼d: The result is a bar invariant, finite q Z½q-linear combination of Fm’s. Now repeat the process to reduce the expression to zero after finitely many steps. &

Deﬁne n # n EZ ¼ Z Z½q;qÀ1EZ½q;qÀ1;

n # n FZ ¼ Z Z½q;qÀ1FZ½q;qÀ1;

cn # cn FZ ¼ Z Z½q;qÀ1FZ½q;qÀ1;

À1 A n where we are viewing Z as a Z½q; q -module so that q acts as 1: For l Zþ; we will # A n # A n # Acn # A n denote the elements 1 El EZ; 1 Ll EZ; 1 Fl FZ and 1 Ul FZ by Elð1Þ; n n cn Llð1Þ; Flð1Þ and Ulð1Þ; respectively. By Lemma 3.30, EZ; FZ and FZ are modules # over UZ :¼ Z Z½q;qÀ1UZ½q;qÀ1: In theiraction on these lattices, the elements ! "# Hi Ki EðrÞ ¼ 1#EðrÞ; F ðrÞ ¼ 1#F ðrÞ; ¼ 1# ð3:31Þ i i i i r r of UZ satisfy the deﬁning relations of the usual generators of the Kostant Z-form for the universal enveloping algebra of type bN: n We note that at q ¼ 1; the deﬁning relations for FZ simplify to the following:

va4va ¼ 0 ðaa0Þ;

va4vb ¼Àvb4va ða4b; a þ ba0Þ;

a va4vÀa ¼ÀvÀa4va þðÀ1Þ v04v0 ðaX1Þ:

cn The action of UZ on FZ is given explicitly by the formulae X

EiFlð1Þ¼ bl;rFlÀdr ð1Þ; ð3:32Þ A n r with lÀdr Zþ sr¼À;Àþ or ÀÀ X

FiFlð1Þ¼ bl;rFlþdr ð1Þ; ð3:33Þ A n r with lþdr Zþ sr¼þ;Àþ or þþ ARTICLE IN PRESS

J. Brundan / Advances in Mathematics 182 (2004) 28–77 55

y A n where ðs1; ; snÞ is the i-signature of l Zþ deﬁned according to (2.30) and ( zðlÞ 1 þðÀ1Þ if sr ¼Àþ; bl;r ¼ 1 otherwise:

bn cn The pairing /:; :S between E and F deﬁned earlier induces a pairing at q ¼ 1 n cn between EZ and FZ with / S LÀw0lð1Þ; Umð1Þ ¼ dl;m ð3:34Þ

A n forall l; m Zþ: Alternatively, as follows immediately from (3.7) and (3.10) taken at q ¼ 1; we have that / S EÀw0lð1Þ; Fmð1Þ ¼ dl;m ð3:35Þ

A n forall l; m Zþ: The ﬁnal theorem of the section gives an explicit formula for the coefﬁcients um;lðqÞ at q ¼ 1:

A n I m ? ? Theorem 3.36. Let l Zþ and p :¼ #l=2 : Choose 1pr1o orpospo os1pn y y y such that lrq þ lsq ¼ 0 for all q ¼ 1; ; p: Let I0 ¼fjl1j; ; jlnjg: For q ¼ 1; ; p; deﬁne Iq and kq inductively according to the following rules: e (1) if lrq 40; let kq be the smallest positive integer with lrq þ kq IqÀ1; and set Iq ¼

IqÀ1,flrq þ kqg; 0 0 e 0 (2) if lrq ¼ 0; let kq and kq be the smallest positive integers with kq; kq IqÀ1; kqokq if 0 0 zðlÞ is even and kq4kq if zðlÞ is odd, and set Iq ¼ IqÀ1,fkq; kqg: y A p n Finally, for each y ¼ðy1; ; ypÞ f0; 1g ; letPRyðlÞ denote the unique element of Zþ p lying in the same Sn-orbit as the weight l þ q¼1 yqkq ðdrq À dsq Þ: Then, X ðzðlÞÀzðRyðlÞÞÞ=2 Ulð1Þ¼ 2 FRyðlÞð1Þ; y

p summing over all y ¼ðy1; y; ypÞAf0; 1g :

Proof. Use the algorithm explained in Section 3.5, (3.32)–(3.33) and induction. See [B3, Theorem 3.34(i)] fora similarargument. & P n ð#lÀzðlÞÞ=2 IzðlÞ=2m Corollary 3.37. For all lAZ ; A n um;lð1Þ¼2 3 : þ m Zþ

Example 3.38. Let us compute Uð5;3;2;1;0;0;À1;À4;À6Þð1Þ using the theorem, recall Example 3.38. We have that p ¼ 2 and ði1; i2; j2; j1Þ¼ð4; 5; 6; 7Þ; then get k1 ¼ 6 0 (hence the entries ð1; À1Þ change to ð7; À7Þ and ðk2; k2Þ¼ð8; 9Þ (hence the entries ARTICLE IN PRESS

56 J. Brundan / Advances in Mathematics 182 (2004) 28–77

ð0; 0Þ change to ð8; À8Þ). Hence

Uð5;3;2;1;0;0;À1;À4;À6Þð1Þ¼Fð5;3;2;1;0;0;À1;À4;À6Þð1ÞþFð7;5;3;2;0;0;À4;À6;À7Þð1Þ

þ 2Fð8;5;3;2;1;À1;À4;À6;À8Þð1Þþ2Fð8;7;5;3;2;À4;À6;À7;À8Þð1Þ:

Forcomparison,

Uð5;3;2;1;0;0;0;À1;À4;À6Þð1Þ¼Fð5;3;2;1;0;0;0;À1;À4;À6Þð1ÞþFð7;5;3;2;0;0;0;À4;À6;À7Þð1Þ

þ 2Fð9;5;3;2;1;0;À1;À4;À6;À9Þð1Þþ2Fð9;7;5;3;2;0;À4;À6;À7;À9Þð1Þ:

Example 3.39. We have by the theorem that:

Uð0;0Þð1Þ¼Fð0;0Þð1Þþ2Fð1;À1Þð1Þ;

Uð0;0;0Þð1Þ¼Fð0;0;0Þð1Þþ2Fð2;0;À2Þð1Þ;

Uð0;0;0;0Þð1Þ¼Fð0;0;0;0Þð1Þþ2Fð1;0;0;À1Þð1Þþ2Fð3;0;0;À3Þð1Þþ4Fð3;1;À1;À3Þð1Þ;

Uð1;À1Þð1Þ¼Fð1;À1Þð1ÞþFð2;À2Þð1Þ;

Uð1;0;À1Þð1Þ¼Fð1;0;À1Þð1ÞþFð2;0;À2Þð1Þ;

Uð1;0;0;À1Þð1Þ¼Fð1;0;0;À1Þð1ÞþFð2;0;0;À2Þð1Þþ2Fð3;1;À1;À3Þð1Þþ2Fð3;2;À2;À3Þð1Þ:

Hence by (3.10),

Eð1;À1Þð1Þ¼Lð1;À1Þð1Þþ2Lð0;0Þð1Þ;

Eð1;0;À1Þð1Þ¼Lð1;0;À1Þð1Þ;

Eð1;0;0;À1Þð1Þ¼Lð1;0;0;À1Þð1Þþ2Lð0;0;0;0Þð1Þ;

Eð2;À2Þð1Þ¼Lð2;À2Þð1Þ;

Eð2;0;À2Þð1Þ¼Lð2;0;À2Þð1ÞþLð1;0;À1Þð1Þþ2Lð0;0;0Þð1Þ;

Eð2;0;0;À2Þð1Þ¼Lð2;0;0;À2Þð1ÞþLð1;0;0;À1Þð1Þ: ARTICLE IN PRESS

J. Brundan / Advances in Mathematics 182 (2004) 28–77 57

4. Character formulae

4.1. Representations of QðnÞ

Now we are ready to introduce the supergroup G ¼ QðnÞ into the picture. For the remainder of the article, we will work over the ground field C; and all objects (superalgebras, superschemes, y) will be defined over C without further mention. We refer the reader to [B1,BK2] for a fuller account of the basic results concerning the representation theory of G summarized here, most of which were proved originally by Penkov [P1]. By definition [BK2, Section 3], G is the functor from the category of commutative superalgebras to the category of groups defined on a superalgebra A so that GðAÞ is the group of all invertible 2n Â 2n matrices of the form

; ð4:1Þ

0 where S is an n Â n matrix with entries in A0% and S is an n Â n matrix with entries in A1% : The underlying purely even group Gev of G is by definition the functorfrom superalgebras to groups with GevðAÞ :¼ GðA0% Þ: In ourcase, GevðAÞ consists of all 0 matrices in GðAÞ of the form (4.1) with S ¼ 0; so GevDGLðnÞ: We also need the Cartan subgroup H of G defined on a commutative superalgebra A so that HðAÞ consists of all matrices in GðAÞ with S; S0 being diagonal matrices, and the negative Borel subgroup B of G defined so that BðAÞ consists of all matrices in GðAÞ with S; S0 being lowertriangular. Let T ¼ Hev be the usual maximal torus of Gev consisting of diagonal matrices. The character group XðTÞ¼HomðT; GmÞ is the free abelian group on generators d1; y; dn; where dr picks out the rth diagonal entry of a diagonal matrix. We will n n always identifyP XðTÞ with Z ; the tuple l ¼ðl1; y; lnÞAZ corresponding to the n A character r¼1 lrdr XðTÞ: The root system associated to Gev is denoted R ¼ þ þ þ n n R ,ðÀR Þ; where R ¼fdr À ds j 1prospngCZ : The dominance ordering on Z is defined by lpm if and only if m À l is a sum of positive roots. This is not the same as the Bruhat ordering % defined in Section 2.3, though we have by Lemma 2.15 that l%m implies lpm: A representation of G means a natural transformation r : G-GLðMÞ forsome vectorsuperspace M; where GLðMÞ is the supergroup with GLðM; AÞ being equal to the group of all even automorphisms of the A-supermodule M#A; foreach commutative superalgebra A: Equivalently, as with group schemes [J1, I.2.8], M is a right k½G-comodule, i.e. there is an even structure map Z : M-M#k½G satisfying the usual comodule axioms. We will usually refer to such an M as a G-supermodule. Let Cn denote the category of all finite-dimensional G-supermodules. Note that we allow arbitrary (not necessarily homogeneous) morphisms in Cn: We write P forthe parity change functor, and denote the dual of a finite-dimensional G-supermodule M Ã t by M : There is another natural duality M/M on Cn; see e.g. [BK2, Section 10] forthe definition. ARTICLE IN PRESS

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There are two sorts of irreducible G-supermodule: either type M if EndGðMÞ is one dimensional or type Q if EndGðMÞ is two dimensional. Forexample, we have the 0 0 natural representation V of G; the vectorsuperspace on basis v1; y; vn; v1; y; vn; 0 # where vr is even and vr is odd. For a superalgebra A; we identify elements of V A with column vectors 0 1 a B 1 C B ^ C B C Xn B C B an C ðv #a þ v0 #a0 Þ2B C: r r r r B 0 C r 1 B a1 C ¼ B C @ ^ A 0 an

Then, the action of GðAÞ on V#A deﬁning the supermodule structure is the obvious action on column vectors by left multiplication. The map

- / 0 0 / J : V V; vr vr; vr À vr ð4:2Þ is an odd automorphism of V as a G-supermodule. Hence, V is irreducible of type Q: 7 7 A n l l1 ln A 1 1 For l Z ; we write x ¼ x1 yxn Z½x1 ; y; xn : Given an H-supermodule M; we let Ml denote its l-weight space with respect to the torus T: Identifying the Weyl group associated to Gev with the symmetric group Sn; the character X l ch M :¼ ðdim MlÞx lAZn of a ﬁnite-dimensional G-supermodule M is naturally Sn-invariant, so is an element 71 71 Sn of the ring Z½x1 ; y; xn of symmetric functions. Forevery lAZn; there is by Brundan and Kleshchev [BK2, Lemma 6.4] a unique irreducible H-supermodule denoted uðlÞ with character 2IðhðlÞþ1Þ=2mxl; where hðlÞ¼ n À zðlÞ denotes the numberof r ¼ 1; y; n forwhich lra0: It is an irreducible H- supermodule of type M if hðlÞ is even, type Q otherwise. We will often regard uðlÞ instead as an irreducible B-supermodule via the obvious epimorphism B-H: Introduce the induced supermodule

0 G 0 H ðlÞ :¼ indB uðlÞ¼H ðG=B; LðuðlÞÞÞ; ð4:3Þ see [B1, Section 2], [BK2, Section 6], forthe detailed construction.The following theorem is due to Penkov [P1, Theorem 4], see also [BK2, Theorem 6.11].

A n 0 A n 0 Theorem 4.4. For l Z ; H ðlÞ is non-zero if and only if l Zþ: In that case H ðlÞ has a unique irreducible submodule denoted LðlÞ: Moreover,

(i) fLðlÞg A n is a complete set of pairwise non-isomorphic irreducible G-super- l Zþ modules; ARTICLE IN PRESS

J. Brundan / Advances in Mathematics 182 (2004) 28–77 59

(ii) LðlÞ is of the same type as uðlÞ; i.e. type M if hðlÞ is even, type Q otherwise; I m (iii) ch LðlÞ¼2 ðhðlÞþ1Þ=2 xl þ ðÃÞ where ðÃÞ is a linear combination of xm for mol; Ã t (iv) LðlÞ DLðÀw0lÞ and LðlÞ DLðlÞ:

4.2. Euler characteristics

Let KðCnÞ denote the Grothendieck group of the superadditive category Cn; in the sense of [BK1, Section 2.3]. So by Theorem 4.4(i), KðCnÞ is the free abelian group on basis f½LðlÞg A n : Since ch is additive on short exact sequences, there is an induced l Zþ map

7 - 1 71 Sn / ch : KðCnÞ Z½x1 ; y; xn ; ½M ch M:

Theorem 4.4(iii) implies that the irreducible characters are linearly independent, hence this map is injective. By Theorem 4.4(iv), the duality Ã on ﬁnite-dimensional Ã G-supermodules induces an involution Ã : KðCnÞ-KðCnÞ with ½LðlÞ ¼½LðÀw0lÞ: On the otherhand, the duality t leaves characters invariant, so gives the identity map at the level of Grothendieck groups. Foreach lAZn; we have the highercohomology supermodules

i i G i H ðlÞ :¼ R indP uðlÞ¼H ðG=P; LðuðlÞÞÞ; ð4:5Þ where P is the largest parabolic subgroup of G to which the B-supermodule uðlÞ can be lifted and LðuðlÞÞ denotes the G-equivariant OG=P-supermodule induced by the P-supermodule uðlÞ; we refer to [B1, Section 4] fordetails. In particular,it is known that each HiðlÞ is finite-dimensional and is zero for ib0; so the Euler characteristic X ½EðlÞ :¼ ðÀ1Þi½HiðlÞ ð4:6Þ iX0 is a well-defined element of the Grothendieck group KðCnÞ: Its character X ch EðlÞ :¼ ðÀ1Þi ch HiðlÞð4:7Þ iX0 can be computed explicitly by a method going back at least to Penkov [P2, Section A n 2.3]. We need to recall the definition of Schur’sP-function pl for l Zþ: 0 1

X B Y À1 C B l 1 þ xi xjC pl ¼ wx@ À1 A; ð4:8Þ 1 À x xj wASn=Sl 1piojpn i li4lj where Sn=Sl denotes the set of minimal length coset representatives for the stabilizer Sl of l in Sn: Note that pl is equal to the Hall–Littlewood symmetric ARTICLE IN PRESS

60 J. Brundan / Advances in Mathematics 182 (2004) 28–77 function 0 1 X B Y C B l xi À txjC plðtÞ¼ wx@ A; ð4:9Þ xi À xj wASn=Sl 1piojpn li4lj evaluated at t ¼À1; see [M, III(2.2),III.8] (actually, Macdonald only describes the case when all liX0; but everything easily extends to liAZ). We note the following combinatorial lemma. P ? y Lemma 4.10. ðx1 þ þ xnÞpl ¼ r plþdr ; where the sum is over all r ¼ 1; ; n such A n a that l þ dr Zþ and moreover lr À 1 if zðlÞ is odd.

n Proof. Let lAZ with l1X?Xln: Following the proof of [M, III(3.2)], one shows that X ? ? nr ðx1 þ þ xnÞplðtÞ¼ ð1 þ t þ þ t Þplþdr ðtÞ; r where the sum is over all r ¼ 1; y; n with lrolrÀ1 if r41; and nr denotes #fs j 1pspn; ls ¼ lr þ 1g: The lemma follows on setting t ¼À1: &

The following theorem is [PS1, Proposition 1], see [B1, Theorem 4.7] fora more detailed exposition.

A n IðhðlÞþ1Þ=2m Theorem 4.11. For each l Zþ; ch EðlÞ¼2 pl:

This shows in particular that ch EðlÞ equals 2IðhðlÞþ1Þ=2mxl þ ðÃÞ where ðÃÞ is a linearcombination of xm for mol: Comparing with Theorem 4.4(iii), we deduce that A n foreach l Zþ; the decomposition numbers dm;l deﬁned from X ½EðmÞ ¼ dm;l½LðlÞ ð4:12Þ A n l Zþ satisfy dm;m ¼ 1 and dm;l ¼ 0 for l4/ m: Hence, f½EðlÞg A n gives anothernatural l Zþ basis for the Grothendieck group KðCnÞ: Also deﬁne the inverse decomposition À1 numbers dm;l from X À1 ½LðmÞ ¼ dm;l½EðlÞ: ð4:13Þ A n l Zþ

À1 À1 Again we have that dm;m ¼ 1 and dm;l ¼ 0 for l4/ m: ARTICLE IN PRESS

J. Brundan / Advances in Mathematics 182 (2004) 28–77 61

4.3. Category On

The Lie superalgebra g of G can be identiﬁed with the Lie superalgebra qðnÞ of all matrices of the form

ð4:14Þ underthe supercommutator ½:; :; where S and S0 are n Â n matrices over C and such 0 a matrix is even if S ¼ 0 orodd if S ¼ 0; see [BK2, Section 4]. We will let er;s resp. 0 er;s; denote the even, resp. odd, matrix unit, i.e. the matrix of the form (4.14) with the rs-entry of S; resp. S0; equal to 1 and all other entries equal to zero. We will 0 0 abbreviate hr :¼ er;r; hr :¼ er;r: Let h be the Cartan subalgebra of g spanned by 0 fhr; hr j 1prpng; and let b be the positive Borel subalgebra spanned by 0 fer;s; er;s j 1prpspng: Note h is the Lie superalgebra of the subgroup H; but b is not the Lie superalgebra of B since that consisted of lower triangular matrices! (We make this perverse choice to avoid an additional twist when inducing from B to G versus from b to g:) n Forany lAZ and an h-supermodule M; we deﬁne the l-weight space Ml of M by

Ml ¼fmAM j him ¼ lim foreach i ¼ 1; y; ng:

When M is an H-supermodule viewed as an h-supermodule in the canonical way, the notion of weight space defined here agrees with the earlier one. Let On denote the category of all finitely generated g-supermodules M that are locally finite dimensional as b-supermodules and satisfy

M ¼ " Ml; lAZn cf. [BGG].By[BK2, Corollary 5.7], we can identify the category Cn with the full subcategory of On consisting of all the ﬁnite-dimensional objects.

Lemma 4.5. Every MAOn has a composition series.

Proof. Since the universal enveloping superalgebra UðgÞ of g is Noetherian, every finitely generated g-supermodule M admits a descending filtration M ¼ M04M14? with each MiÀ1=Mi simple. We need to show this filtration is of finite length in case MAOn: It suffices for this to show that the restriction of M to D g0% glðnÞ has a composition series. Since UðgÞ is a free left Uðg0% Þ-supermodule of finite rank, M is still finitely generated when viewed as a g0% -module, and clearly it is locally finite-dimensional over b0% : Hence the restriction of M to g0% belongs to the analogue of category On for g0% : It is well-known all such g0% -modules have a composition series. & ARTICLE IN PRESS

62 J. Brundan / Advances in Mathematics 182 (2004) 28–77

Foreach lAZn; we define the Verma supermodule # MðlÞ :¼ UðgÞ UðbÞuðlÞAOn; ð4:16Þ where uðlÞ is as defined in the previous subsection but viewed now as a b- supermodule by inflation from h: By the PBW theorem, we have that

Y À1 I m 1 þ x x ch MðlÞ¼2 ðhðlÞþ1Þ=2 xl i j: ð4:17Þ 1 À xÀ1x 1piojpn i j

Also note by its deﬁnition as an induced supermodule that MðlÞ is universal amongst all g-supermodules generated by a b-stable submodule isomorphism to uðlÞ: The following theorem is quite standard and parallels Theorem 4.4.

Theorem 4.18. For every lAZn; MðlÞ has a unique irreducible quotient denoted LðlÞ: Moreover,

(i) fLðlÞglAZn is a complete set of pairwise non-isomorphic irreducibles in category On; (ii) LðlÞ is of the same type as uðlÞ; i.e. type M if hðlÞ is even, type Q otherwise; I m (iii) ch LðlÞ¼2 ðhðlÞþ1Þ=2 xl þ ðÃÞ where ðÃÞ is a linear combination of xm for mol; A n (iv) LðlÞ is ﬁnite-dimensional if and only if l Zþ; in which case it is the same as the supermodule denoted LðlÞ before.

Let KðOnÞ be the Grothendieck group of the category On: By Lemma 4.15 and Theorem 4.18(i), KðOnÞ is the free abelian group on basis f½LðlÞglAZn : Howeverthe b ½MðlÞ’s do not form a basis for KðOnÞ: To get round this problem, let KðOnÞ be the completion of KðOnÞ with respect to the descending ﬁltration ðKd ðOnPÞÞdAZ; where A n n X Kd ðOnÞ is the subgroup of KðOnÞ generated by f½LðlÞg for l Z with r¼1 rlr d: Taking characters induces a well-deﬁned map

b - 71 71 / ch : KðOnÞ Z½½x1 ; y; xn ; ½M ch M:

By Theorem 4.18(iii), the characters of the irreducible supermodules in On are linearly independent, hence the map ch is injective. Moreover, ½MðlÞ equals ½LðlÞ þ ðÃÞ where ðÃÞ is a ﬁnite linearcombination of ½LðmÞ’s for mol: So working b in KðOnÞ; we can also write ½LðlÞ as ½MðlÞ þ ðwÞ where ðwÞ is a possibly inﬁnite linearcombination of ½MðmÞ’s for mol: This shows that the elements f½MðlÞglAZn b form a topological basis for KðOnÞ:

4.4. Central characters

Let Z denote the even center of the universal enveloping superalgebra UðgÞ: n Sergeev [S1] has constructed an explicit set of generators of Z: Foreach lAZ ; let wl ARTICLE IN PRESS

J. Brundan / Advances in Mathematics 182 (2004) 28–77 63 be the central character afforded by the Verma supermodule MðlÞ; so zAZ acts on MðlÞ as scalarmultiplication by wlðzÞ: Also recall the deﬁnition of wtðlÞ from (2.12), which is an element of the weight lattice P of type bN as deﬁned in Section 2.1. As a consequence of Sergeev’s description of Z; see e.g. [BK2, Lemma 8.9(ii),(iv)],we have the following fundamental fact:

n Theorem 4.19. For l; mAZ ; wl ¼ wm if and only if wtðlÞ¼wtðmÞ:

Foreach gAP; let Og denote the full subcategory of On consisting of the objects all of whose irreducible subquotients are of the form LðlÞ for lAZn with wtðlÞ¼g: As a consequence of Theorem 4.19, we have the block decomposition

On ¼ " Og: ð4:20Þ gAP

Moreover, Og is non-zero if and only if g is a non-trivial weight of the tensor space Tn of Section 2.2. Note that given lAZn with wtðlÞ¼g; hðlÞðg; gÞ=2 ðmod 2Þ: Hence, recalling Theorem 4.18(ii), all the irreducible supermodules belonging to the block Og are of the same type. We refer to this as the type of the block Og: type M if ðg; gÞ=2 is even, type Q if ðg; gÞ=2 is odd. The block decomposition (4.20) induces the block decomposition

KðOnÞ¼" KðOgÞð4:21Þ gAP of the Grothendieck group, so here KðOgÞ is the Grothendieck group of the category b b n Og: Let KðOgÞ be the closure of KðOgÞ in KðOnÞ: The elements f½MðlÞg for lAZ b with wtðlÞ¼g form a topological basis for KðOgÞ: In a similarfashion, we have the block decomposition of the category Cn of ﬁnite- dimensional G-supermodules:

Cn ¼ " Cg: ð4:22Þ gAP

This time, Cg is non-zero if and only if g is a non-trivial weight of the dual exterior n power E from Section 3.3. Note moreover that the natural embedding of Cn into On embeds Cg into Og: We also get the block decomposition of the Grothendieck group:

KðCnÞ¼" KðCgÞ: ð4:23Þ gAP

A n The elements f½LðlÞg for l Zþ with wtðlÞ¼g form a basis for KðCgÞ: Q 1ÀxÀ1x i j A À1 Lemma 4.24. 1 i j n À1 Z½½xi xj j 1piojpn can be expressed as an inﬁnite p o p 1þxi xj linear combination of xm’s for mp0 with wtðmÞ¼0: ARTICLE IN PRESS

64 J. Brundan / Advances in Mathematics 182 (2004) 28–77

b M n K Proof. Since theP f½ ðmÞgmAZ ;wtðmÞ¼0 form a topological basis for ðO0Þ; we can A write ½Lð0Þ ¼ mp0;wtðmÞ¼0 am½MðmÞ forsome coefﬁcients am Z: Taking characters using (4.17) gives ! X Y À1 IðhðmÞþ1Þ=2m m 1 þ xi xj 1 ¼ 2 amx : 1 À xÀ1x mp0 1piojpn i j wtðmÞ¼0

The lemma follows. &

A n Theorem 4.25. For l Zþ with wtðlÞ¼g; the Euler characteristic ½EðlÞ belongs to the block KðCgÞ:

A n Proof. Take l Zþ: By Theorem 4.11 and (4.8), we have that 0 1

X B Y À1 Y À1 C IðhðlÞþ1Þ=2m B l 1 À xi xj 1 þ xi xjC ch EðlÞ¼2 wx@ À1 À1 A 1 þ x xj 1 À x xj wASn=Sl 1piojpn i 1piojpn i li¼lj X Y À1 Y À1 IðhðlÞþ1Þ=2m cðwÞ wl 1 À xwi xwj 1 þ xi xj ¼ 2 ðÀ1Þ x À1 À1 : 1 þ x xwj 1 À x xj wASn=Sl 1piojpn wi 1piojpn i li¼lj Q 1ÀxÀ1x By Lemma 4.24, xwl wi wj is a (possibly infinite) linearcombination 1 i j n;li¼lj À1 p o p 1þxwi xwj of xm’s for mpwl with wtðmÞ¼wtðwlÞ¼g: Hence recalling (4.17), ch EðlÞ is a (possibly infinite) linearcombination of ch MðmÞ’s for mAZn with wtðmÞ¼g: b Working instead in KðOnÞ; we have shown that we can write ½EðlÞ as a (possibly n b infinite) linearcombination of ½MðmÞ’s for mAZ with wtðmÞ¼g; i.e. ½EðlÞAKðOgÞ: The theorem follows. &

The theorem immediately implies that the decomposition numbers dm;l deﬁned À1 in (4.12) and the inverse decomposition numbers dm;l deﬁned in (4.13) are zero wheneverwt ðmÞawtðlÞ: In particular:

A n Corollary 4.26. The elements f½EðlÞg for l Zþ with wtðlÞ¼g form a basis for KðCgÞ:

4.5. Translation functors

n Now we are ready to link the Grothendieck group KðCnÞ with the UZ-module EZ constructed in Section 3.6. We deﬁne an isomorphism of abelian groups - n / A þ i : KðCnÞ EZ; ½EðlÞ Elð1Þðl Zn Þ: ð4:27Þ ARTICLE IN PRESS

J. Brundan / Advances in Mathematics 182 (2004) 28–77 65

n Using i; we lift the actions of the generators of UZ on EZ from (3.31) to deﬁne an action of UZ directly on the Grothendieck group KðCnÞ: Note that Corollary 4.26 shows that the block decomposition (4.23) coincides with the usual weight space decomposition of KðCnÞ as a UZ-module. Using Lemma 3.20 specialized at q ¼ 1; we can write down the action of the operators Ei; Fi on KðCnÞ explicitly: X Ei½EðlÞ ¼ cl;r½Eðl À drÞ; ð4:28Þ A n r with lÀdr Zþ; sr¼À;Àþ or ÀÀ

X Fi½EðlÞ ¼ cl;r½Eðl þ drÞ; ð4:29Þ A n r with lþdr Zþ; sr¼þ;Àþ or þþ

A n X y for l Zþ and i 0: Here, ðs1; ; snÞ is the i-signature of l deﬁned according to (2.30) and ( zðlÞ 1 þðÀ1Þ if sr ¼ÀÀ or þþ; cl;r ¼ 1 otherwise:

The goal in the remainder of this subsection is to give a representation theoretic interpretation of the operators Ei and Fi: We need certain ‘‘translation functors’’

i Tr ; Tri : On-On; one foreach iX0; similarto the functorsdefined by Penkov and Serganova [PS2, Section 2.3]. Recalling the decomposition (4.22), it suffices to define the functors i Tr ; Tri on the subcategory Og forsome fixed gAP; since one can then extend - additively to get the definition on On itself. We will write prg : On Og forthe natural projection functor. Given MAOg; let

i # # Ã Tr M ¼ prgÀai ðM VÞ; Tri M ¼ prgþai ðM V Þ; ð4:30Þ recalling from Section 2.1 that aiAP denotes the ith simple root of bN: On a i morphism f : M-N in Og; Tr f and Tri f are the restrictions of the maps f #id: Obviously these functors send ﬁnite dimensional supermodules to ﬁnite-dimensional supermodules, so also give us functors

i Tr ; Tri : Cn-Cn by restriction. ARTICLE IN PRESS

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i Lemma 4.31. For each iX0; the functors Tr and Tri (on either of the categories On or Cn) are both left and right adjoint to each other, hence both are exact functors. Moreover, there are natural isomorphisms

i Ã Ã Ã i Ã ðTr MÞ DTriðM Þ; ðTri MÞ DTr ðM Þ; ð4:32Þ

i t i t t t ðTr MÞ DTr ðM Þ; ðTri MÞ DTriðM Þð4:33Þ

for each MACn:

Proof. The ﬁrst part is a well-known fact about translation functors. For the second part, we obviously have natural isomorphisms ðM#VÞÃDMÃ#V Ã; t t ðM#VÞ DM #V: So it sufﬁces to show that Ã-duality maps Cg into CÀg: and that t-duality maps Cg into itself. This is immediate from Theorem 4.4(iv). &

A n Lemma 4.34. For l Zþ and iX0; and a ﬁnite-dimensional G-supermodule M belonging to the block Cg for some gAP; we have that

( Ei½M if i ¼ 0 and Cg is of type M; ½Tri M¼ 2Ei½M if ia0 or Cg is of type Q;

( Fi½M if i ¼ 0 and Cg is of type M; ½Tri M¼ 2Fi½M if ia0 or Cg is of type Q:

i Proof. We explain the argument for Tr ; the case of Tri being similar. Since the i i functorTr is exact, it induces an additive map also denoted Tr on KðCnÞ: By A n Corollary 4.26, the f½EðlÞg for l Zþ with wtðlÞ¼g form a basis for KðCgÞ: Therefore it sufﬁces to consider Tri½EðlÞ forsuch a l: Observe that ch V ¼ 2ðx1 þ ? þ xnÞ and that the functor ?#V is isomorphic to i the functor "iX0 Tr : So a calculation using Theorem 4.11 and Lemma 4.10 gives that

X " i Tr ½El¼ dl;r½Elþdr ; iX0 A n r with lþdr Zþ ARTICLE IN PRESS

J. Brundan / Advances in Mathematics 182 (2004) 28–77 67 where 8 > 2ifl a À 1; 0; > r > > 1iflr ¼ 0andhðlÞ is even; <> 2iflr ¼ 0andhðlÞ is odd; dl;r ¼ > 0ifl ¼À1 and zðlÞ is odd; > r > > 2iflr ¼À1; zðlÞ is even and hðlÞ is even; :> 4iflr ¼À1; zðlÞ is even and hðlÞ is odd:

Now apply prwtðlÞÀai to both sides and use Theorem 4.25 and (4.29) to deduce that ( Fi½EðlÞ if i ¼ 0andhðlÞ is even; Tri½EðlÞ ¼ 2Fi½EðlÞ if ia0orhðlÞ is odd:

This completes the proof. &

4.6. Crystal structure

i In this section we relate the structure of the supermodules Tr LðlÞ and Tri LðlÞ forcertain weights l to the crystal operators deﬁned in Section 3.4. The methods employed here are essentially the same as those of Penkov and Serganova in [PS2, Section 2.3]. Throughout the subsection we will use the following notation: for X supermodules X and Y; Y will denote some extension of X by Y; and ½M : L will denote the composition multiplicity of an irreducible L in M:

Lemma 4.35. Let l ¼ð1; 0nÀ2; À1Þ: Then ½MðlÞ : Lð0ÞX2:

0 0 0 0 0 0 0 0 0 Proof. Pick a basis v; v for MðlÞl such that h1v ¼ v ; h1v ¼ v; hnv ¼ v ; hnv ¼Àv 0 0 0 and hrv ¼ hrv ¼ 0 for r ¼ 2; y; n À 1: Let

0 0 0 0 0 0 x ¼ e2;1ðe3;2 þ e3;2h3Þyðen;nÀ1 þ en;nÀ1hnÞv; y ¼ h1x:

We claim that /x; yS is a two-dimensional indecomposable b-submodule of MðlÞ: The lemma follows immediately form this, since both x and y are of weight 0. To prove the claim, we proceed by induction on n: The base case n ¼ 2 follows on 0 0 0 0 checking that x ¼ e2;1v is annihilated by e1;2 and e1;2 and that y ¼ h1x ¼ h2x: For n42; we set

0 0 0 0 w ¼ðen;nÀ1 þ en;nÀ1hnÞv ¼ en;nÀ1v þ en;nÀ1v ;

0 0 0 0 0 0 w ¼Àðen;nÀ1 þ en;nÀ1hnÞv ¼ en;nÀ1v À en;nÀ1v : ARTICLE IN PRESS

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0 0 0 0 0 0 0 0 0 Now one checks that h1w ¼ w ; h1w ¼ w; hnÀ1w ¼ w ; hnÀ1w ¼Àw and that hrw ¼ 0 0 y 0 0 hrw ¼ 0 foreach r ¼ 2; ; n À 2; n: Moreover, w; w are annihilated by er;rþ1; er;rþ1 for r ¼ 1; y; n À 1: Now the result follows from the induction hypothesis. &

A n Lemma 4.36. Let l Zþ with lr40 and lr þ ls ¼ 0 for some 1prospn: Then, ( 1 if lr41; ½MðlÞ : Lðl À dr þ dsÞ ¼ 2 if lr ¼ 1:

Proof. We will actually only prove here that the left-hand side is X the right-hand side, since that is all we really need later on. The fact that the left-hand side actually equals the right-hand side can easily be extracted from the proof of Lemma 4.39. First suppose that lr41: By Penkov and Serganova [PS2, Proposition 2.1],

HomgðMðl À dr þ dsÞ; MðlÞÞa0:

This immediately implies that ½MðlÞ : Lðl À dr þ dsÞX1: Now assume that lr ¼ 1; in which case we need to show that ½MðlÞ : Lðl À dr þ dsÞX2: Let p be the parabolic subalgebra of g consisting of all matrices of the form (4.14) such that S and S0 are block upper triangular matrices, with block sizes 1rÀ1; ðs À r þ 1Þ; 1nÀs down the # diagonal. Consider the Verma supermodule MpðlÞ¼UðpÞ UðbÞuðlÞ over p: By Lemma 4.35, it contains the irreducible p-supermodule Lpðl À dr þ dsÞ of highest # D weight l À dr þ ds with multiplicity at least two. Since UðgÞ UðpÞMpðlÞ MðlÞ and # 7 UðgÞ UðpÞLpðl À dr þ dsÞ Lðl À dr þ dsÞ; the lemma follows. &

Lemma 4.37. Let lAZn: Then M :¼ MðlÞ#V has a ﬁltration 0 ¼ M0oM1o?oMn ¼ M such that 8 " a > Mðl þ drÞ PMðl þ drÞ if lr À 1; 0 > > Mðl þ d Þ if l ¼ 0 and hðlÞ even; <> r r " D Mðl þ drÞ PMðl þ drÞ if lr ¼ 0 and hðlÞ odd; Mr=MrÀ1 > > MðlþdrÞ > if lr ¼À1 and hðlÞ even; > MðlþdrÞ :> PMðlþdrÞ MðlþdrÞ " if lr ¼À1 and hðlÞ odd; MðlþdrÞ PMðlþdrÞ for each r ¼ 1; y; n:

Proof. By the tensoridentity,

# # # D # # MðlÞ V ¼ðUðgÞ UðbÞuðlÞÞ V UðgÞ UðbÞðuðlÞ VÞ: ARTICLE IN PRESS

J. Brundan / Advances in Mathematics 182 (2004) 28–77 69

Moreover, VDuðd1Þ"?"uðdnÞ as an h-supermodule. Now the lemma follows from the observation that

8 " a > uðl þ drÞ Puðl þ drÞ if lr À 1; 0 > > uðl þ d Þ if l ¼ 0 and hðlÞ even; <> r r " # D uðl þ drÞ Puðl þ drÞ if lr ¼ 0 and hðlÞ odd; uðlÞ uðdrÞ > > uðlþdrÞ > if lr ¼À1 and hðlÞ even; > uðlþdrÞ :> PuðlþdrÞ uðlþdrÞ " if lr ¼À1 and hðlÞ odd uðlþdrÞ PuðlþdrÞ

as h-supermodules. We prove this in just two of the situations, the rest being similar. First suppose lr ¼À1andhðlÞ is odd. By character considerations uðlÞ#uðdrÞ has just four composition factors, all isomorphic to uðl þ drÞ: Moreover, both uðlÞ and uðd Þ possess odd automorphisms J ; J ; respectively, with J2 ¼À1: The pffiffiffiffiffiffiffi r 1 2 i 7 À1-eigenspaces of the even automorphism J1#J2 of uðlÞ#uðdrÞ decompose it into a direct sum of two factors, and the map J1#1 gives an odd isomorphism between the factors. Hence uðlÞ#uðdrÞ has the structure given. Second suppose lra À 1; 0 and hðlÞ is even. In that case, uðlÞ is of type M; and by character considerations uðlÞ#uðdrÞ has just two composition factors, both isomorphic to uðl þ drÞ which also has type M: Let J be an odd automorphism of uðdrÞ: Then, 1#J is an odd automorphism of uðlÞ#uðdrÞ: If uðlÞ#uðdrÞ was to be a non-split extension of uðl þ drÞ and Puðl þ drÞ; it would possess a nilpotent odd endomorphism, but no odd automorphism. Hence it must split as a direct sum as required. &

# If we apply the exact functorpr wtðlÞÀai to the filtration of MðlÞ V constructed in i A n the lemma, we obtain a filtration of Tr MðlÞ: Suppose in addition that l Zþ; when LðlÞ is a finite-dimensional quotient of MðlÞ: Then, Tri LðlÞ is a finite-dimensional quotient of Tri MðlÞ: So the filtration of Tri MðlÞ just constructed induces a filtration of the quotient Tri LðlÞ all of whose the factors are finite-dimensional quotients of corresponding factors in the filtration of Tri MðlÞ: In particular, the only non-zero factors that can survive on passing to the quotient necessarily have n highest weights belonging to Zþ: We refer to the filtration constructed in this way as the canonical filtration of Tri LðlÞ: Its properties are used repeatedly in the proofs of the next two lemmas.

A n X Lemma 4.38. Suppose l Zþ and i 0 are such that jiðlÞ¼1: Then, Fi½LðlÞ ¼ ½LðF˜iðlÞÞ: ARTICLE IN PRESS

70 J. Brundan / Advances in Mathematics 182 (2004) 28–77

Proof. We actually prove the slightly stronger statement that ( LðmÞ if i ¼ 0andhðlÞ is even; Tri LðlÞD LðmÞ"PLðmÞ if i40orhðlÞ is odd; where m :¼ F˜iðlÞ: The lemma follows from this and Lemma 4.34. The possible cases for l; m are listed explicitly in Section 3.4, in particular we have that m ¼ l þ ds for some 1pspn: By considering the canonical ﬁltration in each of the cases, Tri MðlÞDMðmÞ if i ¼ 0 and hðlÞ is even, Tri MðlÞDMðmÞ"PMðmÞ otherwise. In particular, ½Tri MðlÞ : LðmÞ ¼ 1ifi ¼ 0 and hðlÞ is even, ½Tri MðlÞ : LðmÞ ¼ 2 otherwise. Now, Tri LðlÞ is a quotient of Tri MðlÞ; and moreover it is self-dual underthe duality t by Theorem 4.4(iv) and (4.33). Hence, Tri LðlÞ is necessarily a quotient of LðmÞ if i ¼ 0 and hðlÞ is even, LðmÞ"PðmÞ otherwise. So to complete the proof, we just need to show that ½Tri LðlÞ : LðmÞ ¼ ½Tri MðlÞ : LðmÞ: Suppose fora contradiction that ½Tri LðlÞ : LðmÞo½Tri MðlÞ : LðmÞ: Then there must be some composition factor Lðl0ÞD/ LðlÞ of MðlÞ such that ½Tri Lðl0Þ : LðmÞ40: Hence, ½Tri Mðl0Þ : LðmÞ40 forsome l0ol with wtðl0Þ¼wtðlÞ: Considering the canonical ﬁltration of Tri Mðl0Þ; there must exist some 1prpn 0 0 0 such that m ¼ l þ dspl þ dr: But then l À dr þ dspl ol and wtðl Þ¼wtðlÞ: It is easy to see in each case that there is no such l0; giving the desired contradiction. &

Forthe next lemma, recallthe deﬁnition of jiðlÞAN from (3.22).

A n X Lemma 4.39. Suppose l Zþ and i 0 are such that jiðlÞ¼0: Then, Fi½LðlÞ ¼ 0:

Proof. The possibilities for l are listed explicitly in Section 3.4. In almost all of the i configurations, we get that Tr LðlÞ¼0; hence Fi½LðlÞ ¼ 0; immediately by looking at the canonical filtration. There are just two difficult cases in which we need to argue further. In the first case, i ¼ 0 and l ¼ðy; 1; 0rÀ1; À1; yÞ forsome rX1; where y denote entries different from À1; 0; 1: We let m ¼ðy; 0rþ1; yÞ and n ¼ðy; 1; 0r; yÞ; where the y are the same entries as in l: Also let c ¼ 1ifhðlÞ is even, c ¼ 2ifhðlÞ is odd. By considering the canonical filtration, we see here that ½Tri MðlÞ : LðnÞ ¼ 2c: By Lemma 4.36, ½MðlÞ : LðmÞX2; hence ½Tri MðlÞ : LðnÞX½Tri LðlÞ : LðnÞ þ 2½Tri LðmÞ : LðnÞ: By Lemma 4.38, ½Tri LðmÞ : LðnÞ ¼ c: This shows that ½Tri LðlÞ : LðnÞ ¼ 0 (and also that ½MðlÞ : LðmÞ ¼ 2; completing the proof of Lemma 4.36 in this case). But by the canonical filtration, Tri LðlÞ has a filtration where all the factors are quotients of MðnÞ; so it has to be zero. In the second case, i40 and l ¼ðy; i þ 1; y; Ài À 1; yÞ; where y denote entries different from Ài À 1; Ài; i; i þ 1: We let m ¼ðy; i; y; Ài; yÞ and n ¼ ðy; i þ 1; y; Ài; yÞ; where the y are the same entries as in l: By the canonical filtration, Tri MðlÞDMðnÞ"PMðnÞ; hence ½Tri MðlÞ : LðnÞ ¼ 2: By Lemma 4.36, ½MðlÞ : LðmÞX1; hence ½Tri MðlÞ : LðnÞX½Tri LðlÞ : LðnÞ þ ½Tri LðmÞ : LðnÞ: By ARTICLE IN PRESS

J. Brundan / Advances in Mathematics 182 (2004) 28–77 71

Lemma 4.38, ½Tri LðmÞ : LðnÞ ¼ 2: This shows that ½Tri LðlÞ : LðnÞ ¼ 0 (and also completes the proof of Lemma 4.38 in this case). Hence, since Tri LðlÞ is a quotient of MðnÞ"PMðnÞ; it has to be zero. &

4.7. Injective supermodules

We refer at this point to [J1, I.3] forthe generalfacts about injective modules over a group scheme, all of which generalize to supergroups. In particular, for every G- supermodule M; there is an injective G-supermodule U; unique up to isomorphism, D A n such that socG M socG U: We call U the injective hull of M: For l Zþ; let UðlÞ denote the injective hull of LðlÞ: Any injective G-supermodule M is isomorphic to a direct sum of UðlÞ’s, the numberof summands isomorphicto UðlÞ being equal to the multiplicity of LðlÞ in socG M:

Lemma 4.40. Each UðlÞ is ﬁnite-dimensional.

Proof. Considerthe functorind G :¼ Hom ðUðgÞ; ?Þ: Since UðgÞ is a free right Gev Uðg0% Þ Uðg0% Þ-supermodule of finite rank, this is an exact functor mapping finite- dimensional Gev-supermodules to finite-dimensional G-supermodules. It is right adjoint to the restriction functor resG from the category of G-supermodules to the Gev A n category of Gev-supermodules, so sends injectives to injectives. Now take l Zþ: Since every G -module is injective, we get that indG resG L l is a finite- ev Gev Gev ð Þ dimensional injective G-supermodule. Moreover, the unit of the adjunction gives an embedding of L l into indG resG L l : Hence the injective hull of L l is finite- ð Þ Gev Gev ð Þ ð Þ dimensional. &

Ã Let Cn be the category of all ﬁnite-dimensional injective G-supermodules. The block decomposition (4.22) of Cn induces an analogous block decomposition of the Ã subcategory Cn

Ã " Ã Cn ¼ Cg : ð4:41Þ gAP

Ã Ã Ã Ã Let KðCnÞ (resp. KðCg Þ) be the Grothendieck group of the category Cn (resp. Cg ). By Ã Ã Lemma 4.40, KðC Þ is the free abelian group on basis f½UðlÞg A n ; and KðC Þ is the n l Zþ g A n subgroup generated by the f½UðlÞg for l Zþ with wtðlÞ¼g: b Ã Ã Form the completion KðCnÞ of the Grothendieck group KðCnÞ with respect to the Ã Ã descending filtration ðKdPðCnÞÞdAZ where Kd ðCnÞ is the subgroup generated by A n n X f½UðlÞg for l Zþ with r¼1 rlnþ1Àr d: The important thing in this definition is that vectors of the form ½UðlÞ þ ðÃÞ make sense whenever ðÃÞ is an infinite linear combination of ½UðmÞ’s for m4l: In particular, the following are well-defined ARTICLE IN PRESS

72 J. Brundan / Advances in Mathematics 182 (2004) 28–77

b Ã A n elements of KðCnÞ foreach l Zþ: X F l : dÀ1 U m ; 4:42 ½ ð Þ ¼ Àw0m;Àw0l½ ð Þ ð Þ A n m Zþ recall (4.13). By the unitriangularity of the inverse decomposition numbers, the b Ã elements f½FðlÞg A n give a topological basis for KðC Þ: Note that ½FðlÞ does not l Zþ n Ã in general belong to KðCnÞ itself. / S Ã We deﬁne a pairing :; : between the Grothendieck groups KðCnÞ and KðCnÞ by letting

/½LðÀw0lÞ; ½UðmÞS ¼ dl;m ð4:43Þ

A n / S foreach l; m Zþ: The pairing :; : extends by continuity to give a pairing also / S b Ã denoted :; : between KðCnÞ and KðCnÞ: In that case, by the deﬁnitions (4.12), (4.42) and (4.43), we have that

/½EðÀw0lÞ; ½FðmÞS ¼ dl;m ð4:44Þ

A n foreach l; m Zþ: We record the following lemma which follows from a standard property of injective hulls, see [J1, I.3.17(3)].

Ã Lemma 4.45. Suppose we are given MACn and UACg for some gAP: Then, ( Ã dim HomGðM ; UÞ if Cg is of type M; / M ; U S ½ ½ ¼ 1 Ã 2 dim HomGðM ; UÞ if Cg is of type Q:

cn Now recall the deﬁnition of the UZ-module FZ from Section 3.6. We deﬁne a continuous isomorphism

Ã cn - b Ã / A þ i : FZ KðCnÞ; Flð1Þ ½FðlÞ ðl Zn Þ: ð4:46Þ

The following lemma shows that this map iÃ is the dual map to i from (4.27) with respect to the pairings /:; :S:

/ S / Ã S A Acn Lemma 4.47. ið½MÞ; v ¼ ½M; i ðvÞ for all ½M KðCnÞ and v FZ:

A n Proof. It sufﬁces to check this for ½M¼½EðÀw0lÞ and v ¼ Fmð1Þ for l; m Zþ; when it follows immediately from (3.35) and (4.44). &

cn As in Section 4.5, we lift the action of UZ on FZ to the completed Grothendieck b Ã Ã group KðCnÞ through the isomorphism i : By Lemmas 4.47 and 2.21, we have at ARTICLE IN PRESS

J. Brundan / Advances in Mathematics 182 (2004) 28–77 73 once that

/Ei½L; ½US ¼ /½L; Ei½US; /Fi½L; ½US ¼ /½L; Fi½US; ð4:48Þ

b Ã foreach iX0 and all ½LAKðCnÞ; ½UAKðCnÞ: The next lemma gives a purely b Ã representation theoretic interpretation of the operators Ei and Fi on KðCnÞ; cf. i Lemma 4.31. Forthe statement, recall by Lemma 4.31 that the functorsTr i and Tr send injectives to injectives.

A n X Lemma 4.49. For l Zþ and i 0; there exist injective G-supermodules EiUðlÞ and FiUðlÞ; unique up to isomorphism, characterized by ( EiUðlÞ if i ¼ 0 and hðlÞ is even; Tri UðlÞ D EiUðlÞ"PEiUðlÞ if ia0 or hðlÞ is odd;

( FiUðlÞ if i ¼ 0 and hðlÞ is even; Tri UðlÞ D FiUðlÞ"PFiUðlÞ if ia0 or hðlÞ is odd:

Moreover, Ei½UðlÞ ¼ ½EiUðlÞ and Fi½UðlÞ ¼ ½FiUðlÞ:

A n X Proof. We just explain the proof for Fi: Take l Zþ and i 0: Note uniqueness of FiUðlÞ is immediate by Krull–Schmidt. For existence, we consider three separate cases. i Case 1: If i ¼ 0 and hðlÞ is even, let FiUðlÞ :¼ Tr UðlÞ: Case 2: If hðlÞ is odd, LðlÞ is of type Q; so possesses an odd automorphism J1 with 2 J1 ¼À1: This induces an odd automorphism also denoted J1 of the injective hull UðlÞ: Also the natural representation V possesses the odd automorphism J defined earlier (4.2). The map J #J induces an even automorphism of the summand 1 pffiffiffiffiffiffiffi Tri UðlÞ of UðlÞ#V: Its 7 À1-eigenspaces decompose Tri UðlÞ into a direct sum # of two G-supermodules,ffiffiffiffiffiffiffi and the map 1 J is an odd isomorphism between them. p i Let FiUðlÞ be the À1-eigenspace (say), then Tr UðlÞDFiUðlÞ"PFiUðlÞ as required. Case 3: If hðlÞ is even and i40; then the map 1#J induces an odd automorphism of Tri UðlÞ; hence also of the socle S of Tri UðlÞ: All constituents of S are of type M; so S must decompose as a G-supermodule as SÀ"Sþ with ð1#JÞS7 ¼ S8: This decomposition of the socle induces a decomposition of the injective supermodule i i Tr UðlÞ; say Tr UðlÞ¼MÀ"Mþ where socG M7 ¼ S7: In this case we let FiUðlÞ¼Mþ (say). It remains to show that Fi½UðlÞ ¼ ½FiUðlÞ: To do this, it suffices to prove that / S / S A n ½LðmÞ; Fi½UðlÞ ¼ ½LðmÞ; ½FiUðlÞ forall m Zþ: This is done using Lemmas i 4.45, 4.34, (4.48) and the adjointness of Tr ; Tri from Lemma 4.31. We just explain the argument in the case that hðlÞ is even and i ¼ 0; the othersituations being entirely similar. First note by weight considerations that both sides of the identity we ARTICLE IN PRESS

74 J. Brundan / Advances in Mathematics 182 (2004) 28–77 are trying to verify are zero unless hðmÞ is odd. Now compute:

/ S / S 1/ i S ½LðmÞ; Fi½UðlÞ ¼ Fi½LðmÞ; ½UðlÞ ¼ 2 ½Tr LðmÞ; ½UðlÞ

1 i Ã ¼ 2 dim HomGððTr LðmÞÞ ; UðlÞÞ

1 Ã ¼ 2 dim HomGðTriðLðmÞ Þ; UðlÞÞ

1 Ã i ¼ 2 dim HomGðLðmÞ ; Tr UðlÞÞ

i ¼ /LðmÞ; ½Tr UðlÞS ¼ /LðmÞ; ½FiUðlÞS:

This completes the proof. &

A n X Lemma 4.50. Let l Zþ and i 0: D ˜ (i) If jiðlÞ¼0 and eiðlÞ40 then EiUðlÞ UðEiðlÞÞ: ˜ (ii) If eiðlÞ¼0 and jiðlÞ40 then FiUðlÞ¼UðFiðlÞÞ:

Proof. We just prove (ii), since (i) follows by applying Ã: We know that FiUðlÞ is a direct sum of injective indecomposables. To compute the multiplicity of UðmÞ fora A n given m Zþ; it sufﬁces to compute

/½LðmÞ; ½FiUðlÞS ¼ /½LðmÞ; Fi½UðlÞS ¼ /Fi½LðmÞ; ½UðlÞS:

By block considerations and Lemmas 4.38 and 4.39, that is zero unless m ¼ Àw0F˜iðlÞ; in which case it is one. Hence FiUðlÞ¼UðF˜iðlÞÞ: &

Now we can construct the injective supermodules UðlÞ: If l is typical, there are no problems:

A n 0 Lemma 4.51. Suppose that l Zþ is typical. Then UðlÞ¼H ðlÞ¼LðlÞ:

A n Proof. If l is typical then by Lemma 2.14 there are no other m Zþ with wtðmÞ¼ wtðlÞ: So by the linkage principle [BK2, Theorem 8.10], the induced module H0ðlÞ is actually equal to LðlÞ in this case. Using this and the observation that uðlÞ is an injective H-supermodule for typical l; we get from [BK2, Theorem 7.5] that 1 1 A n ExtGðLðlÞ; LðmÞÞ ¼ ExtGðLðmÞ; LðlÞÞ ¼ 0 forall m Zþ: In particular, LðlÞ is injective, hence LðlÞ¼UðlÞ: &

Note in particular that the lemma shows that UðlÞ is self-dual with respect to the A n duality t in the case that l is typical. Now suppose that l Zþ is not typical. Apply A n A Procedure 3.26 to get m Zþ and an operator Xi fEi; Fig: Since this process reduces l to a typical weight in ﬁnitely many steps, we may assume inductively that UðmÞ has already been constructed, and that UðmÞDUðmÞt: Just like in Lemma 3.27, but ARTICLE IN PRESS

J. Brundan / Advances in Mathematics 182 (2004) 28–77 75 applying Lemma 4.50 in place of Lemma 3.25, we have that UðlÞDXiUðmÞ: Moreover,

t t t UðlÞ DðXiUðmÞÞ DXiðUðmÞ ÞDXiUðmÞDUðlÞ; hence UðlÞ is also self-dual. We obtain in this way an explicit algorithm to construct all the injective indecomposables. As a by-product we see that each UðlÞ is actually self-dual with respect to the duality t; hence is isomorphic to the projective cover of LðlÞ: Now we can prove the main result of the article. It shows that the map iÃ from n Ã (4.46) maps the canonical basis of FZ to the canonical basis of KðCnÞ given by the injective indecomposables, and that the map i from (4.27) maps the canonical basis n of KðCnÞ given by the irreducible supermodules to the dual canonical basis of EZ:

A n Ã Theorem 4.52. For each l Zþ; i ðUlð1ÞÞ ¼ ½UðlÞ and ið½LðlÞÞ ¼ Llð1Þ:

Proof. In view of Lemma 4.47 and the facts that ½LðlÞ is dual to ½UðÀw0lÞ and Ã Llð1Þ is dual to UÀw0lð1Þ; it sufﬁces to prove just that i ðUlð1ÞÞ ¼ ½UðlÞ: If l is A n typical, then Ulð1Þ¼Flð1Þ and ½UðlÞ ¼ ½FðlÞ; since there are no other m Zþ with wtðmÞ¼wtðlÞ: So the result holds for typical weights. The result follows in general Ã because by deﬁnition the map i commutes with the operators Ei; Fi and the algorithm for constructing the UðlÞ’s explained above exactly parallels the algorithm forconstructingthe Ulð1Þ’s from Section 3.5. &

We get by the theorem and (3.7), (3.10) and (3.17), respectively, that X

½EðlÞ ¼ uÀw0l;Àw0mð1Þ½LðmÞ; ð4:53Þ A n m Zþ X ½LðlÞ ¼ lm;lð1Þ½EðmÞ; ð4:54Þ A n m Zþ X ½UðlÞ ¼ um;lð1Þ½FðmÞ; ð4:55Þ A n m Zþ

A n foreach l Zþ: Note ﬁnally that uÀw0l;Àw0mðqÞ¼ul;mðqÞ; since oðFmÞ¼FÀw0m and oðUmÞ¼UÀw0m by Theorem 3.5. So comparing (4.53) with (4.12) gives that dm;l ¼ um;lð1Þ: The Main Theorem stated in the introduction follows from this statement and Theorem 3.36.

4.8. Conjectures

To conclude the article, we make the following conjecture: for each lAZn; we have that X ½MðlÞ ¼ tÀl;Àmð1Þ½LðmÞ: ð4:56Þ mAZn ARTICLE IN PRESS

76 J. Brundan / Advances in Mathematics 182 (2004) 28–77

Comparing (2.24) and Corollary 2.28, this conjecture is equivalent to the statement that X ½LðlÞ ¼ lm;lð1Þ½MðmÞ; ð4:57Þ mAZn b equality in KðOnÞ: The Nl’s in Section 2 correspond to the supermodules NðlÞ # deﬁned by NðlÞ :¼ UðgÞ UðbÞu˜ðlÞ; where u˜ðlÞ denotes the projective cover of uðlÞ in the category of h-supermodules that are semisimple over h0% : Note at least that zðlÞ ½NðlÞ ¼ 2 ½MðlÞ in the Grothendieck group, cf. (2.7) and [BK2, (7.1)]. The Tl’s in Section 2 should correspond to the indecomposable tilting modules TðlÞ in n category On; cf. [B2, Example 7.10]. We recall that for lAZ ; TðlÞ is the supermodule characterized uniquely up to isomorphism by the properties:

(1) TðlÞAOn is indecomposable; (2) Ext1 N m ; T l 0 forall mAZn; On ð ð Þ ð ÞÞ ¼ (3) TðlÞ has a ﬁltration where the subquotients are of the form NðmÞ for mAZn; starting with NðlÞ at the bottom.

Conjecture (4.56) is equivalent to the statement X ½TðlÞ ¼ tm;lð1Þ½NðmÞ; ð4:58Þ mAZn as follows by [B2, (7.12)].

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