W-graphs for Hecke algebras with unequal parameters(II)
aa r X i v : . [ m a t h . R T ] M a r W -GRAPHS FOR HECKE ALGEBRAS WITH UNEQUALPARAMETERS (II) YUNCHUAN YIN
Abstract.
This paper is the continuation of the work in [14]. In that paperwe generalized the definition of W -graph ideal in the weighted Coxeter groups,and showed how to construct a W -graph from a given W -graph ideal in thecase of unequal parameters.In this paper we study the full W -graphs for a given W -graph ideal. Weshow that there exist a pair of dual modules associated with a given W -graphideal, they are connected by a duality map. For each of the dual modules, theassociated full W -graphs can be constructed.Our construction closely parallelsthat of Kazhdan and Lusztig [6, 10, 11], which can be regarded as the specialcase J = ∅ . It also generalizes the work of Couillens [2], Deodhar [3, 4], andDouglass [5], corresponding to the parabolic cases. Introduction
Let (
W, S ) be a Coxeter system and H ( W ) its Hecke algebra over Z [ q, q − ],the ring of Laurent polynomials in the indeterminate q . This is now called theone parameter case (or the equal parameter case) . In [9] Howlett and Nguyenintroduced the concept of a W -graph ideal in ( W, L ) with respect to a subset J of S , where L is the left weak Bruhat order on W . They showed that a W -graphcan be constructed from a given W -ideal, and a Kazhdan-Lusztig like algorithmwas obtained.In particular, W itself is a W -graph ideal with respect to ∅ , and the W -graphobtained is the Kazhdan-Lusztig W -graph for the regular representation of H ( W )(as defined in [6]). More generally, it was shown that if J is an arbitrary subsetof S then D J , the set of distinguished left coset representatives of W J in W , is a W -graph ideal with respect to J and also with respect to ∅ , and Deodhars parabolicanalogues of the Kazhdan-Lusztig construction are recovered.In [14] we generalized the definition of W -graph ideal in the Coxeter groups witha weight function L , we showed that the W -graph can also be constructed from agiven W -graph ideal.In this paper we continue the work in [14], it grows out of our attempt to under-stand the ”full W -graphs” that include W -graphs and their dual ones. The dualityhas appeared in some literatures, for instance, in the original paper [6] Kazhdanand Lusztig implicitly provided a pair of dual bases C and C ′ for the Hecke alge-bras, Deodhar introduced a pair of dual modules M J and f M J in parabolic cases(see [3, 4]). Mathematics Subject Classification.
Primary and secondary 20C08, 20F55.
Key words and phrases.
Coxeter group, Hecke algebra, W -graph, Kazhdan-Lusztig basis,Kazhdan-Lusztig polynomial. The paper is organised as follows. In Section 1 we present some basic conceptsand facts concerning the weighted Coxeter groups, Hecke algebras and W -graphs.In Section 2, we recall the concept of W -graph ideal. In Section 3, we show thatthere exist a pair of dual modules M ( E J , L ) and f M ( E J , L ) that are associatedwith a given W -graph ideal E J , they are connected by a duality map, this inturn can be used for the construction of the dual bases of the W -graphs. Thisconstruction closely parallels the work of Deodhar [3, 4], Douglass [5], where theyfocused primarily upon the parabolic cases.In Section 4 we prove in general the construction of another pair of dual W -graphbases. This part is motivated by Lusztig’s work [11, Ch. 10], the construction isobtained by using the bases of H -modules Hom A ( M, A ) and
Hom A ( f M , A ).In Section 5, in the case W is finite we prove an inversion formula that relatesthe two versions of the relative Kazhdan-Lusztig polynomials, . In the last sectionwe give some examples and remarks.1. preliminaries Let W be a Coxeter group, with generating set S . In this section, we brieflyrecall some basic concepts concerning the general multi-parameter framework ofLusztig [10, 11], which introduces a weight function into Coxeter groups and theirassociated Hecke algebras on which all the subsequent constructions depend.We denote by ℓ : W → N = { , , , · · · } the length function on W with respectto S . Let denote the Bruhat order on W .Let Γ be the totally ordered abelian group which will be denoted additively, theorder on Γ will be denoted by . Let { L ( s ) | s ∈ S } ⊆ Γ be a collection of elementssuch that L ( s ) = L ( t ) whenever s, t ∈ S are conjugate in W . This gives rise to aweight function L : W −→ Γin the sense of Lusztig [10, 11]; we have L ( w ) = L ( s ) + L ( s ) + · · · + L ( s k ) where w = s s · · · s k ( s i ∈ S ) is a reduced expression for w ∈ W . We assume throughoutthat L ( s ) > s ∈ S . (If Γ = Z and L ( s ) = 1 for all s ∈ S , then this is the original ”equalparameter” setting of [6]).Let R ⊆ C be a subring and A = R [Γ] be a free R -module with basis { q γ | γ ∈ Γ } where q is an indeterminant.(The basic constructions in this section are independentof the choice of R and so we could just take R = Z ). The flexibility of R will beuseful once we consider representations of W ). There is a well-defined ring structureon A such that q γ q γ ′ = q γ + γ ′ for all γ, γ ′ ∈ Γ. We denote 1 = q ∈ A . If a ∈ A wedenote by a γ the coefficient of a on q γ so that a = P γ ∈ Γ a γ q γ . If a = 0 we definethe degree of a as the element of Γ equal todeg( a ) = max { γ | a γ = 0 } by convention(see [1]), we set deg 0 = −∞ . So deg : A → Γ ∪ {−∞} satisfiesdeg( ab ) = deg( a ) + deg( b ).Let H = H ( W, S, L ) be the generic Hecke algebra corresponding to (
W, S )with parameters { q L ( s ) | s ∈ S } . Thus H has an A -basis { T w | w ∈ W } and the -GRAPHS FOR HECKE ALGEBRAS WITH UNEQUAL PARAMETERS (II) 3 multiplication is given by the rules(1) T s T w = ( T sw if ℓ ( sw ) > ℓ ( w ) T sw + ( q L ( s ) − q − L ( s ) ) T w if ℓ ( sw ) < ℓ ( w ),Let Γ > γ = { γ ∈ Γ | γ > γ } and denote by A > γ ( or R [Γ > γ ] ) the set of all R -linear combinations of terms q γ where γ > γ . The notations A γ>γ , A γ γ , A γ<γ have a similar meaning.We denote by A A, a a the automorphism of A induced by the automor-phism of Γ sending γ to − γ for any γ ∈ Γ. This extends to a ring involution H H , h h , where X w ∈ W a w T w = X w ∈ W a w T − w − ,a w ∈ A for all w ∈ W , and T s = T − s = T s + ( q − L ( s ) − q L ( s ) ) for all s ∈ S. Definition of W -graph.Definition 1.1. (for equal parameter case see [6] ; for general L see [7] ). A W -graph for H consists of the following data:(a) a base set Λ together with a map I which assigns to each x ∈ Λ a subset I ( x ) ⊆ S ;(b) for each s ∈ S with L ( s ) >
0, a collection of elements { µ sx,y | x, y ∈ Λ such that s ∈ I ( x ) , s / ∈ I ( y ) } ;(c) for each s ∈ S with L ( s ) = 0 a bijection Λ → Λ , x → s.x . These dataare subject to the following requirements. First we require that, for any x, y ∈ Λ and s ∈ S where µ sx,y is defined, we have q L ( s ) µ sx,y ∈ R [Γ > ] and µ sx,y = µ sx,y . Furthermore, let [Λ] A be a free A -module with basis { b y | y ∈ Λ } . For s ∈ S , definean A -linear map(2) ρ s ( b y ) = b s.y if L ( s ) = 0; − q − L ( y ) b y if L ( s ) > , s ∈ I ( y ); q L ( y ) b y + P x ∈ Λ; s ∈ I ( x ) µ sx,y b x if L ( s ) > , s / ∈ I ( y ).Then we require that the assignment T s ρ s defines a representation of H .2. W -graph ideals For each J ⊆ S , let ˆ J = S \ J (the complement of J ) and define W J = h J i , thecorresponding parabolic subgroup of W . Let H J be the Hecke algebra associatedwith W J . As is well known, H J can be identified with a subalgebra of H .Let D J = { w ∈ W | ℓ ( ws ) > ℓ ( w ) for all s ∈ J } , the set of minimal cosetrepresentatives of W/W J . The following lemma is well known. YUNCHUAN YIN
Lemma 2.1. [3, Lemma 2.1(iii)](modified) Let J ⊆ S and s ∈ S , and define D − J,s = { w ∈ D J | ℓ ( sw ) < ℓ ( w ) } ,D + J,s = { w ∈ D J | ℓ ( sw ) > ℓ ( w ) and sw ∈ D J } ,D J,s = { w ∈ D J | ℓ ( sw ) > ℓ ( w ) and sw / ∈ D J } , so that D J is the disjoint union D − J,s ∪ D + J,s ∪ D J,s . Then sD + J,s = D − J,s , and if w ∈ D J,s then sw = wt for some t ∈ J .In this section we shall recall [9, Section 5], with some modification.Let L denote the left weak (Bruhat)order on W . We say x L y if and only if y = zx for some z ∈ W such that ℓ ( y ) = ℓ ( z ) + ℓ ( x ). We also say that x is a suffix of y . The following property of the Bruhat order is useful (see [11, Corollary 2.5],for example). Lemma 2.2.
Let y, z ∈ W and let s ∈ S .(i) Assume that sz < z , then y z ⇐⇒ sy z .(ii) Assume that y < sy , then y z ⇐⇒ y sz . Definition 2.3. If X ⊆ W , let P os ( X ) = { s ∈ S | ℓ ( xs ) > ℓ ( x )for all x ∈ X } .Thus P os ( X ) is the largest subset J of S such that X ⊆ D J . Let E be anideal in the poset ( W, L ); that is, E is a subset of W such that every u ∈ W that is a suffix of an element of E is itself in E . This condition implies that P os ( E ) = S \ E = { s ∈ S | s / ∈ E } . Let J be a subset of P os ( E ), so that E ⊆ D J .In contexts we shall denote by E J for the set E , with reference to J , For each s ∈ S we classify the elements in E J as follows: E − J,s = { w ∈ E J | ℓ ( sw ) < ℓ ( w ) and sw ∈ E J } , E + J,s = { w ∈ E J | ℓ ( sw ) > ℓ ( w ) and sw ∈ E J } , E , − J,s = { w ∈ E J | ℓ ( sw ) > ℓ ( w ) and sw / ∈ D J } , E , + J,s = { w ∈ E J | ℓ ( sw ) > ℓ ( w ) and sw ∈ D J \ E J } . Since E J ⊆ D J it is clear that, for each w ∈ E J , each s ∈ S appears in exactly oneof the following four sets SA ( w ) = { s ∈ S | w ∈ E + J,s } , SD ( w ) = { s ∈ S | w ∈ E − J,s } , W A J = { s ∈ S | w ∈ E , + J,s } and W D J = { s ∈ S | w ∈ E , − J,s } . We call theelements of these sets the strong ascents, strong descents, weak ascents and weakdescents of w relative to E J and J . In contexts where the ideal E J and the set J isfixed we frequently omit reference to J , writing W A ( w ) and W D ( w ) rather than W A J ( w ) and W D J ( w ). We also define the sets of descents and ascents of w by D ( w ) = SD ( w ) ∪ W D ( w ) and A ( w ) = SA ( w ) ∪ W A ( w ). Remark . It follows from 2.1 that
W A J ( w ) = { s ∈ S | sw / ∈ E J and w − sw / ∈ J } ,W D J ( w ) = { s ∈ S | sw / ∈ E J and w − sw ∈ J } . since sw / ∈ E J implies that sw > w (given that E J is an ideal in ( W, L )). Notealso that J = W D J (1). Definition 2.4. [9, Definition 5.1](modified) Let (
W, S ) be a Coxeter group withweight function L such that L ( s ) > s ∈ S , H be the corresponding Hecke -GRAPHS FOR HECKE ALGEBRAS WITH UNEQUAL PARAMETERS (II) 5 algebra. The set E J is said to be a W -graph ideal with respect to J ( ⊆ S ) and L ifthe following hypotheses are satisfied.(i) There exists an A -free H -module M ( E J , L ) possessing an A -basis B = { Γ w | w ∈ E J } , for any s ∈ S and any w ∈ E J we have(3) T s Γ w = Γ sw + ( q L ( s ) − q − L ( s ) )Γ w if w ∈ E − J,s , Γ sw if w ∈ E + J,s , − q − L ( s ) Γ w if w ∈ E , − J,s ,q L ( s ) Γ w − P z ∈ E J z
W, S ) be a Coxeter group with weight function L such that L ( s ) > s ∈ S , H be the corresponding Hecke algebra. There exists an algebra mapΦ : H → H given by Φ( q L ( s ) ) = q L ( s ) for all s ∈ S , and Φ( T w ) = ǫ w T w , wherethe bar is the standard involution in H . Further, Φ = Id and Φ commutes withthe bar involution. Duality theorem.
We now give an equivalent definition of a W -graph ideal, andthe associated module is denoted by f M ( E J , L ). The following theorem essentiallyprovides the duality between the two set ups. Theorem-definition 3.1. (I) With the above notation, let the set E J be a W -graph ideal with respect to J ( ⊆ S ) and L , then the following hypotheses are satis-fied.(i) There exists an A -free H -module f M ( E J , L ) possessing an A -basis e B = { e Γ w | w ∈ E J } , for any s ∈ S and any w ∈ E J we have(4) T s e Γ w = e Γ sw + ( q L ( s ) − q − L ( s ) ) e Γ w if w ∈ E − J,s , e Γ sw if w ∈ E + J,s ,q L ( s ) e Γ w if w ∈ E , − J,s , − q − L ( s ) e Γ w + P z ∈ E J z For w ∈ E J , define η (Γ w ) = ǫ w e Γ w . Extend η to the whole of M ( E J , L ) byΦ-linearity. Let s ∈ S . Then we have,(5) η ( T s Γ w ) = η [Γ sw + ( q L ( s ) − q − L ( s ) )Γ w ] if w ∈ E − J,s ,η (Γ sw ) if w ∈ E + J,s ,η ( − q − L ( s ) Γ w ) if w ∈ E , − J,s ,η ( q L ( s ) Γ w − P z ∈ E J z Apply the function η to both the sides of the formula for Γ y and use the factthat η commutes with the involution and then use the formula for e Γ y . We omit thedetails. (cid:3) The above result can also be proved by the following recursive formulas. Lemma 3.3. [14, Prop. 4.1] Let x, y ∈ E J . If s ∈ S is such that y ∈ E − J,s then R x,y = R sx,sy if x ∈ E − J,s ,R sx,sy + ( q − L ( s ) − q L ( s ) ) R x,sy if x ∈ E + J,s , − q L ( s ) R x,sy if x ∈ E , − J,s ,q − L ( s ) R x,sy if x ∈ E , + J,s . Similarly we have Lemma 3.4. Let x, y ∈ E J . If s ∈ S is such that y ∈ E − J,s then e R x,y = e R sx,sy if x ∈ E − J,s , e R sx,sy + ( q − L ( s ) − q L ( s ) ) e R x,sy if x ∈ E + J,s ,q − L ( s ) e R x,sy if x ∈ E , − J,s , − q L ( s ) e R x,sy if x ∈ E , + J,s . We have the further properties of R x,y . Lemma 3.5. If y ∈ E , − J,s then we have R x,y = ( − q − L ( s ) R sx,y if x ∈ E − J,s , − q L ( s ) R sx,y if x ∈ E + J,s . If y ∈ E , + J,s then we have R x,y = ( q L ( s ) R sx,y if x ∈ E − J,s ,q − L ( s ) R sx,y if x ∈ E + J,s . YUNCHUAN YIN Proof. If y ∈ E , − J,s then T s Γ y = − q − L ( s ) Γ y Applying involution bar on both sides. On the left hand side we have T s Γ y = T s Γ y = [ T s + ( q − L ( s ) − q L ( s ) ] X x ∈ E J R x,y Γ x . while the right hand side is − q − L ( s ) Γ y = − q L ( s ) P x ∈ E J R x,y Γ x .Comparing the coefficients of Γ x in the two expressions, we get the result. Theproof for the case y ∈ E , + J,s is similar with the above. (cid:3) Dual bases C and C ′ . Recall [14, Th.4.4] that the invariants in M ( E J , L ) (respec-tively f M ( E J , L )) form a free A -module with a basis { C w | w ∈ E J } (respectively { e C w | w ∈ E J } ), where C w = P y ∈ E J P y,w Γ y and e C w = P y ∈ E J e P y,w e Γ y .Using the map θ , we obtain a dual basis { C ′ w | w ∈ E J } for the invariants in M ( E J , L ) . Analogously, using the map η we obtain the dual basis { e C ′ w | w ∈ E J } for the invariants in f M ( E J , L ).More precisely, we have: Proposition 3.6. Let C ′ w = θ ( e C w ), e C ′ w = η ( C w ). Then(a) The H -module M ( E J , L ) has a unique basis { C ′ w | w ∈ E J } such that C ′ w = C ′ w for all w ∈ E J , and C ′ w = P y ∈ E J ǫ y e P y,w Γ y . for some elements e P y,w ∈ A > with the following properties:(a1) e P y,w = 0 if y (cid:10) w ;(a2) e P w,w = 1;(a3) e P y,w has zero constant term if y = w and e P y,w − e P y,w = X y For y, w ∈ E J , we write the matrix P = ( P y,w ) , where P y,w are E J -relative Kazhdan-Lusztig polynomials. The formula for C w in [14, Th.4.4] may bewritten as C w = Γ w + X y ∈ E J P y,w Γ y and inverting this gives Γ w = C w + X y ∈ E J Q y,w C y -GRAPHS FOR HECKE ALGEBRAS WITH UNEQUAL PARAMETERS (II) 9 where the elements Q y,w (defined whenever y < w ) are given recursively by(7) Q y,w = − P y,w − X z ∈ E J | y 1) of elements in E J , weset ℓ ( ζ ) = n and P ζ = P z ,z P z ,z · · · P z n − ,z n . z is called the initial element of ζ and z n is called the final element of ζ . For y < w , let τ ( y, w ) denote the set of all E J -chains with y as the initial element and w as the final element.The following results are motivated by Lusztig [11, Ch. 10]. For the sake ofcompleteness we attach the proofs. Proposition 3.7. For any y, w ∈ E J we have Q y,w = X ζ ∈ τ ( y,w ) ( − ℓ ( ζ ) P ζ We have Q y,w ∈ A > with the following properties:(a1) Q y,w = 0 if y (cid:10) w ;(a2) Q w,w = 1; Proof. If ℓ ( w ) − ℓ ( y ) = 1, by Eq.(7) we have Q y,w = − P y,w . The statement is true.Applying induction on ℓ ( w ) − ℓ ( y ) > 1. For any z ∈ E J , y < z < w , in the sum ofEq.(7) we use the induction hypothesis. Q y,z = X ζ ′ ∈ τ ( y,z ) ( − ℓ ( ζ ′ ) P ζ ′ We have Q y,w = − P y,w − X ζ ′ ∈ τ ( y,z ) ( − ℓ ( ζ ′ ) P ζ P z,w = X ζ ∈ τ ( y,w ) ( − ℓ ( ζ ) P ζ where the sequence ζ = ( y, w )( ∈ τ ( y, w )) is with ℓ ( ζ ) = 1 and ( ζ ′ , w )( ∈ τ ( y, w ))with the length ℓ ( ζ ′ ) + 1. The listed properties of Q ′ s are by Eq.(7). The result isproved. (cid:3) We define Q ′ y,w = sgn ( y ) sgn ( w ) Q y,w Proposition 3.8. For any y, w ∈ E J we have Q ′ y,w = P z ; y L z L w Q ′ y,z e R z,w Proof. The triangular matrices Q = ( Q y,w ) , P = ( P y,w ) , R = ( R y,w ) are related by P Q = QP = 1 , P = RP, RR = RR = 1where the bar involution over a matrix is the matrix obtained by applying¯to eachentry. We deduce that QP = 1 = QP = QRP Multiplying on the right by Q and using the fact P Q = 1 we deduce Q = QR .Multiplying on the right by R gives Q = QR Let S be the matrix whose ( y, w )-entry is sgn ( y ) δ y,w . We have S = 1. Note that Q ′ = SQS . By Corollary 3.2 we have R = S e RS . Hence Q ′ = SQS = S ( QR ) S = SQS · SRS = Q ′ e R The result follows. (cid:3) W -graphs for the modules ˆ M and ˆ f M Denote by M := M ( E J , L ) and f M :== f M ( E J , L ). Let ˆ M := Hom A ( M, A ) andˆ f M := Hom A ( f M , A ).Define an left H -module structure on ˆ M by hf ( m ) = f ( hm )(with f ∈ ˆ M , m ∈ M, h ∈ H ) . We define a bar operator ˆ M ˆ M by f ( m ) = f ( m ) (with f ∈ ˆ M , m ∈ M ); in f ( m ) the lower bar is that of M and the upper bar is that of A . h · f ( m ) = hf ( m ) = f ( hm ) = f ( hm ) = f ( hm ) = h · f ( m ) . Hence we have h · f = h · f for f ∈ ˆ M , h ∈ H .In the following contexts we focus on the module ˆ M , and usually omit theanalogous details for ˆ f M .If P is a property we set δ P = 1 if P is true and δ P = 0 if P is false. We write δ x,y instead of δ x = y . The basis of ˆ M . We firstly introduce two bases for the module ˆ M . For any z ∈ E J we define ˆΓ z ∈ ˆ M by ˆΓ z (Γ w ) = δ z,w for any w ∈ E J . Then ˆ B =: { ˆΓ z ; z ∈ E J } isan A -basis of ˆ M .Further, for any z ∈ E J we define D z ∈ ˆ M by D z ( C w ) = δ z,w for any w ∈ E J .Then D := { D z ; z ∈ E J } is an A -basis of ˆ M .Obviously we have D z = X y ∈ E J ,z For any y ∈ E J we haveˆΓ y = X w ∈ E J ,y w R y,w ˆΓ w . -GRAPHS FOR HECKE ALGEBRAS WITH UNEQUAL PARAMETERS (II) 11 Proof. For any x ∈ E J we haveˆΓ y (Γ x ) = ˆΓ y (Γ x )= ˆΓ y ( X x ′ ∈ E J ,x ′ x R x ′ ,x Γ x ′ ) = δ y x R y,x = δ y x R y,x = X w ∈ E J ,y w R y,w ˆΓ w (Γ x ) (cid:3) Theorem 4.2. [14, Th.4.7 ] The basis elements { C v | v ∈ E J } give the module M ( E J , L ) the structure of a W -graph module such that(8) T s C v = q L ( s ) C v + C sv + P z ∈ E J ,sz The H -module ˆ M ( E J , L ) has a unique basis { D z | z ∈ E J } suchthat D z = D z for all z ∈ E J , and D z = P y ∈ E J Q z,y ˆΓ y . for some elements Q z,y ∈ A > with the following properties:(a1) Q z,y = 0 if z (cid:10) y ;(a2) P z,z = 1;(a3) Q z,y has zero constant term if z = y and Q z,y − Q z,y = X z x Let s ∈ S and assume that L ( s ) > 0. The basis elements { D z | z ∈ E J } give ˆ M the structure of a W -graph module such that(9) T s D z = − q − L ( s ) D z + D sz + P z
In the case s ∈ SD ( z ), T s D z ( C w ) = D z ( T s C w ) gives T s D z ( C w ) = D z ( q L ( s ) C w + C sw + P x ∈ E J ,sx Corollary 4.5. For s ∈ S with L ( s ) = 0, z ∈ E J , we have T s D z = D sz if s ∈ SD ( z ) or s ∈ SA ( z ), − D z if s ∈ W D ( z ), D z if s ∈ W A ( z ), The D ′ -basis for M .Theorem 4.6. The H -module ˆ M ( E J , L ) has a unique basis { D ′ z | z ∈ E J } suchthat D ′ z = D ′ z for all z ∈ E J , and D ′ z = P y ∈ E J ǫ y e Q z,y ˆΓ y . where e Q z,y ∈ A > , are the -GRAPHS FOR HECKE ALGEBRAS WITH UNEQUAL PARAMETERS (II) 13 analogous elements in the case of f M .(10) T s D ′ z = q L ( s ) D ′ z + D ′ sz + P z
The modules ˆ M ( D J , L ) and M ( D J , L ) are identical. Proof. For any basis element ˆΓ w of ˆ M ( D J , L ) and element Γ y of M ( D J , L ), wehave T s ˆΓ w (Γ y ) = ˆΓ w ( T s Γ y )= δ y ∈ D − J,s δ w,sy + ( q L ( s ) − q − L ( s ) ) δ y ∈ D − J,s δ w,y + δ y ∈ D + J,s δ w,sy + q L ( s ) δ y ∈ D J,s δ w,y = δ w ∈ D + J,s δ sw,y + ( q L ( s ) − q − L ( s ) ) δ w ∈ D − J,s δ w,y + δ w ∈ D − J,s δ sw,y + q L ( s ) δ w ∈ D J,s δ w,y = ( δ w ∈ D + J,s ˆΓ sw + ( q L ( s ) − q − L ( s ) ) δ w ∈ D − J,s ˆΓ w + δ w ∈ D − J,s ˆΓ sw + q L ( s ) δ w ∈ D J,s ˆΓ w )(Γ y )hence we have T s ˆΓ w = ˆΓ sw if w ∈ D + J,s ,ˆΓ sw + ( q L ( s ) − q − L ( s ) )ˆΓ w if w ∈ D − J,s , q L ( s ) ˆΓ sw if w ∈ D J,s ,The result follows. (cid:3) Corollary 4.8. The H -module M ( D J , L ) has basis { D z | z ∈ D J } , where D z = P y ∈ D J ,z In the case W is finite Let ( W, S ) be a finite Coxeter system and w be the longest element in W . Definethe function π : W → W by π ( w ) = w ww , it satisfies π ( S ) = S and it extends toa C -algebra isomorphism π : C [ W ] C [ W ]. We denote by s = π ( s ). For s ∈ S we have ℓ ( w ) = ℓ ( w s ) + ℓ ( s ) = ℓ ( π ( s )) + ℓ ( π ( s ) w ), hence L ( w ) = L ( w s ) + L ( s ) = L ( π ( s )) + L ( π ( s ) w ) = L ( π ( s )) + L ( w s ) so that L ( π ( s )) = L ( s ). It follows that L ( π ( w )) = L ( w ) for all w ∈ W and thatwe have an A -algebra automorphism π : H H where π ( T w ) = T π ( w ) for any w ∈ W . Lemma 5.1. The H -modules M and f M have basis Γ π = { T w Γ w | w ∈ E J } and e Γ π = { T w e Γ w | w ∈ E J } respectively. Moreover we have η ( T w Γ w ) = ǫ w w T w e Γ w . Proof. Since the involution is square 1 and T w is invertible in H , the statementfollows. Furthermore η ( T w Γ w ) = Φ( T w ) η (Γ w ) = ǫ w T w ǫ w e Γ w = ǫ w w T w e Γ w . (cid:3) In the following, for the sake of convenience we primarily focus on the module M and omit the analogous details for f M , unless it is needed. For any w ∈ E J wedenote by w ′ := w w and Γ πw ′ := T w Γ w ( ∈ M ( E J , L )). Remark Generally w E J = E J . We emphasize that, in the following contexts,the set w E J will be just used as the index set for the objects involved.Direct computation gives the following multiplication rules for the basis Γ π . T s Γ πw ′ = Γ πs w ′ + ( q L ( s ) − q − L ( s ) )Γ πw ′ if w ∈ E + J,s , Γ πs w ′ if w ∈ E − J,s , − q − L ( s ) Γ πw ′ if w ∈ E , − J,s ,q L ( s ) Γ πw ′ − P z ∈ E J z For any y ′ ∈ w E J there exist coefficients R πx ′ ,y ′ ∈ A , defined for x ′ ∈ w E J and x ′ < y ′ , such that Γ πy ′ = P x ′ ∈ w E J R πx ′ ,y ′ Γ πx ′ . If R πx ′ ,y ′ = 0 then x ′ y ′ ; particularly R πy ′ ,y ′ = 1.The proof is trivial.We have further properties of R πx ′ ,y ′ . Lemma 5.3. If y ′ ∈ w E , − J,s then we have R πx ′ ,y ′ = ( − q L ( s ) R πs x ′ ,y ′ if x ′ ∈ w E − J,s , − q − L ( s ) R πs x ′ ,y ′ if x ′ ∈ w E + J,s . If y ′ ∈ w E , + J,s then we have R πx ′ ,y ′ = ( q − L ( s ) R πs x ′ ,y ′ if x ′ ∈ w E − J,s ,q L ( s ) R πs x ′ ,y ′ if x ′ ∈ w E + J,s . Proof. The proof is similar with that of Lemma 3.5. (cid:3) The bases C π for M . The elements R πw ′ ,y ′ , where w ′ , y ′ ∈ w E J , lead to theconstruction of another set of elements P πw ′ ,y ′ and the following basis of M ( E J , L ). Theorem 5.4. The H -module M ( E J , L ) has a unique basis { C πy ′ | y ′ ∈ w E J } such that C πy ′ = C πy ′ for all y ∈ w E J , and C πy ′ = P w ′ ∈ w E J P πw ′ ,y ′ Γ πw ′ . for someelements P πw ′ ,y ′ ∈ A > with the following properties: -GRAPHS FOR HECKE ALGEBRAS WITH UNEQUAL PARAMETERS (II) 15 (a1) P πw ′ ,y ′ = 0 if w ′ (cid:10) y ′ ;(a2) P πy ′ ,y ′ = 1;(a3) P πw ′ ,y ′ has zero constant term if y ′ = w ′ and P πw ′ ,y ′ − P πw ′ ,y ′ = X w ′ For y, w ∈ E J . We have (i) y L w ⇐⇒ w ′ L y ′ ;(ii) R πw ′ ,y ′ = R y,w ; e R πw ′ ,y ′ = e R y,w ;(iii) for any w ′ , y ′ ∈ w E J and w ′ < y ′ we have P πw ′ ,y ′ = X w ′ x ′ y ′ x ′ ∈ w E J P πx ′ ,y ′ R πx,w . e P πw ′ ,y ′ = X w ′ x ′ y ′ x ′ ∈ w E J e P πx ′ ,y ′ e R πx,w Proof. (a) is obvious. We prove (b) by induction on ℓ ( w ). If ℓ ( w ) = 0 then w = 1.We have R y, = δ y, . Now R πw ,w y = 0 unless w L w y . On the other hand wehave w y L w . Hence R πw w,w y = 0 unless w y = w , that is y = 1 in whichcase it is 1. The desired equality holds when ℓ ( w ) = 0. Assume that ℓ ( w ) > 1. Wecan find s ∈ S such that sw < w . The proof of the following cases (a) and (b) issimilar with Lusztig....In the case (a) y ∈ E − J,s . By the induction hypothesis we have R y,w = R sy,sw = R πw sw,w sy = R πs w w,s w y = R πw w,w y In the case (b) y ∈ E + J,s . Using Lemma 3.3, by the induction hypothesis we have R y,w = R sy,sw + ( q − L ( s ) − q L ( s ) ) R y,sw = R πw sw,w sy + ( q − L ( s ) − q L ( s ) ) R πw sw,w y = R πs w ′ ,s y ′ + ( q − L ( s ) − q L ( s ) ) R πs w ′ ,y ′ = R πs w ′ ,s y ′ + ( q − L ( s ) − q L ( s ) ) R πw ′ ,s y ′ = R πw ′ ,y ′ In the Case (c) y ∈ E , − J,s . Using Lemma 3.5 and Lemma 5.3, by the inductionhypothesis we have R y,w = − q L ( s ) R y,sw = − q L ( s ) R πw ( sw ) ,w y = − q L ( s ) R πs w ′ ,y ′ = − q L ( s ) ( − q − L ( s ) R πw ′ ,y ′ ) = R πw ′ ,y ′ . Case (d) y ∈ E , + J,s . Using Lemma 3.5 and 5.3, by the induction hypothesis we have R y,w = q − L ( s ) R y,sw = q − L ( s ) R πs w ′ ,y ′ = R πw ′ ,y ′ . (iii) follows (ii). (cid:3) Proposition 5.6. For any y, w ∈ E J we have Q y,w = ǫ y ǫ w e P πw ′ ,y ′ . (Analogously e Q y,w = ǫ y ǫ w P πw ′ ,y ′ ). Proof. We argue by induction on ℓ ( w ) − ℓ ( y ) > 0. If ℓ ( w ) − ℓ ( y ) = 0 we have y = w and both sides are 1. Assume that ℓ ( w ) − ℓ ( y ) > 0. Subtracting the identity in...from that in ...and using induction hypothesis, we obtain ǫ y ǫ w Q y,w − e P πw ′ ,y ′ = ǫ y ǫ w Q y,w − e P πw ′ ,y ′ The right hand side is in A > ; since it is fixed by the involution bar, it is 0. (cid:3) More precisely, we have the following inversion formulas Corollary 5.7. In the above situation, X z ∈ E J ,x z w ε w ε z P x,z e P πw ′ ,z ′ = δ x,w ; X z ∈ E J ,x z w ε w ε z e P x,z P πw ′ ,z ′ = δ x,w for all x, w ∈ E J . Corollary 5.8. If W is finite, for any y, w ∈ E J we have m sy,w = − ǫ w y ǫ w w m π,s w w,w y , where m sy,w are the elements involved in the multiplication formulas for C -basis, m π,s w w,w y are the analogous in the formulas for C π -basis. Corollary 5.9. If W is finite, for the bases D and C π in M ( D J , L ), and the e D -basisand e C π -basis for f M ( D J , L ) we have T w D z = ǫ w z θ ( e C πw z ) and T w e D z = ǫ w z η ( C πw z ) . Some remarks An example: the dual Solomon modules. In this subsection, let ( W, S ) bea finite Coxeter group system. Assume that L ( s ) > s ∈ S . In [14]we introduced the A -free H -module H C w J C ′ w ˆ J , which is called the Solomonmodule with respect to J and L , and where C w J = ǫ w J X w ∈ W J ǫ w q L ( ww J ) T w = ǫ w J q L ( w J ) X w ∈ W J ǫ w q − L ( w ) T w ; C ′ w ˆ J = X w ∈ W ˆ J q − L ( ww ˆ J ) T w = q − L ( w ˆ J ) X w ∈ W ˆ J q L ( w ) T w . that is, C ′ w ˆ J is the C ′ -basis element corresponding to w ˆ J , the maximal lengthelement of W ˆ J , or c -basis element corresponding to w ˆ J (see [11, Corollary 12.2]). C w J is the C -basis element corresponding to w J . -GRAPHS FOR HECKE ALGEBRAS WITH UNEQUAL PARAMETERS (II) 17 In [14] we showed that H C w J C ′ w ˆ J has basis { T x C w J C ′ ˆ J | x ∈ F J } . This basisadmits the multiplication rules listed in the Definition 2.4, and F J is a W -graphideal with respect to J and weight function L .Similarly, the H -module H C ′ w J C w ˆ J has basis { T x C ′ w J C ˆ J | x ∈ F J } . We caneasily prove that this basis admits the multiplication rules listed in the Definition3.1. We call this the dual module of H C w J C ′ w ˆ J . The Kazhdan-Lusztig construction. Assume that J = ∅ . Then D J = W andthe sets W D J ( w ) and W A J ( w ) are empty for all w ∈ W .(1). If L ( s ) > s ∈ S ), both modules M ( E J , L ) and f M ( E J , L ) are with A -basis ( X w | w ∈ E J ) such that, T s X w = ( X sw if ℓ ( sw ) > ℓ ( w ) X sw + ( q L ( s ) − q − L ( s ) ) X w if ℓ ( sw ) < ℓ ( w ),where the elements X w stand for Γ w or e Γ w . If we let X w = T w for all w ∈ W ,then both modules are the regular module H with weight function L . Thus wecan recover some of Lusztig’s results ( for example, see [11, Ch.5, 6, 10, 11])for theregular case. Deodhar’s construction: the parabolic cases. Let J be an arbitrary subsetof S and L ( s ) = 1 for all s ∈ S , we can now turning to Deodhar’s construction.Set E J := D J , then D J is a W -graph ideal with respect to J , and also it is a W -graph ideal with respect with ∅ .In the latter case we have D ∅ = W , if w ∈ E J then SA ( w ) = { s ∈ S | sw > w and sw ∈ D J } ,SD ( w ) = { s ∈ S | sw < w } ,W D ∅ ( w ) = { s ∈ S | sw / ∈ D ∅ } = ∅ ,W A ∅ ( w ) = { s ∈ S | sw ∈ D ∅ DJ } = { s ∈ S | sw = wt for some t ∈ J } . Let H J be the Hecke algebra associated with the Coxeter system ( W J , J ). Let M ψ = H ⊗ H J A ψ , where A ψ is A made into an H J -module via the homomorphism ψ : H J → A that satisfies ψ ( T u ) = q ℓ ( u ) for all u ∈ W J , it is a A -free with basis B = { b w | w ∈ D J } defined by b w = T w ⊗ 1. This corresponds to M J in [3] in thecase u = q (we note that this is denoted by f M J in [4]).Let M φ = H ⊗ H J A φ , where A ψ is A made into an H J -module via the homo-morphism φ : H J → A that satisfies ψ ( T u ) = ( − q ) − ℓ ( u ) for all u ∈ W J , again it isa A -free with basis B = { b w | w ∈ D J } defined by b w = T w ⊗ 1. This correspondsto M J in [3] in the case u = − M J in [4]).Our module M ( E J , L ) is now essentially reduced to be the module M ψ , while f M ( E J , L ) is reduced to be the module M φ , the only differences being due to ournon-traditional definition of H .In the case D J is a W -graph ideal with respect to J , the discussion is similarwith the above. For more details see [9, Sect. 8]. References [1] C. Bonnafe, Two-sided cells in type B in the asymptotic case , J. Algebra 304 (2006), 216-236.[2] Mich`ele Couillens, G´en´eralisation parabolique des polynˆomes et des bases de Kazhdan-Lusztig ,J. Algebra. (1999), 687–720. [3] V. Deodhar, On some geometric aspects of Bruhat orderings II. The parabolic analogue ofkazhdan-Lusztig polynomials , J. Alg. (2), (1987), 483-506.[4] V. Deodhar, Duality in parabolic set up for questions in Kazhdan-Lusztig theory , J. Alg. ,(1991), 201-209.[5] J. Matthew Douglass, An inversion formula for relative Kazhdan-Lusztig polynomials , Comm.Algebra. (1990), 371–387.[6] D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras , Invent.Math. (1979), 165–184.[7] M. Geck and N. Jacon, Representations of Hecke algebras at roots of unity , Algebra andApplications 15, Springer-Verlag, 2011.[8] M. Geck, PyCox: Computing with (finite) Coxeter groups and Iwahori-Hecke algebras , arXiv:1201.5566v2, 2012.[9] R. B. Howlett and V.Nguyen, W -graph ideals , J.Algebra 361(2012), 188-212.[10] G. Lusztig, Left cells in Weyl groups, Lie Group Representations , I, R. L. R. Herb and J.Rosenberg, eds., Lecture Notes in Math., vol. 1024, Springer-Verlag, 1983, 99-111.[11] G. Lusztig, Hecke algebras with unequal parameters , CRM Monographs Ser. 18, Amer. Math.Soc., Providence, RI, 2003.[12] J.Y.Shi, The Laurent polynomials M sy,w in the Hecke algebra with unequual parameters J.Alg. (2012), 1-19.[13] L. Solomon, A decomposition of the group algebra of finite Coxeter group , J. Alg. (1968),220-239.[14] Y. Yin, W -graphs for Hecke algebras with unequal parameters , Manuscripta Math. DOI:10.1007/s00229-014-0719-1, Nov. 2014, published online., Manuscripta Math. DOI:10.1007/s00229-014-0719-1, Nov. 2014, published online.