Kostka functions associated to complex reflection groups
aa r X i v : . [ m a t h . R T ] S e p Kostka functions associated to complex reflection groups
Toshiaki Shoji
Abstract.
Kostka functions K ± λ , µ ( t ) associated to complex reflection groups area generalization of Kostka polynomials, which are indexed by a pair λ , µ of r -partitions of n (and by the sign + , − ). It is expected that there exists a closerelationship between those Kostka functions and the intersection cohomology as-sociated to the enhanced variety X of level r . In this paper, we study combina-torial properties of K ± λ , µ ( t ) based on the geometry of X . In paticular, we showthat in the case where µ = ( − , . . . , − , µ ( r ) ) (and for arbitrary λ ), K − λ , µ ( t ) has aLascoux-Sch¨utzenberger type combinatorial description. Introduction
In 1981, Lusztig gave a geometric interpretation of Kostka polynomials in thefollowing sense; let V be an n -dimensional vector space over an algebraically closedfield, and put G = GL ( V ). Let P n be the set of partitions of n . Let O λ be theunipotent class in G labelled by λ ∈ P n , and K = IC( O λ , ¯ Q l ) the intersection co-homology associated to the closure O λ of O λ . Let K λ,µ ( t ) be the Kostka polynomialindexed by λ, µ ∈ P n , and e K λ,µ ( t ) = t n ( µ ) K λ,µ ( t − ) the modified Kostka polynomial(see 1.1 for the definition n ( µ )). Lusztig proved that(0.1) e K λ,µ ( t ) = t n ( λ ) X i ≥ dim( H ix K ) t i for x ∈ O µ ⊂ O λ , where H ix K is the stalk at x of the 2 i -th cohomology sheaf H i K of K . (0.1) implies that K λ,µ ( t ) ∈ Z ≥ [ t ].Let P n,r be the set of r -tuple of partitions λ = ( λ (1) , . . . , λ ( r ) ) such that P ri =1 | λ ( i ) | = n (we write | λ ( i ) | = m if λ ( i ) ∈ P m ). In [S1], [S2], Kostka functions K ± λ , µ ( t ) associated to complex reflections groups (depending on the signs + , − ) areintroduced, which are apriori rational functions in t indexed by λ , µ ∈ P n,r . Inthe case where r = 2 (in this case K − λ , µ ( t ) = K + λ , µ ( t )), it is proved in [S2] that K ± λ , µ ( t ) ∈ Z [ t ]. In this case, Achar-Henderson [AH] proved that those (generalized)Kostka polynomials have a geometric interpretation in the following sense; underthe previous notation, consider the variety X = G × V on which G acts naturally.Put X uni = G uni × V , where G uni is the set of unipotent elements in G . X uni is a G -stable subset of X , and is isomorphic to the enhanced nilpotent cone introducedby [AH]. It is known by [AH], [T] that X uni has finitely many G -orbits, which arenaturally parametrized by P n, . They proved in [AH] that the modified Kostkapolynomial e K ± λ , µ ( t ) ( λ , µ ∈ P n, ), defined in a similar way as in the original case, can be written as in (0.1) in terms of the intersection cohomology associated to theclosure O λ of the G -orbit O λ ⊂ X uni .In the case where r = 2, the interaction of geometric properties and combinato-rial properties of Kostka polynomials was studied in [LS]. In particular, it was provedthat in the special case where µ = ( − , µ (2) ) (and for arbitrary λ ∈ P n, ), K λ , µ ( t )has a combinatorial description analogous to Lascoux-Sch¨utzenberger theorem forthe original Kostka polynomials ([M, III, (6.5)]).We now consider the variety X = G × V r − for an integer r ≥
1, on which G acts diagonally, and let X uni = G uni × V r − be the G -stable subset of X . Thevariety X is called the enhanced variety of level r . In [S4], the relationship betweenKostka functions K ± λ , µ ( t ) indexed by λ , µ ∈ P n,r and the geometry of X uni wasstudied. In contrast to the case where r = 1 , X uni has infinitely many G -orbits if r ≥
3. A partition X uni = ` λ ∈ P n,r X λ into G -stable pieces X λ was constructed in[S3], and some formulas expressing the Kostka functions in terms of the intersectioncohomology associated to the closure of X λ were obtained in [S4], though it is apartial generalization of the result of Achar-Henderson for the case r = 2.In this paper, we prove a formula (Theorem 2.6) which is a generalization of theformula in [AH, Theorem 4.5] (and also in [FGT (11)]) to arbitrary r . Combinedthis formula with the results in [S4], we extend some results in [LS] to arbitrary r .In particular, we show in the special case where µ = ( − , . . . , − , µ ( r ) ) ∈ P n,r (and forarbitrary λ ∈ P n,r ) that K − λ , µ ( t ) has a Lasacoux-Sch¨utzenberger type combinatorialdescription. 1. Review on Kostka functions
First we recall basic properties of Hall-Littlewood functions and Kostkapolynomials in the original setting, following [M]. Let Λ = Λ ( y ) = L n ≥ Λ n be thering of symmetric functions over Z with respect to the variables y = ( y , y , . . . ),where Λ n denotes the free Z -module of symmetirc functions of degree n . We put Λ Q = Q ⊗ Z Λ , Λ n Q = Q ⊗ Z Λ n . Let s λ be the Schur function associated to λ ∈ P n .Then { s λ | λ ∈ P n } gives a Z -baisis of Λ n . Let p λ ∈ Λ n be the power sum symmetricfunction associated to λ ∈ P n , p λ = k Y i =1 p λ i , where p m denotes the m -th power sum symmetric function for each integer m > { p λ | λ ∈ P n } gives a Q -basis of Λ n Q . For λ = (1 m , m , . . . ) ∈ P n , define aninteger z λ by(1.1.1) z λ = Y i ≥ i m i m i ! . Following [M, I], we introduce a scalar product on Λ Q by h p λ , p µ i = δ λµ z λ . It isknown that { s λ } form an orthonormal basis of Λ . Let P λ ( y ; t ) be the Hall-Littlewood function associated to a partition λ . Then { P λ | λ ∈ P n } gives a Z [ t ]-basis of Λ n [ t ] = Z [ t ] ⊗ Z Λ n , where t is an indeterminate.Kostka polynomials K λ,µ ( t ) ∈ Z [ t ] ( λ, µ ∈ P n ) are defined by the formula(1.1.2) s λ ( y ) = X µ ∈P n K λ,µ ( t ) P µ ( y ; t ) . Recall the dominance order λ ≥ µ in P n , which is defined by the condition P ij =1 λ j ≥ P ij =1 µ j for each i ≥
1. For each partition λ = ( λ , . . . , λ k ), we define aninteger n ( λ ) by n ( λ ) = P ki =1 ( i − λ i . It is known that K λ,µ ( t ) = 0 unless λ ≥ µ ,and that K λ,µ ( t ) is a monic of degree n ( µ ) − n ( λ ) if λ ≥ µ ([M, III, (6.5)]). Put e K λ,µ ( t ) = t n ( µ ) K λ,µ ( t − ). Then e K λ,µ ( t ) ∈ Z [ t ], which we call the modified Kostkapolynomial.For λ = ( λ , . . . , λ k ) ∈ P n with λ k >
0, we define z λ ( t ) ∈ Q ( t ) by(1.1.3) z λ ( t ) = z λ Y i ≥ (1 − t λ i ) − , where z λ is as in (1.1.1). Following [M, III], we introduce a scalar product on Λ Q ( t ) = Q ( t ) ⊗ Z Λ by h p λ , p µ i = z λ ( t ) δ λ,µ . Then P λ ( y ; t ) form an orthogonal basisof Λ [ t ] = Z [ t ] ⊗ Z Λ . In fact, they are characterized by the following two properties([M, III, (2.6) and (4.9)]);(1.1.4) P λ ( y ; t ) = s λ ( y ) + X µ<λ w λµ ( t ) s µ ( y )with w λµ ( t ) ∈ Z [ t ] , and(1.1.5) h P λ , P µ i = 0 unless λ = µ . We fix a positive integer r . Let Ξ = Ξ( x ) ≃ Λ ( x (1) ) ⊗ · · · ⊗ Λ ( x ( r ) ) bethe ring of symmetric functions over Z with respect to variables x = ( x (1) , . . . , x ( r ) ),where x ( i ) = ( x ( i )1 , x ( i )2 , . . . ). We denote it as Ξ = L n ≥ Ξ n , similarly to the caseof Λ . Let P n,r be as in Introduction. For λ ∈ P n,r , we define a Schur function s λ ( x ) ∈ Ξ n by(1.2.1) s λ ( x ) = s λ (1) ( x (1) ) · · · s λ ( r ) ( x ( r ) ) . Then { s λ | λ ∈ P n,r } gives a Z -basis of Ξ n . Let ζ be a primitive r -th root of unityin C . For an integer m ≥ k such that 1 ≤ k ≤ r , put p ( k ) m ( x ) = r X j =1 ζ ( k − j − p m ( x ( j ) ) , where p m ( x ( j ) ) denotes the m -th power sum symmetric function with respect to thevariables x ( j ) . For λ ∈ P n,r , we define p λ ( x ) ∈ Ξ n C = Ξ n ⊗ Z C by(1.2.2) p λ ( x ) = r Y k =1 m k Y j =1 p ( k ) λ ( k ) j ( x ) , where λ ( k ) = ( λ ( k )1 , . . . , λ ( k ) m k ) with λ ( k ) m k >
0. Then { p λ | λ ∈ P n,r } gives a C -basis ofΞ n C . For a partition λ ( k ) as above, we define a function z λ ( k ) ( t ) ∈ C ( t ) by z λ ( k ) ( t ) = m k Y j =1 (1 − ζ k − t λ ( k ) j ) − . For λ ∈ P n,r , we define an integer z λ by z λ = Q rk =1 r m k z λ ( k ) , wher z λ ( k ) is as in(1.1.1). We now define a function z λ ( t ) ∈ C ( t ) by(1.2.3) z λ ( t ) = z λ r Y k =1 z λ ( k ) ( t ) . Let Ξ[ t ] = Z [ t ] ⊗ Z Ξ be the free Z [ t ]-module, and Ξ C ( t ) = C ( t ) ⊗ Z Ξ be the C ( t )-space. Then { p λ ( x ) | λ ∈ P n,r } gives a basis of Ξ n C ( t ). We define a sesquilinearform on Ξ C ( t ) by(1.2.4) h p λ , p µ i = δ λ , µ z λ ( t ) . We express an r -partition λ = ( λ (1) , . . . , λ ( r ) ) as λ ( k ) = ( λ ( k )1 , . . . , λ ( k ) m ) with acommon m , by allowing zero on parts λ ( i ) j , and define a composition c ( λ ) of n by c ( λ ) = ( λ (1)1 , . . . , λ ( r )1 , λ (1)2 , . . . , λ ( r )2 , . . . , λ (1) m , . . . , λ ( r ) m ) . We define a partial order λ ≥ µ on P n,r by the condition c ( λ ) ≥ c ( µ ), where ≥ is the dominance order on the set of compositions of n defined in a similar way asin the case of partitions. We fix a total order λ ≻ µ on P n,r compatible with thepartial order λ > µ .The following result was proved in Theorem 4.4 and Proposition 4.8 in [S1],combined with [S2, § Proposition 1.3.
For each λ ∈ P n,r , there exist unique functions P ± λ ( x ; t ) ∈ Ξ n Q ( t )( depending on the signs + , − ) satisfying the following properties. (i) P ± λ ( x ; t ) can be written as P ± λ ( x ; t ) = s λ ( x ) + X µ ≺ λ u ± λ , µ ( t ) s µ ( x ) with u ± λ , µ ( t ) ∈ Q ( t ) . (ii) h P − λ , P + µ i = 0 unless λ = µ . P ± λ ( x ; t ) are called Hall-Littlewood functions associated to λ ∈ P n,r . ByProposition 1.3, for ε ∈ { + , −} , { P ε λ | λ ∈ P n,r } gives a Q ( t )-basis for Ξ Q ( t ). For λ , µ ∈ P n,r , we define functions K ± λ , µ ( t ) ∈ Q ( t ) by(1.4.1) s λ ( x ) = X µ ∈ P n,r K ± λ , µ ( t ) P ± µ ( x ; t ) .K ± λ , µ ( t ) are called Kostka functions associated to complex reflection groups sincethey are closely related to the complex reflection group S n ⋉ ( Z /r Z ) n (see [S1,Theorem 5,4]). For each λ ∈ P n,r , by putting n ( λ ) = n ( λ (1) ) + · · · + n ( λ ( r ) ), wedefine an a -function a ( λ ) on P n,r by(1.4.2) a ( λ ) = r · n ( λ ) + | λ (2) | + 2 | λ (3) | + · · · + ( r − | λ ( r ) | . We define modifed Kostka functions e K ± λ , µ ( t ) by(1.4.3) e K ± λ , µ ( t ) = t a ( µ ) K ± λ , µ ( t − ) . Remark 1.5.
In the case where r = 1, P ± λ ( x ; t ) coincides with the original Hall-Littlewood function given in 1.1. In the case where r = 2, it is proved by [S2, Prop.3.3] that P − λ ( x ; t ) = P + λ ( x ; t ) ∈ Ξ[ t ], hence K − λ , µ ( t ) = K + λ , µ ( t ) ∈ Z [ t ]. Moreoverit is shown that K ± λ , µ ( t ) ∈ Z [ t ], which is a monic of degree a ( µ ) − a ( λ ). Thus e K ± λ , µ ( t ) ∈ Z [ t ]. As mentioned in Introduction e K ± λ , µ ( t ) has a geometric interpretation,which imples that K ± λ , µ ( t ), and so P ± λ ( x ; t ) are independent of the choice of the totalorder ≺ on P n,r . In the case where r ≥
3, it is not known whether Hall-Littlewoodfunctions do not depend on the choice of the total order ≺ , whether K ± λ , µ ( t ) arepolynomials in t . 2. Enhanced variety of level r Let V be an n -dimensional vector space over an algebraic closure k of afinite field F q , and G = GL ( V ) ≃ GL n . Let B = T U be a Borel subgroup of G , T amaximal torus and U the unipotent radical of B . Let W = N G ( T ) /T be the Weylgroup of G , which is isomorphic to the symmetric group S n . By fixing an integer r ≥
1, put X = G × V r − and X uni = G uni × V r − , where G uni is the set of unipotentelements in G . The variety X is called the enhanced variety of level r . We considerthe diagonal action of G on X . Put Q n,r = { m = ( m , . . . , m r ) ∈ Z r ≥ | P m i = n } .For each m ∈ Q n,r , we define integers p i = p i ( m ) by p i = m + · · · + m i for i = 1 , . . . , r . Let ( M i ) ≤ i ≤ n be the total flag in V whose stabilizer in G coincideswith B . We define varieties f X m = { ( x, v , gB ) ∈ G × V r − × G/B | g − xg ∈ B, g − v ∈ r − Y i =1 M p i } , X m = [ g ∈ G g ( B × r − Y i =1 M p i ) , and the map π m : f X m → X m by ( x, v , gB ) ( x, v ). We also define the varieties f X m , uni = { ( x, v , gB ) ∈ G uni × V r − × G/B | g − xg ∈ U, g − v ∈ r − Y i =1 M p i } , X m = [ g ∈ G g ( U × r − Y i =1 M p i ) , and the map π m , : f X m , uni → X m , uni , similarly. Note that in the case where m = ( n, , . . . , X m (resp. X m , uni ) coincides with X (resp. X uni ). In that case,we denote f X m , π m , etc. by f X , π , etc. by omitting the symbol m . (Note: here wefollow the notation in [S4], but, in part, it differs from [S3]. In [S3], our π m , π m , are denoted by π ( m ) , π ( m )1 for the consistency with the exotic case). In [S3, 5.3], a partition of X uni into pieces X λ is defined X uni = a λ ∈ P n,r X λ , where X λ is a locally closed, smooth irreducible, G -stable subvariety of X uni . If r = 1 or 2, X λ is a single G -orbit. However, if r ≥ X λ is in general a union ofinfinitely many G -orbits.For m ∈ Q n,r , let W m = S m × · · · × S m r be the Young subgroup of W = S n . For m ∈ Q n,r , we denote by P ( m ) the set of λ ∈ P n,r such that | λ ( i ) | = m i . The (iso-morphism classes of) irreducible representations (over ¯ Q l ) of W m are parametrizedby P ( m ). We denote by V λ an irreducible representation of W m corresponding to λ , namley V λ = V λ (1) ⊗ · · · ⊗ V λ ( r ) , where V µ denotes the irreducible representationof S n corresponding to the partition µ of n . (Here we use the parametrization suchthat V ( n ) is the trivial representation of S n ). The following results were proved in[S3]. Theorem 2.3 ([S3, Thm. 4.5]) . Put d m = dim X m . Then ( π m ) ∗ ¯ Q l [ d m ] is a sem-simple perverse sheaf equipped with the action of W m , and is decomposed as ( π m ) ∗ ¯ Q l [ d m ] ≃ M λ ∈ P ( m ) V λ ⊗ IC( X m , L λ )[ d m ] , where L λ is a simple local system on a certain open dense subvariety of X m . Theorem 2.4 ([S3, Thm. 8.13, Thm. 7.12]) . Put d ′ m = dim X m , uni . (i) ( π m , ) ∗ ¯ Q l [ d ′ m ] is a semisimple perverse sheaf equipped with the action of W m ,and is decomposed as ( π m , ) ∗ ¯ Q l [ d ′ m ] ≃ M λ ∈ P ( m ) V λ ⊗ IC( X λ , ¯ Q l )[dim X λ ] . (ii) We have
IC( X m , L λ ) | X m , uni ≃ IC( X λ , ¯ Q l )[dim X λ − d ′ m ] . For a partition λ , we denote by λ t the dual partition of λ . For λ =( λ (1) , . . . , λ ( r ) ) ∈ P ( m ), we define λ t ∈ P ( m ) by λ t = (( λ (1) ) t , . . . , ( λ ( r ) ) t ). Assumethat λ ∈ P ( m ). We write ( λ ( i ) ) t as ( µ ( i )1 ≤ µ ( i )2 ≤ · · · ≤ µ ( i ) ℓ i ), in the increasing order,where ℓ i = λ ( i )1 . For each 1 ≤ i ≤ r, ≤ j < ℓ i , we define an integer n ( i, j ) by n ( i, j ) = ( | λ (1) | + · · · + | λ ( i − | ) + µ ( i )1 + · · · + µ ( i ) j . Let Q = Q λ be the stabilizer of the partial flag ( M n ( i,j ) ) in G , and U Q the unipotentradical of Q . In particular, Q stabilizes the subspaces M p i . Let us define a variety e X λ by e X λ = { ( x, v , gQ ) ∈ G uni × V r − × G/Q | g − xg ∈ U Q , g − v ∈ r − Y i =1 M p i } . We define a map π λ : e X λ → X uni by ( x, v , gQ ) ( x, v ). Then π λ is a proper map.Since e X λ ≃ G × Q ( U Q × Q i M p i ), e X λ is smooth and irreducible. It is known by [S3,Lemma 5.6] that dim e X λ = dim X λ and that Im π λ coincides with X λ , the closureof X λ in X uni .For λ, µ ∈ P n , let K λ,µ = K λ,µ (1) be the Kostka number. We have K λ,µ = 0unless λ ≥ µ . For λ = ( λ (1) , . . . , λ ( r ) ), µ = ( µ (1) , . . . , µ ( r ) ) ∈ P ( m ), we define aninteger K λ , µ by K λ , µ = K λ (1) ,µ (1) K λ (2) ,µ (2) · · · K λ ( r ) ,µ ( r ) . We define a partial order λ D µ on P n,r by the condition λ ( i ) ≥ µ ( i ) for i = 1 , . . . , r .Hence λ D µ implies that λ , µ ∈ P ( m ) for a commom m . We have K λ , µ = 0 unless λ D µ . Note that λ D µ implies that µ t D λ t . We show the following theorem. Inthe case where r = 2, this result was proved by [AH, Thm. 4.5]. Theorem 2.6.
Assume that λ ∈ P n,r . Then ( π λ ) ∗ ¯ Q l [dim X λ ] is a semisimpleperverse sheaf on X λ , and is decomposed as (2.6.1) ( π λ ) ∗ ¯ Q l [dim X λ ] ≃ M µ E λ ¯ Q K µ t, λ t l ⊗ IC( X µ , ¯ Q l )[dim X µ ] . The rest of this section is devoted to the proof of Theorem 2.6. First weconsider the case where r = 1. Actually, the result in this case is contained in [AH].Their proof (for r = 2) depends on the result of Spaltenstein [Sp] concerning the“Springer fibre” ( π λ ) − ( z ) for z ∈ X λ in the case r = 1. In the following, we give analternate proof independent of [Sp] for the later use. Let Q be a parabolic subgroupof G containing B , M the Levi subgroup of Q containing T and U Q the unipotentradical of Q . (In this stage, this Q is independent of Q in 2.5.) Let W Q be theWeyl subgroup of W corresponding to Q . Let G reg be the set of regular semisimpleelements in G , and put T reg = G reg ∩ T . Consider the map ψ : e G reg → G reg , where e G reg = { ( x, gT ) ∈ G reg × G/T | g − xg ∈ T reg } and ψ : ( x, gT ) x . Then ψ is a finite Galois covering with group W . We alsoconsider a variety e G M reg = { ( x, gM ) ∈ G reg × G/M | g − xg ∈ M reg } , where M reg = G reg ∩ M . The map ψ is decomposed as ψ : e G reg ψ ′ −−−→ e G M reg ψ ′′ −−−→ G reg , where ψ ′ : ( x, gT ) ( x, gM ), ψ ′′ : ( x, gM ) x . Here ψ ′ is a finite Galois cov-ering with group W Q . Now ψ ∗ ¯ Q l is a semisimple local system on G reg such thatEnd( ψ ∗ ¯ Q l ) ≃ ¯ Q l [ W ], and is decomposed as(2.7.1) ψ ∗ ¯ Q l ≃ M ρ ∈ W ∧ ρ ⊗ L ρ , where L ρ = Hom W ( ρ, ψ ∗ ¯ Q l ) is a simple local system on G reg . We also have(2.7.2) ψ ′∗ ¯ Q l ≃ M ρ ′ ∈ W ∧ Q ρ ′ ⊗ L ′ ρ ′ , where L ′ ρ ′ is a simple local system on e G M reg . Hence(2.7.3) ψ ∗ ¯ Q l ≃ ψ ′′∗ ψ ′∗ ¯ Q l ≃ M ρ ′ ∈ W ∧ Q ρ ′ ⊗ ψ ′′∗ L ′ ρ ′ . (2.7.3) gives a decompostion of ψ ∗ ¯ Q l with respect to the action of W Q . Comparing(2.7.1) and (2.7.3), we have (2.7.4) ψ ′′∗ L ′ ρ ′ ≃ M ρ ∈ W ∧ ¯ Q ( ρ : ρ ′ ) l ⊗ L ρ , where ( ρ : ρ ′ ) is the multiplicity of ρ ′ in the restricted W Q -module ρ .We consider the map π : e G → G , where e G = { ( x, gB ) ∈ G × G/B | g − xg ∈ B } ≃ G × B B, and π : ( x, gB ) x . We also consider e G Q = { ( x, gQ ) ∈ G × G/Q | g − xg ∈ Q } ≃ G × Q Q. The map π is decomposed as π : e G π ′ −−−→ e G Q π ′′ −−−→ G, where π ′ : ( x, gB ) ( x, gQ ), π ′′ : ( x, gQ ) x . It is well-known ([L1]) that(2.7.5) π ∗ ¯ Q l ≃ M ρ ∈ W ∧ ρ ⊗ IC( G, L ρ ) . Let B M = B ∩ M be the Borel subgroup of M containing T . We consider thefollowing commutative diagram(2.7.6) G × B B e p ←−−− G × ( Q × B B ) e q −−−→ M × B M B Mπ ′ y y r y π M G × Q Q p ←−−− G × Q q −−−→ M, where under the identification G × B B ≃ G × Q ( Q × B B ), the maps p, e p are definedby the quotient by Q . The map q is a projection to the M -factor of Q , and e q is themap induced from the projection Q × B → M × B M . π M is defined similarly to π replacing G by M . The map r is defined by ( g, h ∗ x ) ( g, hxh − ). (We use thenotation h ∗ x ∈ Q × B B to denote the B -orbit in Q × B containing ( h, x ).) Hereall the squares are cartesian squares. Moreover,(a) p is a principal Q -bundle.(b) q is a locally trivial fibration with fibre isomorphic to G × U Q .Thus as in [S4, (1.5.2)], for any M -equivariant simple pervere sheaf A on M , thereexists a unique (up to isomorphism) simple perverse sheaf A on e G Q such that p ∗ A [ a ] ≃ q ∗ A [ b ], where a = dim Q and b = dim G + dim U Q . By using the cartesian squares in (2.7.6), and by (2.7.2), we see that π ′∗ ¯ Q l ≃ IC( e G Q , ψ ′∗ ¯ Q l ), and π ′∗ ¯ Q l is decomposed as(2.7.7) π ′∗ ¯ Q l ≃ M ρ ′ ∈ W ∧ Q ρ ′ ⊗ IC( e G Q , L ′ ρ ′ ) . By comparing (2.7.4) and (2.7.7), we have(2.7.8) π ′′∗ IC( e G Q , L ′ ρ ′ ) ≃ M ρ ∈ W ∧ ¯ Q ( ρ : ρ ′ ) l ⊗ IC( G, L ρ ) . Note that if ρ = V λ for λ ∈ P n , we have(2.7.9) IC( G, L ρ ) | G uni ≃ IC( O λ , ¯ Q l )[dim O λ − ν G ]by [BM], where ν G = dim U . Hence by restricting on G uni , we have(2.7.10) π ′′∗ IC( e G Q , L ′ ρ ′ )[2 ν G ] | G uni ≃ M λ ∈ P n ¯ Q ( V λ : ρ ′ ) l ⊗ IC( O λ , ¯ Q l )[dim O λ ] . Now assume that W Q ≃ S µ for a partition µ , where we put S µ = S µ × · · · × S µ k if µ = ( µ , . . . , µ k ) ∈ P n . Take ρ ′ = ε the sign representation of W Q .We have(2.8.1) ( V λ : ε ) = ( V λ t : 1 W Q ) = K λ t ,µ , where 1 W Q is the trivial representation of W Q .The restriction of the diagram (2.7.6) to the “unipotent parts” makes sense, andwe have the commutative diagram(2.8.2) G × B U ←−−− G × Q ( Q × B U ) −−−→ M × B M U M y y y G × Q Q uni p ←−−− G × Q uni q −−−→ M uni , where U M is the unipotent radical of B M , and Q uni , M uni are the set of unipotentelements in Q, M , respectively. p , q have similar properties as (a), (b) in 2.7.We consider IC( M, L Mε ) on M , where L Mε is the simple local system on M reg corresponding to ε ∈ W ∧ Q . Then by (2.7.6), we see that p ∗ IC( e G Q , L ′ ε ) ≃ q ∗ IC( M, L Mε ) . By applying (2.7.9) to M , IC( M, L Mε ) | M uni ≃ IC( O ′ ε , ¯ Q l )[dim O ′ ε − ν M ], where O ′ ε is the orbit in M uni corresponding to ε under the Springer correspondence, and ν M is defined similarly to ν G . It is known that O ′ ε is the orbit { e } ⊂ M uni , where e is theidentity element in M . Hence IC( M, L Mε ) | M uni coincides with ¯ Q l [ − ν M ] supportedon { e } . It follows, by (2.8.2)(2.8.3) The restriction of IC( e G Q , L ′ ε ) on G × Q Q uni coincides with i ∗ ¯ Q l [ − ν M ],where i : G × Q U Q ֒ → G × Q Q uni is the closed embedding.We deifne a map π Q : G × Q U Q → G uni by g ∗ x gxg − . Put e G Q = G × Q U Q . Proposition 2.9.
Under the notation as above, (i) π ′′∗ IC( e G Q , L ′ ε )[2 ν G ] | G uni ≃ ( π Q ) ∗ ¯ Q l [dim e G Q ] . (ii) We have ( π Q ) ∗ ¯ Q l [dim e G Q ] ≃ M µ ∈ P n µ ≤ t λ ¯ Q K tλ,µ l ⊗ IC( O λ , ¯ Q l )[dim O λ ] . Proof.
Note that 2 ν G − ν M = 2 dim U Q = dim e G Q . Thus by (2.8.3),(2.9.1) IC( e G Q , L ′ ε )[2 ν G ] | G × Q Q uni ≃ i ∗ ¯ Q l [dim e G Q ] . By applying the base change theorem to the cartesian square G × Q Q uni −−−→ G × Q Q π ′′ y y π ′′ G uni −−−→ G, we obtain (i) from (2.9.1) since π Q = π ′′ ◦ i . Then (ii) follows from (i) by using(2.7.10) and (2.8.1). (cid:3) Returning to the setting in 2.5, we consider the case where r is arbitrary.We fix m ∈ Q n,r , and let P = P m be the parabolic subgroup of G containing B which is the stabilizer of the partial flag ( M p i ) ≤ i ≤ r . Let L be the Levi subgroupof P containing T , and B L = B ∩ L the Borel subgroup of L containing T . Let U L be the unipotent radical of B L . Put M p i = M p i /M p i − for each i , under theconvention M p = 0. Then L acts naturally on M p i , and by applying the definitionof π m , : f X m , uni → X m , uni to L , we can define f X L m , uni ≃ L × B L ( U L × r − Y i =1 M p i ) , X L m , uni = [ g ∈ L g ( U L × r − Y i =1 M p i ) = L uni × r − Y i =1 M p i and the map π L m , : f X L m , uni → X L m , uni similarly. Let Q = Q λ be as in 2.5 for λ ∈ P ( m ). Thus we have B ⊂ Q ⊂ P , and Q L = Q ∩ L is a parabolic subgroup of L containing B L . We consider the followingcommutative diagram(2.10.1) f X m , uni e p ←−−− G × f X P m , uni e q −−−→ f X L m , uni α ′ y y r ′ y β ′ c X Q m , uni b p ←−−− G × f X P,Q m , uni b q −−−→ f X L,Q L m , uni α ′′ y y r ′′ y β ′′ c X P m , uni p ←−−− G × X P m , uni q −−−→ X L m , uni π ′′ y X m , uni , where, by putting P uni = L uni U P (the set of unipotent elements in P ), X P m , uni = [ g ∈ P g ( U × Y i M p i ) = P uni × Y i M p i , c X P m , uni = G × P X P m , uni = G × P ( P uni × Y i M p i ) , f X P m , uni = P × B ( U × Y i M p i ) , c X Q m , uni = G × Q ( Q uni × Y i M p i ) , f X P,Q m , uni = P × Q ( Q uni × Y i M p i ) . f X L,Q L m uni is a similar variety as c X P m , uni defined with respecto to ( L, Q L ), namely, f X L,Q L m , uni = L × Q L (( Q L ) uni × Y i M p i ) . The maps are defined as follows; under the identification f X m , uni ≃ G × B ( U × Q i M p i ), α ′ , α ′′ are the natural maps induced from the inclusions G × ( U × Q M p i ) → G × ( Q uni × Q M p i ) → G × ( P uni × Q M p i ). π ′′ : g ∗ ( x, v ) ( gxg − , g v ). q is defined by ( g, x, v ) ( x, v ), where x → x , v v are natural maps P → L, Q i M p i → Q i M p i . e q is the composite of the projection G × f X P m , uni → f X P m , uni and the map f X P m , uni → f X L m , uni induced from the projection P × ( U × Q M p i ) → L × ( U L × Q M p i ). b q is defined similarly by using the map f X P,Q m , uni → c X L,Q L m , uni induced from the projection P × ( Q uni × Q M p i ) → L × (( Q L ) uni × Q M p i ). p is the quotient by P . e p and b p are also quotient by P under the identifications f X m , uni ≃ G × P f X P m , uni , c X Q m uni ≃ G × P f X P,Q m , uni . β ′ is defined similarly to α ′ and β ′′ is defined similarly to π ′′ . r ′ is the natural map induced from the injection P × ( U × Q M p i ) → P × ( Q uni × Q M p i ), and r ′′ is the natural map induced fromthe map P × Q ( Q uni × Q M p i ) → P uni × Q M p i , g ∗ ( x, v ) ( gxg − , g v ).Put π ′ = α ′′ ◦ α ′ : f X m , uni → c X P m , uni . We have β ′′ ◦ β ′ = π L m , , and the diagram(2.10.1) is the refinement of the diagram (6.3.2) in [S4] (see also the diagram (1.5.1)in [S4]). In particular, the map p is a principal P -bundle, and the map q is alocally trivial fibration with fibre isomorphic to G × U P × Q r − i =1 M p i . Moreover, allthe squares appearing in (2.10.1) are caetesian squares. Hence the diagram (2.10.1)satisfies similar properties as in the diagram (2.8.2).Note that L ≃ G × · · · × G r with G i = GL ( M p i ). Then Q L can be written as Q L ≃ Q × · · · × Q r , where Q i is a parabloic subgroup of G i . We have f X L m , uni ≃ r Y i =1 ( e G i ) uni × V, c X L,Q L m , uni ≃ r Y i =1 ( e G Q i i ) uni × V, X L m , uni ≃ r Y i =1 ( G i ) uni × V, where ( e G i ) uni , ( e G Q i i ) uni , etc. denote the unipotent parts of e G i , e G Q i i , etc. as in (2.8.2).The maps β ′ , β ′′ are induced from the maps ( e G i ) uni → ( e G Q i i ), ( e G Q i i ) uni → ( G i ) uni ,and those maps coincide with the maps π ′ , π ′′ in 2.7 defined with respect to G i . Notethat W Q i ≃ S ( λ ( i ) ) t for each i by the construction of Q = Q λ in 2.5. Put c X Q = G × Q ( U Q × Y M p i ) , f X L,Q L = L × Q L ( U Q L × Y M p i ) , and let i Q : c X Q ֒ → c X Q m , uni , i Q L : f X L,Q L ֒ → f X L,Q L m uni be the closed embeddings. Let π LQ L : f X L,Q L → X L m , uni be the restriction of β ′′ . Let O L µ ≃ O ′ µ (1) × · · · × O ′ µ ( r ) be the L -orbit in X L m , uni , where O ′ µ ( i ) is the G i -oribt in ( G i ) uni × M p i of type ( µ ( i ) , ∅ ). Notethat if we denote by O µ ( i ) the G i -orbit in ( G i ) uni of type µ ( i ) , we have IC( O ′ µ ( i ) , ¯ Q l ) ≃ IC( O µ ( i ) , ¯ Q l ) ⊠ ¯ Q l (the latter term ¯ Q l denotes the constatn sheaf on M p i ). Hencethe decompostion of π LQ L into simple components is described by considering thefactors IC( O µ ( i ) , ¯ Q l ). In particular, by Proposition 2.9, we have(2.10.2) ( π LQ L ) ∗ ¯ Q l [dim f X L,Q L ] ≃ M µ E λ ¯ Q K µ t, λ t l ⊗ IC( O L µ , ¯ Q l )[dim O L µ ] . By using the diagram (2.10.1), we see that b q ∗ ( i Q L ) ∗ ¯ Q l [dim f X L,Q L ] ≃ b p ∗ ( i Q ) ∗ ¯ Q l [dim e X λ ] . It follows, again by using the diagram (2.10.1), we have(2.10.3) ( α ′′ ) ∗ ( i Q ) ∗ ¯ Q l [dim e X λ ] ≃ M µ E λ ¯ Q K µ t, λ t l ⊗ B µ , where B µ is the simple perverse sheaf on c X P m , uni characterized by the property that p ∗ B µ [ a ′ ] ≃ q ∗ IC( O L µ , ¯ Q l )[ b ′ + dim O L µ ]with a ′ = dim P , b ′ = dim G + dim U P + dim Q r − i =1 M p i .On the other hand, by Proposition 1.6 in [S4], we have π ′′∗ A µ ≃ IC( X m , L µ )[ d m ] , where π ′′ : c X P m = G × P ( P × Q i M p i ) → X m is an analogous map to π ′′ , and A µ isa simple perverse sheaf on c X P m such that the restriction of A µ on c X P m , uni coincideswith B µ , up to shift. Thus by Theorem 2.4 (ii), we have(2.10.4) ( π ′′ ) ∗ B µ ≃ IC( X µ , ¯ Q l )[dim X µ ] . Since π λ = π ′′ ◦ α ′′ ◦ i Q , by applying ( π ′′ ) ∗ on both sides of (2.10.3), we obtain theformula (2.6.1). This completes the proof of Theorem 2.6.3. G F -invariant functions on the enhanced varietyand Kostka functions We now assume that G and V are defined over F q , and let F : G → G, F : V → V be the corresponding Frobenius maps. Assume that B and T are F -stable. Then X λ and e X λ have natrual F q -structures, and the map π λ : e X λ → X λ is F -equivariant. Thus one can define a canonical isomorphsim ϕ : F ∗ K λ ∼−→ K λ for K λ = ( π λ ) ∗ ¯ Q l . By using the decomposition in Theorem 2.6, ϕ can be written as ϕ = P µ σ µ ⊗ ϕ µ , where σ µ is the identity map on ¯ Q K µ t, λ t l and ϕ µ : F ∗ L µ ∼−→ L µ isthe isomorphism induced from ϕ for L µ = IC( X µ , ¯ Q l ). (Note that dim X λ − dim X µ is even if µ E λ by [S4, Prop. 4.3], so the degree shift is negligible). We alsoconsider the natural isomorphism φ µ : F ∗ L µ ∼−→ L µ induced from the F q -strucutre of X µ . By using a similar argument as in [S4, (6.1.1)], we see that(3.1.1) ϕ µ = q d µ φ µ , where d µ = n ( µ ). We consider the characteristic function χ L µ of L µ with respectto φ µ , which is a G F -invariant function on X F µ . Take µ , ν ∈ P n,r , and assume that ν ∈ P ( m ). For each z = ( x, v ) ∈ X µ with v = ( v , . . . , v r − ), we define a variety G ν ,z by G ν ,z = { ( W p i ) : x -stable flag | v i ∈ W p i (1 ≤ i ≤ r − ,x | W pi /W pi − : type ν ( i ) (1 ≤ i ≤ r ) } . (3.2.1)If z ∈ X F µ , the variety G ν ,z is defined over F q . Put g ν ,z ( q ) = | G F ν ,z | . Let e K λ,µ ( t ) bethe modified Kostka polynomial indexed by partitions λ, µ . The following result isa generalization of Proposiition 5.8 in [AH]. Proposition 3.3.
Assume that λ , µ ∈ P n,r . For each z ∈ X F µ , we have χ L λ ( z ) = q − n ( λ ) X ν E λ g ν ,z ( q ) e K λ (1) ,ν (1) ( q ) · · · e K λ ( r ) ,ν ( r ) ( q ) . Proof.
Let χ K λ ,ϕ be the characteristic function of K λ with respect to ϕ . By Theorem2.6 together with (3.1.1), we have(3.3.1) χ K λ ,ϕ = X ξ E λ K ξ t , λ t q n ( ξ ) χ L ξ . On the other hand, by the Grothendieck’s fixed point formula, we have χ K λ ,ϕ ( z ) = | π − λ ( z ) F | for z ∈ X F λ . Then if z = ( x, v ) ∈ X F µ ,(3.3.2) | π − λ ( z ) F | = X ν ∈ P n,r | G F ν ,z | Y i | π − λ ( i ) ( x i ) F | , where π λ ( i ) : e O λ ( i ) → O λ ( i ) is a similar map as π λ applied to the case r = 1, byreplacing G by G i = GL ( M p i ), and x i = x | M pi has Jordan type ν ( i ) . It is knownby [L1] that q n ( ξ ( i ) ) χ L ξ ( i ) ( x i ) = e K ξ ( i ) ,ν ( i ) ( q ) for a partition ξ ( i ) of m i . It follows, byapplying (3.3.1) to the case where r = 1, and by the Grothendieck’s fixed pointformula, we have | π − λ ( i ) ( x i ) F | = X ξ ( i ) ≤ λ ( i ) K ξ ( i ) t ,λ ( i ) t e K ξ ( i ) ,ν ( i ) ( q ) . Then (3.3.2) implies that(3.3.3) χ K λ ,ϕ = | π − λ ( z ) F | = X ν ∈ P n,r g ν ,z ( q ) X ξ E λ K ξ t , λ t e K ξ (1) ,ν (1) ( q ) · · · e K ξ ( r ) ,ν ( r ) ( q ) . Since ( K ξ t , λ t ) λ , ξ is a unitriangular matrix with respect to the partial order ξ E λ ,by comparing (3.3.1) and (3.3.3), we obtain the required formula. (cid:3) Remark 3.4.
In general, X µ consists of infinitely many G -orbits. Hence the value g ν ,z ( q ) may depend on the choice of z ∈ X F µ . However, if X µ is a single G -orbit,then X F µ is also a single G F -orbit, and g ν ,z ( q ) is constant for z ∈ X F µ , in which case,we denote g ν ,z ( q ) by g µν ( q ). In what follows, we show in some special cases thatthere exists a polynomial g µν ( t ) ∈ Z [ t ] such that g µν ( q ) coincides with the value at t = q of g µν ( t ). We consider the special case where µ ∈ P ( m ′ ) is such that m ′ i = 0 for i = 1 , . . . , r −
2. In this case, X µ consists of a single G -orbit. In particular, for λ ∈ P n,r , dim H iz IC( X λ , ¯ Q l ) does not depend on the chocie of z ∈ X µ . We definea polynomial IC − λ , µ ( t ) ∈ Z [ t ] byIC − λ , µ ( t ) = X i ≥ dim H iz IC( X λ , ¯ Q l ) t i . The following result was proved in [S4].
Proposition 3.6 ([S4, Prop. 6.8]) . Let λ , µ ∈ P n,r , and assume that µ is as in3.5. (i) Assume that z ∈ X F µ . Then H iz IC( X λ , ¯ Q l ) = 0 if i is odd, and the eigen-values of φ λ on H iz IC( X λ , ¯ Q l ) are q i . In particular, χ L λ ( z ) = IC − λ , µ ( q ) . (ii) e K − λ , µ ( t ) = t a ( λ ) IC − λ , µ ( t r ) . As a corollary, we have the following result, which is a generalization of [AH,Prop. 5.8] (see also [LS, Prop. 3.2]).
Corollary 3.7.
Assume that µ is as in 3.5. (i) There exists a polynomial g µν ( t ) ∈ Z [ t ] such that g µν ( q ) coincides with thevalue at t = q of g µν ( t ) . (ii) We have (3.7.1) e K − λ , µ ( t ) = t a ( λ ) − rn ( λ ) X ν E λ g µν ( t r ) e K λ (1) ,ν (1) ( t r ) · · · e K λ ( r ) , ν ( r ) ( t r ) . Proof.
By Proposition 3.6 (i) and Proposition 3.3, we have(3.7.2) IC − λ , µ ( q ) = q − n ( λ ) X ν E λ g µν ( q ) e K λ (1) ,ν (1) ( q ) · · · e K λ ( r ) ,ν ( r ) ( q )By fixing µ , we consider two sets of functions { IC − λµ ( q ) | λ ∈ P n,r } and { g µν ( q ) | ν ∈ P n,r } . If we notice that e K λ (1) ,ν (1) ( q ) · · · e K λ ( r ) ,ν ( r ) ( q ) = q n ( λ ) for ν = λ , (3.7.2)shows that the transition matrix between those two sets is unitriangular. Hence g µν ( q ) is determined from IC − λ , µ ( q ), and a similar formula makes sense if we replace q by t . This implies (i). (ii) now follows from (3.7.2) by replacing q by t . (cid:3) In what follows, we assume that µ is of the form µ = ( − , . . . , − , ξ ) with ξ ∈ P n . In this case, g µν ( t ) coincides with the polynomial g ξν (1) ,...,ν ( r ) ( t ) obtainedfrom G ξν (1) ,...,ν ( r ) ( o ) discussed in [M, II, 2]. On the other hand, we define a polynomial f ξν (1) ,...,ν ( r ) ( t ) by(3.8.1) P ν (1) ( y ; t ) · · · P ν ( r ) ( y ; t ) = X ξ ∈ P n f ξν (1) ,...,ν ( r ) ( t ) P ξ ( y ; t ) . In the case where r = 2, g ξν (1) ,ν (2) ( t ) coincides with the Hall polynomial, and a simpleformula relating it with f ξν (1) ,ν (2) ( t ) is konwn ([M, III (3.6)]). In the general case, wealso have a formula(3.8.2) g ξν (1) ,...,ν ( r ) ( t ) = t n ( ξ ) − n ( ν ) f ξν (1) ,...,ν ( r ) ( t − ) . The proof is easily reduced to [M, III (3.6)].For partitions λ, ν (1) , . . . , ν ( r ) , we define an integer c λν (1) ,...,ν ( r ) by s ν (1) · · · s ν ( r ) = X λ c λν (1) ,...,ν ( r ) s λ . In the case where r = 2, c λν (1) ,ν (2) coincides with the Littlewood-Richardson coeffi-cient.For λ ∈ P n,r , put(3.8.3) b ( λ ) = a ( λ ) − r · n ( λ ) = | λ (2) | + 2 | λ (3) | + · · · + ( r − | λ ( r ) | . The following lemma is a generalization of [LS, Lemma 3.4].
Lemma 3.9.
Let λ , µ ∈ P n,r , and assume that µ = ( − , . . . , − , ξ ) . Then we have K − λ , µ ( t ) = t b ( µ ) − b ( λ ) X ν E λ f ξν (1) ,...,ν ( r ) ( t r ) K λ (1) ,ν (1) ( t r ) · · · K λ ( r ) ,ν ( r ) ( t r ) , (3.9.1) K − λ , µ ( t ) = t b ( µ ) − b ( λ ) X η ∈ P n c ηλ (1) ,...,λ ( r ) K η,ξ ( t r ) . (3.9.2) Proof.
The formula (3.7.1) can be rewritten as(3.9.3) K − λ , µ ( t ) = t a ( µ ) − a ( λ )+ rn ( λ ) X ν E λ t − rn ( ν ) g ξν (1) ,...,ν ( r ) ( t − r ) K λ (1) ,ν (1) ( t r ) · · · K λ ( r ) ,ν ( r ) ( t r ) . Substituting (3.8.2) into (3.9.3), we obtain (3.9.1). Next we show (3.9.2). One canwrite as s λ ( i ) ( y ) = X ν ( i ) K λ ( i ) ,ν ( i ) ( t ) P ν ( i ) ( y ; t ) . Hence s λ (1) ( y ) · · · s λ ( r ) ( y ) = X ν ∈ P n,r K λ (1) ,ν (1) ( t ) · · · K λ ( r ) ,ν ( r ) ( t ) P ν (1) ( y ; t ) · · · P ν ( r ) ( y ; t )(3.9.4) = X ν ∈ P n,r X ξ ∈ P n f ξν (1) ,...,ν ( r ) ( t ) K λ (1) ,ν (1) ( t ) · · · K λ ( r ) ,ν ( r ) ( t ) P ξ ( y ; t ) . On the other hand, s λ (1) ( y ) · · · s λ ( r ) ( y ) = X η ∈ P n c ηλ (1) ,...,λ ( r ) s η ( y )(3.9.5) = X η ∈ P n c ηλ (1) ,...,λ ( r ) X ξ ∈ P n K η,ξ ( t ) P ξ ( y ; t ) . By comparing (3.9.4) and (3.9.5), we have an equality for each ξ ∈ P n , X η ∈ P n c ηλ (1) ,...,λ ( r ) K η,ξ ( t ) = X ν ∈ P n,r f ξν (1) ,...,ν ( r ) ( t ) K λ (1) ,ν (1) ( t ) , . . . K λ ( r ) ,ν ( r ) ( t ) . Combining this with (3.9.1), we obtain (3.9.2). The lemma is proved. (cid:3)
Let η ′ = λ ′ − θ ′ , η ′′ = λ ′′ − θ ′′ be skew diagrams, where θ ′ ⊂ λ ′ , θ ′′ ⊂ λ ′′ are partitions. We define a new skew diagram η ′ ∗ η ′′ = λ − θ as follows; write thepartitions λ ′ , λ ′′ as λ ′ = ( λ ′ , . . . , λ ′ k ′ ) , λ ′′ = ( λ ′′ , . . . , λ ′′ k ′′ ) with λ ′ k ′ > , λ ′′ k ′′ >
0. Put a = λ ′′ . We define a partition λ = ( λ , . . . , λ k ′ + k ′′ ) by λ i = ( λ ′ i + a for 1 ≤ i ≤ k ′ ,λ ′′ i − k ′ for k ′ + 1 ≤ i ≤ k ′ + k ′′ . Write partitions θ ′ , θ ′′ as θ ′ = ( θ ′ , . . . , θ ′ k ′ ) , θ ′′ = ( θ ′′ , . . . , θ ′′ k ′′ ) with θ ′ k ′ ≥ θ ′′ k ′′ ≥ θ = ( θ , . . . , θ k ′ + k ′′ ), in a similar way as above, by θ i = ( θ ′ i + a for 1 ≤ i ≤ k ′ ,θ ′′ i − k ′ for k ′ + 1 ≤ i ≤ k ′ + k ′′ . We have θ ⊂ λ , and the skew diagram η ′ ∗ η ′′ = λ − θ can be defined.For λ, µ ∈ P n , let SST ( λ, µ ) be the set of semistandard tableaux of shape λ andweight µ . Let λ ∈ P n,r . An r -tuple T = ( T (1) , . . . , T ( r ) ) is called a semistandardtableau of shape λ if T ( i ) is a semistandard tableau of shape λ ( i ) with respect to the letters { , . . . , n } . We denote by SST ( λ ) the set of semistandard tableaux ofshape λ . For λ ∈ P n,r , let e λ be the skew diagram λ (1) ∗ λ (2) ∗ · · · ∗ λ ( r ) . Then T ∈ SST ( λ ) is regarded as a usual semistandard tableau e T associated to the skewdiagram e λ . Assume π ∈ P n . We say that T ∈ SST ( λ ) has weight π if thecorresponidng tableau e T has shape e λ and weight π . We denote by SST ( λ , π ) theset of semistandard tableaux of shape λ and weight π . In [M, I, (9.4)], a bijective map Θ (3.11.1) Θ : SST ( e λ , π ) ∼−→ a ν ∈ P n ( SST ( e λ , ν ) × SST ( ν, π ))was constructed, where SST ( e λ , ν ) is the set of tableau T such that the associatedword w ( T ) is a lattice permutation (see [M, I, 9] for the definition). Under theidentification SST ( e λ , π ) ≃ SST ( λ , π ), the subset SST ( λ , ν ) of SST ( λ , ν ) is alsodefined. Then we can regard Θ as a bijection with respect to the set SST ( λ , π )(and SST ( λ , ν )).In the case where r = 2, it is shown in [LS, Cor. 3.9] that | SST ( λ , ν ) | coincideswith the Littlewood-Richardson coefficient c νλ (1) ,λ (2) . A similar argument can beapplied also to the general case, and we have Corollary 3.12.
Assume that λ ∈ P n,r , ν ∈ P n . Then we have | SST ( λ , ν ) | = c νλ (1) ,...,λ ( r ) . For a semistandard tableau S , the charge c ( S ) is defined as in [M,III, 6]. It is known that Lascoux-Sch¨utzenberger Theorem ([M, III, (6.5)]) gives acombinatorial description of Koskta polynomials K λ,µ ( t ) in terms of sesmistandardtableaux,(3.13.1) K λ,µ ( t ) = X S ∈ SST ( λ,µ ) t c ( S ) . In the case where r = 2, a similar formula was proved for K λ , µ ( t ) in [LS, Thm.3.12], in the special case where µ = ( − , µ ′′ ). Here we consider K λ , µ ( t ) for general r .Assume that λ ∈ P n,r and ξ ∈ P n . For T ∈ SST ( λ , ξ ), we write Θ ( T ) = ( D, S )with S ∈ SST ( ν, ξ ) for some ν . we define a charge c ( T ) of T by c ( T ) = c ( S ). Wehave the following theorem. Note that the proof is quite similar to [LS]. Theorem 3.14.
Let λ , µ ∈ P n,r , and assume that µ = ( − , . . . , − , ξ ) . Then K − λ , µ ( t ) = t b ( µ ) − b ( λ ) X T ∈ SST ( λ ,ξ ) t r · c ( T ) . Proof.
We define a map Ψ :
SST ( λ , ξ ) → ` ν ∈ P n SST ( ν, ξ ) by T S , where Θ ( T ) = ( D, S ). Then by Corollary 3.12, for each S ∈ SST ( ν, ξ ), the set Ψ − ( S ) has the cardinality c ξλ (1) ,...,λ ( r ) , and by definition, any T ∈ Ψ − ( S ) has the charge c ( T ) = c ( S ). Hence X T ∈ SST ( λ ,ξ ) t c ( T ) = X ν ∈ P n X S ∈ SST ( ν,ξ ) c νλ (1) ,...,λ ( r ) t c ( S ) = X ν ∈ P n c νλ (1) ,...,λ ( r ) K ν,ξ ( t ) . The last equality follows from (3.13.1). The theorem now follows from (3.9.2). (cid:3)
Corollary 3.15.
Under the assumption of Theorem 3.14, we have K − λ , µ (1) = | SST ( λ , ξ ) | . In the rest of this section, we shall give an alternate description ofthe polynomial g µν ( t ) in the case where µ = ( − , . . . , − , ξ ). For ν ∈ P n,r , put R ν ( x ; t ) = P ν (1) ( x (1) ; t r ) · · · P ν ( r ) ( x ( r ) ; t r ). Then { R ν | ν ∈ P n,r } gives a basis ofΞ n [ t ]. We define funtions h µν ( t ) ∈ Q ( t ) by the condition that(3.16.1) R ν ( x ; t ) = X µ ∈ P n,r h µν ( t ) P − µ ( x ; t ) . The following formula is a generalization of Proposition 4.2 in [LS].
Proposition 3.17.
Assume that µ = ( − , . . . , − , ξ ) . Then h µν ( t ) = t a ( µ ) − a ( ν ) g µν ( t − r ) . Proof.