The path model, the quantum Frobenius map and Standard MonomialTheory

Peter Littelmann1

Departement de MathematiquesUniversite Louis Pasteur et Institut Universitaire de France7, rue Rene DescartesF-67084 Strasbourglittelma@math.u-strasbg.fr

Introduction

The aim of this article is to give an introduction to the theory of path modelsof representations and its associated bases. The starting point for the theory was aseries of articles in which Lakshmibai, Musili and Seshadri initiated a program toconstruct a basis for the space H0(G/B,L!) with some particularly nice geometricproperties. Here we suppose that G is a reductive algebraic group defined over analgebraically closed field k, B is a fixed Borel subgroup, and L! is the line bun-dle on the flag variety G/B associated to a dominant weight. The purpose of theprogram is to extend the Hodge-Young standard monomial theory for the groupGL(n) to the case of any semisimple linear algebraic group and, more generally, toKac-Moody algebras. Apart from the independent interest in such a construction,the results have important applications to the combinatorics of representations aswell as to the geometry of Schubert varieties. Inspired by the classical theory ofYoung tableaux for representations of GLn(C), they generalized the notion of atableau to all classical groups and some exceptional groups and a!ne Kac-Moodyalgebras. For the geometric applications note that standard monomial theory pro-vides proofs of the vanishing theorems for the higher cohomology of e"ective linebundles on Schubert varieties, explicit bases for the rings of invariants in classicalinvariant theory, a proof of Demazure’s conjecture, normality of Schubert varieties,another proof of the good filtration property, determination of the singular locusof Schubert varieties [9], deformation of SL(n)/B into a toric variety [2], etc.

We provide a di"erent access to standard monomial theory which avoids com-pletely all case by case consideration. In the first section we recall the theory ofpath models. The approach we take here is somewhat di"erent from [16,17]: Westart with lattice paths and develop from this point of view the notion of an a!nepath and the definition of the root operators e", f". We have provided completeproofs as far as it was made necessary by the new approach. In section 2 andsection 7, we discuss some applications of the combinatorics of the path model tocharacter theory (shrinking of characters and crystalline excellent filtration).

In section 4 we construct a basis of H0(G/B,L!) which is indexed in a canon-ical way by the L-S paths of shape !. The main tool in our construction is the

1Supported by TMR-Grant ERB FMRX-CT97-0100.

1

2

Frobenius map [23] for quantum groups at roots of unity (section 3). Having inmind the applications, one could say that in our approach we replace the “alge-braic” Frobenius splitting of Schubert varieties [25,26,27] by the “representationtheoretic” Frobenius.

The applications to the geometry of Schubert varieties are discussed in sec-tion 5 and 6. For example, the basis is compatible with the restriction mapH0(G/B,L!) ! H0(X,L!) to a Schubert variety X , and it has the “standardmonomial property”. I.e. for !, µ dominant, there exists a simple combinatorialrule to choose out of the set of all monomials p#p$ " H0(G/B,L!+µ) of ba-sis elements p# " H0(G/B,L!) and p$ " H0(G/B,Lµ), a subset, the standardmonomials, which forms a basis of H0(G/B,L!+µ). Other consequences of thetheory are that intersections of unions of Schubert varieties are scheme theo-retically reduced, we get a new proof of the Demazure character formula, thevanishing theorem for higher cohomology groups of dominant line bundles andSchubert varieties are projectively normal. In the last section we provide a repre-sentation theoretic application of the basis, we provide a new proof of the fact thatH0(G/B,L!) # H0(G/B,Lµ) admits a good filtration. All the proofs are charac-teristic free and work over the ring R obtained from Z by adjoining all roots ofunity. We restrict ourself here for simplicity to the finite dimensional case, but allthe results hold, with the obvious adaptions to the infinite dimensional case, forsymmetrizable Kac-Moody algebras.

The author wishes to thank the Newton Institute in Cambridge for its hospi-tality.

1. Some combinatorics

Let G be a connected complex reductive algebraic group G. Fix a Borel subgroupB $ G and a maximal torus H $ G, and let X = X(H) be the weight lattice ofH . We denote by XR := X #Z R the corresponding real vector space. We considernow “weights with tails”. More precisely, let # be the set of rectifiable paths inXR starting at the origin and ending in an integral weight. Two paths "1,"2 areconsidered as identical if there exists a nondecreasing, surjective, continuous map# : [0, 1] ! [0, 1] (a reparameterization) such that "1 = "2 % #:

# := {" : [0, 1] &! XR | " rectifiable, "(0) = 0, "(1) " X}/(reparameterization).

For simplicity, we will mainly consider the subset #Q $ # of piecewise linear pathshaving only rational turning points.

The aim of this section is to give an introduction to the combinatorics of thepath model of a representation. The notion of an a!ne path is new and is slightlymore general then the notion of L-S paths introduced in [16]. The advantage is thatthe definition is less technical then the definition of L-S paths. Though we havetried to give complete proofs whenever it seemed appropriate, we have skipped theproofs of those parts which are not used later in the basis construction. We referto [16,17,20] for detailed proofs.

3

Example 1. By abuse of notation, we write also ! for the path ! : [0, 1] ! XR,t '! t!, which connects the origin with a weight ! " X by a straight line.

By the concatenation " := "1 ("2 of two paths "1,"2 we mean the path definedby:

"(t) :=

!"1(2t), if 0 ) t ) 1/2

"1(1) + "2(2t & 1), if 1/2 ) t ) 1.

The concatenation is an associative operation, so # is a monoıde with the trivialpath I : [0, 1] ! XR, t '! 0, as the unit element. Note that the concatenation ofpaths in #Q is again a path in #Q, so #Q $ # is a submonoıde.

Example 2. Lattice paths: Let H $ GLn be the subgroup of diagonal matrices,note that H is a maximal torus in GLn. Denote by $i the projection of a diagonalmatrix onto its i-th entry. The weight lattice X = X(H) is then of rank n withbasis the $i, i = 1, . . . , n, i.e., X = Z$1 * . . . * Z$n and XR + Rn.

By a lattice path in XR we mean a concatenation of the ±$i, i = 1, . . . , n, sosuch a path is of the form " = (±$i1) ( (±$i2) ( . . . ( (±$ir ). Denote by L $ #the submonoıde of all lattice paths, and let L+ $ L be the submonoıde of latticepaths that are concatenations only of the $i, i = 1, . . . , n.

The picture below (n = 2) shows the image of the lattice path:

" = $1 ( (&$2) ( (&$2) ( (&$1) ( (&$1) ( (&$1) ( $2 ( $1 ( $2 ( $2 ( $2 ( $1 ( $1 ( $1 ( $1

$2"#

&&&&! $1

&&&&&&&!

Example 3. Let H ! $ SLn be the subgroup of diagonal matrices of determinant1. As above, let $i denote the projection of such a matrix onto its i-th entry aswell as the corresponding path $i : [0, 1] ! XR, t '! t$i, that joins the origin withthe weight by a straight line. Recall that X = Z$1 + . . . + Z$n, but the sum is nolonger direct because $1 + . . . + $n = 0.

We denote by L+ $ # the submonoıde of all lattice paths of the form " =$i1 ( $i2 ( . . . ( $ir . Note that for G = SLn every weight in X is the endpoint ofsome " " L+.

Example 4. Suppose G = SL3. Consider the set D(2, 1) := {T1, T2, . . . , T7, T8}of all semi-standard Young tableaux (i.e., the entries are strictly increasing in thecolumns and non-decreasing in the rows) of shape (2, 1):

D(2, 1) :=

$

T1 :=1 12

, T2 :=1 13

, T3 :=1 22

, T4 :=1 23

,

T5 :=1 32

, T6 :=1 33

, T7 :=2 23

, T8 :=2 33

%

.

4

We associate to a tableau T a lattice path "T " L+, which is a concatenation ofthe $i according to the entries of the tableaux. We read the entries of the tableaurow-wise, from the right to the left in each row, and we start with the top row:

"T1 := $1 ( $1 ( $2, "T2 := $1 ( $1 ( $3, "T3 := $2 ( $1 ( $2, "T4 := $2 ( $1 ( $3,"T5 := $3 ( $1 ( $2, "T6 := $3 ( $1 ( $3, "T7 := $2 ( $2 ( $3, "T8 := $3 ( $2 ( $3.

The endpoint "T (1) of the associated path is the weight of the correspondingtableau.

Example 5. Suppose G = SLn. The general rule to associate a path to a semi-standard Young tableau T is the following: Numerate the boxes b1, . . . , b% of thetableau row-wise, from the right to the left in each row, and we start with the toprow. Let nj be the entry of the j-th box. We define the path associated to thetableau T to be the concatenation "T := $n1 ( . . .( $n!

. In this way we can identifythe semi-standard Young tableaux with a special class of lattice paths.

The connection between the combinatorics of lattice paths (or rather the wordalgebra) and tableaux has been studied extensively, see for example [8,13].

Example 6. A!ne paths Let G be an arbitrary connected complex reductivealgebraic group. It turns out that in the general case it is not su!cient to consideronly lattice paths. The a!ne paths can be thought of as a generalization of latticepaths.

We fix a Weyl group invariant scalar product (·, ·) on on XR. Denote by $the root system of G, by $+ the subset of positive roots, and for % " $+ let%" := 2%/(%,%) be its co-root. We write & , µ for two weights &, µ " X if thedi"erence is a sum of positive roots, and for % " $ we write % , 0 if % " $+. Form " Z denote by H

m& the a!ne hyperplane:

Hm& := {& " XR | (&,%") = m}.

Denote by H the union of the H0& and by H the union of the a!ne hyperplanes

Hm& :

H =&

&#0

H0& , H =

&

&#0,m$Z

Hm& .

Recall that we can characterize the simple roots among the positive roots by thefollowing property: ' , 0 is a simple root if and only if s"(() , 0 for all ( , 0,( -= '. In other words: ' is simple if and only if for any ( , 0 the propertys"(() . 0 implies ( = '.

Let R be a subset of the set of simple roots. We say that % " R is simple if forany ( " R the properties s&(() . 0 and &s&(() " R implies ( = %.

Let &, µ be rational weights. We say that µ is obtained from & by a simplebending with respect to a positive root % if (&,%") < 0, s&(&) = Cµ for someC > 0, and % is simple (in the sense above) among the positive roots such that(&, () ) 0.

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For example, if ' is a simple root and (&,') < 0, then (&, s"(&)) is a simplebending.

Suppose " : [0, 1] ! XR, " " #Q, is a path with rational turning points,so we can find rational weights &1, . . . , &r such that (up to reparameterization)" = &1 ( . . . ( &r.

Definition 1. The path " is called an a!ne path if the following conditions aresatisfied:i) All turning points lie in H.ii) Suppose P = &1 + . . .+ &s is a turning point and P -" X . Then there exist some

positive roots %1, . . . ,%q such that P "'q

i=1 Hmi

&ifor appropriate m1, . . . , mq "

Z, and &s+1 is obtained from &s by a sequence of simple bendings: (&s, s&1(&s))with respect to %1, . . . , and (s&q!1 . . . s&1(&s), &s+1) with respect to %q.

Roughly speaking, the conditions imply that the path can change its directiononly in a point P lying in the set H, and, if P is not an integral weight, then itis only allowed to change the direction by applying simple bendings. Note thatthese reflections correspond to roots that are associated to the a!ne hyperplanescontaining P .

A priori, the property of being an a!ne path depends on the chosen parameteri-zation. If we speak of an a!ne path " = &1(. . .(&r, then we assume always that thepath is a!ne with respect to this parameterization. In particular, all the turningpoints Pi = &1 + . . . + &i are elements of H.

To explain the “usefulness” of the simple bending, note the following “geomet-ric” consequence for turning points with respect to simple roots.

Lemma 1. Let " = &1(. . .(&r be an a!ne path and denote by Pi = &1+. . .+&i thei-th turning point. Fix a simple root ' and suppose that " changes in Pi its directionrelative to ' such that either (&i,'") < 0 and (&i+1,'") / 0, or (&i,'") = 0 and(&i+1,'") > 0. Then Pi " H

m" for some m " Z.

Proof. This is obvious if Pi " X . We may hence assume that Pi -" X , so Pi

satisfies condition ii). Suppose that µ is obtained from & by a simple bending and(&,'") < 0 and (µ,'") / 0, or (&,'") = 0 and (µ,'") > 0. We will show that thebending is of the form µ = Cs"(&). Since &i+1 is obtained from &i by a sequenceof simple bendings, one of them has to correspond to the reflection s". By thedefinition of an a!ne path, this implies Pi " H

m" for some m " Z and finishes

hence the proof.Let % , 0, % -= ' be such that µ = s&(&) is obtained from & by a simple

bending. Suppose first (&,'") < 0, and (µ,'") / 0. Since (&,%") < 0 we can have

(µ,'") = (s&(&),'") = (& & (&,%")%,'") = (&,'") & (&,%")(%,'") / 0

only if (%,'") > 0. But this implies s&(') . 0. It follows that &s&(') , 0 and

(&,&s&('")) = &(s&(&),'

") = &(µ,'") ) 0,

which contradicts the assumption that % is simple in the set of ( , 0 such that(&, () ) 0.

The same arguments apply also to the case (&,'") = 0 and (µ,'") > 0. !

6

Example 7.i) If µ1, . . . , µr " X is a sequence of integral weights, then " := µ1 (µ2 ( . . .(µr is

an a!ne path. In particular, the lattice paths defined in Example 2 and 3 area!ne paths.

ii) If ", ) are a!ne paths, then " ( ) is again an a!ne path. So the set of a!nepaths forms a submonoıde of #.

iii) If necessary, then we can introduce more “turning points” on an a!ne path.Let " := &1 ( &2 ( . . . ( &r be an a!ne path. Suppose 1 ) i ) r and c, c! > 0are such that c + c! = 1 and P := &1 + . . . + &i%1 + c&i " H. Then "! :=&1 ( &2 ( . . . ( &i%1 ( c&i ( c!&i ( &i+1 ( . . . ( &r and " have the same image, and"’ is an a!ne path: all turning points except P are admissible by assumption.Further, P " H, so condition i) is satisfied, and ii) is trivially satisfied becausethe path does not change its “direction” in P .

Example 8. A way to produce new paths out of given ones is by stretching them:For n " N and " " # let n" be the path obtained by stretching " by the factorn, i.e., (n")(t) := n("(t)) for t " [0, 1]. Note that if " is an a!ne path, then n"is a!ne too. Further, if ) " #Q is an arbitrary path, then we can always find ann " N such that n) is an a!ne path. For example, we could choose n such thatall turning points of n) are integral weights.

The evaluation map Char : # ! X , " '! "(1), is a map of monoıdes:

Char("1 ( "2) = ("1 ( "2)(1) = "1(1) + "2(1) = Char("1) + Char("2).

Let Z[#] be the (non-commutative) Z-algebra generated by #, and let Z[X ] be the(commutative) group algebra over the weight lattice. We extend the evaluationmap to a map of Z-algebras:

Char : Z[#] &! Z[X ],k(

i=1

ai"i '!k(

i=1

aie#i(1).

Example 9. For G = SL3 and D := D(2, 1), let S be the formal sum)

T$D "T "Z[#] over all semi-standard Young tableaux of shape (2, 1) (Example 4). The image)

T$D e#T (1) of S by Char is the character of the adjoint representation of G.

Example 10. Suppose now G = SLn. Recall that the irreducible finite dimen-sional complex representations of g are in bijection with partitions p = (p1, . . . , pn)of length ) n (i.e. p1, . . . , pn is a weakly decreasing sequence of non-negative in-tegers). Let V (p) be the corresponding irreducible representation, and let D(p)be the set of all semi-standard Young tableau of shape p, i.e., the tableau hasp1 boxes in the first row, p2 boxes in the second row, etc. It is well-known thatthe combinatorics of tableaux and the representation theory of SLn are closelyrelated: The character of V (p) is the sum

)D(p) e'(T ), where &(T ) denotes the

weight of the tableaux T (= the endpoint of the associated lattice path "T ). So interms of lattice paths we find: CharV (p) =

)T$D(p) e#T (1)

7

Fix a simple root '. To get character formulas for representations as the sumof endpoints of a set of paths, we define linear operators e" and f" on Z[#]. Moreprecisely, these operators are maps # ! # 0 {0} which will be extended linearlyto all of Z[#].

Example 11. Suppose G = SLn. We will characterize the paths associated tosemi-standard Young tableaux by defining operators on L+ 0 {0}. For a simpleroot ' = $i & $i+1 and a path " = $j1 ( . . . ( $j!

set

h"(t) := *{k|jk = i, 1 ) k ) t}& *{k|jk = i + 1, 1 ) k ) t}, for 0 ) t ) +.

Let m# := min{h"(t)|0 ) t ) +} be the minimal value of the h"(t), note thatm# ) 0 because h"(0) = 0. We define now operators ei, fi on L+ 0 {0} accordingto the value m#.

We set ei(0) = fi(0) := 0.If m# = 0, then set ei(") := 0. Otherwise fix t minimal such that h"(t) = m#.

The minimality of t implies that jt = i+1, so " = $j1 (. . .($jt!1($i+1($jt+1(. . .($j!.

We set:ei(") := $j1 ( . . . ( $jt!1 ( $i ( $jt+1 ( . . . ( $j!

.

If m# = h"(+), then set fi(") := 0. Otherwise fix t maximal such that h"(t) = m#.The maximality of t implies that jt+1 = i, so " = $j1 ( . . . ( $jt ( $i ( $jt+2 ( . . . ( $j!

.We set:

fi(") := $j1 ( . . . ( $jt!1 ( $i+1 ( $jt+1 ( . . . ( $j!.

For a partition p = (p1, . . . , pn) set

"p := $1 ( . . . ( $1* +, -p1

( $2 ( . . . ( $2* +, -p2

( . . . ( $n ( . . . ( $n* +, -pn

,

and denote by B(p) $ L+ the smallest subset containing "p such that B(p)0{0}is stable under the operators ei, fi, 1 ) i ) n & 1. We leave it as an exercise forthe reader to verify that

B(p) = {"T | T semi-standard Young tableau of shape p}

For the action of the operators on tableaux for other classical groups and relatedcombinatorial problems we refer to [7,12,14,18,22], for the connection to Gelfand-Tsetlin patterns and its generalizations we refer to [1,22].

Example 12. We extend the definition of the operators to a!ne paths. Let " =&1 ( &2 ( . . . ( &r be an a!ne path. Fix a simple root ', and let m ) 0 be minimalsuch that H

m" 1 Im" -= 2. Recall that any turning point relative to a simple root

lies on an a!ne hyperplane corresponding to this root (Lemma 1), so the imageof " lies in the a!ne halfspace H

m,+" := {& " XR | (&,'") / m}.

If m = 0, then set e"(") := 0. Otherwise, let j be minimal such that &1 +. . . + &j " H

m" , and let 1 ) i ) j be maximal such that &1 + . . . + &i " H

m+1" .

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(We may assume that such a turning point exists by Example 7, and note that(&i+1,'"), . . . , (&j ,'") < 0 by Lemma 1.) We define

e"(") := &1 ( . . . ( &i ( s"(&i+1) ( . . . ( s"(&j) ( &j+1 ( . . . ( &r.

The definition of the operators f" is similar. If m = ("(1),'"), then we set f"(") :=0. Otherwise let i be maximal such that &1 + . . . + &i " H

m" , and let i ) j ) r be

minimal such that &1+ . . .+&j " Hm+1" . (We may assume that such a turning point

exists by Example 7, and note that (&i+1,'"), . . . , (&j ,'") > 0 by Lemma 1.) Wedefine

f"(") := &1 ( . . . ( &i ( s"(&i+1) ( . . . ( s"(&j) ( &j+1 ( . . . ( &r.

Note if e"(") -= 0, then e"(")(1) = "(1) + ', and if f"(") -= 0, then f"(")(1) ="(1) & '. In particular, e"("), f"(") " #Q 0 {0}.

Note if G = SLn and " is a lattice path, then f"(") respectively e"(") is thesame path as defined in Example 11.

It remains to point out that the new paths (if di"erent from 0) are again a!ne.We will prove this only for the path e"(") = µ1 ( . . . (µr, the arguments for f"(")are similar.

Let P% = &1 + . . . + &% and Q% = µ1 + . . . + µ% be the turning points of the path" and e"("). Let i and j be as in the definition of e" above. Then P% = Q% for1 ) + ) i, P% = Q% & ' for j ) + ) r and Q% is obtained from P% by an a!nereflection. Since H is stable under these operations we see that the Q% " H. If theturning point is an integral weight, then nothing is to prove. So suppose in thefollowing that the turning point is not an integral weight.

Further, (µ%, µ%+1) = (&%, &%+1) for + < i and j < +, so µ%+1 is obtained from µ%by a sequence of simple bendings.

For i < + < j we have (µ%, µ%+1) = (s"(&%), s"(&%+1)). Suppose !, & are rationalweights such that (!,'), (),'") > 0 or (!,'), (),'") < 0, and C& = s&(!) isobtained from ! by a simple bending. It is then easy to see that s"()) is obtainedfrom s"(!) by a simple bending with respect to the root s"(%). It follows thats"(&%+1) is obtained from s"(&%) by a sequence of simple bendings.

Note that (µi, µi+1) = (&i, s"(&i+1)). Since &i+1 is obtained from &i by a se-quence of simple bendings and (&i+1, s"(&i+1)) is a simple bending, it follows thatµi+1 is obtained from µi by a sequence of simple bendings.

It remains to consider the pair (µj , µj+1) = (s"&j , &j+1). We know that &j

is obtained from &j+1 by a sequence of simple bendings, let %1, . . . ,%q be thecorresponding positive roots. Since (&j ,'") ) 0 and (&j+1,'") > 0, we have seenin the proof of Lemma 1 that at least one of the %% has to be equal to '. Supposethat 1 ) + ) q is minimal with this property, one sees then easily that the rootss"(%), . . . , sa(%%%1),%%+1, . . . ,%q provide a sequence of simple bendings for thepair (s"&j , &j+1). !

Since all turning points relative to the direction of a simple root lie on an a!nehyperplane H

m" , it follows easily that if ", ) are a!ne paths, then e"(" ( )) =

(e"") ( ) or " ( (e")). Similarly, f"(" ( )) = (f"") ( ) or " ( (f")). More precisely,a simple calculation shows that:

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Lemma 2. If ", ) are a!ne paths, then e"(" ( )) = " ( (e")) if there exists ann > 0 such that en

") -= 0 but fn"" = 0, and it is equal to (e"") ( ) otherwise.

Similar, f"(" ( )) = (f"") ( ) if there exists an n > 0 such that fn"" -= 0 but

en") = 0, and it is equal to " ( (f")) otherwise.

We generalize the operators now to arbitrary piecewise linear paths. The def-inition of these operators have been inspired by the work of Kashiwara on crys-tal bases, see [4,5]. Note that the operators defined on a!ne paths “commute”with stretching, i.e. n

.f"(")

/= fn

" (n") respectively n.e"(")

/= en

"(n"), andwith “shrinking”, i.e., if 1

n" is also an a!ne path, then 1n

.fn" (")

/= f"( 1

n") and1n

.en"(")

/= e"( 1

n").We have seen that if " " #Q, then we can always find an n such that n" is

a!ne. We define

f"(") :=1

nfn" (n"), e"(") :=

1

nen"(n").

Since on a!ne paths stretching/shrinking commutes with the operators, the defi-nition is independent on the choice of n.

We provide now a di"erent description of the operators, we leave it as an exerciseto verify that the definitions coincide. For a path " " #Q let h# be the functiondefined by:

h# : [0, 1] ! R, t '! ("(t),'"),

and let m# := min h# be the minimal value attained by the function h#. If h#(1)&m# / 1, then fix t0 " [0, 1] maximal such that h#(t0) = m# and let t1 be minimalsuch that h#(t) / m# + 1 for t / t1. Denote by #+ : [0, 1] ! R the function:

#+(t) :=

012

13

0, if t " [0, t0];

min{h#(s) & m# | s " [t, t1]}, if t " [t0, t1];

1, if t " [t1, 1].

Figure 1. The functions h!(t) and !+(t)

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Definition 2. If h#(1) & m# < 1, then set f"(") := 0. Otherwise let f"(") " #Q

be the path defined by f"(")(t) := "(t) & #+(t)'.

The definition of the operator e" is similar. If m# ) &1, then fix t0 " [0, 1]maximal such that h#(t) / m# + 1 for t ) t0 and let t1 be minimal such thath#(t) = m#. Denote by #% : [0, 1] ! R the function:

#%(t) :=

012

13

0, if t " [0, t0];

max{m# & h#(s) + 1 | s " [t0, t]}, if t " [t0, t1];

1, if t " [t1, 1].

Definition 3. If m# > &1, then set e"(") := 0. Otherwise let e"(") " #Q be thepath defined by e"(")(t) := "(t) + #%(t)'.

A chamber C $ XR is a connected component of XR & H, and an alcove A is aconnected component of XR & H. The closure C of C respectively the closure Aof A is taken in the usual topology on Rn. The closure of the chamber C0 := {& "XR | (&,%") > 0 3% , 0} is called the dominant Weyl chamber. The followingproperties follow easily from the definition of the operators:

Lemma 3.a) If e"" -= 0, then e"(")(1) = "(1) + ' and f"(e"") = ".b) If f"" -= 0, then f"(")(1) = "(1) & ' and e"(f"") = ".c) Let , be half the sum of the positive roots. Then e"" = 0 for all simple roots if

and only if ,+Im" (the image of " shifted by ,) is completely contained in theinterior C0 of the dominant Weyl chamber C0.

d) Let m, n be maximal such that ema " -= 0 resp. fn

a " -= 0. Then n&m = ("(1),'"),and

m = max{a " Z | a ) |m#|}, n = max{a " Z | a ) ("(1),'") & m#}.

e) The stretching of paths commutes with the operators, i.e., n(e"") = en"(n") and

n(f"") = fn" (n").

Extend the operators linearly to the algebra Z[#Q]. For a path " denote byB(") $ #Q the set of all paths one gets from " by applying the operators. TheZ-span ZB(") $ Z[#Q] is then the minimal submodule which is stable under theoperators and contains ".

Example 13. Recall that we denote by µ " X also the straight line joining theorigin with the weight µ. For a simple root ' set n := (µ,'"). It is easy to see thatif n / 0, then e"(µ) = 0 and fn

" (µ) = s"(µ), and if n < 0, then f"(µ) = 0 ande%n" (µ) = s"(µ). As a consequence we find all the paths w(µ), w " W , in B(µ).

Example 14. By part e) of the lemma we have a well defined map B(") -!B(n"), ) '! n). We can hence identify B(") with the subset of B(n") obtainedby applying only n-th powers of the operators to n".

11

Proposition 1. The set B("), " " #Q, is finite.

Proof. Let " : [0, 1] ! XR, " = &1 ( . . . ( &p be an a!ne path. By Example 7 wemay add turning points if necessary and assume that all “parts” of the path arecompletely contained in the closure of some alcove. I.e., we may assume that forall j = 1, . . . , q, there exists a (not necessarily unique) alcove Aj such that theimage of the path t '! t&j , shifted by the weight &1 + . . .+&j%1, is contained in Aj .In particular, Im" $

4j Aj . So we can think of the path as a sequence of alcoves

Aj together with a segment Sj $ Aj joining two points p1, p2 " Aj & Aj .Suppose f"(") = µ1(. . .(µp -= 0. If we want to choose alcoves for this path with

the same properties as above, then, by the construction of the operators, we knowthat there exist i, j such that we may take for f"(") the alcoves A% for 1 ) + ) i,the alcoves A% + ' (i.e., shifted by the root ') for j ) + ) p, and s",m(A%) fori < + ) j (and appropriate m). Here s",m denotes the a!ne reflection at the a!nehyperplane H

m" . The same arguments apply also to a path of the form e"(").

It follows that we can view the set B(") as a subset of the set of paths we getby rearranging a given finite set of “marked” alcoves. Recall that a face F of analcove A is the intersection A1 H

m1&1

1 . . .1 Hmr

&rof A with some a!ne hyperplanes

Hm1&1

, . . . , Hmr

&r" H. So applying the operators is like rearranging the alcoves such

that they are still joined along common faces, and the segments of two such joinedalcoves have each an endpoint in the common face, and these endpoints match.

The moves which are allowed for the “rearrangements” are a!ne reflectionsand translation with respect to simple roots. Since these rearrangements haveto be connected to the origin (i.e., the first segment has the origin as one of theendpoints), we see that there are only a finite number of possibilities. In particular,B(") is a finite set.

Now suppose " " #Q is an arbitrary path. In Example 14 we have seen that wecan identify B(") with a subset of B(n"). If we choose n such that n" is a!ne(Example 8), then B(n") is finite by the arguments above, and hence B(") is afinite set. !

Remark 1. The proof shows that we can think of the “a!ne” paths as a kind ofdomino game in the a!ne space XR. The pieces are closed alcoves together withmark: a segment joining two points in A & A. One rule to join two pieces alonga face is of course that the the endpoints of the segments should match, but, ifthe “direction” of the segment changes, then one has to assure that the changeis obtained by a sequence of “allowed” simple bendings. This definition assuresthat the root operators f", e" translate in this language into operators on a givenarrangement of marked “pieces”, i.e., the alcoves.

Lemma 4. For " " #Q there exists an ) " #Q such that , + Im ) $ C0 andB()) = B("). If " is a!ne, then there exists an a!ne ) such that Im ) $ C0 andB()) = B(")

Proof. We know by Lemma 3 a),b): B(") = B()) for ) " B("), so we may replaceB(") by B(e"") if e"" -= 0. Since B(") is finite, we may hence assume that

12

e"" = 0 for all simple roots. By Lemma 3, this is equivalent to Im ) + , $ C0. If" is a!ne, then e"(") = 0 means that Im" 1 H

%1" = 2. So Im" lies in the a!ne

halfspace H0,+" (see Example 12), and hence Im ) $ C0. !

Theorem 1.a) Let B $ # be a finite subset such that the span ZB $ Z[#] is stable under the

root operators, and let "1, . . . ,"s " B be the paths such that Im"j +, $ C0. Thesum CharB :=

)$$B(#) e$(1) is the sum of the characters of the finite dimen-

sional irreducible representations of G of highest weight !1 := "1(1), . . . ,!s :="s(1).

b) For a dominant weight ! " X+ denote by ! : [0, 1] ! XR also the path t '!t!. Then CharB(!) is the character of the irreducible representation V (!) ofhighest weight !.

Remark 2. The part b) of the theorem above holds in much more generality, butin the rest of this paper we need only the weaker statement above. Note that b)is true for any set of paths of the form B("), where " is a path that such thatits image is completely contained in the dominant Weyl chamber (see [17] for aproof).

Proof. By the Weyl character formula we have to show: ($(w) := the sign of w)

(

#$B

e#(1) =s(

i=1

CharV (!i) =s(

i=1

. (

w$W

$(w)ew(!i+()//. (

w$W

$(w)ew(()/.

To prove the latter is equivalent to prove that

(

#$B,w$W

$(w)e#(1)+w(() =s(

i=1

. (

w$W

$(w)ew(!i+()/.

The terms on both sides are semi-invariant with respect to the action on the Weylgroup on Z[X ]. So to prove the equality above, it is su!cient to compare the termscorresponding to dominant weights:

(

#$B,w$W,#(1)+w(()$X+

$(w)e#(1)+w(() =s(

i=1

e!i+(.

Let B! be the set of pairs (", w) such that " " B and w " W are such that w(,)+"(1) " X+ and (", w) -= ("i, id) for some i. In other words: B! is the set of pairs(", w) occurring in the sum on the left above which have the property that Im"+w(,) is not completely contained in the interior of the dominant Weyl chamber.So to prove part a) is equivalent to prove that

)(#,w)$B" $(w)e#(1)+w(() = 0.

Let p " C0 & C0 be a point lying on a face of the dominant Weyl chamber.Denote by B!(p) the subset of paths (", w) " B! such that "(t0) + w(,) = p forsome t0 " [0, 1] and "(t) + w(,) lies in the interior of the dominant Weyl chamber

13

for t0 < t ) 1. Since B! is the disjoint union of such B!(p), to prove a) it su!cesto prove

)(#,w)$B"(p) $(w)e#(1)+w(() = 0 for such a point p.

We define an involution I on B!(p). For (", w) " B!(p) let t0 " [0, 1] be suchthat "(t0) + w(,) = p and "(t) + w(,) lies in the interior of the dominant Weylchamber for t0 < t ) 1. Let ' be a simple root such that (p,'") = 0 and setn := (w(,),'"). Suppose first n > 0. Since (w(,),'") + ("(t0),'") = (p,'") = 0,we know that the minimum of the function t ! ("(t),'") is smaller or equal to&n. By Lemma 3 this implies that "! := en

"(") -= 0. Consider the pair ("!, s"w).Since "(t)+w(,) lies in the interior of the dominant Weyl chamber for t0 < t ) 1,by the definition of the operator e" we know that "!(t) = "(t) + n' for t / t0 andhence

w(,) + "(t) = s"w(,) + "!(t) for t / t0.

In particular, ("!, s"w) " B!(p). So for n > 0 we define I(", w) := ("!, s"w).If n < 0, then similar arguments show that I(", w) := (f%n

" ", s"w) " B!(p)and w(,) + "(t) = s"w(,) + f%n

" "(t) for t / t0. It is easy to check that I is aninvolution. Since $(w)e#(1)+w(() + $(s"w)e#

"(1)+s"w(() = 0 for ("!, s"w) = I(", w),it follows that

)(#,w)$B"(p) $(w)e#(1)+w(() = 0.

It remains to prove part b). By a) we have to show that the only path " " B(!)such that Im" + , $ C0 is the path " = !. Since ! is an a!ne path (Example 7),all paths in B(!) are a!ne.

Let ) = µ1 ( . . . ( µr " B(!) be an arbitrary path. For all i = 1, . . . , r we canfind an element wi " W/W! such that µi = ciwi(!). Suppose w1 / . . . / wr. It iseasy to check that then the corresponding Weyl group elements for f"), e") havethe same property. Since the starting path ! = id(!) has obviously this property,it follows that all elements in B(!) have the property that the associated sequenceof elements in W/W! is decreasing in the Bruhat order.

We have seen in the proof of Lemma 4 that the condition Im"+, $ C0 impliesfor an a!ne path in fact Im" $ C0. Suppose " = &1 ( . . . ( &p " B(!), thenIm" $ C0 implies that C&1 = ! for some C > 0. But this implies for the path "that the associated sequence of elements in W/W! starts with w1 = id, so wi = idfor all i and hence " = !. !

As noted above, the second part holds more generally for arbitrary paths havingits image in the dominant Weyl chamber. Let now ", ) be a!ne paths having itsimage in the dominant Weyl chamber. Denote by ! := "(1) and µ := )(1) theendpoints. The set of all concatenations of paths in B(") and B()) is denoted byB(") ( B()). By Lemma 2, B(") ( B()) 0 {0} is stable under the root operators.So we get:

Proposition 2. B(")(B()) =4$" B("()!), where )! is running over all elements

in B()) such that the image of " ( )! is contained in the dominant Weyl chamber.

The character formula implies the following generalization of the Littlewood-Richardson rule:

14

Corollary 1. V (!) # V (µ) +5$" V (! + )!(1), where )! is running over all el-

ements in B()) such that the image of " ( )! is contained in the dominant Weylchamber.

We have already pointed out in Remark 2 that the character formula holds forall sets of the form B("), where " " #Q is such that its image is contained in thedominant Weyl chamber. This independence on the chosen path has in fact a muchdeeper background. Denote by G(") the associated colored directed graph havingas vertices the set B("), and we put an arrow )

"&!)! with color a simple root '

between two paths ), )! " B(") if f"()) = )!. A proof of the following theoremcan be found in [17] or [20]. The theorem implies that the graph is isomorphic tothe crystal graph of the corresponding representation (see [6] or [3]).

Theorem 2. Suppose ","! " #Q are such that Im", Im "! are contained in thedominant Weyl chamber. The map " ! "! extends to an isomorphism of coloreddirected graphs G(") ! G("!) if and only if "(1) = "!(1).

2. Contracting characters

There are many ways to transform the path model of a representation suchthat the new set of paths is stable under the operators and hence corresponds toa character of a representation. One method is that of contracting or shrinkingthe paths. This section is not really needed in the following, but the results areinteresting on its own and have been the motivation for the construction of thebasis in the next section.

For a dominant weight ! " X+ let V (!) be the corresponding irreducible com-plex representation of G. Fix a natural number + " N, and let , " X be half thesum of the positive roots. By the contracted character CharV (!)

1! we mean the

(Weyl group invariant) sum

CharV (!)1! :=

(

µ$%X

dimVµ(!)eµl .

Proposition 3. CharV (!)1! is a nonnegative linear combination of characters of

irreducible representations:

CharV (!)1! =

(

'$X+

m'!,% CharV (&).

The multiplicity m'!,% with which the character of V' occurs in the sum above isequal to the multiplicity of V ((+ & 1),+ +&) in the tensor product V ((+ & 1),) #V (!).

Proof. As before, denote by ! also the path joining the origin with ! by a straightline, and let B(!) be the corresponding set of paths obtained by applying the rootoperators. The character of V (!) is given by

CharV (!) =(

$$B(!)

e$(1).

15

If )(1) " +X , then the path ()/+) : [0, 1] ! XR, t '! )(t)/+, is again an element of#Q (but not necessarily a!ne). Denote by B(!)% the set of paths ) " B(!) suchthat )(1) " +X , and let

B(!) := {)/+ | ) " B(!)%}

be the set of contracted paths. We have obviously: CharV (!)1! =

)$$B(!) e$(1).

The root operators commute with “stretching”: For all k " N and all ) " # wehave f"(k)) = fk

"()) and e"(k)) = ek"()). Since B(!)% 0 {0} is stable under the

+-th powers of the root operators, it follows that the set B(!)0{0} is stable underthe root operators. The character formula implies:

(

$$B(!)

e$(1) =(

$$B(!)+

CharV ()(1)),

where B(!)+ $ B(!) is the subset of paths ) such that Im ) + , is contained inC0. The latter is equivalent to say that Im(+)) + +, is contained in C0. Since +) isa!ne, this is equivalent to say that Im(+&1),(+) is contained in C0. By the tensorproduct formula this implies that the multiplicity m'!,% is equal to the multiplicityof V ((+ & 1),+ +&) in V ((+& 1),) # V (!). !

We describe now a di"erent version of the contraction procedure which, as wewill see later, has a representation theoretic interpretation. We fix a numerationof the simple roots '1, . . . ,'n, let A = (ai,j)1&i,j&n be the Cartan matrix of g

and denote by At the transposed matrix. Fix d = (d1, . . . , dn), di " N, minimalsuch that the matrix (diai,j) is symmetric. We denote by d the smallest commonmultiple of the dj , and we set d = (d1, . . . , dn), where di := d/di. The form (·, ·)is chosen such that ('"

i , h) := 4'i, h5/di.The transposed matrix At = (ai,j), ai,j := aj,i is the Cartan matrix of the

root system $t dual to the root system $ of g. To realize $t as a root system inH', note that the di are minimal with the property that (djai,j) is a symmetricmatrix. The triple (H,%,%") defined by (i := 'i/di " H' and ("i := di'"

i " H is arealization of At, i.e., (1, . . . , (n is a basis of a root system isomorphic to $t, andthe ("i form a set of co-roots. Denote by Xt the weight lattice for the dual rootsystem:

Xt = {! " H' | 3i : 4!, ("i 5 " Z} = {! " H' | 3i : di4!,'"i 5 " Z}.

It follows immediately from the definition: dXt $ X $ Xt.Denote by g the Lie algebra of G. Suppose now ! " Xt is a dominant weight

for the semisimple Lie algebra gt associated to the Cartan matrix At, let V (!)t bethe corresponding complex irreducible highest weight representation. Fix a naturalnumber l divisible by d and set + := +/d. Let ,t " Xt be half of the positive rootsfor the root system of gt.

16

Proposition 4. Char(V (!)t)% :=)

µ$%X dimVµ(!)teµ

! is a nonnegative linearcombination of characters of irreducible g-representations:

Char(V (!)t)% =(

'$X+

m'!,% CharV (&).

The multiplicity m'!,% is equal to the multiplicity of V t(+,& ,t + +&) in the tensor

product of gt-representations V t(+,& ,t) # V t(!).

Proof. We attach to a gt-object a t to distinguish it from the corresponding g-object. Note that the dominant Weyl chamber in XR = Xt

R is for both the same.Let ! be the path that joins the weight ! with the origin by a straight line. Denoteby Bt be the set of paths obtained from ! by applying the root operators f)i , e)i .The character of V t(!) is equal to

)$$Bt e$(1).

Denote by B% the subset of Bt of paths ending in a point of the lattice +X , andlet B be the set of paths {)/+ | ) " B%}. We will show that the set B 0 {0} isstable under the root operators f"i , e"i .

Since ("i = di'"i and (i = 'i/di, it follows by the definition of the operators that

fdi)i

= f"i . By assumption we know that + is divisible by d and hence fdi%/d)i = f %"i

.

If ) " B%, and f %"i()) -= 0, then f %"i

()) = fdi%/d)i ()) " Bt, and the endpoint of

the path is )(1) & +'i " +X . This proves: f %"i()) " B%, and therefore: B 0 {0} is

stable under the root operators f"i , i = 1, . . . , n. The proof for the operator e"i

is similar.Now the same arguments as in the proof of Proposition 3 apply. We have to look

for those paths ) " B such that ,+ Im ) is in contained in C0, but this conditionis equivalent to the condition that +,+ Im(+)) stays in C0. Since +) is a!ne (forgt), this condition is equivalent to Im

.(+,& ,t) ( +)

/is contained in C0. The same

arguments as above prove hence the multiplicity formula. !

3. The quantum Frobenius map

We use the same notation as in section 2. For a dominant weight ! " Xt,+ letN(!) be the Weyl module for the quantum group Uv(gt) at a 2+-th root of unity.The subspace

5µ$%X N(!)µ admits in a natural way a U(g)-action, which gives

a representation theoretic interpretation of Proposition 4. We assume throughoutthe rest of the article that + is divisible by 2d.

Let Uq(gt) be the quantum group associated to gt over the field Q(q), withgenerators E)i , F)i , K)i and K%1

)i. We use the usual abbreviations

[n]i :=qdin & q%din

qdi & q%di

, [n]i! := [1]i · · · [n]i,

6n

m

7

i

:=[n]i!

[m]i![n & m]i!,

where the latter is supposed to be zero for n < m. We write sometimes Ei, Ki, . . .for E)i , K)i, . . . and

qi := qdi = q(#i,#i)

t

2 ,

6Ki; c

p

7:=

p8

s=1

Kiqdi(c%s+1) & K%1i qdi(%c+s%1)

qdis & q%dis.

17

Let Uq,A be the form of Uq defined over the ring of Laurent polynomials A :=Z[q, q%1]. We denote by R the ring A/I, where I is the ideal generated by the2+-th cyclotomic polynomial, and set Uq,R := Uq,A #A R.

Similarly, let U+q (respectively U%

q ) be the subalgebra generated by the Ei (Fi),

and denote by U+q,A (respectively U%

q,A) the subalgebra of Uq,A generated by the

divided powers E(n)i := E

(n)i

[n]i(F (n)

i := F(n)i

[n]i). Let U+

q,R be the algebra U+q,A #A R,

and denote by U%q,R the algebra U%

q,A #A R.We use a similar notation for the enveloping algebra U(g). To distinguish be-

tween the generators of U(g) and Uq(gt), we denote the generators of U(g) byX", H", Y" or Xi, Hi, Yi. Let U = U(g) be the enveloping algebra of g definedover Q, let UZ be the Kostant-Z-form of U , set UR := UZ #Z R, etc. We fix thesymmetric bilinear form (·, ·) on H such that ('"

i , h) = 4'i, h5/di, and let (·, ·)t bedefined accordingly. We use the forms to identify H with H', it will be clear fromthe context which form we use.

Denote by v the image of q in R. Set +i := %di

d , then, by the definition of d, +iis minimal such that

+i((i, (i)t

2= +idi = +i

d

di" +Z.

Let N(!) be the simple Uq(gt)-module of highest weight ! " Xt,+, fix an A-latticeN(!)A := Uq,Am! in N(!) by choosing a highest weight vector m! " N(!). SetN(!)R := N(!)A #A R, then N(!)R is an Uq,R-module such that its character isgiven by the Weyl character formula. Consider the weight space decomposition:

N(!)R =9

µ$Xt

N(!)R,µ and set N(!)%R :=9

µ$%X

N(!)R,µ.

The subspace N(!)%R is stable under the subalgebra of Uq,R generated by the E(n%i)i

and F (n%i)i : If µ " +X , then so is µ ± n+i(i = µ ± ndi%

d (i = µ ± n+'i.

Theorem 3. The map

X(n)i '! E(n%i)

i |N(!)!

R

, Y (n)i '! F (n%i)

i |N(!)!

R

,

:Hi + m

n

;'!

6Ki; m+i

n+i

7|N(!)!

R

,

extends to a representation map UR ! EndR N(!)%R.

Some remarks to the proof. One has to prove that the map is compatible with theSerre relations. For U+

R and U%R , this is a direct consequence of the higher order

quantum Serre relations ([23], Chapter 7). For a detailed proof see [23], section35.2.3. For the proof that also the remaining Serre relations hold see [15].

Let N =5

µ$Xt Nµ be a finite dimensional Uq(gt)-module with a weightspace decomposition. If N admits a Uq,A(gt)-stable A-lattice NA such that NA =5

µ$Xt NA,µ (where NA,µ := NA 1 Nµ), then we denote for any A-algebra R by

18

NR the Uq,R(gt)-module NA#A R. We have a corresponding weight space decom-position NR =

5µ$Xt NR,µ.

The same arguments as above show that we can make N %R :=5

µ$%X NR,µ intoan UR(g)-module by the same construction. Let S be the antipode, the actionof Uq,R(gt) on the dual module N'

R := HomR(NR, R) is given by: (uf)(m) :=f(S(u)(m)) for u " Uq,R(gt) and f " N'

R. It is easy to check:

Proposition 5. The map UR ! EndR

.N %R

/'defined by

X(n)i f(m) := f(S(E(n%i)

i )m), Y (n)i f(m) := f(S(F (n%i)

i )m),

and.Hi+k

n

/f(m) := f(S(

<Ki;k%i

n%i

=)m), is the representation map corresponding to

the dual representation of the representation of UR(g) on N %R.

4. A basis associated to LS-paths

We use the same notation as before. Let ! be a dominant weight, and let B(!)be the corresponding set of paths obtained from the straight line joining ! and theorigin. This set is called the set of LS-paths (Lakshmibai-Seshadri paths) of shape!. Note if " = &1 ( . . . ( &r (where &i -= &i+1), then we can find .1, . . . , .r " W/W!and 0 < a1 < a2 < . . . < ar = 1 such that " = a1.1(!) ( (a2 & a1).2(!) ( . . . ((1 & ar%1).r(!). We denote the path " by (.1, . . . , .r; 0, a1, . . . , ar = 1), and leti(") := .1 be the element corresponding to the initial direction of the path.

Denote by R the ring obtained by adjoining all roots of unity to Z. For all + " Nsuch that 2d divides +, fix an embedding R -! R.

If k is an algebraically closed field and chark = 0, then we consider k as anR-module by the inclusion R -! R $ k. If chark = p > 0, then we consider kas an R-module by extending the canonical map Z ! k to a map R ! k. Fora dominant weight ! denote by V (!)R = V (!)Z #Z R the corresponding Weylmodule for UR(g) over the ring R.

By the previous section we know that we have the following sequence of inclu-sions of R-modules, where the top row is an inclusion of Uq,R(gt)-modules, thebottom row is an inclusion of UR(g)-modules:

N(+!)R -! N(!)R # . . . # N(!)R* +, -%

6 6

V (!)R -!.N(+!)R

/%-!

.N(!)R # . . . # N(!)R* +, -

%

/%

The inclusions induce restriction maps for the corresponding dual modules:.N(!)R # . . . # N(!)R* +, -

%

/'! N(+!)'

R

7 7.(N(!)R # . . . # N(!)R* +, -

%

)%/'

!.(N(+!)R)%

/'! V (!)'

R

19

We use these maps to define some special vectors in V (!)'R

. Fix a highest weight

vector m! in N(!) such that N(!)A := Uq,A(gt)m! and N(!)R = N(!)A #A

R. We define for . " W/W (!) a canonical extremal weight vector m* of weight.(!) as follows: Fix a reduced decomposition . = si1 · · · sir . According to thisdecomposition let n1, . . . , nr be defined by

nr := 4'"ir

,!5, nr%1 := 4'"ir!1

, sir (!)5, . . . , n1 := 4'"i1 , si2 · · · sir (!)5.

We set m* := F (n1)i1

. . . F (nr)ir

m(!). The fact that m* is independent of the choiceof the decomposition follows from the quantum Verma identities.

Denote by b* " N(!)'R

the corresponding dual weight vector of m* of weight&.(!). We define in the same way extremal weight vectors v* " V (!)Z and p* "V '(!)Z.

Let " = (.1, . . . , .s; 0, a1, . . . , 1) be an L-S path of shape !. Suppose + is minimalwith the property that 2d divides + and +ai " Z for all i = 1, . . . , s. Then we canassociate to " the vector

b# := b*s # . . . # b*s* +, -%(1%as!1)

# . . . # b*2 # . . . # b*2* +, -%(a2%a1)

# b*1 # . . . # b*1* +, -%a1

" (N(!)'R

)(%.

Definition 4. We call the image of b# in V (!)'R

the path vector associated to ",and we denote it by p#. By abuse of notation, we denote by p# as well its imagein V (!)'k = V (!)'

R#R k for any algebraically closed field k.

The vector p# depends only on the path " = (.1, . . . ; 0, a1, . . . , 1) (and thechoice of m! " N(!)A). We use the following partial order on the set of L-S pathsB(!) of shape !: If ) = (/1, . . . ; 0, b1, . . . , 1) " B(!) is another L-S paths, then wewrite " / ) if .1 > /1, or .1 = /1 and a1 > b1, or .1 = /1, a1 = b1 and .2 > /2,etc.

The following theorem has been proved in [15]:

Theorem 4. B(!) := {p# | " " B(!)} is a basis of V (!)'k of H-eigenvectors ofweight &"(1). Further, let "1,"2, . . . be a numeration of the L-S paths such that"i > "j implies i > j, and denote by V (!)'k(j) the subspace spanned by the p#i ,i / j. The flag V(!)'k is Uk(g)+-stable:

V(!)'k : V (!)'k = V (!)'k(1) 8 V (!)'k(2) 8 V (!)'k(3) 8 . . .

Idea of the proof: We associate to a path " = (.1, . . . ; 0, a1, . . . , 1) " B(!) avector v# " V (!)Z of the form

v# := Y (n1)"i1

· · ·Y (nr)"ir

v!,

where v(!) is a highest weight vector, .1 = s"i1· · · s"ir

is a reduced decomposition,and the sequence s(") = (n1, . . . , nr) is given by an algorithm explained below.

20

We show that the vectors v# form a basis of the lattice V (!)Z. Further, we willsee that p#(v$) = 0 for ) -/ " and p#(v#) is a root of unity, which implies thenthat the path vectors p# form a basis of V (!)'k for any algebraically closed field k.Note that the basis given by v# depends on several non-canonical choices.

The construction of the vectors v# : We explain now the construction and theproperties of the vectors v# mentioned above. We assume always that 2d divides+ and + := +/d as before. For & " Xt let m' " N(!)R,' be a weight vector. Denoteby “,” the usual partial order on the set of weights. We say m'1 # . . . # m'

!<

m!1 # . . . # m!!

if there exists a j such that &i = !i for all i < j and &j , !j . If

" is such that +ai " Z for all i = 1, . . . , s, then we associate to " the vector

m# := m*1 # . . . # m*1* +, -%a1

#m*2 # . . . # m*2* +, -%(a2%a1)

# . . .# m*s # . . . # m*s* +, -%(1%as!1)

" (N(!)R)(%.

This will be the leading term of the vector v#. Note if " > ) and +bi " Z for) = (/1, . . . ,/t, 0, b1, . . . , 1), then m# > m$.

The set up for this procedure has been inspired by the article of K. N. Raghavanand P. Sankaran [28]. For " = (.1, . . . ; 0, a1, . . . , 1) fix a reduced decomposition.1 = si1 . . . sir , the sequence s(") will depend on the chosen reduced decomposi-tion. Fix j minimal such that si1.j > .j , and set j = r + 1 if si1.j ) .j for all j.It is easy to see that the path ) = (si1.1, . . . , si1.j%1, .j , . . . , .r; 0, a1, . . . , 1) is anL-S path of shape ! (it is understand that we omit aj%1 if si1.j%1 = .j).

It follows that )(1)&"(1) is an integral multiple of the simple root 'i1 . Let n1 "N be such that )(1) & "(1) = n1'i1 . Note that si1.1 = si2 . . . sir is a reduced de-composition, and si1.1 < .1. Suppose we have already defined s()) = (n2, . . . , nr)(where s(id; 0, 1) is the empty sequence). We define the sequence for " to be theone obtained by adding n1 to the sequence for ):

Definition 5. We denote by s(") the sequence (n1, n2, . . . , nr), and we associate

to " the vector v# := Y (n1)"i1

· · ·Y (nr)"ir

v! " V (!)Z. We call i(") := .1 the initialdirection of ".

The vector v# depends on the choice of the reduced decomposition. By con-struction, we know that v# is a weight vector of weight "(1). The most importantstep in the proof is the following lemma which describes the image of v# under the

inclusion V (!)R -! N(!)(%R . The proof is a simple calculation, for details see [15],Lemma 3.

Lemma 5. If +ai " Z for all i = 1, . . . , s, then there exists an h " N such that

v# = vhm# +(

m'1 # . . . # m'!

where the m'j " N(!)R are weight vectors such that m# > m'1 # . . . # m'!.

Choose now + big enough such that +ai " Z for all " " B(!), so for all " " B(!)

we have a leading term (Lemma 5) m# " N(!)(%R of v# " V (!)R -! N(!)(%R . Since

21

the m# are obviously linearly independent, the same is true for the v#. By theWeyl character formula we know hence the the v#, " " B(!), span an R-lattice inV (!)R of the desired rank. Note that the set {m# | " " B(!)} can be extended to

an R-basis of N(!)(%R . It follows easily from this that the v# form hence an R-basisof V (!)R. But since the v# are in the Kostant lattice by construction, it follows:

Lemma 6.

a) B(!) := {v# | " " B(!)} is a basis of V (!)Z.

b) X(n)" v# =

)#>$ a$v$.

Proof. It remains to prove part b). Suppose again that + is big enough such that

+aj " Z for all paths " " B(!). If we apply a generator X(n)" to v# and express

the result as a linear combination of tensor products of weight vectors in N(!)R:

X(n)" v# =

(m'1 # . . . # m'

!

then it is easy to see by Lemma 5 that m# > m'1 # . . . # m'!

for such vectorsoccurring in the expression. Since the m$ are linearly independent, in an expression

of X(n)" v# as a linear combination in the v$:

X(n)" v# =

(a$v$,

the coe!cient a$ has to vanish for all ) -< ". !

Proof of Theorem 4. Lemma 5 implies that p#(v$) = 0 if ) -/ " and p#(v$) is aroot of unity. We can hence find an upper triangular matrix, with roots of unitieson the diagonal, that transforms the dual basis of B(!) into B(!). It follows thatB(!) is a basis of V (!)'k for any algebraically closed field k. The second part ofLemma 6 proves the claim about the U+

k (g)-stable flags. !

During the rest of this section we will prove a slightly stronger version of Theo-rem 4. For . " W/W (!) denote by V (!)Z(.) $ V (!)Z the submodule U+

Z (g)v* . Re-call that V (!)Z(.) can also be described as the submodule spanned by all vectors of

the form Y (n1)"i1

· · ·Y (nr)"ir

v!, where the ni are arbitrarily chosen and . = s"i1· · · s"ir

is a reduced decomposition. As an immediate consequence we see: v# " V (!)Z(.)if i(") ) . .

Theorem 5.

a) The set {v# | " | " " B(!), i(") ) .} is a basis of V (!)Z(.).b) V (!)Z(.) is a direct summand of V (!)Z.c) The restrictions {p#|V (!)k(*) | " " B(!), i(") ) .}, form a basis of V (!)'k(.),

and if i(") -) . , then p#|V (!)k(*) 9 0.

Proof. The second and third part is a simple consequence of Lemma 5 and parta) of the theorem. Lemma 6 proves that the span of the v#, i(") ) . , is a U+

Z (g)-stable submodule of V (!)Z(.). Since v* = v$ for ) = (. ; 0, 1) we know in addition

22

that v* is in this submodule, which implies that V (!)Z(.) is equal to the span ofthe v#, i(") ) . . !

Denote by &" the Demazure operator on the group ring Z[X ]:

&"(eµ) :=

eµ+( & es"(µ+()

1 & e%"e%(

By the Demazure formula for path models (see [16]) we get:

Corollary 1. CharV (!)Z(.) = &i1 . . .&ire! for any reduced decomposition . =

si1 . . . sir .

5. Schubert varieties

We apply now the results above to the geometry of Schubert varieties. The re-sults obtained above have consequences which seem to be as powerful as the Frobe-nius splitting introduced in the articles of Mehta, Ramanan and Ramanathan (seefor example [25,26,27]). As a consequence of the path basis one gets the normalityof Schubert varieties, the vanishing theorems, the reducedness of intersections ofunions of Schubert varieties etc. For detailed proofs see [15] and [21]. In fact, theproofs run along the same line as the proofs in [11]. But since the construction ofthe basis is not anymore part of the inductive procedure, these arguments can beapplied in a straight forward way. Note that the cases discussed in [11] are specialcases of the situation discussed in this and the preceding section. It is easy to seethat for fundamental weights of classical type the basis constructed here and in[11] are the same.

Let k be an algebraically closed field, we will omit the subscript k wheneverthere is no confusion possible. Let G be the simply connected semisimple groupcorresponding to g, and, according to the choice of the triangular decomposition ofg, let B $ G be a Borel subgroup. Fix a dominant weight ! and let P 8 B be theparabolic subgroup of G associated to !. It is well-known that the space of globalsections %(G/P,L!) of the line bundle L! := G :P k(!) is, as G-representation,isomorphic to V (!)'. Let # : G/P -! P(V (!)) be the corresponding embedding.

For . " W/W (!) denote by X(.) $ G/P the Schubert variety. Let Y =4ri=1 X(.i) be a union of Schubert varieties. By abuse of notation, we denote

by L! and p# also the restrictions L!|Y and p#|Y . Recall that the linear span ofthe a!ne cone over X(.) in V (!) is the submodule V (!)(.). The restriction map%(G/P,L!) ! %(X(.),L!) induces hence an injection V (!)'(.) -! %(X(.),L!).We call a path vector p# standard on Y if i(") ) .i for at least one 1 ) i ) r.Denote B(!)(Y ) the set of standard path vectors on Y .

Theorem 6.

a) B(!)(Y ) is a basis of %(Y,L!).b) p#|Y 9 0 if and only if i(") -) .i for all i = 1, . . . , r.

As an immediate consequence we get:

23

Corollary.a) The restriction map %(G/P,L!) ! %(Y,L!) is surjective.b) For any reduced decomposition . = si1 . . . sir , Char%(X(.),L!)' is given by the

Demazure character formula Char%(X(.),L!)' = &i1 . . .&ire!.

The proof of the theorem is by induction on the dimension and the number ofirreducible components of maximal dimension. Let Y, Y1, Y2 be unions of Schubertvarieties. During the induction procedure one proves in addition:

Theorem 7.

i) Hi(Y,L!) = 0 for i / 1.ii) X(.) is a normal variety.iii) The scheme theoretic intersection Y1 1 Y2 is reduced.

6. Standard Monomials

Let !1, . . . ,!r be some dominant weights, set ! =)!i, and fix . " W/W (!).

For each i let .i be the image of . in W/W!i . A module V (!) (without specifyingthe underlying ring) is always meant to be a module over an algebraically closedfield. The inclusion V (!) -! V (!1) # . . . # V (!r) induces an inclusion V* (!) -!V (!1)(.1) # . . . # V (!r)(.r), and hence in turn the restriction map V (!1)'(.1) #. . . # V (!r)'(.r) ! V (!)'(.).

We write "i respectively "! for the path t '! t!i respectively t '! t!. Denoteby Bi respectively B(!) the set of L-S paths of shape !i respectively !.

Denote by B1 ( . . . ( Br the set of concatenations of all paths in B1, . . . , Br.The set of paths B1 ( . . . ( Br 0 {0}is stable under the root operators. Denote byB("1 ( . . . ( "r) the smallest subset such that B("1 ( . . . ( "r)0 {0} is stable underthe root operators and contains "1 ( . . . ( "r.

A monomial )1 ( . . . ( )r " B1 ( . . . ( Br is called standard if it an element ofB("1 ( . . . ( "r).

A more combinatorial way of defining i()1 ( . . . ( )r) is via the defining chain.Let first ) := )1 ( . . . ( )r be an arbitrary element of B1 ( . . . ( Br, where )i =(. i

1, . . . , . isi

; 0, a1, . . . , 1). If the !i belong to di"erent faces of the dominant Weylchamber, then there is no way of comparing the . i

j using the Bruhat order.Choose lifts wi

j 9 . ij mod W!i with respect to the projection maps W/W! !

W/W!i for 1 ) i ) r, 1 ) j ) si. If one can choose the wij such that we have for

w := (w11 , . . . , wr

sr):

w11 / w1

2 / . . . / w1s1

/ w21 / w2

2 / . . . / wr1 / wr

2 / . . . / wrsr

.

then we call w a defining chain for ).Such a defining chain exists if and only if ) is standard, i.e. ) " B1 ( . . . ( Br

[18]. Note that if ! = !i for all i, then we get in this way a very simple criteria for) to be standard: ) is standard i"

.11 / .1

2 / . . . / .1s1

/ .21 / .2

2 / . . . / .r1 / .r

2 / . . . / .rsr

.

24

The defining chain (if it exists) is in general not necessarily unique. We introducea partial ordering on these chains by saying w / w! if wi

j / w!ji for all i, j. There

exists a unique minimal and a unique maximal defining chain. If ) is standard,then we set i()) := w1

1 for the minimal defining chain of ). See [18] for a proofthat the two notions are the same.

Definition 6. Let )1, . . . , )r be L-S paths of shape !1, . . . ,!r. A monomial ofpath vectors p$1 · · · p$r is called standard if the concatenation )1(. . .()r is standard.The standard monomial is called standard with respect to . if i()1 ( . . . ( )r) ) . .

For a detailed proof we refer to [15] and [21].

Theorem 8. The set of standard monomials form a basis of V (!)', and the setof monomials standard with respect to . form a basis for V (!)'(.).

Remark 3. We have seen that the p$, ) not standard with respect to . , form abasis of the kernel of the restriction map Vk(!)' ! Vk(!)'(.). This not true ingeneral for standard monomials if the !i have di"erent stabilizers. The reasonfor this is that the partial order on the tensor products of weight vectors is notanymore related to i()1 ( . . . ( )r). Let H be the Cartan subalgebra of diagonalmatrices of trace zero of g = sl4, let $i be the character that projects a diagonalmatrix onto its i-th entry, and let 0i = $1+ . . .+$i be the i-th fundamental weight.Set !1 := 01, )1 := (s2s1; 0, 1), "1 := (s1; 0, 1), and set !2 := 03, )2 := (s3; 0, 1),"2 := (s2s3; 0, 1). The concatenations )1 ( )2 and "1 ( "2 are standard, )1 ( )2 >"1 ( "2, but i()1 ( )2) = s2s1s3, whereas i("1 ( "2) = s1s2s3. These two are notcompatible. It is easy to check that p#1p#2(v$1'$2) -= 0, so the restriction of p#1p#2to Vk(01 + 03)(s1s2s3) does not vanish though i("1 ( "2) -) s1s2s3.

Let Y be a union of Schubert varieties in G/Q and let L!,Lµ be base pointfree line bundles on G/Q. As an immediate consequence of the Basis Theorem forstandard monomials we get:

Corollary.i) The product map %(Y,L!) # %(Y,Lµ) ! %(Y,L!+µ) is surjective.ii) The product map Sn%(Y,L!) ! %(Y,Ln!) is surjective.

7. Good filtrations I

The following combinatorial construction is part of a work in progress togetherP. Magyar and V. Lakshmibai [10]. We fix in the following !, µ " X+ and . "W/W!. Let !, µ also denote the paths that join the origin with ! respectively µby a straight line, and let B(!)* be the set of L–S paths ) = (.1, . . . , .r; 0, . . . , 1)such that .1 ) . . Recall that CharH0(X(.),L!) is given by

)$$B(!)$

e$(1), and

the path vectors p$|X(*), ) " B(!)* , form a basis of H0(X(.),L!).The decomposition formula (see section 1) implies that

B(!) ( B(µ) = B(! ( )1) 0 . . . 0 B(! ( )s),

where )1, . . . , )s runs through the set of all paths in B(µ) such that Im(!()) $ C0.Since the paths are a!ne, this is equivalent to say that !()1, . . . ,!()s runs throughthe set of all paths in B(!) ( B(µ) such that e"(! ( )) = 0 for all simple roots.

25

The formula for the action of the root operators on a concatenation of pathsimplies that ! ( B(µ)* $ B(!) ( B(µ) is a subset stable under the e". Denote byD(µ, i) or D(µ, )i) the subset:

D(µ, )i) := {) " B(µ)* | ! ( ) " B(! ( )i)}.

By section 1, this definition is equivalent to

D(µ, )i) := {) " B(µ)* | ; i1, i2, i3, . . . such that en1"i1

en2"i2

en3"i3

. . . (! ( )) = ! ( )i}

Since the union of the B(! ( )i) is disjoint we have obviously:

Lemma 7. ! ( B(µ)* =4s

i=1 ! ( D(µ, i)

To describe the character of such a subset we attach to each D(µ, k), 1 )k ) s, a dominant weight &k and a union of Schubert varieties Yk such that)$$D(µ,k) e!+$(1) is the character CharH0(Yk,L'k

).

The obvious candidat for the dominant weight is:

Definition 7. &k := !+ )k(1)

The definition of Yk is more involved. Let F1, . . . , Fr be the open proper subfacesof the dominant Weyl chamber (-= interior of the chamber) met by !+ )k([0, 1]),and counted with multiplicity. I. e., there exist 0 ) i1 ) j1 ) . . . ) ir ) jr ) 1 suchthat )k is linear on [i%, j%], !+)k(]i%, j%[) $ F% if i% < j% respectively !+)k(i%) " F%if i% = j%, and !+)k(t) is an element of the interior of the dominant Weyl chamberfor t -"

4r%=1[i%, j%].

Since !+)k([i%, j%]) $ F %, if i% < j% and !+)k(i%) -" F%, then we may assume thati%%1 = j%%1 = i%, and if !+)k(j%) -" F%, then we may assume that i%+1 = j%+1 = j%.

Similarly, if i% = j% = i%+1, then we may assume that i%+1 < j%+1, and ifj%%1 = i% = j%, then we may assume that i%%1 < j%%1 .

Denote by W% the stabilizer in W of the face F%. We set 1r+1 := &k, and wedefine inductively 1%, 1 ) + ) r, to be the unique weight in W%(1%+1) such that41%,'"5 ) 0 for all simple roots ' such that s"(F%) = F%.

We associate now to the triple (!, D(µ, )k), &k) a union of Schubert varieties.Let Qk 8 B be the parabolic subgroup such that its Weyl group WQk

is equal toW'k

, and suppose )k = (.1, . . . ; 0, a1, . . . ).

Definition 8. If ! is a regular dominant weight, then let 2k " W/W'kbe the

unique element such that 2k(&k) = 11 and set Yk := X(2k) $ G/Qk.If ! is a singular weight (and hence ! " F1), then let 2k " W/W'k

be theunique element such that 2k(&k) = 12 and set Yk :=

4X(si1 · · · sit2k) $ G/Qk,

where the union runs over all sequences i1, . . . , it such that sij (F1) = F1 for allj = 1, . . . , t and .1 ) sit.1 ) . . . ) si1 · · · sit.1 ) . .

Let $% be the set of simple roots ( such that s)(F%) = F%.

26

Theorem 9.a) CharH0(Yk,L'k

) =)$$D(µ,k) e!+$(1)

b) D(µ, k) =

>) " B(µ)* | "! ( ) = fn1

)i1· · · f

np1)ip1* +, -

)$!1

fm1)j1

· · · fnp2)jp2* +, -

)$!2

. . . f q1)!1

· · · f qpr)!pr* +, -

)$!r

("! ( )k)?

where the ( " $1 run over all sequences such that .1 ) s)ip1.1 ) . . . )

s)i1· · · s)ip1

.1 ) . in W/W'k.

Proof. We show first that the second part of the theorem implies part a). Let& " X+ be an arbitrary dominant weight. In [16] we have shown that if s". > . ,then

B(&)s"* 0 {0} =&

n)0

>fn") | ) " B(&)*

?,

and B(&)* 0 {0} is stable under the operator f" if s". ) . . Let now " be a pathsuch that "(1) = & and Im" is contained in the dominant Weyl chamber C0, anddefine B(")* to be the subset

B(")* :=>) " B(") | ) = fn1

"i1· · · fnq

"iq(")

?,

where . = s"i1· · · s"iq

is a reduced decomposition. The graph isomorphism (The-orem 2) implies that CharB(")* = CharB(&)* , B(")* 0 {0} is stable under theoperator f" if s". ) . , and B(")s"* 0 {0} =

4n)0{f

n") | ) " B(")*} if s". > . .

It follows that the sum)

e!+$(1) running over the elements on the right hand sideof b) is equal to CharH0(Yk,L'k

), so b) implies a).It remains to prove b). Since (! + )k([0, 1])) 1 Fr -= 2, it follows by Lemma 2

that if ' " $r, then f"("! ( )k) = "! ( (f")k), and )k di"ers from f")k only fort / jr. The same argument implies that

fm1)j1

· · · fnp2)jp2* +, -

)$!2

. . . f q1)!1

· · · f qpr)!pr* +, -

)$!r

("! ( )k) = "! ( (fm1)j1

· · · f qpr)!pr

)k),

and the latter di"ers from )k only for t / j2 > 0. In particular, these paths are ofthe form )! = (.1, . !2, . . . ), where the .1 is the same as the “first direction” for thepath )k, so )! " B(µ)* .

If ! is a regular weight, then j1 > 0, so the same arguments as above applyto prove that the paths obtained on the right hand side of b) are of the form)!! = (.1, . !!2 , . . . ), in particular, )!! " B(µ)* .

If ! is a singular weight, then the condition on the sequences (i1 , . . . , (ip1" $1

imply that the paths on the right hand side of b) are of the form )!! = (. !!1 , . !!2 , . . . ),where . !!1 ) . , so )!! " B(µ)* .

It follows that the right hand side of b) is a subset of D(µ, k). It remains to provethat the two sets are equal. Fix a path ) " D(µ, k). Let t0 be maximal such that

27

!+ )(t0) " C0. If t0 = 1, then ) = )k and nothing is to prove. Otherwise, let F bethe face such that !+)(t0) " F . By the maximality of t0, there exists a simple root' such that s"(F ) = F and 4!+)(t),'"5 < 0 for t "]t0, t0 + $] for some 1 < $ > 0.It follows by Lemma 1 and Lemma 3 that e"(! ( )) -= 0. Let m be maximal suchthat em

" (! ( )) -= 0, by the choice of t0 we know that em" (! ( )) = ! ( (em

" )), andem" )(t) = )(t) for 0 ) t ) t0. By repeating the procedure with other simple roots'ij such that sij (F ) = F , we get a path )! = emt

"it· · · em1

"i1) such that:

fm1"i1

· · · fmt"it

(! ( )!) = ! ( ), )(t) = )!(t) for 0 ) t ) t0, sij (F ) = F, 1 ) j ) t,

and 4!+)(t),'"5 / 0 for t " [0, t0+$] for some 1 < $ > 0. So !+)! stays “longer”in C0 then !+ ). Note that the path )! still meets the face F .

We proceed now in the same way with )!: Let t!0 be maximal such that !+)(t!0) "C0. If t!0 = 1, then )! = )k and ) is hence an element of the set defined on theright hand side of b). Otherwise, let F ! be the face such that ! + )!(t!0) " F !.By the maximality of t!0, there exists a simple root ' such that s"(F !) = F ! and4! + )!(t),'"5 < 0 for t "]t!0, t

!0 + $] for some 1 < $ > 0. It follows as above that

e"(! ( )!) -= 0.We can find simple roots (jk

such that sjk(F !) = F ! and )!! = ens

)js· · · en1

)j1)! is

such that:

fm1"i1

· · · fmt"it

.fn1)j1

· · · fns)js

(! ( )!!)/

= fm1"i1

· · · fmt"it

(! ( )!) = ! ( ),

and 4!+ )!!(t), ("5 / 0 for t " [0, t!0 + $] for some 1 < $ > 0. Note that )!!(t0) " Fand s"ij

(F ) = F , and )!!(t!0) " F ! and s)jk(F !) = F !.

Proceeding in the same way with )!!, we see that after a finite number of stepswe get a path ) such that Im!() is contained in C0 chamber, so ) is equal to )k forsome 1 ) k ) s. Further, we get a sequence t0 < t1 < . . . < 1 such that )k(t0) " F ,)k(t!0) " F !, . . . , and sequences of simple roots 'i1 , . . . ,'it , (j1 , . . . , (js , . . . , suchthat s"ij

(F ) = F , s)jk(F !) = F !, . . . , and

fm1"i1

· · · fmt"it

.fn1)j1

· · · fns)js

.. . . (! ( )k) . . .

//= ! ( ). (1)

This proves part b) of the theorem for the case where ! is a regular dominantweight. If ! is a singular weight, then we have two cases to consider: Suppose firstt0 -= 0. In this case F = Fi (in the definition of Yk) for some i / 2, so (1) satisfiesautomatically the condition on the sequence of simple roots from $1 because thesequence is empty in this case.

Next suppose t0 = 0 and hence ! " F and F = F1. If ) = (21, . . . ), then . / 21

because ) " B(µ)* . By the very definition of the sequence 'i1 , . . . ,'it " $1 weknow that

. / 21 > s"it21 > s"it!1

s"it21 > . . . > .1 = s"i1

· · · s"it21,

where )k = (.1, . . . ) (see [16], the action of the root operators on L-S paths). Itfollows that this sequence satisfies the condition in b), so we have proved part b)of the theorem also in this case. !

28

8. Good filtrations II

This section is a generalization of the results proved in [19]. In characteristic0 we know that the tensor product of two irreducible modules decomposes intothe direct sum of irreducible modules, and the multiplicities are described by thetensor product rule given in section 1. In positive characteristic one knows [24] thatthe module H0(G/B,L!) # H0(G/B,Lµ) admits a G-stable filtration such thatall subquotients are again isomorphic to H0(G/B,L') for some dominant weight&. We will give a proof of this “good filtration”-property using the path bases. Theidea of the proof is very similar to that in [19], but because of the more generalresults on B-stable flags (Theorem 4), the proof is much less technical then theone in [19].

We use the same notation as in the preceding sections. We consider the rep-resentation, quantum groups and enveloping algebras over an algebraically closedfield k. We fix in the following a dominant weight ! " X+ and a total order onW/W! which refines the Bruhat order. We denote by /t the corresponding in-duced total order on the set of L-S paths B(!) (see section 4). Let "1,"2, . . . be anumeration of the L-S paths such that "i /t "j if and only if i / j. By Theorem 4we know that the flag

V(!)' : V (!)' = V (!)'(1) 8 V (!)'(2) 8 V (!)'(3) 8 . . .

is U(g)+-stable, where V (!)'(j) denotes the subspace spanned by the p#i , i / j.Let µ " X+ be a dominant weight, and let "i1 , . . . ,"is " B(!) be the paths

such that µ + Im("ij ) is completely contained in the dominant Weyl chamber.Denote by Fr the subspace of V (!)' spanned by all p#j , j / ir, and by Ur the

subspace spanned by all p#j , j > ir. If Ur -= Fr+1, then denote by !Ur/Fr+1 thevector bundle G :B Ur/Fr+1 on G/B associated to the B-module Ur/Fr+1.

Theorem 10. If Ur -= Fr+1, then Hi(G/B,Lµ # !Ur/Fr+1) = 0 for all i / 0.

Corollary. The tensor product H0(G/B,Lµ) # H0(G/B,L!) admits a G-stablefiltration

W1 = H0(G/B,Lµ) # H0(G/B,L!) 8 W2 8 . . . 8 Ws 8 Ws+1 = 0

such that the subquotient Wj/Wj+1 is isomorphic to H0(G/B,Lµ+#ij(1)) for all

j = 1, . . . , s.

Proof of the corollary. The vanishing of the higher cohomology groups implies thatwe have an isomorphism

H0(G/B,Lµ) # H0(G/B,L!) + H0(G/B,Lµ # !H0(G/B,L!)),

and Hi(G/B,Lµ # !H0(G/B,L!)) = 0 for i / 1. The theorem above implies thatthe B-module k%µ # H0(G/B,L!) admits a filtration such that the subquotient

29

is either a one-dimensional B-module corresponding to the weight &µ& "ij (1), sothe corresponding line bundle has as cohomology groups H0(G/B,Lµ+#ij

(1)) and

Hi(G/B,Lµ+#ij(1)) = 0 for i / 1, or the subquotient is a module such that all

cohomology groups of the associated vector bundle on G/B vanish.It follows that the filtration of the B-module k%µ # H0(G/B,L!) induces a

G-stable filtration W1 8 . . . 8 Ws 8 Ws+1 = 0 such that the subquotients areisomorphic to H0(G/B,Lµ+#ij

(1)) for all j = 1, . . . , s. !

Proof of the theorem. Suppose r is such that Fr+1 -= Ur, and let ) " B(!) be suchthat p$ " Ur but p$ -" Fr+1. It follows that µ+Im ) is not completely contained inthe dominant Weyl chamber. Fix t0 (> 0) minimal such that (µ + )(t0),'") = &1for some simple root '. Denote by '()) the set of paths )! " B(!) such that)!(t) = )(t) for all 0 ) t ) t0. By the choice of the total order we know thatp$" " Ur but -" Fr+1 for all )! " '()). Note that if n is maximal such thatfn" (µ ( )!) -= 0, then f j

")! " '()) for j = 0, . . . , n, and if m is maximal such that

em" (µ ( )!) -= 0, then ej

")! " '()) for j = 0, . . . , m & 1.

Fix )1, . . . , )p such that {" " B(!) | p# " Ur, p# -" Fr+1} is the disjoint unionof the '()i). We may assume that the numeration is such that )p /t . . . /t )1.We refine the filtration Ur 8 Fr+1 by defining Ur,j as the span of all p# such that" / )j or " " '()j). We get a B-stable filtration

Ur = Ur,1 8 Ur,2 8 . . .Ur,p 8 Ur,p+1 := Fr+1.

To prove the theorem, it su!ces to prove that Hi(G/B,Lµ # !Ur,j/Ur,j+1) = 0 forall i / 0 and j = 1, . . . , p. Note that the images of the p$, ) " '()j), form a basisof Ur,j/Ur,j+1.

Let Ur,j be the span of all vectors v$ such that ) ) )j or ) " '()j), and let Ur,0

be the span of all vectors v$ such that ) ) "ir . The restriction map V '! ! U'

r,j

induces an isomorphism of B-modules Ur,j/Ur,j+1 ! (Ur,j/Ur,j%1)'.Let SL2(') $ G be the subgroup corresponding to the simple root ', and

denote by T" its maximal torus contained in T and by B" its Borel subgroupcontained in B. Denote by P (') = SL2(')B the associated minimal parabolicsubgroup.

We show in Lemma 8 (see below) that Ur,j/Ur,j%1 admits a B-stable filtrationsuch that the unipotent radical of P (') acts trivially on the subquotients, andthe B"-module structure on the subquotients comes from an SL2(')-structure,twisted by the T"-character )j(t0). Note that this finishes the proof: Ur,j/Ur,j%1

admits a filtration such that the subquotients are isomorphic to k$(t0)#W for someSL2(')-modules W , so Ur,j/Ur,j+1 admits a filtration such that the subquotientsare isomorphic to k%$(t0) # W ', which implies for the associated vector bundle:

Hi(SL2(')/B",Lµ+$(t0) # @W ') + Hi(SL2(')/B",Lµ+$(t0)) # W ' = 0 3 i / 0,

because (µ+ )(t0),'") = &1. Using the filtration and the Leray spectral sequenceassociated to the projection G/B ! G/P ('), it follows that Hi(G/B,Lµ+$(t0) #

!Ur,j/Ur,j+1) = 0 for all i / 0. !

30

It remains to define the filtration and the U(sl2('))-structure. We consider thefollowing more general situation: Let ) = (.1, . . . , .r; 0, a1, . . . , 1) be an L-S pathof shape ! and fix a simple root '. Suppose t0 " [0, 1] is such that ()(t0),'") " Zand (.m(!),'") < 0, where m is such that am%1 < t0 ) am.

Denote by '0 the set of L-S paths " such that )(t) = "(t) for 0 ) t ) t0, and let' be the paths such that either " " '0 or ) /t ". Let V and V ! be the subspacesspanned by the vectors v# , " " ' respectively v#, " " ' & '0. It is easily seenthat both subspaces are U+

k (g)-stable, so V0 := V/V ! is a B-module, with basisthe images v#, " " '0, of the vectors v#.

We define a twisted action of the operators e", f" on '00{0}. Set )1 := )|[0,t0],i.e., )1(s) := )(st0) for s " [0, 1]. A path " " '0 can hence be written as " = )1("1.We define:

f"(") := )1 ( (f"("1)), e"(") := )1 ( (e"("1)).

Note that the definition makes sense because ()(t0),'") " Z. Further, it is easyto see by [16], Lemma 3.1, that e"("), f"(") " '0 0 {0}. Set

+(") := ("(1),'") & min{("1(s),'")|s " [0, 1]},

then the number of nonzero elements in . . . , e2"("), e"("),", f"("), . . . is +(") + 1.

We call +(") the twisted '-string length of ". The same arguments as in thefirst section apply to see that

)#$"0 e#(1)%$(t0) is the character of an Uk(sl2('))-

module.

Lemma 8. V0 admits a B-stable filtration such that the unipotent radical of P (')acts trivially on the subquotients, and the B"-module structure of the subquotientsis induced by an SL2(')-structure, twisted by the character )(t0).

Proof. Let R be the ring obtained by adjoining all roots of unity to Z (section 4).

We realize V0R

as a submodule of NR(!)(%(1%t0), where + is chosen appropriately

as in section 4. For . " W/W! denote by NR(!)0(.) the U+q,R

(gt)-submodule of

NR(!)(.) spanned by all weight vectors of weight -= .(!). Let M and M ! be the

U+R

(gt)-submodules of NR(!)(% spanned by the sum of the following submodules:

M : =m%1(

j=1

%(aj%aj!1)(

i=1

NR(!)(%aj!1+i%1 # NR(!)(.j) # NR(!)(%(1%aj!1)%i

+

%(t0%am!1)(

i=1

NR(!)(%am!1+i%1 # NR(!)(.m) # NR(!)(%(1%am!1)%i

M ! : =m%1(

j=1

%(aj%aj!1)(

i=1

NR(!)(%aj!1+i%1 # NR(!)0(.j) # NR(!)(%(1%aj!1)%i

+

%(t0%am!1)(

i=1

NR(!)(%am!1+i%1 # NR(!)0(.m) # NR(!)(%(1%am!1)%i

31

Note that V ! $ M ! and V $ M . The quotient M/M ! has as basis the images ofall tensor products of weight vectors of the form

v(%a1*1 # . . . v(%(t0%am!1)

*m # m,

where m is an arbitrary weight vector in NR(!)(%(1%t0). Again by looking atthe leading terms, one sees that the morphism V ! M/M0 induces an U+

q,R(g)-

equivariant inclusion V0R-! M/M0.

Since U+q,R

(gt) acts trivially on the “first” part of the tensor product of the basis

elements of M/M !, we can identify M/M ! as U+q,R

(gt)-module with NR(!)(%(1%t0).

The inclusion V0 -! NR(!)(%(1%t0) is hence U+q,R

(g)-equivariant.

Write V0R

as a direct sum V0,1

R* . . .*V0,s

R, where each V0,j

R= *k$ZV0

R(µj +k')

for some weight µj . Each of these modules is U+R

(sl2('))-stable, and, by choosing

an appropriate numeration of the weights, we may assume that *j)iV0,j

Ris U+

R(g)-

stable for all i = 1, . . . , s. Note that the unipotent radical of P (') operates triviallyon *j)iV

0,j

R/ *j)i+1 V

0,j

R, and, as U+

R(sl2('))-module, the quotient is isomorphic

to V0,j

R.

It remains to study the U+R

(sl2('))-module structure of these subquotients. By

the isomorphism mentioned above, we may hence assume that V0,j

R= V0

R, and

'0 consists only of paths of weight µ + k', k " N (so µ is the “lowest” weightoccurring in V0

R).

Note that NR(!)(%(1%t0) is an Uq,R(gt)-module, so it is in particular a mod-

ule for Uq,R(() - the quantum group generated by E(m)) , F (m)

) and the<K# ;c

p

=,

where ( is the simple root of gt corresponding to '. So we can consider again(NR(!)(%(1%t0))%,", the sum of weight spaces corresponding to weights that are ofthe form +µ, where µ " Xt is a weight that is integral for '.

The same arguments as above show that (NR(!)(%(1%t0))%," has a UR(sl2('))-

module structure. Note that the image of V0R

in NR(!)(%(1%t0) is contained in

(NR(!)(%(1%t0))%,", and, looking again at the “leading terms”, we see that the im-age is, as R-submodule, a direct summand, with basis given by the v#. Further, theU+

R(sl2('))-structure on V0

Ris the restriction of the action on (NR(!)(%(1%t0))%,".

So suppose " " '0 is a path such that "(1) = µ, the lowest weight occurring inV0

R. Then f"" = 0 and m := +("), the twisted '-string length, is maximal. Note

that m = &(µ & )(t0),'").

By UR(sl2('))-representation theory, we know that X(m)" v# -= 0, and, by weight

consideration ('0 provides the character of V0R), X(m+1)

" v# = 0. By sl2-theory, this

implies in turn Y"v# = 0. It follows that v#, X"v#, . . . , X(m)" v# is the basis of an

UR(sl2('))-submodule of V0R, which is a direct summand of (NR(!)(%(1%t0))%,".

32

Denote by V0,m

Rthe span of the vectors X(j)

" v#, " " '0, "(1) = µ and 0 ) j ) m.

This is an UR(sl2('))-submodule, and a direct summand of (NR(!)(%(1%t0))%,".

Let '0low be the subset of paths such that f"" = 0, and let '0

low(j) be the subsetof those of twisted '-string length j (= m& 2k for some k " N). Suppose we havealready defined an UR(sl2('))-submodule V0,s+2

Rfor the '0

low(j), j / s + 2, which

is a direct summand of (NR(!)(%(1%t0))%,". Fix a basis B1 of V0,s+2

R(µ+(m&s)'/2)

and complete it to a basis B(µ+(m&s)'/2) of V0R(µ+(m&s)'/2). By UR(sl2('))

theory, we know that the vectors X(i)" b, i = 0, . . . , s, b " B(µ + (m & s)'/2), are

all linearly independent, and, if b -" B1, then we know by weight considerations

that X(s+1)" b " V0,s+2

R. So after replacing b by a linear combination of b with some

element from V0,s+2

R(µ+(m&s)'/2) if necessary, we may assume that X(s+1)

" b = 0.

The vectors b, X"b, . . . , X(s)" b provide hence a basis for an UR(sl2('))-submodule,

which, by construction, is a direct summand as R-module.This inductive procedure provides the desired UR(sl2('))-module structure,

which, by base change, provides the SL2(')-module structure for any algebraicallyclosed field. !

Remark. The same arguments can be used to prove more generally that for any

Schubert variety X , the representation H0(G/B,Lµ # !H0(X,L!)) admits a goodfiltration.

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