DEMAZURE OPERATORS FOR COMPLEX REFLECTION GROUPS G(e,e,n)

SUT Journal of Mathematics Vol. 34, No. 2 (1998), 179{196

DEMAZURE OPERATORS FOR

COMPLEX REFLECTION GROUPS G(e
e
n)

Konstantinos Rampetas

(Received November 11, 1998)

Abstract This paper is a continuation of the work in RS], where we

stud-ied Demazure operators for the imprimitive complex reection group f W = G(e
1
n) and constructed a homogeneous basis of the coinvariant algebra S

f

W. In this paper, we study a similar problem for the reection subgroup W = G(e
e
n) of

f

W. We prove, by assuming certain conjectures, that the

operators w (

w 2 W) are linearly independent over the symmetric algebra S(V). We dene a graded space HW in terms of Demazure operators, and we

show that the coinvariant algebraS

W is naturally isomorphic to H

W. Then we

can dene a homogeneous basis of SW parametrized byw2W.

AMS 1991 Mathematics Subject Classi
cation. Primary 20H15, Secondary 20F55, 51F15.

Key words and phrases. Complex reection groups, Demazure operators.

x

1. Introduction

Let f

W = G(e
1
n) be the imprimitive complex reection group isomorphic

to S n n(Z=eZ) n, regarded as a subgroup of GL(V) with V
=C n. (Here S n

denotes the symmetric group of degreen). LetS f

W be the coinvariant algebra

of f

W, i.e. the quotient of the symmetric algebra S(V) by the ideal

gener-ated by the non-constant homogeneous f

W-invariant polynomials. In BM1],

K. Bremke and G. Malle constructed a length functionn: f W ! N satisfying the propertyP w2 f W t n(w) = P f W( t), where P f W(

t) is the Poincare polynomial

associated with the graded algebra S f

W. In RS], we de ned a Demazure

op-erator w for each w2

f

W, which is an endomorphism on S(V) reducing the
The author gratefully acknowledges nancial support by the Japanese Ministry of

Edu-cation.

180 KONSTANTINOS RAMPETAS

grading by n(w), and constructed a basis of S f W parametrized by w 2 f W by making use off w jw2 f Wg.

In this paper, we consider the groupW =G(e
e
n), which is a subgroup of f

W of indexe, isomorphic toS n

n(Z=eZ)

n;1. The length function

`:W !N,

satisfying the propertyP w2W

t `(w)=

P

W(

t), was constructed by BM2], where P

W(

t) is the Poincare polynomial associated with the coinvariant algebraS W

of W. We recall the de nition of Demazure operators. For each  2 V,

let s

 be the complex reection with eigenvector

. A Demazure operator  : S(V)!S(V) is de ned by ( f) = f;s ( f) 
for f 2S(V):

We de ne an operator w for each

w2W as follows. It is known by BM2]

that there exists a system of representatives N of the left cosets W=S n

sat-isfying the property that `(w 0 w 0 0) = `(w 0) + `(w 0 0) for w 0 2 N, w 0 0 2 S n. We de ne w 0 for w 0

2 N as a certain product of various   for

s

 2 W.

On the other hand, the operator w 00 for

w 0 0

2 S

n is already de ned by the

theory of Demazure operators for nite Coxeter groups. Then we de ne, for

w=w 0 w 0 0 2W (w 0 2N,w 0 0 2S n) the operator w by w = w 0 w 00. In the case of f

W, the crucial step for the proof of the main result is to show that the

operatorsf w

jw2 f

Wgare linearly independent overS(V). In our situation,

we can prove (Theorem 3.10) that the operators f w

0jw 0

2Ng are linearly

independent over S(V). It is also known by the general theory that the

op-erators f w 00 jw 0 0 2S n

g are linearly independent over S(V). We expect that f

w

jw2Wgare linearly independent overS(V). In our paper, we prove this

by assuming certain conjectures, (3.12.1) and (3.12.2), concerning the prop-erty of w

0 ( w

0

2 N). Our main result asserts that a similar theorem as in

the case of f

W holds for W, assuming the above conjectures. More precisely,

let D

W be the subspace of the dual space of

S(V) generated by" w(

w2W),

where":S(V)!C is the evaluation at 0. Then we can show (Theorem 3.25)

thatf" w jw2Wggives a basis of D W, and that S W is naturally isomorphic

to the dual space of D W.

The conjecture (3.12.1) is related to the evaluation of w 1 (

w

1 is the longest

element in W with respect to `) at certain polynomial, and is veri ed to be

true (Theorem 3.14) under the assumption that e  n. This theorem leads

to the following interesting characterization of w 1. Let J be the operator on S(V) de ned by J = P w2W " W( w)w, where " W : W ! f1g is the

sign character of W. Let Q be the product of all eigenvectors of reections

contained in W. Assume that e  n. Then  w

1 is expressed (Proposition

3.18) as w1 = dQ

;1

DEMAZURE OPERATORS 181

x

2. Preliminaries

2.1.

Let V be the unitary space C

n with standard basis x 1
x 2
:::, x n. Let f

W =G(e
1
n) be the imprimitive complex reection group contained in GL(V). The group f W is generated by ft
s 2

s n g, where s i is a reection permutingx i and x

i;1, and t is a complex reection of order

e, which sends x

1 to x

1 and leaves all the other x

i unchanged. (Here

 is a xed primitive e-th root of unity).

LetW =G(e
e
n) be the subgroup of f

W of indexegenerated by reections S = fs 1
s 2

s n g of order 2, where s 1 = ts 2 t ;1 sends x 1 to  ;1 x 2 and x 2 to x 1. Note that

W is the Weyl group of type D n if

e = 2, and W is the

dihedral group of order 2e ifn= 2.

Let S(V) =  i0

S i(

V) be the symmetric algebra on V, where S i(

V)

de-notes the i-th homogeneous part of S(V). The group W acts naturally on S(V) and we denote by I

W the ideal of

S(V) generated by the W-invariant

homogeneous elements of S(V) of strictly positive degree. The coinvariant

algebra associated withW is de ned as S W =

S(V)=I

W, which has a natural

gradingS W =  i0 S i

W inherited from that of

S(V). The Poincare polynomial P

W(

t) is de ned by the formula P W( t) = X i0 dimC( S i W) t i : The groupf

W acts on S(V), and the coinvariant algebra S f

W and the Poincare

polynomialP f

W(

t) associated with f

W are de ned similarly.

2.2.

In BM1], Bremke and Malle constructed a length function n : f

W ! N by making use of a certain root system, and showed that the sum P w2 f W t n(w) coincides with P f W(

t). In BM2], they de ned a dierent type of

length function ` : f

W ! N, (the function `

2 in the notation of BM2]), in

terms of an alternative root system and showed that the restriction of ` on

W satis es the formulaP w2W

t `(w) =

P

W(

t). Note that the subgroup of W

generated by S 0 = fs 2

s n g is identi ed with S n. The restriction of ` on S

n coincides with the usual length function of S

n with respect to S

0.

They found a system of left coset representatives N of W=S

n having nice

properties with respect to the length function`onW as follows. For 0<a e,

1 i nwe de ne an element of f W by w(a
i) = ( s i s 2 t a if 0 <a e=2
s i s 2 t a s 2 s i if e=2<a e. (2.2.1)

182 KONSTANTINOS RAMPETAS

It is known by Lemma 1.10 in BM2] that the length of the elementw(a
i)

is given as `(w(a
i)) = ( (i;1)(2a;1) if 0<a e=2
(i;1)(2e;2a) ife=2<a e. (2.2.2) Put N =fw(a 1
1)w(a n
n)j 1 a i e
n X i=1 a i 0 (mode)g

They proved the following fact.

Proposition 2.3 (BM2, Cor.1.16, Prop. 2.6]).

The setN is a system of

representatives for the left cosets W=S

n satisfying the following.

(i) For w 0 2N, w 00 2S n, we have `(w 0 w 0 0) = `(w 0) + `(w 0 0) : (ii) If w 0 2 N is given as w 0 = w(a 1
1)w(a n
n), then `(w 0) = P n i=2 `(w(a i
i)). (Note that `(w(a 1
1)) = 0 by (2.2.2)).

2.4.

Let s  be the reection in

W with eigenvector  2 V. (Here we

assume that the eigenvalue attached to  is not equal to 1). We de ne an

operator  : S(V)!S(V) by the formula ( f) = f ;s ( f) 
(f 2S(V)):

We call  a Demazure operator on

S(V). Demazure operators are de ned

for complex reection groups in general. In the case of nite Coxeter groups, there exists a well established theory for Demazure operators by BBG], D]. In the case of (non-real) nite complex reection groups, not much is known. In RS], we studied Demazure operators for the group f

W, and showed that

the structure of the coinvariant algebraS f

W is described in terms of Demazure

operators, as in the case of Coxeter groups, by constructing a certain (non-canonical) basis of S

f

W. Here we take up a similar problem for the group W.

We give some properties of Demazure operators. We have the following. 2 = 0
(2.4.1) ( fh) =  ( f)h+f ( h)
(2.4.2)

DEMAZURE OPERATORS 183

forf
h2S(V). Iff 2S(V) iss

-invariant, then (

f) = 0. Now letS(V) W

be the subalgebra of S(V) consisting of the W-invariant elements. Then it

follows from (2.4.2) that ( fh) =f ( h) for f 2S(V) W : (2.4.3) In particular, we have ( I W) I

W and  induces an operation on S W.

2.5.

Let S n be the subgroup of W as in 2.2. Then (S n
S 0) is a Coxeter

system, with associated length function ` : S n

! N. Hence, by the general

theory of Demazure operators for nite Coxeter groups, we have the following facts. Letw=s i 1 s i 2 s i k ( s i 2S 0) be a reduced expression of w2S n. Then we de ne w = i1  i k
(2.5.1) where i =  i with  i = x i ;x

i;1. It is known that the operator w is

independent of the choice of the reduced expression. (See, for example H, IV, Prop. 1.7]).

Letw

0 be the longest element in S n. We de ne a polynomial Q 0 by Q 0 = Q i>j( x i ;x

j). The following facts are known.

Proposition 2.6 (H, IV, Prop. 1.6]).

w0( Q

0) = 1.

Proposition 2.7 (H, IV, Cor. 2.3]).

For anyw
w 0 2W such that `(w) `(w 0), we have  w 0 w ;1 w0 = ww 0 w0.

Note that the condition`(w) `(w

0) is dropped in the statement of

Corol-lary 2.3 in H].

x

3. Demazure operators for

G(e
e
n)

3.1.

From now on we identify S(V) with the polynomial algebra C x 1
:::
x n] with indeterminates x i. The group W = G(e
e
n) acts on C x 1
:::
x n] as in 2.1.

For i = 2
3
:::
n we de ne inductively the element s 0 i as follows Let s 0 2 = s 1 and s 0 i = s i;1 s i s 0 i;1 s i s i;1. Then s 0

iis the complex reection of order 2,

which sendsx ito x i;1, and x i;1to  ;1 x

i. We note that if we put y i =  ;1=2 x i and y i;1 =  1=2 x

i;1, then we can regard s 0 i as a permutation of y i, y i;1. We

de ne two operators s i,  s 0 i on S(V) by the formulas si( f) = f ;s i( f) x i ;x i;1
 s 0 i( f) = f ;s 0 i( f)  ;1=2 x ; 1=2 x
(f 2S(V)): (3.1.1)

184 KONSTANTINOS RAMPETAS

Then the following two formulas hold: s i( x a i x b i;1) = " X x j i x a+b;1;j i;1
(3.1.2) s 0 i( x a i x b i;1) = " (2a;1)=2 X  ;j x j i x a+b;1;j i;1

where in both formulas the sum is taken over j such that minfa,bg j

maxfa,bg;1, and " = 1 (resp. " = ;1) if a > b, (resp. a < b). The rst

formula is contained in RS], and the second one is obtained from the rst by changing the variablesx

i 7!y i, x i;1 7!y i;1.

Fori= 2

n, we de ne operators  (a)

i ,  (a)

i

0 in the following way

(a) i =  s 0 i s i | {z } a;factors
 (a) i 0 =  s i s 0 i | {z } a;factors : (3.1.3)

3.2.

In order to study the above operators in a more detailed way, we need to evaluate them at various polynomials. For this we prepare some notation. Leta,bbe two positive integers such that 1 a b. We put

c(a
b) = (;1) a+1=2] a;1 Y j=1 ( (b;j)=2 ; ;(b;j)=2 )

where a] denotes the smallest integer which does not exceed a. We have c(a
b) = ;1 if a = 1. The following two lemmas will be used in our later

discussion.

Lemma 3.3.

Let a,b be integers such that 1 a b.

(i) Assume that a<b. Then we have

(a) i ( x b i;1) = ( c(a
b)(x b;a i + x b;a i;1) + f
if a is odd
c(a
b)(y b;a i + y b;a i;1) + f
if a is even, (a) i 0 ( x b i;1) = ( (;1) a;1  ;b=2 c(a
b)(y b;a i + y b;a i;1) + f
ifa is odd, (;1) a;1  ;b=2 c(a
b)(x b;a i + x b;a i;1) + f
ifa is even,

where in each case,f denotes a polynomial divisible by x i x i;1= y i y i;1.

(ii) Assume that a=b. Then we have

(a) i ( x a i;1) = c(a
a)
(a) i 0 ( x a i;1) = ( ;1) a;1  ;a=2 c(a
a):

DEMAZURE OPERATORS 185

Proof. We prove only the formula (i). The proof of (ii) is similar, and simpler. We show the rst formula in (i). The case where a = 1 is straightforward

from (3.1.2). The following two formulas are obtained by using the de nition of s

i,  s

0

i and the fact that y i=  ;1=2 x i and y i;1=  1=2 x i;1. s 0 i( x b;a+1 i + x b;a+1 i;1 ) = (  (b;a+1)=2 ; ;(b;a+1)=2)( y b;a i + y b;a i;1) + f 1
s i( y b;a+1 i + y b;a+1 i;1 ) = (  ;(b;a+1)=2 ; (b;a+1)=2)( x b;a i + x b;a i;1) + f 1
where f 1 is a polynomial divisible by x i x i;1 = y i y

i;1. We also notice that

since x i x i;1 = y i y

i;1 is stable by the reections s i and s 0 i, if a polynomial f is divisible by x i x i;1 = y i y

i;1, then so are s i( f) and  s 0 i( f). The rst

formula in (i) follows from the above formulas by induction on a. Next we

show the second formula in (i). If we note that x b i;1 =  ;b=2 y b i;1, it is easy

to see that (a) i

0 ( y

b

i;1) coincides with the polynomial which is obtained from

(a) i ( x b i;1) by replacing x i
x i;1 by y i
y i;1, by replacing  by  ;1, and then by multiplying by

;b=2. Hence the second formula follows immediately from

the rst one.

Next we compute the values (a) i ( x b i) and  (a) i 0 ( x b i). By (3.1.2) we see that si( x b i) = ; si( x b i;1)
 s 0 i( y b i) = ; s 0 i( y b i;1) : Therefore we have s 0 i( x b i) =  b=2 s 0 i( y b i) =; b=2 s 0 i( y b i;1) =; b s 0 i( x b i;1) :

This implies that the value (a) i ( x b i) (resp.  (a) i 0 ( x b i)) coincides with ; (a) i ( x b i;1) (resp. ; b (a) i 0 ( x b

i;1)). Therefore as a corollary to Lemma 3.3

we obtain the following result.

Lemma 3.4.

Let a
b as in Lemma 3.3.

(i) Assume that a<b. Then we have

(a) i ( x b i) = ( ;c(a
b)(x b;a i + x b;a i;1) + f if a is odd
;c(a
b)(y b;a i + y b;a i;1) + f if a is even, (a) i 0 ( x b i) = ( (;1) a  b=2 c(a
b)(y b;a i + y b;a i;1) + f if a is odd, (;1) a  b=2 c(a
b)(x b;a i + x b;a i;1) + f if a is even.

(ii) Assume that a=b. Then we have (a) i ( x a i) = ;c(a
a)
(a) i 0 ( x a i) = ( ;1) a  a=2 c(a
a):

3.5.

We x an integer a0. We dene, for 2in, an operator  i a] on S(V) by the formula i a] = ( (a) 2 0  (a) i 0 if a1
1 ifa= 0. The operator i

a] reduces the grading by (i;1)a. For each a0, we dene

a polynomial g ia( x) of degree (i;1)a by g ia( x) = (x 1 x i;1) a. Then the

following lemma holds.

Lemma 3.6.

Assume that a  1. Let  i a] , g ia( x) be dened as above. Then i a](g ia) = f(;1) a;1  ;a=2 c(a
a)g i;1 : In particular, i a](g ia) 6 = 0 for 1ae;1.

Proof. First we note that the operator (a) i

0 a ects only the variables x

i and x

i;1 and leaves all the others unchanged. Therefore we have

i a](g ia) = ( x 1 x i;2) a (a) i 0 ( x a i;1) : (3.6.1)

But we have (a) i 0 ( x a i;1) = ( ;1) a;1  ;a=2 c(a
a) by Lemma 3.3 (ii).

Hence the right hand side of (3.6.1) can be written as g

i;1a with  = (;1)

a;1

 ;a=2

c(a
a). Repeating this procedure for the operators

(a) (i;1) 0

 (a) 2

0 we obtain the result.

3.7.

Let M = 0
e;1] n;1 (

n;1 copies of the interval 0
e;1]). For

each = ( 2

 n) 2M, we dene an operator   on S(V) by = n  n]  2  2] :

Also for 2 M we dene a polynomialP ( x) by P  = Q n i=2 g ii. Let  = ( 2

 n) ,  = ( 2

 n)

2 M. We dene a total order >  on M

by  2 =  2
:::
 i;1 =  i;1 and  i >  i for some

i 1: Then we have the

Proposition 3.8.

Let
2M. Then there exists a non-zero elementc  2C such that ( P ) = ( c  if =, 0 if >:

Proof. First we note that j  j] leaves g i i = ( x 1 x i;1) i invariant for j < i. In fact,  j 

j] consists of various products of the operators

s2

 sj
 s 0 2 
 s 0

j and these operators leave g

ii invariant, since s j and s 0 j stabilize x j;1 x j = y j;1 y

j (in the notation of 3.1).

First assume that  = . Then by Lemma 3.6  i  i]( g i i) is a non-zero

constant for each i. Combining with the above remark, we see that

( P ) = n Y i=2 i  i]( g i i)

and the right hand side is a non-zero constant, which we write asc .

Next assume that >. Then there exists isuch that  2 =  2
:::
 i;1 =  i;1and  i > i. Then we have ( P ) = c n  n]  i  i]( n Y j=i g j j)

with some c 2 C ;f0g by a similar argument as in the previous case. But

then i  i]( n Y j=i g j j) = ( n Y j=i+1 g j j) i  i]( g i i)
and i  i]( g i i) = 0, since  i 

i] reduces the degree by (

i;1)

i, which is

bigger than the degree ofg

ii. Hence ( P

) = 0.

3.9.

LetD

W be the subalgebra of EndC

S(V) generated by  s( s2S) and  ( 2V), where :

S(V)!S(V) denotes the multiplication by the vector . ThenD

W becomes a left

S(V)-module. We also note that for any w2W

the endomorphismwonS(V) is contained inD

W, since s = 1 ;   2D W for any s  2 S. Since  s 0 i = w s 0 2 w ;1 for some w 2 S n, we see that  s 0 i

(2 i n) are also contained inD

W. Therefore  

2 D

W for any

2M.

As a corollary to Proposition 3.8 we have the following theorem. The proof is immediate from Proposition 3.8.

Theorem 3.10.

The set f 

j2 Mg of operators in D

W is linearly

3.11.

In the case of f

W = G(e
1
n), the operator 

w was constructed

in RS] for each w 2 f

W by making use of a particular reduced expression of w. Here 

w is an operator which reduces the grading by

n(w). In our case,

the operators  with

2M are not directly related to the elements of W.

However, one gets a bijection between the setf 

j2Mgand the set N in W as follows. For each 0<ae, we set

'(a) = (

2a;1 if 0<ae=2

2e;2a ife=2<ae :

Then the map'gives rise to a bijection from the set 1
e] to the set 0
e;1],

and one can dene a bijection 'e : N ! M by 'e(w) = ('(a 2)

:::
'(a n)).

Hence the setf 

j2Mgis in bijection with the setN. It is easily checked,

by using (2.2.2), that if 2M corresponds to w 2N, then 

 reduces the

degree by `(w).

3.12.

In the case of f

W, it was shown in RS, Prop. 2.14] that D f

W is

a free S(V)-module with basis f w

j w 2 f

Wg. In order to obtain a similar

result for W, we try to construct operators w for any

w 2 W. In view of

Proposition 2.3, any elementw2W can be expressed uniquely asw=w 0 w 0 0, withw 0 2N
w 0 0 2S nwith `(w) =`(w 0)+ `(w 0 0). We now dene  w ( w2W) by w=  w 00, where 2Mis given by='e(w

0). (Note that the operator

w

00 corresponding to w

0 0

2S

n is dened without ambiguity, see 2.5).

We know, by Theorem 3.10, that the set f 

j  2 Mg is linearly

inde-pendent over S(V). It is also known that the set f w 00 j w 0 0 2 S n g is

lin-early independent over S(V). We expect that the set f w

j w 2 Wg gives

rise to a basis of D

W. In what follows, we show that this conjecture is

re-duced to some properties of . Here we prepare some notation. For each  2 M we dene the length `() by `() = `(w

0) whenever  corresponds to w 0 2 N. Hence `(w) = `() +`(w 0 0) if

w 2 W corresponds to the pair

(
w 00)

2MS

n. For each integer

c1, we putM c =

f2Mj`() =cg.

For each polynomialP  (

2M) given in 3.7, we dene its average e P  over S n by e P  = P 2S n (P ). Note that ( e P ) is a constant if 
 2 M c for some c. Let  0 = ( e;1

e;1)2 M. Then 

0 is the longest element in Mwith`(

0) =

n(n;1)(e;1)=2. We consider the following two statements.

(3.12.1)  0( e P  0) is a non-zero constant.

(3.12.2) For any integerc1, the matrix ( (

e

P

))2M

c is non-singular.

We don't know whether these two statements hold in a full generality for

W. It is veried that (3.12.1) holds whenever en, which will be discussed

for small e. Note that (3.12.1) is a special case of (3.12.2), since the set M c

consists of a single element 0 if

c=`( 0).

3.13.

In order to look at e P

 more precisely, we shall extend the parameter

set M to N n;1. For each  = ( 2

 n) 2 N n;1, we dene a polynomial F n( ) byF n( ) = Q n i=2 g i i. Hence if 2 M, F n( ) coincides with P . We put e F n( ) = P 2Sn (F n( )).

For each i(1in), let i=  1 2  i i+ 1 i+ 2  n 1 2  n i i+ 1  n;1  2S n : Then f 1

n

g is a complete set of representatives of the right cosets S n;1 nS n. For each  = ( 2

 n) 2 N n;1, we dene  (i) 2 N n;2, (2in;1) by  (i)= (  2

 i;1
 i+  i+1
 i+2

 n) : Also we put  (1) = (  3

 n) 2 N n;2 and  (n) = (  2

 n;1) 2 N n;2.

Then it is easy to see that

i( F n( )) = ( F n;1(  (i)) x b i () n if 1 in;1, F n;1(  (n)) (x 1 x n;1)  n if i=n, (3.13.1) whereb i( ) = i+1+ + nfor

i= 1

n;1. It follows from (3.13.1) that X 2Sn;1 i F n( ) = ( e F n;1(  (i)) x b i () n if 1 in;1, e F n;1(  (n)) (x 1 x n;1)  n if i=n.

Hence we have a recursive formula,

e F n( ) = n;1 X i=1 e F n;1(  (i)) x b i () n + e F n;1(  (n))( x 1 x n;1) n : (3.13.2) LetM 0 = 0
e;1]

n;2be the set corresponding to the situation in

G(e
e
n;1). Then for  = ( 2

 n) 2 M, the operator   can be written as  = n  n]  0 with  0 = (  2

 n;1) 2 M 0. By applying   to the formula (3.13.2), we obtain ( e F n( )) = n;1 X i=1 n  n](  0( e F n;1(  (i))) x b i () n ) + n  n](  0( e F n;1(  (n))) (x 1 x n;1)  n) : (3.13.3)

By making use of the formula (3.13.3), we can compute the value  0(

e

P



0)

Theorem 3.14.

Assume that e  n. Then  0( e P 0) = c 0, where c 0 is given as in Proposition 3.8. Proof. Since  0 = ( e;1

e;1) 2 M,   0 can be written as   0 = n;1 e;1]  0 0, where  0 0 = ( e;1

e;1) 2 M

0. First we note the

following (3.14.1) Let= ( 2

 n) 2N n;1. Assume that  i

0 (mode;1) for all i, and that e;1< P i  i <e(e;1). Then we have  0( e F n( )) = 0.

We prove (3.14.1) by induction on n. We apply the formula (3.13.3) with  = 

0. Note that if

 satises the assumption of (3.14.1), then  (i) (2



in;1) above also satises the same condition. Hence (3.13.3) implies, by

induction hypothesis, that  0( e F n( )) =  n e;1](  0 0( e F n;1(  (1))) x b1() n ) + n e;1](  0 0( e F n;1(  (n))) (x 1 x n;1) ) :

Here we may assume that (1)=  0 0or  (n) =  0 0, since both of   0 0( e F n;1(  (1))) and  0 0( e F n;1( 

(n))) are zero, otherwise. But if  (1) =  0 0, then e F 1(  (n)) = e P  0 0, and   0 0( e P  0

0) is a constant. The same argument holds for the case 

(n) = 

0

0. Therefore, in order to prove (3.14.1), we have only to show that

(3.14.2) n e;1]x b1() n = 0, (3.14.3) n e;1](x 1 x n;1) n = 0 :

The left hand side of (3.14.2) can be computed by making use of the formula in Lemma 3.4. In particular, it is divisible by c(e;1
b

1(

)). We claim that c(e;1
b

1(

)) = 0. In fact, by our assumption, b 1( ) =  2 + + n can be written as b 1(

) = d(e;1) for some d such that 1 <d < e. Then there

exists j (1  j e;2) such that b 1(

);j 0 (mode). This implies that c(e;1
b

1(

)) = 0, and (3.14.2) holds. (3.14.3) can be proved in a similar way,

by replacingb 1(

) by

n, and by using Lemma 3.3. Hence (3.14.1) is proved.

We now prove the theorem. We compute  0( e P  0) by applying (3.13.3) with 0 = . Then (i) 0 (2

in;1) satises the condition in (3.14.1), since

(n;1)(e;1)<e(e;1) by our assumption. Hence, by applying (3.14.1), the

terms corresponding to  (i) (2

in;1) vanish. It follows that

0( e P 0) = n e;1]x (n;1)(e;1) n   0 0( e P  0 0) + n e;1](x 1 x n;1) e;1   0 0( e P  0 0) :

But the rst term of the sum goes to 0 by applying (3.14.2) with= 0. Since (x 1 x n;1) e;1 = g

ne;1, the second term coincides with c



0, by

3.15.

Let w 0 2 S n be as in 2.5, and let w 1 2 W be the element in W corresponding to ( 0
w 0) 2 MS n. Then w

1 is the longest element in W

with `(w 1) =

en(n;1)=2 = N, where N is the number of reections in W.

Let Q 0 be as in 2.5. Then e P 0 Q 0 is a polynomial of degree N. Since e P  is S n-invariant, and w0( Q 0) = 1 by Proposition 2.6, we have  0 w 0( e P  0 Q 0) =  0( e P  0) = c  0 : (3.15.1)

Before stating the next result, we prepare a simple lemma.

Lemma 3.16.

Let ":S(V) !C denotes the evaluation at 0. Let I

W be the

ideal of S(V) dened in 2.3. Then for any w2W we have "

w( I

W) = 0

Proof. Letf be an element ofI

W. Then f can be written as f = X i u i f i
with u i 2 S(V), f i 2 S(V) W, where f

i is homogeneous of positive degree.

Then applying w to f, we obtain w( f) = X w( u i) f i
since f i is W-invariant. Here  w( u i) f

i is a polynomial without a constant

term. This implies that " w(

f) = 0 and the lemma follows.

3.17.

Let " W :

W ! f1g be the sign character of W. Let Q be the

polynomial inCx 1 
x n] dened by Q= Q i>j( x e i ;x e j) : Then degQ=N,

and up to scalar,Qcoincides with the product of the eigenvectors attached to

all the reections inW. It is easy to see that Q generates a one-dimensional

representation of W a ording " W. We dene an operator J : S(V) ! S(V) by J = X w2W " W( w)w:

Then J is a projection on the "

W-isotypic subspace of

S(V). We have the

following remarkable result, although it is not used in the later discussion. Note that it is an analogue of H, IV, Prop. 1.6].

Proposition 3.18.

Assume thaten. Then there exists a non-zero constant d such that  w 1 = dQ ;1 J:

Proof. It is known that S

W is a regular

W-module, and S N

W a ords the sign

representation ofW. Hence we have S N( V) = (I W) N+ CQ
where (I W) N = I W \S N( V). Now e P 0 Q 0 2 S N( V), and (3.15.1) implies,

in view of Lemma 3.16, that e P  0 Q 0 = 2 I

W. Hence there exists a non-zero

constant c 0 2 C such that Q c 0 e P  0 Q 0 (mod I W). In particular, we have w1( Q) = c with c = c 0 c

0, by Theorem 3.14. Since w1 and Q

;1

J are S(V)

W-endomorphisms of

S(V), both of them are determined by the

restric-tion toS N(

V). Hence, by comparing the value atQ, we see that  w

1 = dQ

;1

J

withd=c=jWj. This proves the proposition.

3.19.

We now return to the condition (3.12.2). We deduce several prop-erties of the operators w by assuming this condition. Note that for any 
2M c, the polynomial w0( e P  Q 0) is a constant. We denote byA c the matrix (w0( e P  Q

0))2Mc, under a suitable order,

for a given integer c  0. Then since   w 0( e P  Q 0) =  ( e P ) by a similar

argument as in (3.15.1), we see that

(3.19.1) Assume that (3.12.2) holds for W. Then the matrix A

c is

non-singular.

We have the following lemma.

Lemma 3.20.

Assume that (3.12.2) holds for W. Then the operators f

w

j2M
w2S

n

g are linearly independent over S(V).

Proof. We consider the dependence relation

X

w

a(
w)

w = 0

(3.20.1)

on S(V), where a(
w)2S(V). By induction on the length `(w) of w2S n,

we may assume that a(
w

0) = 0 for any w 0 2 S n such that `(w 0) < `(w)

and for2M. Multiplying  w

;1

w0 to the equation (3.20.1) from the right,

and by making use of Proposition 2.7 together with induction hypothesis, we obtain X 2M a(
w) w0 = 0 : (3.20.2)

We show that a(
w) = 0 by induction on the length of M. Assume that a( 0
w) = 0 for any  0 2 M such that `( 0)

< c. We evaluate the equation

(3.20.2) at e P  Q 0 for  2 M c. Note that   w 0( e P  Q 0) = 0 if `() > c.

Hence the non-zero contribution only comes from the terms corresponding to

2M

c. We consider such equations for all

2M

c. Then it is regarded as a

linear equation with variables a(
w) (2M

c), and with coecient matrix A

c. Since the matrix A

c is non-singular by (3.19.1), we see that

a(
w) = 0

for any 2M

c. This proves the lemma.

We can now prove the following proposition, which is analogous to propo-sition 2.14 in RS].

Proposition 3.21.

Assume that (3.12.2) holds. Then the algebra D W is a

free S(V)-module with basis f w

jw2Wg.

Proof. Let K be the quotient eld ofS(V). The operator   on

S(V) can be

extended to an operator on K. We consider the subalgebra D K W of End C K dened by D K W = K S(V) D W. Since dimK D K W jWj, Lemma 3.20 implies that (3.21.1) The setf w jw2Wg gives a basis of D K W as a K-vector space.

By a similar argument as in the proof of Lemma 2.14 in RS], the proof of the proposition is reduced to showing the following lemma.

Lemma 3.22.

Let  be a d-product of  s (

s 2 S). Then  can be written

as  = X w2W a ww

where a(w) are elements in S(V) satisfying the following conditions. ( a w= 0 if `(w)<d
a w 2S `(w);d( V) if `(w)d. (3.22.1)

We prove Lemma 3.22. Here we recall that any w 0 ( w 0 2 W) can be written as w 0 =  w with 2M, w 2S n. Hence by (3.21.1)  can be expressed as  = X 2M w2Sn a(
w) w
(3.22.2) with a(
w) 2 K. We write a(
w) = a w 0 if w 0 2 W corresponds to (
w).

We shall prove thata(
w) satises the condition (3.22.1) by induction on the

length`() ofM, and on the length `(w) ofS

n. We x w2 S

n and assume

that (3.22.1) is veried for any a( 0
w 0) such that  0 2M and that w 0 2S n with `(w 0)

< `(w). Also we assume that it is veried for any a( 0

that`( 0)

<cfor an integer c0. We show that a(
w) satises (3.22.1) for

any 2M

c. By multiplying  w

;1

w0 on both sides of (3.22.2) from the right,

we have w ;1 w0 = X 2M a(
w) w0 + X  0 w 0 a( 0
w 0 ) 0 w 00
(3.22.3)

where in the second sum, 

0 runs over all the elements in

M, and w 0 in S n such that `(w 0) < `(w). Here w 0 0 2 S n is given by w 00 = w 0 w ;1 w 0 with `(w 0 0) = `(w 0) ;`(w) +`(w

0). We evaluate the equation (3.22.3) at e P  Q 0, with 2M

c, which is a polynomial of degree

c+`(w

0). Then the non-zero

contribution in the rst sum comes from the terms corresponding to2M 1,

where

M

1 =

f2Mj`()cg:

First assume that c+`(w) < d. Then for any  2 M

1, we have

`() + `(w) < d. Hence by induction hypothesis, we have a(
w) = 0 for  2 M

1 such that

`() < c. On the other hand, again by induction hypothesis, a( 0
w 0)  0 w 00( ~ P  Q

0) is a homogeneous polynomial of degree

c+`(w); d<0.

This means that there are no contributions from the terms in the second sum, and we have w ;1 w0( e P  Q 0) = X 2M c a(
w) w 0( ~ P  Q 0) : Since d +`(w ;1 w 0) > `() + `(w 0), we have  w ;1 w0( e P  Q 0) = 0. This

implies thata(
w) = 0 for any 2M

c, since the matrix A

c is non-singular

by (3.19.1). Next assume thatc+`(w)d. Take2Msuch that `()<c.

Then by induction hypothesis,a(
w) is a homogeneous polynomial of degree `()+`(w);dfor such, if it is positive, anda(
w) = 0 if`()+`(w);d <0.

Hencea(
w) w 0( e P  Q

0) is a homogeneous polynomial of degree

c+`(w); d, if it is non-zero. On the other hand, by a similar argument as before we see

that the term in the second suma( 0
w 0)  0 w 00( ~P  Q 0) is also a homogeneous

polynomial of degreec+`(w);d, if it is non-zero. Moreover,  w ;1 w 0( e P  Q 0)

is a homogeneous polynomial of the same degree. Since the matrixA

cis a

non-singularC-matrix, we see that a(
w) is a homogeneous polynomial of degree c+`(w);d for any 2M

c. This shows that

a(
w) satises the condition

in (3.22.1). The lemma is now proved and the proposition follows.

The following lemma can be proved in a similar way as Lemma 2.16 in RS], in view of RS, Remark 2.10].

Lemma 3.23.

Let P be a homogeneous polynomial of degree N. Let I be a

graded ideal of S(V) containing I

W, but not containing

P. Then I =I W.

3.24

Let S(V)

 be the graded vector space dened by S(V)  = i0 S i( V) , where S i( V)

 denotes the dual space of S i( V) overC. We have a natural pairing<
>:S(V)S(V)  !C, <u
f >=f(u). Let ":S(V) !C

denote the evaluation at 0. Then for each  2 D

W we can regard " as an element in S(V) . Let  D W be the subspace of S(V)  generated by " with  2 D W. Let H

W be the dual space of  D

W. Then we have a natural

map c:S(V) ! H

W, which sends

u 2 S(V) to the restriction to D

W of the

map <u
>:S(V) ! C. We can now state the main theorem, which is an

analogue of RS. Th. 2.18].

Theorem 3.25.

Assume that the conjectures (3.12.1) and (3.12.2) hold for

W. Then there exists a unique graded C-algebra structure on H

W such that c induces an isomorphism S W  = H W. The set f" w jw 2 Wg gives a basis of

the C-vector space D

W. In particular, if we denote by fX w jw2Wg the dual basis off" w

jw2Wg, the map c can be described, foru2S(V), as c(u) = X w2W " w( u)X w :

Proof. It follows from proposition 3.21 thatf" w

jw2Wggives rise to a basis

of D

W. Since dim S

W =

jWj, in order to prove the theorem it is enough to

prove that Kerc=I

W. Since D

W has a structure of a right

S(V)-module, we

see that Kercis a graded ideal ofS(V). It also follows from Lemma 3.16 that I

W

 Kerc. Now (3.12.1) asserts that   0 w 0( e P  0 Q 0) 6 = 0 (see (3.15.1)). Hence e P  0 Q

0 is a polynomial with deg e P  0 Q 0 =

N, which is not contained in I. Then one can apply Lemma 3.23 with P =

e

P

0 Q

0 and we conclude that I =I

W. This proves the theorem.

References

BGG] I.N. Bernstein, I.M. Gelfand and S.I. Gelfand Schubert cells and cohomology of the spaceG=P, Russian Math. Surveys28 (1973), 1{26. Also in

\Repre-sentation theory", London Math. Soc. Lecture Note Series69, pp. 115{140,

Cambridge Univ. Press, Cambridge 1982.

BM1] K. Bremke and G. Malle, Reduced words and a length function forG(e1n),

Indag. Math. 8(1997), 453-469.

BM2] K. Bremke and G. Malle, Root systems and length functions, Geometriae Dedicata 72 (1998), 83-97.

D] M. Demazure, Invariants sym
etriques des groupes de Weyl et torsion, Inv. Math. 21(1973), 287-301.

196 KONSTANTINOS RAMPETAS

H] H.L. Hiller, Geometry of Coxeter groups, Research Notes in Mathematics, No.54, Pitman, Boston 1982.

RS] K. Rampetas and T. Shoji, Length functions and Demazure operators for

G(e
1
n),I and II, to appear in Indag. Math.

Konstantinos Rampetas

Department of Mathematics, Scinecne University of Tokyo Noda, Chiba 278-8510, Japan