Abstract
This paper is a continuation of [RS]. In this second part, we study Demazure operators for W = G(e, 1, n). A Demazure operator A,, : S(V) + S(V) is defined by A,,(f) = cc' (J'-s,,(f)) for a complex reflection s,, with an eigenvector a E V. We define for each u' E W, an operator A,,. : S(V) + S(V), which reduces the degree by n(w), as a product of various A,, associated to some reduced expression of M?. Note that in contrast to the case of Coxeter groups, our operator A,,. depends on a particular choice of a reduced expression. Let VW be the subalgebra of Endc S(V) generated by A,, (s,, E S) and the multiplication by an element ,!? E V. Let E : S(V) + C be the evaluation at 0, and set VDw = {&A ( A E D,w}. Then one can show that EA,,.

by W-invariant homogeneous elements of positive degree. The coinvariant algebra SW is defined as SW = S( V)/Z W, which has a natural grading SW = @;>sSb inherited from the grading of S( I'). The Poincare polynomial Pw(q) which appeared in Section 1 is defined by the formula Let s,, be a reflection in W with an eigenvector o E I'. (Here we assume that the eigenvalue attached to o is not equal to 1). One can define an operator A,, :
S(f') + S(v) by 4,(f) = 4f -&l(f)).
forJ' E S(V). We call A,, a Demazure operator on S(V). The above construction of Demazure operators is a natural generalization to the case of complex reflection groups of the corresponding notion in the case of Coxeter groups. It is a classical result ([D] , [BGG] , see also [HI) that in the case where W is a Coxeter group, the coinvariant algebra SU; has a natural basis parametrized by elements of W, and that it is described in terms of Demazure operators. It gives a geometric interpretation of the formula C,,,Fw q'("') = Pw(q), where P(N~) is the length function of the Coxeter group W In this section, we construct, for W = G(e, l:n), a certain basis of SW parametrized by elements of W, by making use of Demazure operators. 
,(fh) =0,(h) forf E S( V)w and I? E S(V).
In particular, we have A,(Zw) 2 ZW and A,, induces an operation on SW.
2.3. We identify S(V) with the polynomial ring C [xi, ,x,,] in n indeterminates. The action of W on C[xl, . . . ,x,,] is given as follows; for each s; E S (2 < i I n), s; permutes xi and xi_ I and leaves the other X, invariant. For si = t, t sends si to <XI and leaves the other .Xj invariant, where < is a primitive e-th root of unity. For each s E S we choose an eigenvector o E V by N = xi -xi_ 1 or cy = xi according to the case where s,, = s; or s,, = t. We define a Demazure operator A,s by A,v = A, for s = s,,. We shall now construct, for each 1~' E W, a Demazure operator A,,. : S(V) 4 S(V) as follows. Let t, (2 I i < n) be a reflection as given in (1.21). We also put tl = t as before. We define A, = A,, by choosing an eigenvector o = xi, and set A.,, = AZ-2A.5s for each i = 2,. . ,n. To each w,,(k,a) given in ( Note that since A,(k,u) reduces the degree of the grading of S(V) by n(w,,(k. a)), the operator A,,. reduces the degree by n(r~).
Remark 2.4. The above decomposition of w gives rise to a reduced expression of IV. Hence in the case where e = 2, i.e., in the case where W is a Weyl group of type B,,, the construction of A,,, coincides with the usual definition of Demazure operators (see, eg. [HI). However, in contrast to the case of Coxeter groups, A,,. depends on the expression of w. It should be noted that even the braid relation
AszAtAszA, = A,As2AlAs2
does not hold in general. Our definition of A,,. is based on a particular choice of a reduced expression of H' (and its modification).
2.5.
In order to study the operators A,. more precisely, we need to evaluate A,, at various polynomials fin C[xi , . . ,x,,]. First we note the following formula, which is obtained directly from the definition. where E = 1 (resp. E = -1) if a > b (resp. a < b). We prepare some notation here. For positive integers, a, 6, set Proof. Letfb = A,,, . L$,, (,f). First we show (2.6.1) Assume that n(f') # 0. Then the polynomial ,I; contains the term _v;'-'x;J.. .x;-'x;', and only this term gives a non-trivial contribution to the value n(j).
Moreover we have A,,, ...L\,,A:'(X[m' ..,_ uip'.x;') =c(e-2)xp'c(u).
We show (2.6.1). Let
for 2 5 i 5 k, and set f;' = As,(J). Then fx' is a homogeneous polynomial of degree e -2. It is easy to see from (2.5.1) that except the term xl --2 all the monomials of degree e -2 are annihilated by applying n: 2. On the other hand, by using (2.5.2), we see that if h is a monomial of degree e -1 such that n,,(h) contains the term .Y; -2, then h coincides either with xl-' or .x;:; up to scalar. It follows thatfk must contain the term .x{~ ' or _Y[: I and that only these two terms give non-trivial contributions to the value n(j). Hence fj_ , contains the term xl-'$1: or _Y& s. But since c(2e -3, e -2) = c(e -3, e -2) = 0 by (2.5). Therefore we have only to consider the term xi-'x;: t, and by the similar argument as above, applied to &, , = n:;:;n,, , , we see that,fk ~ I contains the term _x{ '_Y;: ,! or x{-'.Y;af76:e4bb:d472:9fe5:10c0:37a3:766a:5b99. or .x;-' . . .x;-lx;'-'. In fact, let h; be a monomial in g; for 2 I i 5 k. Then by (2.6.1) A,!(h;) must contain a monomial h' which is divisible exactly by .Y: -I. Such a monomial coincides with h;_ I. Then by (2.52) c;_ I satisfies the condition in (iii) and (iv). In the case where i = 2, h' = hi coincides with xt-' xpp '.x-f'. .Hence cl = U.
The coefficient of hi is also easily determined by (2.5.2). This shows (2.6.3). The converse statement is also easily verified, and the lemma follows. 0
A similar argument as in the proof of (2.6.2) shows the following lemma.
Lemma 2.7. Let A = A,(k,a) with a = 0. Let d = n(wn(k, a)) = n -k. and let
if the following holds. First we show that (2.9.1)
Moreover, in this case we hnve
A
,](Q,,) = ne . c(e -2)" '
The polynomial Qn can be written as
We can write,f, as fi = .x~.x,~'~,~ . x;'I, where In fact, since J(F) =fI(h) forf E S( V)w and similarly for A,,.,,, it is enough to compare A,,, and Jon Si( V) for i 5 N. Since J is divisible by Q, we have J = 0 on S'(V) for i < N. The similar fact holds also for A,,.,,. Hence, by comparing the values at Q using Proposition 2.9, we get (2.10.1).
Let
A={X=(X,... .,A,) E Z" IO < A, < ie-1).
Then Wis in bijection with the set 4 in the following way. Any element IL' E W can be written uniquely as )t' = uln(k,l, a,). . WI (/cl, ~1). Put X; = n(w;(k;, ai)). Then X = (Xi, A,,) is an element of il and the correspondence MOW X gives the required bijection. We denote 11' = crx if IV E W corresponds to X E n. We define a total order (a lexicographical order) on 4 as follows; let X = (X,. . , A,,). A' = (A(. , A:,) b e elements in 4. We say X > X' if Xi = x;;...,x;_, =X(_, and X, > X,! for some i. This induces a total order on W, which is also denoted by 11: > M". More generally, we consider the set j = N" containing the set A. The total order on 4 can be extended to a total ordering on j naturally. For each x= (Xl,... . A,,) E A, let 1x1 = C X;. We define a polynomial
of degree 1x1 by PA = x,? . . xf I. The following proposition is crucial for our later discussion.
Proposition 2.12. Let X E A and X' E 2. Then there exists a non-zero element CJ, in C such that
Proof. First we note the following. 
.Y,:>,' A,(.x,~~)
for some c E C, by the same argument as above. But then Ai(xlf:) = 0 since Ai reduces the degree by A;, which is greater than A,!. This shows that A,,.,(P,r) = 0 as asserted. 0 2.13. Let VW be the subalgebra of Endc(S( V)) generated by A,y (s E S) and c).*(cL E V), where a* : S(V) -+ S(V) denotes the multiplication by a. Then 2)~ becomes a left S( V)-module. Note that for any M' E W the endomorphism it' on S(V) is contained in VW, since s,, = 1 -a*A,, on S(V) for s,, E S. Since A,, = tr~A,w~' for some u' E W, we see that A, (1 < i < n) are also contained in VW. Hence A,,. E VW for any u' E W. The following proposition is an analogue of the corresponding result in the case of Coxeter groups (cf. [H, IV, Proposition 1.81).
Proposition 2.14. The algebra VW is a,free S( V)-module Ic,ith basis A,, (II' E W).

Proof. Let K be the quotient field of S( V). Then Watts naturally on K, and the
Demazure operator A,, on S(V) can be extended to an operator on K. We denote by 27; the subalgebra of Endc K obtained by 23; = K @s(vl VW. Then VL turns out to be a K-vector space via the left multiplication of K. First we note that (2.14.1) The set {A,,. 1 w E W} gives a K-basis of Vi,.
In fact, it is easy to see that V, K is generated by it' E W as a K-vector space. Hence dimKV$ 5 1 WI. On the other hand, Proposition 2.12 shows that A,,. (\c E W) are linearly independent over K. Our assertion follows from this. We now show that {A,,. / w E W} gives a basis of VW. Since A,,. are linearly independent over K, they are also linearly independent over S(V). We show that any element in 27~ is written as a linear combination of A,,. with coefficients in S( I'). If we note the relation
it is enough to verify this for A E V W, where A is a product of various A,.
Therefore in order to prove the proposition, we have only to show the following lemma.
Lemma 2.15. Let A be a d-product of A, for s E S (i.e., A is an operator reducing the degree by d). Then A can be written as where a,,. are elements of S( V) satisfying thefollowing.
We shall prove the lemma. By (2.14. l), one can express A as
A = C u,l.A,,.. (a,,, E K). vt w
We show (2.15.1) by induction on the total order of W Take ~I'A E W, and assume that (2.15.1) holds for any u' < MIA. Let PA be the polynomial of degree n(wx) given in (2.11). We have since A,,.(P,) = 0 if \C > M'A by Proposition 2.12, and since Alr(P,) = 0 if an > n(w,,). We consider the coefficient qC appearing in the expression. This implies that ,Y + LL' < X + X' < X0 and that ILLI + Ip'l = N. Now by induction on X + X' with respect to the total order on A, we may assume that XJ,,, = 0. Then it follows from (2.21.2) that XxXx, = 0. Hence (2.21.1) holds. We now prove the proposition. If X' < X*, then X + X' < X + X* = X0. Hence X,X,, = 0 by (2.21.1). Now assume that X' = X*. Then We consider the subalgebra HT of Hw consisting of WJ-invariant elements.
The following proposition describes the structure of H$". 

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