Page 1

arXiv:math/0208061v1 [math.QA] 8 Aug 2002

MACDONALD FUNCTIONS ASSOCIATED TO

COMPLEX REFLECTION GROUPS

TOSHIAKI SHOJI

Department of Mathematics

Science University of Tokyo

Noda, Chiba 278-8510, Japan

To Robert Steinberg

Abstract. Let W be the complex reflection group Sn⋉ (Z/eZ)n. In the author’s

previous paper [S1], Hall-Littlewood functions associated to W were introduced. In

the special case where W is a Weyl group of type Bn, they are closely related to Green

polynomials of finite classical groups. In this paper, we introduce a two variables

version of the above Hall-Littlewood functions, as a generalization of Macdonald

functions associated to symmetric groups. A generalization of Macdonald operators

is also constructed, and we characterize such functions by making use of Macdonald

operators, assuming a certain conjecture.

0. Introduction

Macdonald functions Pλ(x;q,t), which were introduced by I.G. Macdonald [M2]

in 1987, are two variables versions of Hall-Littlewood functions Pλ(x;t). Those Hall-

Littlewood functions and Macdonald functions are parametrized by partitions λ of

n. Since partitions of n parameterize irreducible characters of the symmetric group

Sn, we may say that these functions are associated to symmetric groups. On the

other hand, Hall-Littlewood functions are closely related to Green polynomials of a

finite general linear group GLn(Fq). In this direction, partitions λ of n occur as

unipotent classes of GLn(Fq). Unipotent classes of other finite classical groups such

as Sp2n(Fq),SO2n+1(Fq) have more complicated patterns. Lusztig introduced in [L2]

(unipotent) symbols, as a generalization of the notion of partitions, to describe such

unipotent classes in connection with Springer representations of Weyl groups. He also

introduced in [L1] a notion of symbols to parameterize unipotent characters of finite

classical groups.

In [S1], the author constructed Hall-Littlewood functions associated to complex

reflection groups W ≃ Sn⋉ (Z/eZ)n. In [S2], [S3], some related topics are discussed.

Our Hall-Littlewood functions are parametrized by e-tuples of partitions (which pa-

rameterize irreducible characters of W), or rather by various types of e-symbols. In the

case where e = 2, W is the Weyl group of type Bn. In this case, Hall-Littlewood func-

tions attached to unipotent symbols are closely related to Green functions of Sp2n(Fq)

or SO2n+1(Fq). It is also expected that our Hall-Littlewood functions attached to

symbols have some connection with unipotent characters.

1

Page 2

2 T. SHOJI

This paper is an attempt to generalize Hall-Littlewood functions P±

is an e-symbol) to the two variables version P±

We call such functions P±

original Macdonald functions, they are characterized as simultaneous eigenfunctions

of various Macdonald operators. We also construct Macdonald operators having (con-

jecturally) good properties. However, we note that our construction of Macdonald

operators works only for a special type of symbols (in the case where e = 2, this is

exactly the symbols used to parameterize unipotent characters), though Macdonald

functions can be constructed for any type of symbols. In the case of original Macdon-

ald operators, the representation matrix with respect to the basis of Schur functions

is a triangular matrix, with distinct eigenvalues. In our case, the matrix of Macdonald

operators turns out to be a block triangular matrix, where the blocks correspond to

families of symbols. One can conjecture that the matrices appearing in the diago-

nal blocks have no common eigenvalues. Assuming this conjecture, we show that the

Macdonald operator characterizes Macdonald functions, not as eigenfunction, but as

a unique solution of linear systems attached to the above diagonal blocks.

The properties of Macdonald functions discussed in this paper are just a part of

those established for the original Macdonald functions. We hope to discuss more about

them in a subsequent paper.

Λ(x;t) (where Λ

Λ(x;q,t), just as in the case of Pλ(x;q,t).

Λ(x;q,t) Macdonald functions associated to W. In the case of

1. Symmetric functions with two parameters

1.1.

An e-tuple of partitions α = (α(0),...,α(e−1)) is called an e-partition. We

define the size |α| of α by |α| =?e−1

parts αk. We denote by Pnthe set of partitions of n, and Pn,ethe set of e-partitions

of size n. Let W be the complex reflection group Sn⋉ (Z/eZ)n. Then the set of

conjugacy classes in W is in one to one correspondence with the set Pn,e.

Let us fix a sequence of positive integers m = (m0,...,me−1), and consider inde-

terminates x(k)

j

(0 ≤ k ≤ e−1,1 ≤ j ≤ mk). We denote by x = xmthe whole variables

(x(k)

root of unity in C. For each integer r ≥ 1 and i such that 0 ≤ i ≤ e − 1, put

k=0|α(k)|, where |α(k)| is the size of the partition

α(k). For a partition α : α1≥ α2≥ ··· ≥ αk≥ 0, let l(α) be the number of non-zero

j), and also denote by x(k)the variables x(k)

1,...,x(k)

mk. Let ζ be a primitive e-th

p(i)

r(x) =

e−1

?

j=0

ζijpr(x(j)),

where pr(x(j)) denotes the r-th power sum symmetric function with respect to the

variables x(j). We put p(i)

with α(k): α(k)

1

≥ ··· ≥ α(k)

r (x) = 1 for r = 0. For an e-partition α = (α(0),...,α(e−1))

mk, we define a function pα(x) by

pα(x) =

e−1

?

k=0

mk

?

j=1

p(k)

α(k)

j

(x).

Page 3

MACDONALD FUNCTIONS3

Next, we define the Schur function sα(x) and monomial symmetric functions

mα(x) associated to α by

sα(x) =

e−1

?

k=0

sα(k)(x(k)),mα(x) =

e−1

?

k=0

mα(k)(x(k)),

where sα(k)(x(k)) (resp. mα(k)(x(k)) denotes the usual Schur function (resp. monomial

symmetric function) associated to the partition α(k)with respect to the variables

x(k). These are the symmetric functions associated to complex reflection groups W ≃

Sn⋉ (Z/eZ)n, as given in [M1, Appendix B].

1.2. Put Sm= Sm0×···×Sme−1. We denote by Ξm=?e−1

has a structure of a graded ring Ξm=?

the inverse limit

Ξi= lim

←

m

k=0Z[x(k)

1,...,x(k)

mk]Smk

j). Ξm

the ring of symmetric polynomials (with respect to Sm) with variables x = (x(k)

i≥0Ξi

m, where Ξi

mconsists of homogeneous

symmetric polynomials of degree i, together with the zero polynomial. We consider

Ξi

m

with respect to homomorphisms ρm′,m: Ξi

m′

?

j

tions. Schur functions sα(x) with infinitely many variables x(k)

as elements in Ξnwith n = |α|, and the set {sα(x) | α ∈ Pn,e} forms a Z-basis of

Ξn. Similarly, {mα(x) | α ∈ Pn,e} gives a Z-basis of Ξn. Put ΞC= C ⊗ Ξ. Then

{pα(x) | α ∈ Pn,e} gives rise to a basis of ΞC.

m′ → Ξi

m, where m′= (m′

0,...,m′

e−1) with

k= mk+ l for some integer l ≥ 0, and ρm′,mis induced from the homomorphism

kZ[x(k)

m′

leaving the other x(k)

1,...,x(k)

k] →?

kZ[x(k)

invariant. Ξ =?

1,...,x(k)

mk] given by sending x(k)

i

to 0 for i > mk, and

i≥0Ξiis called the space of symmetric func-

1,x(k)

2 ... are regarded

1.3.

Let q,t be independent indeterminates and let F = C(q,t) be the field of

rational functions in q,t. We consider the F-algebra of symmetric functions ΞF =

F ⊗ZΞ with coefficients in F. Let α = (α(0),...,α(e−1)) be an e-partition. For each

partition α(k): α(k)

1

≥ α(k)

2

≥ ···, put

zα(k)(q,t) =

l(α(k))

?

j=1

1 − ζkqα(k)

1 − ζktα(k)

j

j

.

We then define zα(q,t) ∈ F by

zα(q,t) = zα

e−1

?

k=0

zα(k)(q,t), (1.3.1)

where zαis the order of the centralizer of wαin W (a representative of the conjugacy

class of W corresponding to α ∈ Pn,e). Explicitly, zαis given as follows. For α ∈ Pn,e,

put l(α) =?e−1

k=0l(α(k)). For a partition α = (1n1,2n2,...), put zα =?

i≥1inini!.

Then zα= el(α)?e−1

k=0zα(k).

Page 4

4 T. SHOJI

We define a sesquilinear form on ΞF by

?pα,pβ? = δα,βzα(q,t) (1.3.2)

for α,β ∈ Pn,e. Let x = (x(k)

many variables, and we define an infinite product of x and y by

j), y = (y(k)

j) (0 ≤ k ≤ e−1) be two sequences of infinitely

Π(x,y;q,t) =

e−1

?

e−1

?

k=0

∞

?

∞

?

i,j=1

∞

?

∞

?

r=0

1 − tx(k−r−1)

i

1 − x(k−r)

y(k)

y(k)

jqr

ijqr

(1.3.3)

=

k=0

i,j=1

r=0

1 − tx(k)

1 − x(k)

iy(k+r+1)

j

iy(k+r)

j

qr

qr

.

(Here the upper indices of the variables in the formula should be read

that in the case where q = 0, the product Π(x,y;q,t) reduces to the product Ω(x,y;t)

introduced in [S1, 2.5], (or rather [S2, (5.7.1)], see the remark there).

mod e). Note

Lemma 1.4. Π(x,y;q,t) has the following expansion.

Π(x,y;q,t) =

?

α

zα(q,t)−1pα(x)pα(y)

where α runs over all the e-partitions of any size.

Proof. Taking the log on both sides of the first formula of (1.3.3),

logΠ(x,y;q,t)

=

?

k

?

i,j

∞

?

r=0

∞

?

m=1

?1

m(x(k−r)

i

y(k)

jqr)m−tm

m(x(k−r−1)

i

y(k)

jqr)m

?

.

By making use of the equation

1

e

e−1

?

a=0

(ζkζ−k′)a= δk,k′,

the above formula can be written as

logΠ(x,y;q,t) =

e−1

?

a=0

?

−tm

emζa(r+1)ζa(k−r−1)ζ−ak′(x(k−r−1)

m≥1

?

r≥0

?

k,k′,i,j

?1

emζarζa(k−r)ζ−ak′(x(k−r)

i

y(k′)

j

qr)m

i

y(k′)

j

qr)m

?

.

Page 5

MACDONALD FUNCTIONS5

It follows that

logΠ(x,y;q,t) =

e−1

?

e−1

?

e−1

?

a=0

?

?

∞

?

m≥1

?

?

1

em·1 − ζatm

r≥0

?

em·1 − ζatm

k,k′,i,j

1 − ζatm

em

ζarqrm(x(k)

iy(k′)

j

)mζakζ−ak′

=

a=0

m≥1

k,k′,i,j

1

1 − ζaqmζakζ−ak′(x(k)

iy(k′)

j

)m

=

a=0

m=1

1 − ζaqmp(a)

m(x)p(a)

m (y).

Hence we have

Π(x,y;q,t) =

e−1

?

?

a=0

∞

?

zα(q,t)−1pα(x)pα(y).

m=1

exp

?1

em·1 − ζatm

1 − ζaqmp(a)

m(x)p(a)

m (y)

?

=

α

1.5. For a fixed y(k)

j, we define a function g(k)

m,±(x;q,t) as the coefficient of (y(k)

j)m

in

?

i

?

r≥0

1 − tx(k∓r∓1)

i

1 − x(k∓r)

y(k)

y(k)

jqr

ijqr

=

?

m≥0

g(k)

m,±(x;q,t)(y(k)

j)m

(1.5.1)

and put, for each α = (α(k)

j) ∈ Pn,e,

gα,±(x;q,t) =

?

j,k

g(k)

α(k)

j

,±.

Then we have

Π(x,y;q,t) =

?

α

gα,+(x;q,t)mα(y) =

?

α

mα(x)gα,−(y;q,t). (1.5.2)

In fact, by comparing the first formula of (1.3.3) and (1.5.1), we have

Π(x,y;q,t) =

?

?

k,j

?

j

α(k)

≥0

g(k)

α(k)

j

,+(x;q,t)(y(k)

j)α(k)

j

=

α

gα,+(x;q,t)mα(y).

This shows the first equality. If we compare the second formula of (1.3.3) and (1.5.1)

by replacing x and y, the second equality is obtained in a similar way.

Page 6

6 T. SHOJI

Now by using a similar argument as in [M1, VI, 2.7], we see that

?gα,+(x;q,t),mβ(x)? = ?mα(x),gβ,−(x;q,t)? = δα,β. (1.5.3)

In particular, the functions gα,±(x;q,t) form a basis of ΞF dual to mα.

{gn(x;q,t) | n ≥ 0} are algebraically independent over F, and ΞF= F[g1,g2,...].

The following lemma can be proved in a similar way as [M1, VI, 2.13] by using

(1.5.2) and (1.5.3).

Hence

Lemma 1.6. Let E±: ΞF → ΞF be two F-linear operators. Then the following

conditions are equivalent.

(i) ?E+f,g? = ?f,E−g? for any f,g ∈ ΞF

(ii) E+

the x variables, and similarly for y.

xΠ(x,y;q,t) = E−

yΠ(x,y;q,t), where the suffix x indicates the action of E±on

1.7.

We shall give an explicit form of the function gα,±(x;q,t). For this, we

prepare some notation. For µ = (µ0,...,µe−1) ∈ Ze

≥0, put

f(k,i)

µ,±(x;q,t) =

e−1

?

a=0

µa

?

j=1

x(k∓a)

i

− tx(k∓a∓1)

i

1 − qej

qe(j−1)

.

Let Mmbe the set of sequences µ = (µ(1),µ(2),...) such that µ(i)∈ Ze

?

Proposition 1.8. For each m ≥ 0, we have

?

≥0and that

i|µ(i)| = m. For µ = (µ1,µ2,...,µe) ∈ Ze

≥0, put n(µ) =?

?

j(j − 1)µj. Then

g(k)

m,±(x;q,t) =

µ∈Mm

i

f(k,i)

µ(i),±(x;q,t)qn(µ(i)). (1.8.1)

Proof. By [M1, I, §2, Ex. 5], the following identity of formal power series is known.

∞

?

i=0

1 − bqit

1 − aqit=

?

m≥0

?m

i=1

?

a − bqi−1

1 − qi

?

tm.

Substituting this into

A =

?

r≥0

1 − tx(k∓r∓1)

i

1 − x(k∓r)

yqr

i

yqr

=

e−1

?

a=0

?

r≥0

1 − tx(k∓a∓1)

i

1 − x(k∓a)

yqre+a

i

yqre+a

.

with a = x(k∓a)

i

,b = tx(k∓a∓1)

i

, t = qay, q = qe, we see that

A =

e−1

?

?

a=0

∞

?

µa=0

µa

?

µ,±(x;q,t)qn(µ)y|µ|.

j=1

x(k∓a)

i

− tx(k∓a∓1)

i

1 − qej

qe(j−1)

(qay)µa

=

µ∈Ze

≥0

f(k,i)

Page 7

MACDONALD FUNCTIONS7

It follows that

?

i≥1

?

r≥0

1 − tx(k∓r∓1)

i

1 − x(k∓r)

y(k)

y(k)

jqr

ijqr

=

?

m≥0

?

µ∈Mm

?

i≥1

f(k,i)

µ(i),±(x;q,t)qn(µ(i))(y(k)

j)m.

By comparing this with (1.5.1), we obtain the required formula.

Remark 1.9.

[S1, 2.2]. By using a similar, but much simpler arguments as above, one obtains an

alternative expression of q(k)

g(k)

m,±(x;0,t) coincides with the function q(k)

m,±(x;t) introduced in

m,±(x;t) as follows.

q(k)

m,±(x;t) =

?

µ∈Pm

|Sµ|−1?

w∈Sµ

w

?l(µ)

i=1

?

(x(k)

i

− tx(k∓1)

i

)(x(k)

i)µi−1

?

,

where Sµ is the stabilizer of µ in Sm.

x(k)

i

(1 ≤ i ≤ m), and Smacts on both variables).

The notion of symbols was introduced in [S1]. (Although a more general

setting was discussed in [S2], we do not use it in the discussion below. We remark that

similar symbols were also considered by G. Malle in [Ma]). Let m = (m0,...,me−1) be

as before. We denote by Z0,0

n

= Z0,0

Pn,esuch that each α(k)is written ( as an element in Zmk) in the form α(k): α(k)

··· ≥ α(k)

r ≥ s ≥ 0 and define an e-partition Λ0= Λ0(m,s,r) = (Λ(0),...,Λ(e−1)) as follows.

(Here we are considering finite variables

i,x(k∓1)

1.10.

n(m) the set of e-partitions α = (α(0),...,αe−1)) ∈

1

≥

mk≥ 0. We express α as α = (α(k)

j) in matrix form. Let us fix integers

Λ(0): (m0− 1)r ≥ ···2r ≥ r ≥ 0,

Λ(i): s + (mi− 1)r ≥ ··· ≥ s + 2r ≥ s + r ≥ s

(1.10.1)

for i = 0,...,e − 1. We denote by Zr,s

Λ = α + Λ0, where α ∈ Z0,0

Λ = α+Λ0, and call it the e-symbol of type (r,s) corresponding to α. We often denote

the symbol Λ = (Λ(0),...,Λ(e−1)) in the form Λ = (Λ(k)

for k = 0,...,e − 1.

Put m′= (m0+1,...,me−1+1), and define a shift operation Zr,s

by associating Λ′= (Λ′

Λ′

Λ = Λ(α), Λ′is obtained as Λ′= α+Λ0(m′,s,r), where α is regarded as an element

of Z0,0

?

with n = 0.

Two elements Λ and Λ′in¯Zr,s

if there exist representatives in Zr,s

multiplicities. The set of symbols which are similar to a fixed symbol is called a family

in Zr,s

n = Zr,s

n(m) the set of e-partitions of the form

and the sum is taken entry-wise. We write Λ = Λ(α) if

n

j) with Λ(k): Λ(k)

1

> ··· > Λ(k)

mk

n(m) → Zr,s

n(m), where

n(m′)

0,...,Λ′

e−1) ∈ Zr,s

k= (Λk+ r) ∪{s} for k = 0,...,e−1. In other words, for

n(m) to Λ = (Λ0,...,Λe−1) ∈ Zr,s

0= (Λ0+ r) ∪{0}, and Λ′

n(m′) by adding 0 in the entries of α. We denote by¯Zr,s

m′Zr,s

Pn,ecoincides with the set¯Z0,0

n

the set of classes in

n(m′) under the equivalence relation generated by shift operations. Note that

n. Also note that Λ0is regarded as a symbol in Zr,s

n

n are said to be similar, and are written as Λ ∼ Λ′,

n(m) such that all the entries of them coincide with

n.

Page 8

8 T. SHOJI

We define a function a : Zr,s

n → Z≥0, for Λ ∈ Zr,s

?

n, by

?

a(Λ) =

λ,λ′∈Λ

min{λ,λ′} −

µ,µ′∈Λ0

min{µ,µ′}. (1.10.2)

The function a on Zr,s

a on¯Zr,s

regard the a-function as a function on Z0,0

n is invariant under the shift operation, and it induces a function

n. Clearly, the a-function takes a constant value on each family in Zr,s

by using the bijection Z0,0

n. We

nn ≃ Zr,s

n.

1.11.

Hall-Littlewood functions P±

were introduced in [S1]. We shall now construct a two parameter version of Hall-

Littlewood functions. Let us introduce a total order α ≺ β on Z0,0

a(β) whenever α ≺ β and that each family in Z0,0

The following proposition is easily obtained by a similar argument as in Remark

4.9 in [S1] (i.e., a generalization of Gram-Schmidt orthogonalization process) if one

notices that Π(x,y;0,t) coincides with Ω(x,y;t) in [S1, 2.5].

Λ(x;t) and Q±

Λ(x;t) attached to symbols Λ

n

such that a(α) ≥

n

forms an interval.

Proposition 1.12. There exists a unique function P±

satisfying the following two properties.

(i) P±

Λ(x;q,t) ∈ ΞF for Λ ∈ Zr,s

n

Λ(x;q,t) for Λ = Λ(α) can be expressed in terms of sβ(x) as

P±

Λ= sα+

?

β

u±

α,βsβ,

with u±

Λ,P−

α,β∈ F, where u±

Λ′? = 0 unless Λ ∼ Λ′.

α,β= 0 unless β ≺ α and β ?∼ α.

(ii) ?P+

We then define Q±

Λ(x;q,t) as the dual of P∓

Λ(x;q,t), i.e., by the property that

?P+

Λ,Q−

Λ′? = ?Q+

Λ,P−

Λ′? = δΛ,Λ′.

P±

flection groups W (with respect to symbols in Zr,s

Λ(x;q,t), Q±

Λ(x;q,t) are called the Macdonald functions associated to complex re-

n).

Remark 1.13. (i) The orthogonality relations of Macdonald functions given above

imply, by [M1, VI, 2.7], that

Π(x,y;q,t) =

?

?

Λ

P+

Λ(x;q,t)Q−

Λ(y;q,t)(1.13.1)

=

Λ

Q+

Λ(x;q,t)P−

Λ(y;q,t).

(ii) In the case where q = 0, the scalar product given in (1.3.2) coincides with the

one given in [S1, 4.7]. Then by Proposition 4.8 in [S1], one sees that

P±

Λ(x;0,t) = P±

Λ(x,t),Q±

Λ(x;0,t) = Q±

Λ(x;t),(1.13.2)

where the right hand sides are the Hall-Littlewood functions defined in [S1].

Page 9

MACDONALD FUNCTIONS9

(iii) In the case where q = t, the scalar product in (1.3.2) coincides with the usual

scalar product on the space ΞQ, where the Schur functions form an orthonormal basis

of it. Hence Proposition 1.12 implies that

P±

Λ(α)(x;t,t) = Q±

Λ(α)(x;t,t) = sα(x).(1.13.3)

2. Macdonald operators

2.1. The original Macdonald functions related to symmetric groups are character-

ized as the simultaneous eigenfunctions of Macdonald operators (see [M1, VI]). In this

section, we shall construct certain operators which can be viewed as a generalization

of Macdonald operators to the case of W. Here we restrict ourselves to the case where

symbols are of the type (r,s) with r = 1 and s = 0. (We note that the arguments

in this section can not be applied to other types of symbols. See Remark 2.9.) In

particular, Λ0= δ, where δ = (δ(0),...,δ(e−1)) with δ(k)= (mk− 1,...,1,0). Hence,

in the case where e = 2 (i.e., W is the Weyl group of type Bn) with m1= m0+1, these

symbols are exactly the ones used to parameterize unipotent characters of finite clas-

sical groups Sp2n(Fq) or SO2n+1(Fq) by Lusztig [L1]. Each family F in Z1,0

a unique element ΛF= (Λ(k)

n

contains

j) with the property

Λ(0)

1

≥ Λ(1)

1

≥ ··· ≥ Λ(e−1)

1

≥ Λ(0)

2

≥ Λ(1)

2

≥ ··· .(2.1.1)

Such an element is called a special symbol associated to the family F. The set of

families is in bijection with the set of special symbols. Special symbols are regarded

as partitions of N =?Λ(k)

be families in Z1,0

n

and ΛF,ΛF′ be special symbols corresponding to them. We put

F < F′if ΛF < ΛF′ with respect to the dominance order on PN. Recall that for

λ = (λi),µ = (µi) ∈ PN, the dominance order λ < µ is defined by the condition that

j

by (2.1.1).

We shall define a partial order on the set of families in Z1,0

n. Let F and F′

k

?

i=1

λi≤

k

?

i=1

µi, (1 ≤ k ≤

?

j

mj).

We define a partial order on Z1,0

For λ = (λi) ∈ PN, put

n

by inheriting the partial order on the set of families.

n(λ) =

?

i≥1

(i − 1)λi.

Then, for each Λ in a family F, the value a(Λ) is given as α(Λ) = n(ΛF) − n(Λ0),

where ΛF is regarded as an element in PM. In particular, we have a(Λ) > a(Λ′) if

Page 10

10 T. SHOJI

Λ < Λ′. As before, by using the bijection Z0,0

on Z0,0

n

≃ Z1,0

n, we consider the partial order

n, which will be denoted by the same symbol.

2.2. In order to construct Macdonald operators, we shall start with finitely many

variables x = xm. We consider the expansion of Π(x,y;q,t) in the case of finitely

many variables. Assume that x = (x(k)

Let us denote by Pmthe set of e-partitions α = (α(k)

mα(x) = 0 unless α ∈ Pm, and those non-zero mα(x) form a basis of Ξm,F. The

same is true for Schur functions. Also by a similar argument as in [M1] we see that

gα(x;q,t) such that α ∈ Pmform an F-basis of Ξm,F.

Now by substituting x(k)

j

= y(k)

j

= 0 for j > mk, we have a finite version of (1.5.2),

i.e., for x = xm,y = ym, we have

?

This enables us to define a scalar product on Ξm,F by

j) = xmis as in 1.2 and put Ξm,F= F ⊗ZΞm.

j) such that l(α(k)) ≤ mk. Then

Π(x,y;q,t) =

α∈Pm

gα,+(x;q,t)mα(y) =

?

α∈Pm

mα(x)gα,−(y;q,t). (2.2.1)

?gα,+,mβ? = ?mα,gβ,−? = δα,β. (2.2.2)

We now consider the restriction of the functions P±

one sees that {P±

version of Proposition 1.12 holds, and P±

determined by these properties.

Λto Ξm,F. By Proposition 1.12,

Λ(α)| α ∈ Pm} form a basis of Ξm,F. Moreover, the finite variables

Λ∈ ΞFis obtained as the limit of P±

Λ∈ Ξm,F

2.3. Let

I = I(m) = {i =

i0

...

ie−1

∈ Ze| 1 ≤ ik≤ mk}.

For each i ∈ I and u ∈ F, we define an F-linear operator T±

T±

by ux(k∓1)

ik∓1

for k = 0,...,e−1. (Here we understand that ie= io). More generally, for

each r such that 1 ≤ r ≤ M1= mink{mk}, we define Iras the set of J = {i1,...,ir}

consisting of ik∈ I such that any two ikhave no common entries (i.e., ij−ikdoes not

contain 0 entries for each j ?= k as vectors in Ze). Then we define, for each J ∈ Ir,

an operator T±

commute with each other, and so T±

Let Z = Z(m) be the set of sequences β = (β(k)

with β(k)

i

∈ Z≥0. For each β ∈ Z, we denote by [β] the element in Z obtained from β

by permuting the entries inside each row, so that each row is arranged in decreasing

order. We often regard β as a matrix, and denote its k-th row by β(k). Then the set

J is regarded as a subset of the set of indices {(k,j) | 0 ≤ k ≤ e − 1,1 ≤ j ≤ mk} of

(β(k)

u,i: F[x] → F[x] by

u,if = f′, where f′is a polynomial obtained from f by replacing the variables x(k)

ik

u,J: F[x] → F[x] by T±

u,J=?r

k=1T±

u,ik. Note that T±

u,ikin the product

u,Jdoes not depend on the order of the product.

j) (0 ≤ k ≤ e − 1,1 ≤ j ≤ mk)

j)

Page 11

MACDONALD FUNCTIONS 11

We put, for each β ∈ Z and J ∈ Ir,

?β,J? =

?

(k,j)∈J

β(k)

j.

Smacts naturally on Z and on Ir, respectively, and this pairing is Sm-invariant. We

also note that the action of Smon Iris transitive.

The operation of T±

permuting the entries of β so that T±

the monomial?

aβ(x) =

q,Jon x = (x(k)

i) also induces an action β ?→ βJ±on Z by

q,Jxβ= q?β,J?xβJ±, where, as usual, xβdenotes

i,k(x(k)

i)β(k)

i . For each β ∈ Z, we define a function aβ(x) by

?

w∈Sm

ε(w)w(xβ)

We now define, for each 1 ≤ r ≤ M1, an F-linear operator Dr

±(q,t) on F[x] by

Dr

±(q,t) = aδ(x)−1?

w∈Sm

ε(w)

?

J∈Ir

xw(δ)J±t?w(δ),J?T±

q,J. (2.3.1)

Then Dr

±(q,t) can be written as

Dr

±(q,t) =

?

J∈Ir

A±

J(x;t)T±

q,J, (2.3.2)

with

A±

J(x;t) = aδ(x)−1?

= aδ(x)−1T±

w∈Sm

ε(w)xw(δ)J±t?w(δ),J?

t,Jaδ(x)

?

(k,j)∈J

= tr(r−1)/2

e−1

k=0

?

(k,i)/ ∈J

x(k)

i

x(k)

− tx(k∓1)

− x(k)

j′

ij

, (2.3.3)

where we write J = {i1,...,ir}, and take (k ∓ 1,j′) ∈ iaif (k,j) ∈ iafor a = 1,...,r.

The formulas (2.3.2) and (2.3.3) are the analogue of (3.4)rand (3.5)rin [M, VI, 3].

First we show that

Lemma 2.4.

(i) For α ∈ Pm, we have

Dr

±(q,t)mα(x) =

?

β

?

J∈Ir

t?δ,J?q?β,J?s(β+δ)J±−δ(x),

where β ∈ Z runs over all the row permutations of α.

Page 12

12 T. SHOJI

(ii) For α ∈ Pm, we have,

Dr

±(q,t)mα(x) =

?

β∈Z0,0

n

br,±

α,β(q,t)mβ(x)

with br,±

Dr

α,β(q,t) ∈ F, where br,±

±is an operator on the space Ξm,F.

α,β(q,t) = 0 unless β ∼ α or β < α. In particular,

Proof. For each β ∈ Z0,0

n, we have

Dr

±(q,t)xβ= aδ(x)−1

?

w1∈Sm

ε(w1)

?

J∈Ir

t?w1(δ),J?q?β,J?x(β+w1(δ))J±.

If we replace β by w2(β) for w2∈ Smand put w2= w1w, the term (β + w1(δ))J±

(resp. ?β,J? ) is replaced by w1((w(β) + δ)J′±) (resp. ?w(β),J′? ), respectively, with

J′= w−1

1(J). It follows that for α ∈ Z0,0

n, we have

Dr

±(q,t)mα(x) = |Sα|−1aδ(x)−1

?

t?δ,J′?q?w(α),J′?xw1((w(α)+δ)J′±)

w,w1∈Sm

?

?

ε(w1)

×

J′∈Ir

= |Sα|−1?

w∈Sm

J∈Ir

t?δ,J?q?w(α),J?s(w(α)+δ)J±−δ(x).

This proves (i).

Next we show (ii). Since δ = Λ0by our assumption, α + δ coincides with the

symbol Λ(α) for α ∈ Z0,0

to a family strictly smaller than the family containing α. On the other hand, if w = 1,

(α+δ)J±is obtained from the symbol α+δ by permuting some entries, and s(α+δ)J±−δ

coincides with ±sγ, where γ + δ is obtained from (α + δ)J±by rearranging the rows

in decreasing order. It follows that γ + δ is contained in the same family as α + δ.

Hence, for β ∈ Z0,0

n

in the expression of (ii), we see that β < α if w ?= 1 and β ∼ α if

w = 1.

n. We note that if w ?= 1, then the symbol w(α) + δ belongs

Next we show

Lemma 2.5. The operators Dr

±are adjoint each other, i.e., we have

?Dr

+f,g? = ?f,Dr

−g?,(f,g ∈ Ξm,F). (2.5.1)

Proof. By Lemma 1.6, (2.5.1) is equivalent to the formula

Π−1(Dr

+)xΠ = Π−1(Dr

−)yΠ. (2.5.2)

Page 13

MACDONALD FUNCTIONS 13

But for J ∈ Ir, we have

Π−1(T+

q,J)xΠ =

e−1

?

e−1

?

k=0

?

?

j≥1

?

?

(k,i)∈J

1 − x(k)

1 − tx(k)

iy(k)

iy(k+1)

j

j

,

Π−1(T−

q,J)yΠ =

k=0

i≥1

(k,j)∈J

1 − x(k)

1 − tx(k−1)

iy(k)

j

y(k)

ji

.

It follows that both of Π−1(Dr

the proof of (2.5.2), we may assume that q = t. In other words, we have only to prove

(2.5.1) under the assumption that q = t.

Now assume that q = t. Since

+)xΠ and Π−1(Dr

−)yΠ are independent of q. Hence in

T±

t,J(xw(δ)f) = xw(δ)J±t?w(δ),J?T±

t,Jf

for any polynomial f ∈ F[x], we have

Dr

±(t,t)f = a−1

δ

?

J∈Ir

T±

t,J(aδf).

It follows that for any α ∈ Z0,0

n,

Dr

±(t,t)sα= a−1

δ

?

t?α+δ,J?s(α+δ)J±−δ.

J∈Ir

T±

t,J(aα+δ)

=

?

J∈Ir

As before, s(α+δ)J±−δ coincides with ±sβ, where β + δ = [(α + δ)J±] (under the

notation in 2.3). Now in the case where q = t, {sα(x) | α ∈ Pm} is an orthonormal

basis of Ξm,F. It follows that

?Dr

+(t,t)sα,sβ? =

?

J

εJ+t?α+δ,J?, (2.5.3)

where J runs over all the elements in Irsuch that [(α + δ)J+] coincides with β + δ,

and εJ = (−1)l(w)with w ∈ Sm such that [(α + δ)J+] = w((α + δ)J+). But if

β + δ = w((α + δ)J+), we have α + δ = w−1((β + δ)J′−) = [(β + δ)J′−] with

J′= w(J). Also in this case,

?β + δ,J′? = ?w(α + δ)J′+,J′? = ?w(α + δ),J′? = ?α + δ,J?.

Since εJ+= εJ′−, we see that the right hand side of (2.5.3) is equal to

?

J′

εJ′−t?β+δ,J′?,

Page 14

14 T. SHOJI

where J′∈ Irruns over all the elements such that [(β + δ)J′−] = α + δ. Clearly this

coincides with ?sα,Dr

−(t,t)sβ?. So the lemma is proved.

2.6.

We fix a total order ≺ on Z0,0

the partial order <. Let Br

the formula in Lemma 2.4 (ii) with respect to the total order ≺. We consider Br

as a block matrix with respect to the equivalence relation α ∼ β, and denote it as

(Br,±

Lemma 2.4 (ii) implies that Br

Macdonald functions P±

Macdonald functions attached to symbols in a fixed family behaves as an eigenfunction

for Macdonald operators, where the eigenvalues should be replaced by the diagonal

blocks Br,±

n

as in 1.11, so that it is compatible with

α,β) be the matrix consisting of the coefficients in

±= (br,±

±

F,F′), where Br,±

F,F′ is the submatrix of Br

±is lower triangular as a block matrix. We consider the

Λconstructed via ≺. The following result shows that the set of

±corresponding to the families F,F′. Then

F,Fof B±.

Proposition 2.7. Let P±

Λ(α). Then we have

Λ(x;q,t) ∈ Ξm,F be Macdonald functions attached to Λ =

Dr

±P±

Λ=

?

β∼α

br,±

α,βP±

Λ(β),

where the coefficients br,±

α,βare the same as in Lemma 2.4 (ii).

Proof. By Proposition 1.12, P±

Λcan be written as

P±

Λ= mα+

?

β≺α,β?∼α

u′

α,βmβ

with u′

α,β∈ F. It follows, by Lemma 2.4 (ii), that one can write as

?

Hence, for each Λ′such that Λ′≺ Λ and that Λ′?∼ Λ, we have

Dr

±P±

Λ=

β∼α

br,±

α,βP±

Λ(β)+

?

Λ′≺Λ,Λ′?∼Λ

c±

Λ,Λ′P±

Λ′.

?Dr

+P+

Λ,P−

Λ′? = c+

Λ,Λ′.

On the other hand thanks to Lemma 2.5, we have

?Dr

+P+

Λ,P−

Λ′? = ?P+

Λ,Dr

−P−

Λ′? = 0

since Dr

that c+

−P−

Λ,Λ′ = 0 and the proposition holds for the + case. The − case is similar.

Λ′ is a linear combination of P−

Λ′′, where Λ′′∼ Λ′or Λ′′≺ Λ′. It follows

In view of Proposition 2.7, it is important to know the diagonal part of B±. By

lemma 2.4, the matrix B±

F,Fis described as follows.

Page 15

MACDONALD FUNCTIONS 15

Lemma 2.8. For α,β ∈ Z0,0

n

such that β ∼ α, we have

br,±

α,β(q,t) =

?

J∈Ir

[ΛJ±]=Λ(β)

εΛ,J±(tq−1)?δ.J?q?Λ,J?,

where Λ = α+δ ∈ Z1,0

n

and εΛ,J±= (−1)l(w)for w ∈ Smsuch that [ΛJ±] = w(ΛJ±).

Remark 2.9. The results in this section work only for a special type of symbols. It

seems to be difficult to extend the definition of Macdonald operators in (2.3.1) directly

to a more general case. A naive idea for the general situation is to replace δ by Λ0in the

definition (2.3.1). Then one gets some operator related to the symbols associated with

Λ0. However, the thus obtained operator does not preserve the set of polynomials in x,

since it involves the factor aΛ0(x)−1. If one leaves the denominator aδ(x) unchanged,

and replaces δ in all other places, then the operator preserves polynomials, but it does

not preserve the degree of them, and is not so useful.

3. A characterization of Macdonald functions

3.1.

We write the operator Dr

operators Dr

the case where r = 1, one can modify D1

compatible with ρm′,m. Let us define an operator E±

±as Dr

m,±to indicate the dependence on m. The

m,±are not compatible with the restriction homomorphisms ρm′,m. In

m,±as discussed in [M1], so that they are

m= E±

m(q,t) on Ξm,F by

E±

m= t−MD1

m,±−

?

i∈I

t?δ,i?−M. (3.1.1)

We show the following lemma.

Lemma 3.2. The operators E±

mare compatible with ρm′,m, i.e., we have

ρm′,m◦ E±

m′ = E±

m◦ ρm′,m

for m′= (m0+ 1,...,me−1+ 1).

Proof. Let us define another operator?E±

m: Ξm,F→ Ξm,F by

?

?E±

m= t−MD1

m,±−

i∈I

t?δ,i?−Ma−1

δT±

1,iaδ.

Then for each α ∈ Pm, we have

?E±

mmα=

?

β

?

i∈I

(q?β,i?− 1)t−?iks(β+δ)i±−δ, (3.2.1)

where β ∈ Z runs over all the row permutations of α, and i = (i0,...,ie−1). In

fact, by applying [M1, VI, 4] one can write mα =?

βsβ with β as above. Since

a−1

δT±

1,iaδsβ= s(β+δ)i±−δ, we obtain (3.2.1).

Page 16

16T. SHOJI

We claim that?E±

(3.2.2)

mis compatible with ρm′,m, i.e.,

ρm′,m◦?E±

n(m) and let α′∈ Z0,0

m′ =?E±

m◦ ρm′,m.

Recall that the map ρm′,m: Ξm′,F → Ξm,F is defined by substituting x(k)

k = 0,...,e − 1. Take α ∈ Z0,0

by adding 0’s to the last part of α. We consider the expression of?E±

We note that s(β′+δ′)i±−δ′ goes to zero under ρm′,mif there exists some k such that

β(k)

j ≤ mk. Hence if β(k)

whose entries are all non-zero, and so s(β′+δ′)i±−δ′ goes to zero.

It follows that in the expression of?E±

Z(m) into Z(m′) by adding 0 to the last part of β ∈ Z(m). Then those β′are

identified, under the embedding Z0,0

Now I(m) is also embedded into I(m′) in the same way. Take i ∈ I(m′) such that

i ?∈ I(m). If i is not equal to i1=t(m0+ 1,...,me−1+ 1), then s(β′+δ′)i±−δ′ goes to

zero under ρm′,mby the same reason as before. If i = i1, then ?β′,i? = 0. In any case,

such i does not give a contribution, and we may only consider i ∈ I(m) in (3.2.1).

Hence (3.2.2) holds.

Let? H±be the coefficient matrix of?E±

F = F′or F′< F. Then by comparing (3.2.1) with Lemma 2.4 (i) on the diagonal

parts, we see that

mk+1= 0 for

n(m′) be the element obtained

m′mα′ as given in

(3.2.1). Let δ′be the element for m′corresponding to δ for m, and take β′∈ Z(m′).

mk+1?= 0 for β′= (β(k)

j). In fact, if we write β′+ δ′= (γ(k)

mk+1?= 0 for some k, then (β′+δ′)i±= (γ′(k)

j), then γ(k)

j) contains a row γ′(k′)

j

> 0 for

m′mα′ in (3.2.1), we may only consider β′(a

row permutation of α′) such that the last column consists of zeros. One can embed

n(m) ֒→ Z0,0

n(m′), with a row permutation β of α.

mmαin terms of mβ. Then by Lemma 2.4

F,F′), where? H±

?

(ii),? H±can be expressed as a block matrix? H±= (? H±

F,F′ = 0 unless

? H±

F,F= t−MB±

F,F−

i∈I

t?δ,i?−M.

On the other hand, if we write? H±

F,F= (h±

α,β)α,β∈F, Proposition 2.7 together with

(3.1.1) implies that

E±

mP±

Λ(α)=

?

β∼α

h±

α,βP±

Λ(β).

But by (3.2.2), the matrix? H±

F,Fdoes not depend on the shift operation under m′→ m.

This shows that the operator E±

mis compatible with ρm′,m. The lemma is proved.

3.3. By Lemma 3.2, one can define an operator E±= E±(q,t) on ΞFas the limit

of E±

also satisfies the adjointness property, i.e., we have

m. Since E±

msatisfies a similar formula as given in Lemma 2.5, the operator E±

?E+f,g? = ?f,E−g?,(f,g ∈ ΞF). (3.3.1)

Page 17

MACDONALD FUNCTIONS 17

By Lemma 2.4 (ii), one can write, for each α ∈ Pn,e,

E±(q,t)mα(x) =

?

β∈Z0,0

n

h±

α,β(q,t)mβ(x) (3.3.2)

with h±

and write it as a block matrix H±= (H±

In the case of symmetric groups, the matrix H±is a triangular matrix, with dis-

tinct eigenvalues. This property was used to characterize Macdonald functions as

eigenfunctions of Macdonald operators. As an analogy, it is likely that the following

property holds for the diagonal parts of the block matrix B±.

α,β(q,t) ∈ F, where h±

α,β(q,t) = 0 unless β ∼ α or β < α. Let H±= (h±

F,F′) as in 2.6.

α,β),

Conjecture A. Let F,F′be any distinct families in Z0,0

and H±

n. Then the matrices H±

F,F

F′,F′ have no common eigenvalues (according to the sign + or −, respectively).

We have verified the conjecture in the case where e = 2, and n ≤ 5.

3.4.

Before giving a characterization of Macdonald functions in terms of Mac-

donald operators, we prepare an easy lemma. Let A = (aij) (resp. B = (bij)) be a

square matrix of degree m (resp. n), and let C = (Cα,β)1≤α,β≤nbe a block matrix of

size mn, consisting of blocks Cα,βof size m, defined by

?

−bα,βIm

Cα,β=

A − bα,αIm

if β = α,

otherwise.

We consider a matrix equation AX = XB, where X = (xij) is a m × n matrix of

unknown variables. Then this equation can be regarded as a system of linear equations

with respect to the mn variables {xij}, whose coefficient matrix is given by the matrix

C. Moreover, if B is a triangular matrix, then C is block wise triangular, and so

detC ?= 0 if and only if det(A − bα,αIm) ?= 0 for 1 ≤ α ≤ n. Hence we have the

following lemma.

Lemma 3.5. Under the above notation, the following are equivalent.

(i) The matrix equation AX = XB has a unique solution X = 0.

(ii) detC ?= 0.

(iii) The matrices A and B have no common eigenvalues.

We now show the following.

Theorem 3.6. Suppose that Conjecture A holds. Then the Macdonald functions P±

ΞF are characterized by the following two properties.

Λ∈

P±

Λ= mα+

?

α,βP±

β<α

w±

α,βmβ, (3.6.1)

E±P±

Λ=

?

β∼α

h±

Λ(β). (3.6.2)

In particular, P±

Λare determined independently from the choice of the total order ≺.

Page 18

18T. SHOJI

Proof. Let {Fi| i ∈ I} be the set of families in Z0,0

index set I, according to the total order ≺ on Z0,0

where Hij= H±

omit them). Let X = (w±

ΞF, and write it as X = (Xij). By Proposition 1.12, X is block wise lower triangular,

with identity diagonal blocks. Moreover, by Proposition 2.7, we see that XHX−1= G,

where G = (Gij) is a block diagonal matrix with diagonal blocks Gii= Hii. In order

to prove the theorem, we have only to show the following.

n. We give a total order ≺ on the

n, and write H±as H±= (Hij)i,j∈I,

Fi,Fj. (Since the following discussion is independent of the sign {±}, we

α,β) be the transition matrix between basis {mα} and {P±

Λ} of

(3.6.3) Let X = (Xij) be a block wise lower triangular matrix, with identity diagonal

blocks, such that XHX−1= G. Then X is determined uniquely, and Xij= 0 unless

i = j or Fj< Fi.

We show (3.6.3). The equation XHX−1= G can be written as

HiiXij− XijHjj=

?

j≺k≺i

XikHkj

(3.6.4)

for any pair j ≺ i. By backwards induction on j, we may assume that Xikare already

determined for j ≺ k ≺ i. Then (3.6.4) determines Xijuniquely, by Conjecture A and

Lemma 3.5. Now suppose that Fj ?< Fi. Again by induction, we may assume that

Xik = 0 unless Fk < Fi. Since Hkj = 0 unless Fj < Fk by (3.3.2), we must have

HiiXij−XijHjj= 0. This implies that Xij= 0 by Lemma 3.5, and we obtain (3.6.3).

Thus the theorem is proved.

Remark 3.7.

P±

and so the Kostka functions K±

This answers the questions posed in [S1, Remark 4.5, (ii)] and in [GM, Remark 2.4],

modulo the truth of the conjecture.

The Hall-Littlewood function P±

Λ(x;0,t). Hence it satisfies similar formulas as (3.6.1), (3.6.2). In particular, P±

α,β(x;t) do not depend on the choice of the total order.

Λ(x;t) given in [S1] coincides with

Λ(x;t),

3.8. We give here some examples of the matrices B±

rank cases with e = 2. Assume that W is the Weyl group of type C2. Then the

symbols and families are given as follows.

F1=??3 0

,

0

F,Fand H±

F,Ffor some small

0

??,F2=??2 1 0

,

2

2 1

??,

??,

F3=??2 0

1

??2 1

??1 0

which correspond, in this order, to the double partitions of 2,

(2;−),(−;11), (1;1),(11;−),(−;2)

Page 19

MACDONALD FUNCTIONS 19

respectively. In the tables below, we express the matrices B±

they are independent of the sign ±. We have

F,F,H±

F,Fas BF,HF, since

BF1= (1),BF2= (t3q + tq),BF3=

0

q

q

0

tq2

−tq

0 tq2

−q2

.

Up to C5, only 3-element or 1-element families occur. The Weyl group of type C6

contains a unique 10-element family, which is given as follows,

F4=??4 2 0

,

4 1

3 1

?

?

,

?4 2 1

?3 2 0

3 0

?

?

,

?4 3 0

?3 2 1

2 1

?

?

,

?4 3 1

?3 1 0

2 0

?

?

,

?4 3 2

?2 1 0

1 0

?

??.

,

?4 1 0

3 2

,

4 0

,

4 2

,

4 3

The corresponding double partitions of 6 are, in this order, as follows.

(21;21), (211;2), (22;11), (221;1), (222;−),

(2;22), (11;31), (111;3), (1;32),(−;33).

The matrix BFis given by

Throughout the above examples, HFis given by

BF=

0

q

q

0

0

t2q3

0

0

q

−q2

−t2q2

−t2q4

0

t2q3

0

0 tq2

−tq3

−tq

tq2

0

0

−tq3

tq4

0

0

tq2

−tq

−tq3

tq2

0

0

0

0

t3q4

−t3q3

t3q4

0

−t3q3

0

−t2q2

0

0

q

tq2

−tq3

00

0

t2q3

−t2q2

0

0

0

−t2q4

−t2q2

t2q

t2q3

0

t2q3

q

0

tq2

tq2

0

−t2q4

−tq3

0

t3q4

0

−t3q3

t3q2

0

q

0

−q2

q3

t2q3

0

tq2

tq2

t3q4

0

0

t2q3

t2q3

−t2q2

0

t3q4

tq2

−tq

0

t2q3

t2q3

−t2q2

−q2

0

t3q4

0

−tq3

.

HF= t−(2m+1)BF− t−(2m+1)(1 − tm)(1 − tm+1)

(1 − t)2

for symbols of the shape m = (m+1,m), (so, m = 1 for F1,F3, and m = 2 for F2,F4,

respectively). Note that BFis not necessarily symmetric. However, if we put q = t, it

turns out to be symmetric since D1is a self adjoint operator, and the representation

matrix with respect to the orthonormal basis of Schur functions coincides with the

diagonal blocks (BF).

3.9. In the case of symmetric groups, Macdonald operators are commuting with

each other since they are simultaneously diagonalizable. In our case, Proposition 2.7

shows that Dr

they are commuting with each other if and only if the matrices Br,±

with for r = 1,...,M1(for a fixed F). As the following examples show, one might

expect that Br,±

±are simultaneously diagonalizable in the sense of block matrices. So

F,Fare commuting

F,Fare commuting with each other in general.

Page 20

20T. SHOJI

First consider a simple example. Let F3= {Λ1,Λ2,Λ3} be the 3-element family

of C2as in 3.8. We consider its m times shifts F(m)

?x + m, y + m, m − 1, ..., 0

for Λi=?x y

Br

3

= {Λ′

1,Λ′

2,Λ′

?

3}, where

Λ′

i=

z + m, m − 1, ..., 0

z

?. Then the matrices Br

F3are given as

F3= aB1

F3+ b

with

a = t2m?

J′

t?δ,J′?,b =

?

J

t?δ,J?,

where J ∈ Irruns over the elements of the form J = {ij1,...,ijr} for ij=?j+1

implies that Br

In the following, we discuss some related results, i.e., we show that when e = 2,

and q = t, then the operators Drare commuting with each other. First we prepare a

lemma. (Since we deal with the case where e = 2, we omit the sign ± in the discussion

below.)

j

?with

2 ≤ j ≤ m + 1, and J′∈ Ir−1runs over the elements having similar properties. This

F3are all commuting with for 1 ≤ r ≤ m.

Lemma 3.10. Assume that e = 2. Then, for each r,r′, there exists a bijective map

ϕ : Ir× Ir′ → Ir× Ir′ satisfying the following properties: let J ∈ Ir,J′∈ Ir′ and put

ϕ(J,J′) = (K,K′). For each α ∈ Z, we have

(i) αJJ′ = αK′K,

(ii) ?α,J?+?αJ,J′? = ?α,K′?+?αK′,K?.

Proof. Take J ∈ Ir,J′∈ Ir′. Let J0be the subset of J consisting of i =?a

similar properties. We define an equivalence relation on J0by connecting i =?a

denote by {JC| C ∈ C} the set of equivalence classes in J0. Then the class JChas the

following form.

JC=??a1

where?ai

J′

bi

where p = 0 (resp. q = k + 1) if there exists?a0

b

?such that

?,i′=

there exists i′=?x

?c

y

?∈ J′with x = a or y = b, and let J′

∈ J0when there exists

y

0be the subset of J′having

b

d

??x

?

∈ J′such that (x,y) = (a,d) or (x,y) = (c,b). We

c

?

,

?a2

b1

?

,...,

?ak−1

bk−2

?

,

?ak

bk−1

?

,

?d

bk

??,

bi

?∈ J′for 1 ≤ i ≤ k. We put

C=??ai

?

| p ≤ i ≤ q?,

?∈ J′(resp.

b0

?ak+1

bk+1

?∈ J′) such that

b0= c (resp. ak+1= d), and p = 1,q = k otherwise.

Page 21

MACDONALD FUNCTIONS 21

For each C ∈ C, we define the sets KC,K′

Since J,J′are subsets of the index set of elements in Z, JC, etc. induce permutations

on the entries of elements in Z. We denote by xC,x′

in SMcorresponding to JC,J′

that

x′

(3.10.1)

Cas follows:

KC=

?

JC

J′

J′

??a0

?a3

if p = 1,q = k,

if p = 1,q = k + 1,

if p = 0,q = k,

C

C

b1

?

,...,

?ak

bk+1

??

?

if p = 0,q = k + 1.

K′

C=

??a2

JC

JC

J′

C

c

,

b1

?

,...,

?ak

bk−2

,

?

d

bk−1

??

if p = 1,q = k,

if p = 1,q = k + 1,

if p = 0,q = k,

if p = 0,q = k + 1.

C(resp. yC,y′

C) the permutations

C(resp. KC,K′

C), respectively. Then it is easy to check

C◦ xC= yC◦ y′

C.

We put

K = (J − J0) ∪

?

C∈C

KC,K′= (J′− J′

0) ∪

?

C∈C

K′

C.

Components of K (resp. K′) are mutually disjoint, and we see that K ∈ Ir,K′∈ Ir′.

We now define the map ϕ : Ir× Ir′ → Ir× Ir′ by ϕ(J,J′) = (K,K′). Then one can

check that ϕ2= id, and so ϕ is a bijection. The assertion (i) follows from (3.10.1).

To show (ii), it is enough to verify the formula in the case where J = JC,J′=

J′

p = 1,q = k or p = 0,q = k + 1. Then one can check by a direct computation that

C,K = KC,K′= K′

C. The assertion is clear when KC= J′

C,K′

C= JC. Assume that

?α,JC? = ?αK′

C,KC?,?αJC,J′

C? = ?α,K′

C?.

Hence the formula holds in these cases also, and the lemma follows.

Proposition 3.11. Assume that e = 2, and q = t. Then the operators Dr(t,t) are

commuting with each other for r = 1,...,M1.

Proof. As remarked in 3.9, it is enough to show that the matrices Br

muting with each other for r = 1,...,M1. By Lemma 2.4 (i), the part corresponding

to the diagonal block in the expression of Dr(t,t)sαis given as

?

where Λ = α + δ. Let X be the part of Dr′(t,t)Dr(t,t)sα corresponding to the

diagonal blocks. Then we have

?

F= Br,±

F,Fare com-

J∈Ir

t?Λ,J?sΛJ−δ,

X =

J∈Ir

εJt?Λ,J??

J′′∈Ir′

t?[ΛJ],J′′?s[ΛJ]J′′−δ.

Page 22

22T. SHOJI

For each J, there exists w ∈ Smsuch that [ΛJ] = w(ΛJ) and εJ is given by εJ =

(−1)l(w). Then if we put J′= w−1(J′′), we have ?[ΛJ],J′′? = ?ΛJ,J′?, and [ΛJ]J′′ =

w(ΛJJ′). Hence one can write

?

where εJ,J′ = (−1)l(w′)for w′∈ Sm such that [ΛJJ′] = w′(ΛJJ′). Now by apply-

ing Lemma 3.10 for α = Λ, we see that X coincides with the diagonal part for

Dr(t,t)Dr′(t,t)sα. The proposition is proved.

X =

J∈Ir

?

J′∈Ir′

εJ,J′t?Λ,J?+?ΛJ,J′?s[ΛJJ′]−δ,

References

[GM] M. Geck and G. Malle; On special pieces in the unipotent variety, Experimental Math. 8

(1999),281–290.

G. Lusztig; Characters of reductive groups over a finite field, Annals of Math. Studies 107,

Princeton University Press, Princeton, N.J., 1984.

G. Lusztig; Intersection cohomology complexes on a reductive group, Invent. Math. 75

(1984), 205–272.

I.G. Macdonald; Symmetric functions and Hall Polynomials, second edition. Clarendon

Press. Oxford 1995.

I.G. Macdonald; Commuting differential operators and zonal spherical functions. Springer

Lecture Notes, 1271 (1987), 189–200.

G. Malle; Green functions, special pieces, unipotent classes. In The Proceedings of “Rep-

resentation theory of finite and algebraic groups”, Ed. by N. Kawanaka et.al., Osaka Univ.

(2000), 154-160.

T. Shoji; Green functions associated to complex reflection groups. J. Algebra. 245, (2001),

650–694.

T. Shoji; Green functions associated to complex reflection groups, II. To appear in J. Alge-

bra.

T. Shoji; Green functions attached to limit symbols, preprint.

[L1]

[L2]

[M1]

[M2]

[Ma]

[S1]

[S2]

[S3]