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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

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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).

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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).

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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

?

.

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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.