xiv INTRODUCTIO N
These result s an d conjecture s i n turn prompte d analogou s conjecture s [M ] fo r
an arbitrar y roo t syste m R. T o state thes e in their simples t form , assum e that R i s
reduced an d irreducible , an d fo r each a G R le t e
a
denot e the corresponding forma l
exponential, a s i n Chapte r II , §3 . The n th e constan t ter m i n th e expansio n o f th e
product
(6) II(
1)Qee-—
±
(where a s befor e k i s a positiv e integer ) shoul d b e
(
7) n
( k d i
where d\ ,..., d
r
ar e th e degree s o f th e fundamenta l polynomia l invariant s o f th e
Weyl grou p o f R; and , mor e generally , th e constan t ter m (i.e. , involvin g q but n o
exponentials) i n
(8) J ] (e
a:q)k(qe-a;q)k
aeR+
should b e
kdi
k
(9) n
2 = 1
where[™ ] is the (/-binomia l coefficien t o r Gaussia n polynomial , define d b y
= (q;q)n/(q;q)
r
(q;q)n-r
Clearly (9 ) implie s (7 ) b y settin g q 1. Moreover , whe n th e roo t syste m R i s of
type A
n
-i, th e degree s di ar e 2,3,...,n , an d s o (9 ) reduce s t o th e specia l cas e of
(5) i n whic h k\ = k
n
= k. Whethe r th e genera l for m o f (5) , wit h n distinc t
parameters ki, ha s counterpart s fo r othe r roo t system s i s still a n ope n question .
The combinatorialist s mad e som e progres s toward s establishin g (9 ) o n a case -
by-case basis—se e [K ] fo r th e classica l roo t systems , [GG ] fo r F4 an d [H ] fo r
G2—but wer e unabl e t o handl e th e case s o f th e exceptiona l roo t system s
EQ,E^,
and E%. The first unifor m proof , fo r al l roo t system s R, o f the conjectur e (7 ) (th e
case q = 1 of (9) ) wa s found b y E. Opda m [O] , by exploiting the shif t operator s h e
and Heckma n ha d constructe d a s par t o f thei r theor y o f hypergeometri c function s
and Jacob i polynomial s allude d t o earlie r i n thi s introduction ; an d finally (9 ) wa s
proved i n ful l generalit y b y Cheredni k [C ] in 1995.
In the context o f the theory of orthogonal polynomials attached t o root system s
developed i n Chapter II , the conjectur e (9 ) essentially appear s a s the simples t cas e
(A = 0 ) o f th e nor m formul a (8.3') . (Th e functio n A (wit h t = q k) define d i n
Chapter II , §4 , is not quit e th e sam e a s (8 ) above , bu t i t i s not difficul t t o switc h
from on e to th e other. )
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