Since the operators are self-adjoint with respect to a certain scalar product, Mac-
donald polynomials are multivariate orthogonal polynomials. From this point of
view, they generalize various classical orthogonal polynomials.
These root system polynomials are connected with earlier work of Macdonald
on spherical functions for p-adic groups, which in the present context are obtained
by specializing various parameters to 0. On the other hand taking a suitable limit
as the parameters approach 1 one obtains the multivariate Jacobi polynomials that
had been previously studied by G. Heckman and E. Opdam
and which in
turn generalize the characters and spherical functions of the corresponding compact
Lie groups. Thus Macdonald's results can be seen as a manifestation of Barish-
Chandra's "Lefschetz principle". This principle, which was one of the guiding
philosophies of Barish-Chandra's work, asserts that representation theoretic results
for an algebraic group over a field should have analogues for the same group over
other fields. In a certain sense Macdonald polynomials "see" the representation
theory of the group
G (F)
for "every field
2. Macdonald Conjectures
Many of the basic properties of Macdonald polynomials were initially formulated
as conjectures by Macdonald. These include the constant term formula, the norm
formula, the duality/symmetry property. A great deal of research in recent years
has been focused on proving these conjectures.
For the Jacobi limit these conjectures were proved by E. Opdam [Opl] by
the technique of shift operators. Subsequently, I. Cherednik
[Cl], [C2]
proved the
Macdonald conjectures for all reduced root systems. Cherednik's approach involved
his theory of double affine Heeke algebras which is one of the major developments
in this area. In the non-reduced BCn-case, Macdonald polynomials are known as
Koornwinder polynomials
and they can be viewed as the multivariate ana-
logue of the celebrated Askey-Wilson polynomials. In this case the Macdonald
conjectures were proved by S. Sahi in
following earlier work of J.F. van Diejen
Macdonald's latest book
gives an exposition of all these results.
Another set of conjectures was formulated by Macdonald in the type
ting, see
These conjectures are known as the "integrality" and "positivity"
conjectures, and are concerned with the expansion of these polynomials in terms of
other bases of symmetric functions, e.g. the monomial basis. Macdonald made sep-
arate conjectures for Jack polynomials and for symmetric Macdonald polynomials.
It has recently been discovered that in the case of Jack polynomials, Jack himself
had conjectured some of these properties in an unpublished manuscript
before his death. In the case of Jack polynomials, both conjectures were proved
by F. Knop and S. Sahi in [KnS2]. The "integrality" conjecture for Macdonald
polynomials was established in six different papers which appeared roughly at the
same time.
The positivity conjecture for Macdonald polynomials proved to be much harder.
Garsia and Haiman [ G H] generalized this to a conjecture for the dimension of a
certain doubly-graded Sn-modules, which came to be known as the n! conjecture.
M. Haiman established a spectacular connection between Macdonald poly-
nomials and the geometry of the Hilbert scheme of points in the plane, following
a suggestion of C. Procesi. This enabled Haiman to prove the n! conjecture, as
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