4. Other directions
There are several areas of mathematics where Macdonald polynomials make a nat-
ural appearance. To give a complete and historically accurate description of these
areas would require a much longer article, and considerably more expertise than
we possess. We shall be content here with a brief mention of some of the themes
and some of the key names in those areas. Hopefully this will help the interested
reader to track down further results and interconnections.
Macdonald polynomials appear in the context of the exactly solvable quantum
Calogero-Sutherland model [Su] and its generalizations by Olshanetsky-Perelomov
[OP], Ruijsenaars
and others. This field is closely related to the study of an
ideal gas by Haldane
and Shastry [Sh]. Considerable work in this area has
been carried out by T. Baker and P. Forrester
Another circle of ideas involving Macdonald polynomials centers around the
theory of vertex operator algebras, W-algebras, and conformal blocks. We refer the
reader to papers by Frenkel and Reshetikhin
The theory of symmetric functions and representations of the symmetric group
plays a big role in algebraic combinatorics. We refer the reader to papers by Las-
coux, Leclerc and Thibon on the subject [LLT, LT].
Jack and Macdonald polynomials are also intimately connected with the study
of random phenomena on the symmetric group, such as random partitions and
random permutations. We refer the reader to various papers by Vershik-Kerov and
The subject of harmonic analysis on the affine Heeke algebra has been advanced
considerably by the work of E. Opdam
We also refer the reader to papers
by I. Cherednik and J. Stokman in this area.
About these proceedings
The first part of these proceedings consists of material of historical significance,
including some previously unpublished texts. We include here biographical notes
on Jack by B. Sleeman and on Hall, Littlewood and Macdonald by A. Morris. We
also include
reprints of the original papers of Littlewood and Jack, and notes on
Hall's (unpublished) results by I. Macdonald. Finally we print, in its entirety, a
recently discovered manuscript of Jack together with comments by I. Macdonald.
The second part of these proceedings consists of original contributions to the
subject in the form of refereed research papers. For the reader's convenience we
briefly describe the mathematical background for some of these papers. As before,
the purpose is to give the interested reader an opportunity to follow up on some of
the ideas mentioned in the papers. We lack the space and the expertise to provide
a complete and historically accurate exposition of the various subjects.
In 1974 T.H. Koornwinder wrote a series of papers
dedicated to the or-
thogonal symmetric polynomials of type

He constructed several shift
operators and derived explicit series representations for these polynomials in two
variables. These results were generalized by E. Opdam. In
A.M. Kirillov and
M. Noumi obtained explicit parameter preserving lowering and raising operators for
Macdonald polynomials of the type
thereby generalizing the previous results
for Jack polynomials due to L. Lapointe and L. Vinet.
Using Heckman-Opdam's
theory of multivariate hypergeometric func-
tions, 0. Chalykh,
Styrkas and A. Veselov [VSC] proved that the quantum
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