5. ABOUT THESE PROCEEDINGS xiii
Calogero-Sutherland model is algebraically integrable for integer values of the pa-
rameters. Generalization of this result to Macdonald operators is due to P. Etingof
and K. Styrkas
[ES].
A. Sergeev discovered the relation of the super Jack polyno-
mials introduced in [KOO] with the deformed quantum Calogero-Moser systems
and Lie superalgebras
[Ser].
Further generalizations including the difference case
and super Macdonald polynomials have been investigated by A. Sergeev and A.
Veselov (to appear in this volume) who have shown that these polynomials are the
joint eigenfunctions of certain difference operators on algebraic varieties.
P. Etingof
and A.A. Kirillov, Jr. have shown in [EKl], [EK2] how Macdonald
polynomials for the root system of type An could be interpreted in terms of the
representation theory of quantum groups. Namely, Macdonald polynomials arise as
traces of certain natural intertwining operators, which generalizes the description
of Schur functions as traces of irreducible SLn-modules. This leads, in particular,
to elegant proofs of various Macdonald polynomials identities, such as inner prod-
uct and symmetry identities. It also, in the affine case, leads to natural elliptic
extensions of Macdonald theory.
Asymptotic properties of Macdonald polynomials were investigated by G. 01-
shanski, in collaboration with S. Kerov and A. Okounkov. In particular, the analog
of the Vershik-Kerov asymptotics for the characters of the symmetric and unitary
groups for the case of Jack polynomials were obtained in [KOO] and [002], respec-
tively. Remarkably, the same type of asymptotics continues to hold, with minimal
and very natural modifications. Recently, J .F. van Diejen suggested a general ap-
proach to deriving asymptotics of a class of multivariate orthogonal polynomials as
the degree tends to infinity and applied it to Jack polynomials.
Another connection between interpolation and Macdonald polynomials arose
recently in the work ofT. Miwa and his collaborators. In [F JMMl] and [F JMM2]
they showed, that certain ideals in the algebra of symmetric functions which are
of interests in the representation theory of affine Lie algebras have a linear basis of
Macdonald polynomials.
V.B. Kuznetsov, V.V. Mangazeev and E.K. Sklyanin have recently completed
the long-standing task of factorizing Jack polynomials [KS, KMS] by advancing
the theories of separation of variables and Backlund transformations for quantum
integrable systems.
The French group based mainly in Marne-la-Vallee have over the years made
considerable contributions to algebraic combinatorics in general and to Macdonald
polynomials
[LLT], [LT]
in particular.
Elementary proofs of Macdonald conjectures are by now available for the clas-
sical root systems, see [M2] for the An-case and [R] for the general BCn-case. In
recent work (math.QA/0309252) E. Rains constructed a family of elliptic biorthog-
onal functions generalizing the Koornwinder polynomials.
R. Gustafson has discovered a method of evaluating many important hyper-
geometric integrals
[Gus]
which are intimately connected to Jack and Macdonald
polynomials.
In the joint work with M. Lassalle
[LS],
M. Schlosser recently presented an
explicit analytic formula for Macdonald polynomials. This was obtained from a re-
cursion for Macdonald polynomials being derived from inverting the Pieri formula.
M. Lassalle gave an elementary proof of the expansion formula for Macdonald poly-
nomials in terms of 'modified complete' symmetric functions.
CONTEMPORARY MATHEMATICS
xiii
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