3. VARIANTS OF MACDONALD POLYNOMIALS
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well as the related
(n
+
1)n-l
conjecture on the dimension of the space of diagonal
harmonics
[H2].
3. Variants of Macdonald polynomials
As explained above, Macdonald polynomials generalize characters of compact group
and, like these characters, they are symmetric (i.e. invariant with respect to the
Weyl group action). It was therefore somewhat surprising when the study of these
symmetric polynomials gave rise to a natural family of
non-symmetric
polynomials.
These polynomials were first introduced in the Jacobi setting by E. Opdam
[Op2],
who credits the definition to G. Heckman. In turn, Heckman was moti-
vated by the work of Cherednik who had expressed the Macdonald operators as
symmetric polynomials in certain commuting first order operators. These Chered-
nik operators are trigonometric analogs of operators first considered by C. Dunkl
[Du].
The nonsymmetric Macdonald polynomials are defined to be the simultane-
ous eigenfunctions of these Cherednik operators.
The discovery of the non-symmetric polynomials led to substantial simplifica-
tions in the theory of Macdonald polynomials. This was crucial in the proof of the
integrality and positivity conjectures for Jack polynomials in
[KnS2].
Generalizing
the ideas in that paper to arbitrary root system, Cherednik
[C3]
formalized the
theory of intertwiners and used them to give alternate proofs of some of the Mac-
donald conjectures. Although the non-symmetric polynomials are very useful and
natural in the Macdonald theory, they remain somewhat mysterious. For certain
special values of the parameters they have been identified with Demazure char-
acters of basic representations of affine Kac-Moody Lie algebras by Y. Sanderson
[San]
for type
A,
and by B. Ion
[I]
for arbitrary root systems. However for general
parameters their representation-theoretic meaning is still obscure.
Another class of polynomials which turned out to be closely connected to Mac-
donald polynomials are the so-called
interpolation
polynomials. These polynomials
were first defined by S. Sahi
[S4],
in connection with joint work with B. Kostant
on the Capelli identity. They are symmetric inhomogeneous polynomials, depend-
ing on several parameters, and defined by fairly simple vanishing properties. In
the special case when the parameters form an arithmetic progression, F. Knop and
S. Sahi proved in
[KnS1]
that the top degree terms of the interpolation polynomial
is the usual Jack polynomial. A similar result also holds for Macdonald polynomials
[S1, Kn1].
Many results for Jack and Macdonald polynomials, both symmetric and non-
symmetric, continue to hold for the interpolation polynomials. Indeed some of
the results are easier to prove in the inhomogeneous setting because of the strong
uniqueness result for these polynomials. Results for the homogeneous polynomials
can then be deduced by considering the top homogeneous terms. Considerable
work on these polynomials was done by A. Okounkov who obtained combinatorial
and integral formulas for these polynomials, and also defined their analogs in the
BCn
setting. It turns out that special values of interpolation polynomials are
the coefficients in the series expansion of the Jack polynomial about the point
x
=
(1, ... , 1)
[001].
Analogous results are true for symmetric and non-symmetric
Macdonald polynomials, and in the
BCn
setting one obtains a multivariable analog
of the hypergeometric series representing the Askey-Wilson polynomials
[01, 02,
S2, Kn2].
CONTEMPORARY MATHEMATICS
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