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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