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Séminaire Lotharingien de Combinatoire, B45a (2000), 40 pp.

# Ian G. Macdonald

#
Orthogonal Polynomials Associated with Root Systems

**Abstract.**
Let *R* and *S* be two irreducible root systems spanning
the same vector space and having the same Weyl group *W*, such that
*S* (but not necessarily *R*) is reduced. For each such pair
(*R,S*)
we construct a family of *W*-invariant orthogonal polynomials in
several variables, whose coefficients are rational functions of
parameters
*q,t*_{1},*t*_{2},...,*t*_{r},
where *r* (= 1, 2 or 3) is the
number of *W*-orbits in *R*. For particular values of these
parameters, these polynomials give the values of zonal spherical
functions on real and *p*-adic symmetric spaces. Also when
*R*=*S* is
of type *A*_{n}, they conincide with the symmetric polynomials
described in I. G. Macdonald, *Symmetric Functions and Hall
Polynomials*, 2nd edition, Oxford University Press (1995),
Chapter VI.

### Foreword

The text of the paper is that of my 1987 preprint with
the above title. It is now in many ways a period piece, and I have
thought it best to reproduce it unchanged. I am grateful to Tom
Koornwinder and Christian Krattenthaler for arranging for its
publication in the Séminaire Lotharingien de Combinatoire.
I should add that the subject has advanced considerably in the
intervening years. In particular, the conjectures in Section 12
are now theorems. For a sketch of these later developments the reader may
refer to my booklet *"Symmetric functions and orthogonal
polynomials"*, University Lecture Series Vol. 12, American
Mathematical Society (1998), and the references to the literature
given there.

Ian G. Macdonald, November 2000

Received: August 21, 2000; Accepted: August 21, 2000.

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