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,t1,t2,...,tr,
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 An, 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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