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Special Functions Associated with Root Systems: A First Introduction for Non-Specialists

A survey is given about recent developments in special functions associated with root systems. The paper addresses a general mathematical audience and it does not assume earlier knowledge of either special functions or root systems. The emphasis is on multivariable orthogonal polynomials: the Heckman-Opdam Jacobi polynomials and Maconald's q-analogues of these. In this paper I present a brief survey of the active area of Special Functions associated with Root Systems. The article is intended for a general mathematical audience. It will not suppose prerequisites on either special functions or root systems. It will also skip many technical details. Some early developments in this area took place during the seventies ((21), (40)). During the last ten years important break-throughs were made by Heckman and Opdam (14), (35), (36), (15), I. G. Macdonald (29), (30), C. F. Dunkl (10) and I. Cherednik (4), (5), (6), (7). A lot of the motivation for the subject of this paper comes from analysis on semisimple Lie groups. Spherical functions on Riemannian symmetric spaces of the compact or non- compact type can be written as special functions depending on parameters which assume only special discrete values. In the one-variable cases these special functions were classical, also for parameter values without group theoretic interpretation, but in the more-variable cases they were new. In the case of group theoretic interpretation, many properties of these special functions, as well as associated harmonic analysis, immediately follow by group theoretic arguments. The case of more general parameter values yields the special functions associated with root systems. Properties derived in the group case can still be formulated in the general case, but now as conjectures rather than theorems. This paper describes some of the progress which has been made in proving these conjectures. For convenience, I will restrict to the polynomial (compact) case, with Bessel functions as a sole exception. (For an introduction to the non-polynomial case see (37), (16).) Neither will I discuss the recent work on commuting operators with elliptic functions as coecients.