Cambridge Tracts in Mathematics. 157. Cambridge: Cambridge University Press. x, 175 p. £ 35.00; $ 48.00 (2003).
This is a beautiful book, treating in a concise and clear way the recent developments concerning the connection between orthogonal polynomials in several variables and root systems in two or more parameters. The reader should be warned: it is not a simple book! A good knowledge of algebra, geometry of root systems and Weyl groups is indispensable. Combined with the ability for formula manipulation and perseverence, the reader is sure to get a better understanding of orthogonal polynomials situated high in the hierarchy of the generalized Askey scheme [the Askey-Wilson polynomials and the Jacobi polynomials due to Heckman and Opdam]. The first four chapters treat the basic properties of affine root systems and a complete characterization thereof, the extended affine Weyl group, the Artin braid group, the double braid group and the affine and double affine Hecke algebra. After these first chapters, that can be seen as a unified foundation, the main [deep] results on orthogonal polynomials follow in Chapter 5. Finally, in Chapter 6, the author treats the case of a rank $1$ affine root system and makes the results, derived previously, explicit, leading to $q$-ultraspherical (Roger) polynomials and the Askey-Wilson polynomials.