Root Systems and Special Functions

The proposal is in the area of the theory of hypergeometric functions associated to root systems, and their q-deformations. A number of key results have been obtained by the investigator in the course of previous NSF supported research. These results have led to the resolution of various conjectures of Dunkl, Macdonald, Koornwinder and others. It is proposed to examine some of the outstanding problems in this area, which have to do with certain positivity conjectures of Macdonald, Stanley, and the proposer. The research described above should lead to a better understanding of several different areas of mathematics including algebra, representation theory, combinatorics, analysis, and multivariate statistics.