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.