Mathematical Sciences: Symmetry & Orthogonal Polynomials

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940142 Dunkl This award supports mathematical research on problems arising in the theory of special functions emphasizing underlying group structures and extensions to several variables. The analysis involves the study of root systems using a technique a parametrized algebra of differential-difference operators associated to finite reflection groups. There is an intertwining operator which relates these algebras for different parameter values. An important part of the project will consist of finding integral formulas for this operator for Weyl groups by means of techniques based on the associated compact Lie groups. Progress in this topic would lead to more detailed information, such as asymptotics, about the Bessel functions of Coxeter groups and certain generalizations of the Fourier transform. Work will also continue on singular polynomials of Coxeter groups, at topic which connects representation theory, monodromy of differential equations with rational singularities on the reflecting hyperplanes of the groups, Hecke algebras and the intertwining operator. Approximation theory and special functions has fruitful connections with many branches of applied mathematics such as differential equations, operator theory, numerical integration, continued fractions and control theory to name a few. These subjects have close connections with special functions and related symmetry groups which provide a natural synergism between the theoretical and applied aspects of this research. ***